PyPy (Posts about z3)

A DSL for Peephole Transformation Rules of Integer Operations in the PyPy JIT

CF Bolz-Tereick — Wed, 23 Oct 2024 15:00:00 GMT

As is probably apparent from the sequence of blog posts about the topic in the last year, I have been thinking about and working on integer optimizations in the JIT compiler a lot. This work was mainly motivated by Pydrofoil, where integer operations matter a lot more than for your typical Python program.

In this post I'll describe my most recent change, which is a new small domain specific language that I implemented to specify peephole optimizations on integer operations in the JIT. It uses pattern matching to specify how (sequences of) integer operations should be simplified and optimized. The rules are then compiled to RPython code that then becomes part of the JIT's optimization passes.

To make it less likely to introduce incorrect optimizations into the JIT, the rules are automatically proven correct with Z3 as part of the build process (for a more hands-on intro to how that works you can look at the knownbits post). In this blog post I want to motivate why I introduced the DSL and give an introduction to how it works.

Motivation

This summer, after I wrote my scripts to mine JIT traces for missed optimization opportunities, I started implementing a few of the integer peephole rewrite that the script identified. Unfortunately, doing so led to the problem that the way we express these rewrites up to now is very imperative and verbose. Here's a snippet of RPython code that shows some rewrites for integer multiplication (look at the comments to see what the different parts actually do). You don't need to understand the code in detail, but basically it's in very imperative style and there's quite a lot of boilerplate.

def optimize_INT_MUL(self, op):
    arg0 = get_box_replacement(op.getarg(0))
    b0 = self.getintbound(arg0)
    arg1 = get_box_replacement(op.getarg(1))
    b1 = self.getintbound(arg1)

    if b0.known_eq_const(1):
        # 1 * x == x
        self.make_equal_to(op, arg1)
    elif b1.known_eq_const(1):
        # x * 1 == x
        self.make_equal_to(op, arg0)
    elif b0.known_eq_const(0) or b1.known_eq_const(0):
        # 0 * x == x * 0 == 0
        self.make_constant_int(op, 0)
    else:
        for lhs, rhs in [(arg0, arg1), (arg1, arg0)]:
            lh_info = self.getintbound(lhs)
            if lh_info.is_constant():
                x = lh_info.get_constant_int()
                if x & (x - 1) == 0:
                    # x * (2 ** c) == x << c
                    new_rhs = ConstInt(highest_bit(lh_info.get_constant_int()))
                    op = self.replace_op_with(op, rop.INT_LSHIFT, args=[rhs, new_rhs])
                    self.optimizer.send_extra_operation(op)
                    return
                elif x == -1:
                    # x * -1 == -x
                    op = self.replace_op_with(op, rop.INT_NEG, args=[rhs])
                    self.optimizer.send_extra_operation(op)
                    return
            else:
                # x * (1 << y) == x << y
                shiftop = self.optimizer.as_operation(get_box_replacement(lhs), rop.INT_LSHIFT)
                if shiftop is None:
                    continue
                if not shiftop.getarg(0).is_constant() or shiftop.getarg(0).getint() != 1:
                    continue
                shiftvar = get_box_replacement(shiftop.getarg(1))
                shiftbound = self.getintbound(shiftvar)
                if shiftbound.known_nonnegative() and shiftbound.known_lt_const(LONG_BIT):
                    op = self.replace_op_with(
                            op, rop.INT_LSHIFT, args=[rhs, shiftvar])
                    self.optimizer.send_extra_operation(op)
                    return
        return self.emit(op)

Adding more rules to these functions is very tedious and gets super confusing when the functions get bigger. In addition I am always worried about making mistakes when writing this kind of code, and there is no feedback at all about which of these rules are actually applied a lot in real programs.

Therefore I decided to write a small domain specific language with the goal of expressing these rules in a more declarative way. In the rest of the post I'll describe the DSL (most of that description is adapted from the documentation about it that I wrote).

The Peephole Rule DSL

Simple transformation rules

The rules in the DSL specify how integer operation can be transformed into cheaper other integer operations. A rule always consists of a name, a pattern, and a target. Here's a simple rule:

add_zero: int_add(x, 0)
    => x

The name of the rule is add_zero. It matches operations in the trace of the form int_add(x, 0), where x will match anything and 0 will match only the constant zero. After the => arrow is the target of the rewrite, i.e. what the operation is rewritten to, in this case x.

The rule language has a list of which of the operations are commutative, so add_zero will also optimize int_add(0, x) to x.

Variables in the pattern can repeat:

sub_x_x: int_sub(x, x)
    => 0

This rule matches against int_sub operations where the two arguments are the same (either the same box, or the same constant).

Here's a rule with a more complicated pattern:

sub_add: int_sub(int_add(x, y), y)
    => x

This pattern matches int_sub operations, where the first argument was produced by an int_add operation. In addition, one of the arguments of the addition has to be the same as the second argument of the subtraction.

The constants MININT, MAXINT and LONG_BIT (which is either 32 or 64, depending on which platform the JIT is built for) can be used in rules, they behave like writing numbers but allow bit-width-independent formulations:

is_true_and_minint: int_is_true(int_and(x, MININT))
    => int_lt(x, 0)

It is also possible to have a pattern where some arguments needs to be a constant, without specifying which constant. Those patterns look like this:

sub_add_consts: int_sub(int_add(x, C1), C2) # incomplete
    # more goes here
    => int_sub(x, C)

Variables in the pattern that start with a C match against constants only. However, in this current form the rule is incomplete, because the variable C that is being used in the target operation is not defined anywhere. We will see how to compute it in the next section.

Computing constants and other intermediate results

Sometimes it is necessary to compute intermediate results that are used in the target operation. To do that, there can be extra assignments between the rule head and the rule target.:

sub_add_consts: int_sub(int_add(x, C1), C2) # incomplete
    C = C1 + C2
    => int_sub(x, C)

The right hand side of such an assignment is a subset of Python syntax, supporting arithmetic using +, -, *, and certain helper functions. However, the syntax allows you to be explicit about unsignedness for some operations. E.g. >>u exists for unsigned right shifts (and I plan to add >u, >=u, <u, <=u for comparisons).

Here's an example of a rule that uses >>u:

urshift_lshift_x_c_c: uint_rshift(int_lshift(x, C), C)
    mask = (-1 << C) >>u C
    => int_and(x, mask)

Checks

Some rewrites are only true under certain conditions. For example, int_eq(x, 1) can be rewritten to x, if x is known to store a boolean value. This can be expressed with checks:

eq_one: int_eq(x, 1)
    check x.is_bool()
    => x

A check is followed by a boolean expression. The variables from the pattern can be used as IntBound instances in checks (and also in assignments) to find out what the abstract interpretation of the JIT knows about the value of a trace variable (IntBound is the name of the abstract domain that the JIT uses for integers, despite the fact that it also stores knownbits information nowadays).

Here's another example:

mul_lshift: int_mul(x, int_lshift(1, y))
    check y.known_ge_const(0) and y.known_le_const(LONG_BIT)
    => int_lshift(x, y)

It expresses that x * (1 << y) can be rewritten to x << y but checks that y is known to be between 0 and LONG_BIT.

Checks and assignments can be repeated and combined with each other:

mul_pow2_const: int_mul(x, C)
    check C > 0 and C & (C - 1) == 0
    shift = highest_bit(C)
    => int_lshift(x, shift)

In addition to calling methods on IntBound instances, it's also possible to access their attributes, like in this rule:

and_x_c_in_range: int_and(x, C)
    check x.lower >= 0 and x.upper <= C & ~(C + 1)
    => x

Rule Ordering and Liveness

The generated optimizer code will give preference to applying rules that produce a constant or a variable as a rewrite result. Only if none of those match do rules that produce new result operations get applied. For example, the rules sub_x_x and sub_add are tried before trying sub_add_consts, because the former two rules optimize to a constant and a variable respectively, while the latter produces a new operation as the result.

The rule sub_add_consts has a possible problem, which is that if the intermediate result of the int_add operation in the rule head is used by some other operations, then the sub_add_consts rule does not actually reduce the number of operations (and might actually make things slightly worse due to increased register pressure). However, currently it would be extremely hard to take that kind of information into account in the optimization pass of the JIT, so we optimistically apply the rules anyway.

Checking rule coverage

Every rewrite rule should have at least one unit test where it triggers. To ensure this, the unit test file that mainly checks integer optimizations in the JIT has an assert at the end of a test run, that every rule fired at least once.

Printing rule statistics

The JIT can print statistics about which rule fired how often in the jit-intbounds-stats logging category, using the PYPYLOG mechanism. For example, to print the category to stdout at the end of program execution, run PyPy like this:

PYPYLOG=jit-intbounds-stats:- pypy ...

The output of that will look something like this:

int_add
    add_reassoc_consts 2514
    add_zero 107008
int_sub
    sub_zero 31519
    sub_from_zero 523
    sub_x_x 3153
    sub_add_consts 159
    sub_add 55
    sub_sub_x_c_c 1752
    sub_sub_c_x_c 0
    sub_xor_x_y_y 0
    sub_or_x_y_y 0
int_mul
    mul_zero 0
    mul_one 110
    mul_minus_one 0
    mul_pow2_const 1456
    mul_lshift 0
...

Termination and Confluence

Right now there are unfortunately no checks that the rules actually rewrite operations towards "simpler" forms. There is no cost model, and also nothing that prevents you from writing a rule like this:

neg_complication: int_neg(x) # leads to infinite rewrites
    => int_mul(-1, x)

Doing this would lead to endless rewrites if there is also another rule that turns multiplication with -1 into negation.

There is also no checking for confluence (yet?), i.e. the property that all rewrites starting from the same input trace always lead to the same output trace, no matter in which order the rules are applied.

Proofs

It is very easy to write a peephole rule that is not correct in all corner cases. Therefore all the rules are proven correct with Z3 before compiled into actual JIT code, by default. When the proof fails, a (hopefully minimal) counterexample is printed. The counterexample consists of values for all the inputs that fulfil the checks, values for the intermediate expressions, and then two different values for the source and the target operations.

E.g. if we try to add the incorrect rule:

mul_is_add: int_mul(a, b)
    => int_add(a, b)

We get the following counterexample as output:

Could not prove correctness of rule 'mul_is_add'
in line 1
counterexample given by Z3:
counterexample values:
a: 0
b: 1
operation int_mul(a, b) with Z3 formula a*b
has counterexample result vale: 0
BUT
target expression: int_add(a, b) with Z3 formula a + b
has counterexample value: 1

If we add conditions, they are taken into account and the counterexample will fulfil the conditions:

mul_is_add: int_mul(a, b)
    check a.known_gt_const(1) and b.known_gt_const(2)
    => int_add(a, b)

This leads to the following counterexample:

Could not prove correctness of rule 'mul_is_add'
in line 46
counterexample given by Z3:
counterexample values:
a: 2
b: 3
operation int_mul(a, b) with Z3 formula a*b
has counterexample result vale: 6
BUT
target expression: int_add(a, b) with Z3 formula a + b
has counterexample value: 5

Some IntBound methods cannot be used in Z3 proofs because their control flow is too complex. If that is the case, they can have Z3-equivalent formulations defined (in every case this is done, it's a potential proof hole if the Z3 friendly reformulation and the real implementation differ from each other, therefore extra care is required to make very sure they are equivalent).

It's possible to skip the proof of individual rules entirely by adding SORRY_Z3 to its body (but we should try not to do that too often):

eq_different_knownbits: int_eq(x, y)
    SORRY_Z3
    check x.known_ne(y)
    => 0

Checking for satisfiability

In addition to checking whether the rule yields a correct optimization, we also check whether the rule can ever apply. This ensures that there are some runtime values that would fulfil all the checks in a rule. Here's an example of a rule violating this:

never_applies: int_is_true(x)
    check x.known_lt_const(0) and x.known_gt_const(0) # impossible condition, always False
    => x

Right now the error messages if this goes wrong are not completely easy to understand. I hope to be able to improve this later:

Rule 'never_applies' cannot ever apply
in line 1
Z3 did not manage to find values for variables x such that the following condition becomes True:
And(x <= x_upper,
    x_lower <= x,
    If(x_upper < 0, x_lower > 0, x_upper < 0))

Implementation Notes

The implementation of the DSL is done in a relatively ad-hoc manner. It is parsed using rply, there's a small type checker that tries to find common problems in how the rules are written. Z3 is used via the Python API, like in the previous blog posts that are using it. The pattern matching RPython code is generated using an approach inspired by Luc Maranget's paper Compiling Pattern Matching to Good Decision Trees. See this blog post for an approachable introduction.

Conclusion

Now that I've described the DSL, here are the rules that are equivalent to the imperative code in the motivation section:

mul_zero: int_mul(x, 0)
    => 0

mul_one: int_mul(x, 1)
    => x

mul_minus_one: int_mul(x, -1)
    => int_neg(x)

mul_pow2_const: int_mul(x, C)
    check C > 0 and C & (C - 1) == 0
    shift = highest_bit(C)
    => int_lshift(x, shift)

mul_lshift: int_mul(x, int_lshift(1, y))
    check y.known_ge_const(0) and y.known_le_const(LONG_BIT)
    => int_lshift(x, y)

The current status of the DSL is that it got merged to PyPy's main branch. I rewrote a part of the integer rewrites into the DSL, but some are still in the old imperative style (mostly for complicated reasons, the easily ported ones are all done). Since I've only been porting optimizations that had existed prior to the existence of the DSL, performance numbers of benchmarks didn't change.

There are a number of features that are still missing and some possible extensions that I plan to work on in the future:

All the integer operations that the DSL handles so far are the variants that do not check for overflow (or where overflow was proven to be impossible to happen). In regular Python code the overflow-checking variants int_add_ovf etc are much more common, but the DSL doesn't support them yet. I plan to fix this, but don't completely understand how the correctness proofs for them should be done correctly.
A related problem is that I don't understand what it means for a rewrite to be correct if some of the operations are only defined for a subset of the input values. E.g. division isn't defined if the divisor is zero. In theory, a division operation in the trace should always be preceded by a check that the divisor isn't zero. But sometimes other optimization move the check around and the connection to the division gets lost or muddled. What optimizations can we still safely perform on the division? There's lots of prior work on this question, but I still don't understand what the correct approach in our context would be.
Ordering comparisons like int_lt, int_le and their unsigned variants are not ported to the DSL yet. Comparisons are an area where the JIT is not super good yet at optimizing away operations. This is a pretty big topic and I've started a project with Nico Rittinghaus to try to improve the situation a bit more generally.
A more advanced direction of work would be to implement a simplified form of e-graphs (or ae-graphs). The JIT has like half of an e-graph data structure already, and we probably can't afford a full one in terms of compile time costs, but maybe we can have two thirds or something?

Acknowledgements

Thank you to Max Bernstein and Martin Berger for super helpful feedback on drafts of the post!

A Knownbits Abstract Domain for the Toy Optimizer, Correctly

CF Bolz-Tereick — Sat, 03 Aug 2024 14:00:00 GMT

After Max' introduction to abstract interpretation for the toy optimizer in the last post, I want to present a more complicated abstract domain in this post. This abstract domain reasons about the individual bits of a variable in a trace. Every bit can be either "known zero", "known one" or "unknown". The abstract domain is useful for optimizing integer operations, particularly the bitwise operations. The abstract domain follows quite closely the tristate abstract domain of the eBPF verifier in the Linux Kernel, as described by the paper Sound, Precise, and Fast Abstract Interpretation with Tristate Numbers by Harishankar Vishwanathan, Matan Shachnai, Srinivas Narayana, and Santosh Nagarakatte.

The presentation in this post will still be in the context of the toy optimizer. We'll spend a significant part of the post convincing ourselves that the abstract domain transfer functions that we're writing are really correct, using both property-based testing and automated proofs (again using Z3).

PyPy has implemented and merged a more complicated version of the same abstract domain for the "real" PyPy JIT. A more thorough explanation of that real world implementation will follow.

I'd like to thank Max Bernstein and Armin Rigo for lots of great feedback on drafts of this post. The PyPy implementation was mainly done by Nico Rittinghaus and me.

Contents:

Motivation

In many programs that do bit-manipulation of integers, some of the bits of the integer variables of the program can be statically known. Here's a simple example:

x = a | 1
...
if x & 1:
    ...
else:
    ...

After the assignment x = a | 1, we know that the lowest bit of x must be 1 (the other bits are unknown) and an optimizer could remove the condition x & 1 by constant-folding it to 1.

Another (more complicated) example is:

assert i & 0b111 == 0 # check that i is a multiple of 8
j = i + 16
assert j & 0b111 == 0

This kind of code could e.g. happen in a CPU emulator, where i and j are integers that represent emulated pointers, and the asserts are alignment checks. The first assert implies that the lowest three bits of i must be 0. Adding 16 to such a number produces a result where the lowest three bits are again all 0, therefore the second assert is always true. So we would like a compiler to remove the second assert.

Both of these will optimizations are doable with the help of the knownbits abstract domain that we'll discuss in the rest of the post.

The Knownbits Abstract Domain

An abstract value of the knownbits domain needs to be able to store, for every bit of an integer variable in a program, whether it is known 0, known 1, or unknown. To represent three different states, we need 2 bits, which we will call one and unknown. Here's the encoding:

one	unknown	knownbit
0	0	0
1	0	1
0	1	?
1	1	illegal

The unknown bit is set if we don't know the value of the bit ("?"), the one bit is set if the bit is known to be a 1. Since two bits are enough to encode four different states, but we only need three, the combination of a set one bit and a set unknown is not allowed.

We don't just want to encode a single bit, however. Instead, we want to do this for all the bits of an integer variable. Therefore the instances of the abstract domain get two integer fields ones and unknowns, where each pair of corresponding bits encodes the knowledge about the corresponding bit of the integer variable in the program.

We can start implementing a Python class that works like this:

from dataclasses import dataclass

@dataclass(eq=False)
class KnownBits:
    ones : int
    unknowns : int

    def __post_init__(self):
        if isinstance(self.ones, int):
            assert self.is_well_formed()

    def is_well_formed(self):
        # a bit cannot be both 1 and unknown
        return self.ones & self.unknowns == 0

    @staticmethod
    def from_constant(const : int):
        """ Construct a KnownBits corresponding to a constant, where all bits
        are known."""
        return KnownBits(const, 0)

    def is_constant(self):
        """ Check if the KnownBits instance represents a constant. """
        # it's a constant if there are no unknowns
        return self.unknowns == 0

We can also add some convenience properties. Sometimes it is easier to work with an integer where all the known bits are set, or one where the positions of all the known zeros have a set bit:

class KnownBits:
    ...

    @property
    def knowns(self):
        """ return an integer where the known bits are set. """
        # the knowns are just the unknowns, inverted
        return ~self.unknowns

    @property
    def zeros(self):
        """ return an integer where the places that are known zeros have a bit
        set. """
        # it's a 0 if it is known, but not 1
        return self.knowns & ~self.ones

Also, for debugging and for writing tests we want a way to print the known bits in a human-readable form, and also to have a way to construct a KnownBits instance from a string. It's not important to understand the details of __str__ or from_str for the rest of the post, so I'm putting them into a fold:

KnownBits from and to string conversions

class KnownBits:
    ...

    def __repr__(self):
        if self.is_constant():
            return f"KnownBits.from_constant({self.ones})"
        return f"KnownBits({self.ones}, {self.unknowns})"

    def __str__(self):
        res = []
        ones, unknowns = self.ones, self.unknowns
        # construct the string representation right to left
        while 1:
            if not ones and not unknowns:
                break # we leave off the leading known 0s
            if ones == -1 and not unknowns:
                # -1 has all bits set in two's complement, so the leading
                # bits are all 1
                res.append('1')
                res.append("...")
                break
            if unknowns == -1:
                # -1 has all bits set in two's complement, so the leading bits
                # are all ?
                assert not ones
                res.append("?")
                res.append("...")
                break
            if unknowns & 1:
                res.append('?')
            elif ones & 1:
                res.append('1')
            else:
                res.append('0')
            ones >>= 1
            unknowns >>= 1
        if not res:
            res.append('0')
        res.reverse()
        return "".join(res)

    @staticmethod
    def from_str(s):
        """ Construct a KnownBits instance that from a string. String can start
        with ...1 to mean that all higher bits are 1, or ...? to mean that all
        higher bits are unknown. Otherwise it is assumed that the higher bits
        are all 0. """
        ones, unknowns = 0, 0
        startindex = 0
        if s.startswith("...?"):
            unknowns = -1
            startindex = 4
        elif s.startswith("...1"):
            ones = -1
            startindex = 4
        for index in range(startindex, len(s)):
            ones <<= 1
            unknowns <<= 1
            c = s[index]
            if c == '1':
                ones |= 1
            elif c == '?':
                unknowns |= 1
        return KnownBits(ones, unknowns)

    @staticmethod
    def all_unknown():
        """ convenience constructor for the "all bits unknown" abstract value
        """
        return KnownBits.from_str("...?")

And here's a pytest-style unit test for str:

def test_str():
    assert str(KnownBits.from_constant(0)) == '0'
    assert str(KnownBits.from_constant(5)) == '101'
    assert str(KnownBits(5, 0b10)) == '1?1'
    assert str(KnownBits(~0b1111, 0b10)) == '...100?0'
    assert str(KnownBits(1, ~0b1)) == '...?1'

An instance of KnownBits represents a set of integers, namely those that match the known bits stored in the instance. We can write a method contains that takes a concrete int value and returns True if the value matches the pattern of the known bits:

class KnownBits:
    ...

    def contains(self, value : int):
        """ Check whether the KnownBits instance contains the concrete integer
        `value`. """
        # check whether value matches the bit pattern. in the places where we
        # know the bits, the value must agree with ones.
        return value & self.knowns == self.ones

and a test:

def test_contains():
    k1 = KnownBits.from_str('1?1')
    assert k1.contains(0b111)
    assert k1.contains(0b101)
    assert not k1.contains(0b110)
    assert not k1.contains(0b011)

    k2 = KnownBits.from_str('...?1') # all odd numbers
    for i in range(-101, 100):
        assert k2.contains(i) == (i & 1)

Transfer Functions

Now that we have implemented the basics of the KnownBits class, we need to start implementing the transfer functions. They are for computing what we know about the results of an operation, given the knowledge we have about the bits of the arguments.

We'll start with a simple unary operation, invert(x) (which is ~x in Python and C syntax), which flips all the bits of at integer. If we know some bits of the arguments, we can compute the corresponding bits of the result. The unknown bits remain unknown.

Here's the code:

class KnownBits:
    ...

    def abstract_invert(self):
        # self.zeros has bits set where the known 0s are in self
        return KnownBits(self.zeros, self.unknowns)

And a unit-test:

def test_invert():
    k1 = KnownBits.from_str('01?01?01?')
    k2 = k1.abstract_invert()
    assert str(k2) == '...10?10?10?'

    k1 = KnownBits.from_str('...?')
    k2 = k1.abstract_invert()
    assert str(k2) == '...?'

Before we continue with further transfer functions, we'll think about correctness of the transfer functions and build up some test infrastructure. To test transfer functions, it's quite important to move being simple example-style unit tests. The state-space for more complicated binary transfer functions is extremely large and it's too easy to do something wrong in a corner case. Therefore we'll look at property-based-test for KnownBits next.

Property-based Tests with Hypothesis

We want to do property-based tests of KnownBits, to try make it less likely that we'll get a corner-case in the implementation wrong. We'll use Hypothesis for that.

I can't give a decent introduction to Hypothesis here, but want to give a few hints about the API. Hypothesis is a way to run unit tests with randomly generated input. It provides strategies to describe the data that the test functions expects. Hypothesis provides primitive strategies (for things like integers, strings, floats, etc) and ways to build composite strategies out of the primitive ones.

To be able to write the tests, we need to generate random KnownBits instances, and we also want an int instance that is a member of the KnownBits instance. We generate tuples of (KnownBits, int) together, to ensure this property. We'll ask Hypothesis to generate us a random concrete int as the concrete value, and then we'll also generate a second random int to use as the unknown masks (i.e. which bits of the concrete int we don't know in the KnownBits instance). Here's a function that takes two such ints and builds the tuple:

def build_knownbits_and_contained_number(concrete_value : int, unknowns : int):
    # to construct a valid KnownBits instance, we need to mask off the unknown
    # bits
    ones = concrete_value & ~unknowns
    return KnownBits(ones, unknowns), concrete_value

We can turn this function into a hypothesis strategy to generate input data using the strategies.builds function:

from hypothesis import strategies, given, settings

ints = strategies.integers()

random_knownbits_and_contained_number = strategies.builds(
    build_knownbits_and_contained_number,
    ints, ints
)

One important special case of KnownBits are the constants, which contain only a single concrete value. We'll also generate some of those specifically, and then combine the random_knownbits_and_contained_number strategy with it:

constant_knownbits = strategies.builds(
    lambda value: (KnownBits.from_constant(value), value),
    ints
)

knownbits_and_contained_number = constant_knownbits | random_knownbits_and_contained_number

Now we can write the first property-based tests, for the KnownBits.contains method:

@given(knownbits_and_contained_number)
def test_contains(t):
    k, n = t
    assert k.contains(t)

The @given decorator is used to tell Hypothesis which strategy to use to generate random data for the test function. Hypothesis will run the test with a number of random examples (100 by default). If it finds an error, it will try to minimize the example needed that demonstrates the problem, to try to make it easier to understand what is going wrong. It also saves all failing cases into an example database and tries them again on subsequent runs.

This test is as much a check for whether we got the strategies right as it is for the logic in KnownBits.contains. Here's an example output of random concrete and abstract values that we are getting here:

110000011001101 ...?0???1
...1011011 ...1011011
...1001101110101000010010011111011 ...1001101110101000010010011111011
...1001101110101000010010011111011 ...100110111010100001?010?1??1??11
1000001101111101001011010011111101000011000111011001011111101 1000001101111101001011010011111101000011000111011001011111101
1000001101111101001011010011111101000011000111011001011111101 1000001101111101001011010011111101000011000111????01?11?????1
1111100000010 1111100000010
1111100000010 ...?11111?00000??
110110 110110
110110 ...?00?00????11??10
110110 ??0??0
...100010111011111 ...?100?10111??111?
...1000100000110001 ...?000?00000??000?
110000001110 ...?0?0??000?00?0?0000000?00???0000?????00???000?0?00?01?000?0??1??
110000001110 ??000000???0
1011011010000001110101001111000010001001011101010010010001000000010101010010001101110101111111010101010010101100110000011110000 1011011010000001110101001111000010001001011101010010010001000000010101010010001101110101111111010101010010101100110000011110000
...1011010010010100 ...1011010010010100
...1011111110110011 ...1011111110110011
101000011110110 101000011?10?1?
100101 ?00?0?

That looks suitably random, but we might want to bias our random numbers a little bit towards common error values like small constants, powers of two, etc. Like this:

INTEGER_WIDTH = 64
# some small integers
ints_special = set(range(100))
# powers of two
ints_special = ints_special.union(1 << i for i in range(INTEGER_WIDTH - 2))
# powers of two - 1
ints_special = ints_special.union((1 << i) - 1 for i in range(INTEGER_WIDTH - 2))
# negative versions of what we have so far
ints_special = ints_special.union(-x for x in ints_special)
# bit-flipped versions of what we have so far
ints_special = ints_special.union(~x for x in ints_special)
ints_special = list(ints_special)
# sort them (because hypothesis simplifies towards earlier elements in the list)
ints_special.sort(key=lambda element: (abs(element), element < 0))

ints = strategies.sampled_from(ints_special) | strategies.integers()

Now we get data like this:

1110 1110
...10000000000000000001 ...10000??0??0000??00?1
1 ??0??0000??00?1
1 ?
...10101100 ...10101100
110000000011001010111011111111111111011110010001001100110001011 ...?0?101?
110000000011001010111011111111111111011110010001001100110001011 ??00000000??00?0?0???0??????????????0????00?000?00??00??000?0??
...1011111111111111111111111111 ...?11?11??
...1011111111111111111111111111 ...?0??????????????????????????
0 ...?0??????????????????????????
101101 101101
111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111
10111 10111
...101100 ...1?111011?0
101000 ?001010?0
101000 ?0?000
110010 110010
...100111 ...100111
1111011010010 1111011010010
...1000000000000000000000000000000000000 ...1000000000000000000000000000000000000

We can also write a test that checks that the somewhat tricky logic in __str__ and from_str is correct, by making sure that the two functions round-trip (ie converting a KnownBits to a string and then back to a KnownBits instance produces the same abstract value).

@given(knownbits_and_contained_number)
def test_hypothesis_str_roundtrips(t1):
    k1, n1 = t1
    s = str(k1)
    k2 = KnownBits.from_str(s)
    assert k1.ones == k2.ones
    assert k1.unknowns == k2.unknowns

Now let's actually apply this infrastructure to test abstract_invert.

When are Transfer Functions Correct? How do we test them?

Abstract values, i.e. instances of KnownBits represent sets of concrete values. We want the transfer functions to compute overapproximations of the concrete values. So if we have an arbitrary abstract value k, with a concrete number n that is a member of the abstract values (i.e. k.contains(n) == True) then the result of the concrete operation op(n) must be a member of the result of the abstract operation k.abstract_op() (i.e. k.abstract_op().contains(op(n)) == True).

Checking the correctness/overapproximation property is a good match for hypothesis. Here's what the test for abstract_invert looks like:

@given(knownbits_and_contained_number)
def test_hypothesis_invert(t):
    k1, n1 = t1
    n2 = ~n1 # compute the real result
    k2 = k1.abstract_invert() # compute the abstract result
    assert k2.contains(n2) # the abstract result must contain the real result

This is the only condition needed for abstract_invert to be correct. If abstract_invert fulfils this property for every combination of abstract and concrete value then abstract_invert is correct. Note however, that this test does not actually check whether abstract_invert gives us precise results. A correct (but imprecise) implementation of abstract_invert would simply return a completely unknown result, regardless of what is known about the input KnownBits.

The "proper" CS term for this notion of correctness is called soundness. The correctness condition on the transfer functions is called a Galois connection. I won't go into any mathematical/technical details here, but wanted to at least mention the terms. I found Martin Kellogg's slides to be quite an approachable introduction to the Galois connection and how to show soundness.

Implementing Binary Transfer Functions

Now we have infrastructure in place for testing transfer functions with random inputs. With that we can start thinking about the more complicated case, that of binary operations. Let's start with the simpler ones, and and or. For and, we can know a 0 bit in the result if either of the input bits are known 0; or we can know a 1 bit in the result if both input bits are known 1. Otherwise the resulting bit is unknown. Let's look at all the combinations:

and
input1: 000111???
input2: 01?01?01? 
result: 00001?0??

class KnownBits:
    ...

    def abstract_and(self, other):
        ones = self.ones & other.ones # known ones
        knowns = self.zeros | other.zeros | ones
        return KnownBits(ones, ~knowns)

Here's an example unit-test and a property-based test for and:

def test_and():
    # test all combinations of 0, 1, ? in one example
    k1 = KnownBits.from_str('01?01?01?')
    k2 = KnownBits.from_str('000111???')
    res = k1.abstract_and(k2)     # should be: 0...00001?0??
    assert str(res) ==   "1?0??"

@given(knownbits_and_contained_number, knownbits_and_contained_number)
def test_hypothesis_and(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    k3 = k1.abstract_and(k2)
    n3 = n1 & n2
    assert k3.contains(n3)

To implement or is pretty similar. The result is known 1 where either of the inputs is 1. The result is known 0 where both inputs are known 0, and ? otherwise.

or
input1: 000111???
input2: 01?01?01? 
result: 01?111?1?

class KnownBits:
    ...

    def abstract_or(self, other):
        ones = self.ones | other.ones
        zeros = self.zeros & other.zeros
        knowns = ones | zeros
        return KnownBits(ones, ~knowns)

Here's an example unit-test and a property-based test for or:

def test_or():
    k1 = KnownBits.from_str('01?01?01?')
    k2 = KnownBits.from_str('000111???')
    res = k1.abstract_or(k2)     # should be:  0...01?111?1?
    assert str(res) ==   "1?111?1?"

@given(knownbits_and_contained_number, knownbits_and_contained_number)
def test_hypothesis_or(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    k3 = k1.abstract_or(k2)
    n3 = n1 | n2
    assert k3.contains(n3)

Implementing support for abstract_xor is relatively simple, and left as an exercise :-).

Addition and Subtraction

invert, and, and or are relatively simple transfer functions to write, because they compose over the individual bits of the integers. The arithmetic functions add and sub are significantly harder, because of carries and borrows. Coming up with the formulas for them and gaining an intuitive understanding is quite tricky and involves carefully going through a few examples with pen and paper. When implementing this in PyPy, Nico and I didn't come up with the implementation ourselves, but instead took them from the Tristate Numbers paper. Here's the code, with example tests and hypothesis tests:

class KnownBits:
    ...

    def abstract_add(self, other):
        sum_ones = self.ones + other.ones
        sum_unknowns = self.unknowns + other.unknowns
        all_carries = sum_ones + sum_unknowns
        ones_carries = all_carries ^ sum_ones
        unknowns = self.unknowns | other.unknowns | ones_carries
        ones = sum_ones & ~unknowns
        return KnownBits(ones, unknowns)

    def abstract_sub(self, other):
        diff_ones = self.ones - other.ones
        val_borrows = (diff_ones + self.unknowns) ^ (diff_ones - other.unknowns)
        unknowns = self.unknowns | other.unknowns | val_borrows
        ones = diff_ones & ~unknowns
        return KnownBits(ones, unknowns)


def test_add():
    k1 = KnownBits.from_str('0?10?10?10')
    k2 = KnownBits.from_str('0???111000')
    res = k1.abstract_add(k2)
    assert str(res) ==   "?????01?10"

def test_sub():
    k1 = KnownBits.from_str('0?10?10?10')
    k2 = KnownBits.from_str('0???111000')
    res = k1.abstract_sub(k2)
    assert str(res) ==   "...?11?10"
    k1 = KnownBits.from_str(    '...1?10?10?10')
    k2 = KnownBits.from_str('...10000???111000')
    res = k1.abstract_sub(k2)
    assert str(res) ==   "111?????11?10"

@given(knownbits_and_contained_number, knownbits_and_contained_number)
def test_hypothesis_add(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    k3 = k1.abstract_add(k2)
    n3 = n1 + n2
    assert k3.contains(n3)

@given(knownbits_and_contained_number, knownbits_and_contained_number)
def test_hypothesis_sub(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    k3 = k1.abstract_sub(k2)
    n3 = n1 - n2
    assert k3.contains(n3)

Now we are in a pretty good situation, and have implemented abstract versions for a bunch of important arithmetic and binary functions. What's also surprising is that the implementation of all of the transfer functions is quite efficient. We didn't have to write loops over the individual bits at all, instead we found closed form expressions using primitive operations on the underlying integers ones and unknowns. This means that computing the results of abstract operations is quite efficient, which is important when using the abstract domain in the context of a JIT compiler.

Proving correctness of the transfer functions with Z3

As one can probably tell from my recent posts, I've been thinking about compiler correctness a lot. Getting the transfer functions absolutely correct is really crucial, because a bug in them would lead to miscompilation of Python code when the abstract domain is added to the JIT. While the randomized tests are great, it's still entirely possible for them to miss bugs. The state space for the arguments of a binary transfer function is 3**64 * 3**64, and if only a small part of that contains wrong behaviour it would be really unlikely for us to find it with random tests by chance. Therefore I was reluctant to merge the PyPy branch that contained the new abstract domain for a long time.

To increase our confidence in the correctness of the transfer functions further, we can use Z3 to prove their correctness, which gives us much stronger guarantees (not 100%, obviously). In this subsection I will show how to do that.

Here's an attempt to do this manually in the Python repl:

>>>> import z3
>>>> solver = z3.Solver()
>>>> # like last blog post, proof by failing to find counterexamples
>>>> def prove(cond): assert solver.check(z3.Not(cond)) == z3.unsat
>>>>
>>>> # let's set up a z3 bitvector variable for an arbitrary concrete value
>>>> n1 = z3.BitVec('concrete_value', 64)
>>>> n1
concrete_value
>>>> # due to operator overloading we can manipulate z3 formulas
>>>> n2 = ~n1
>>>> n2
~concrete_value
>>>> 
>>>> # now z3 bitvector variables for the ones and zeros fields
>>>> ones = z3.BitVec('abstract_ones', 64)
>>>> unknowns = z3.BitVec('abstract_unknowns', 64)
>>>> # we construct a KnownBits instance with the z3 variables
>>>> k1 = KnownBits(ones, unknowns)
>>>> # due to operator overloading we can call the methods on k1:
>>>> k2 = k1.abstract_invert()
>>>> k2.ones
~abstract_unknowns & ~abstract_ones
>>>> k2.unknowns
abstract_unknowns
>>>> # here's the correctness condition that we want to prove:
>>>> k2.contains(n2)
~concrete_value & ~abstract_unknowns ==
~abstract_unknowns & ~abstract_ones
>>>> # let's try
>>>> prove(k2.contains(n2))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 1, in prove
AssertionError
>>>> # it doesn't work! let's look at the counterexample to see why:
>>>> solver.model()
[abstract_unknowns = 0,
 abstract_ones = 0,
 concrete_value = 1]
>>>> # we can build a KnownBits instance with the values in the
>>>> # counterexample:
>>>> ~1 # concrete result
-2
>>>> counter_example_k1 = KnownBits(0, 0)
>>>> counter_example_k1
KnownBits.from_constant(0)
>>>> counter_example_k2 = counter_example_k1.abstract_invert()
>>>> counter_example_k2
KnownBits.from_constant(-1)
>>>> # let's check the failing condition
>>>> counter_example_k2.contains(~1)
False

What is the problem here? We didn't tell Z3 that n1 was supposed to be a member of k1. We can add this as a precondition to the solver, and then the prove works:

>>>> solver.add(k1.contains(n1))
>>>> prove(k2.contains(n2)) # works!

This is super cool! It's really a proof about the actual implementation, because we call the implementation methods directly, and due to the operator overloading that Z3 does we can be sure that we are actually checking a formula that corresponds to the Python code. This eliminates one source of errors in formal methods.

Doing the proof manually on the Python REPL is kind of annoying though, and we also would like to make sure that the proofs are re-done when we change the code. What we would really like to do is writing the proofs as a unit-test that we can run while developing and in CI. Doing this is possible, and the unit tests that really perform proofs look pleasingly similar to the Hypothesis-based ones.

First we need to set up a bit of infrastructure:

INTEGER_WIDTH = 64

def BitVec(name):
    return z3.BitVec(name, INTEGER_WIDTH)

def BitVecVal(val):
    return z3.BitVecVal(val, INTEGER_WIDTH)

def z3_setup_variables():
    # instantiate a solver
    solver = z3.Solver()

    # a Z3 variable for the first concrete value
    n1 = BitVec("n1")
    # a KnownBits instances that uses Z3 variables as its ones and unknowns,
    # representing the first abstract value
    k1 = KnownBits(BitVec("n1_ones"), BitVec("n1_unkowns"))
    # add the precondition to the solver that the concrete value n1 must be a
    # member of the abstract value k1
    solver.add(k1.contains(n1))

    # a Z3 variable for the second concrete value
    n2 = BitVec("n2")
    # a KnownBits instances for the second abstract value
    k2 = KnownBits(BitVec("n2_ones"), BitVec("n2_unkowns"))
    # add the precondition linking n2 and k2 to the solver
    solver.add(k2.contains(n2))
    return solver, k1, n1, k2, n2

def prove(cond, solver):
    z3res = solver.check(z3.Not(cond))
    if z3res != z3.unsat:
        assert z3res == z3.sat # can't be timeout, we set no timeout
        # make the model with the counterexample global, to make inspecting the
        # bug easier when running pytest --pdb
        global model
        model = solver.model()
        print(f"n1={model.eval(n1)}, n2={model.eval(n2)}")
        counter_example_k1 = KnownBits(model.eval(k1.ones).as_signed_long(),
                                       model.eval(k1.unknowns).as_signed_long())
        counter_example_k2 = KnownBits(model.eval(k2.ones).as_signed_long(),
                                       model.eval(k2.unknowns).as_signed_long())
        print(f"k1={counter_example_k1}, k2={counter_example_k2}")
        print(f"but {cond=} evaluates to {model.eval(cond)}")
        raise ValueError(solver.model())

And then we can write proof-unit-tests like this:

def test_z3_abstract_invert():
    solver, k1, n1, _, _ = z3_setup_variables()
    k2 = k1.abstract_invert()
    n2 = ~n1
    prove(k2.contains(n2), solver)

def test_z3_abstract_and():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    k3 = k1.abstract_and(k2)
    n3 = n1 & n2
    prove(k3.contains(n3), solver)

def test_z3_abstract_or():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    k3 = k1.abstract_or(k2)
    n3 = n1 | n2
    prove(k3.contains(n3), solver)

def test_z3_abstract_add():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    k3 = k1.abstract_add(k2)
    n3 = n1 + n2
    prove(k3.contains(n3), solver)

def test_z3_abstract_sub():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    k3 = k1.abstract_sub(k2)
    n3 = n1 - n2
    prove(k3.contains(n3), solver)

It's possible to write a bit more Python-metaprogramming-magic and unify the Hypothesis and Z3 tests into the same test definition.¹

Cases where this style of Z3 proof doesn't work

Unfortunately the approach described in the previous section only works for a very small number of cases. It breaks down as soon as the KnownBits methods that we're calling contain any if conditions (including hidden ones like the short-circuiting and and or in Python). Let's look at an example and implement abstract_eq. eq is supposed to be an operation that compares two integers and returns 0 or 1 if they are different or equal, respectively. Implementing this in knownbits looks like this (with example and hypothesis tests):

class KnownBits:
    ...

    def abstract_eq(self, other):
        # the result is a 0, 1, or ?

        # if they are both the same constant, they must be equal
        if self.is_constant() and other.is_constant() and self.ones == other.ones:
            return KnownBits.from_constant(1)
        # check whether we have known disagreeing bits, then we know the result
        # is 0
        if self._disagrees(other):
            return KnownBits.from_constant(0)
        return KnownBits(0, 1) # an unknown boolean

    def _disagrees(self, other):
        # check whether the bits disagree in any place where both are known
        both_known = self.knowns & other.knowns
        return self.ones & both_known != other.ones & both_known

def test_eq():
    k1 = KnownBits.from_str('...?')
    k2 = KnownBits.from_str('...?')
    assert str(k1.abstract_eq(k2)) == '?'
    k1 = KnownBits.from_constant(10)
    assert str(k1.abstract_eq(k1)) == '1'
    k1 = KnownBits.from_constant(10)
    k2 = KnownBits.from_constant(20)
    assert str(k1.abstract_eq(k2)) == '0'

@given(knownbits_and_contained_number, knownbits_and_contained_number)
def test_hypothesis_eq(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    k3 = k1.abstract_eq(k2)
    assert k3.contains(int(n1 == n2))

Trying to do the proof in the same style as before breaks:

>>>> k3 = k1.abstract_eq(k2)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "knownbits.py", line 246, in abstract_eq
    if self._disagrees(other):
  File "venv/site-packages/z3/z3.py", line 381, in __bool__
    raise Z3Exception("Symbolic expressions cannot be cast to concrete Boolean values.")
z3.z3types.Z3Exception: Symbolic expressions cannot be cast to concrete Boolean values.

We cannot call abstract_eq on a KnownBits with Z3 variables as fields, because once we hit an if statement, the whole approach of relying on the operator overloading breaks down. Z3 doesn't actually parse the Python code or anything advanced like that, we rather build an expression only by running the code and letting the Z3 formulas build up.

To still prove the correctness of abstract_eq we need to manually transform the control flow logic of the function into a Z3 formula that uses the z3.If expression, using a small helper function:

def z3_cond(b, trueval=1, falseval=0):
    return z3.If(b, BitVecVal(trueval), BitVecVal(falseval))

def z3_abstract_eq(k1, k2):
    # follow the *logic* of abstract_eq, we can't call it due to the ifs in it
    case1cond = z3.And(k1.is_constant(), k2.is_constant(), k1.ones == k2.ones)
    case2cond = k1._disagrees(k2)

    # ones is 1 in the first case, 0 otherwise
    ones = z3_cond(case1cond, 1, 0)

    # in the first two cases, unknowns is 0, 1 otherwise
    unknowns = z3_cond(z3.Or(case1cond, case2cond), 0, 1)
    return KnownBits(ones, unknowns)

def test_z3_abstract_eq_logic():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    n3 = z3_cond(n1 == n2) # concrete result
    k3 = z3_abstract_eq(k1, k2)
    prove(k3.contains(n3), solver)

This proof works. It is a lot less satisfying than the previous ones though, because we could have done an error in the manual transcription from Python code to Z3 formulas (there are possibly more heavy-handed approaches where we do this transformation more automatically using e.g. the ast module to analyze the source code, but that's a much more complicated researchy project). To lessen this problem somewhat we can factor out the parts of the logic that don't have any conditions into small helper methods (like _disagrees in this example) and use them in the manual conversion of the code to Z3 formulas.²

The final condition that Z3 checks, btw, is this one:

If(n1 == n2, 1, 0) &
~If(Or(And(n1_unkowns == 0,
           n2_unkowns == 0,
           n1_ones == n2_ones),
       n1_ones & ~n1_unkowns & ~n2_unkowns !=
       n2_ones & ~n1_unkowns & ~n2_unkowns),
    0, 1) ==
If(And(n1_unkowns == 0, n2_unkowns == 0, n1_ones == n2_ones),
   1, 0)

Making Statements about Precision

So far we have only used Z3 to prove statements about correctness, i.e. that our abstract operations overapproximate what can happen with concrete values. While proving this property is essential if we want to avoid miscompilation, correctness alone is not a very strong constraint on the implementation of our abstract transfer functions. We could simply return Knownbits.unknowns() for every abstract_* method and the resulting overapproximation would be correct, but useless in practice.

It's much harder to make statements about whether the transfer functions are maximally precise. There are two aspects of precision I want to discuss in this section, however.

The first aspect is that we would really like it if the transfer functions compute the maximally precise results for singleton sets. If all abstract arguments of an operations are constants, i.e. contain only a single concrete element, then we know that the resulting set also has only a single element. We can prove that all our transfer functions have this property:

def test_z3_prove_constant_folding():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    k3 = k1.abstract_invert()
    prove(z3.Implies(k1.is_constant(),
                     k3.is_constant()), solver)

    k3 = k1.abstract_and(k2)
    prove(z3.Implies(z3.And(k1.is_constant(), k2.is_constant()),
                     k3.is_constant()), solver)

    k3 = k1.abstract_or(k2)
    prove(z3.Implies(z3.And(k1.is_constant(), k2.is_constant()),
                     k3.is_constant()), solver)

    k3 = k1.abstract_sub(k2)
    prove(z3.Implies(z3.And(k1.is_constant(), k2.is_constant()),
                     k3.is_constant()), solver)

    k3 = z3_abstract_eq(k1, k2)
    prove(z3.Implies(z3.And(k1.is_constant(), k2.is_constant()),
                     k3.is_constant()), solver)

Proving with Z3 that the transfer functions are maximally precise for non-constant arguments seems to be relatively hard. I tried a few completely rigorous approaches and failed. The paper Sound, Precise, and Fast Abstract Interpretation with Tristate Numbers contains an optimality proof for the transfer functions of addition and subtraction, so we can be certain that they are as precise as is possible.

I still want to show an approach for trying to find concrete examples of abstract values that are less precise than they could be, using a combination of Hypothesis and Z3. The idea is to use hypothesis to pick random abstract values. Then we compute the abstract result using our transfer function. Afterwards we can ask Z3 to find us an abstract result that is better than the one our transfer function produced. If Z3 finds a better abstract result, we have a concrete example of imprecision for our transfer function. Those tests aren't strict proofs, because they rely on generating random abstract values, but they can still be valuable (not for the transfer functions in this blog post, which are all optimal).

Here is what the code looks like (this is a little bit bonus content, I'll not explain the details and can only hope that the comments are somewhat helpful):

@given(random_knownbits_and_contained_number, random_knownbits_and_contained_number)
@settings(deadline=None)
def test_check_precision(t1, t2):
    k1, n1 = t1
    k2, n2 = t2
    # apply transfer function
    k3 = k1.abstract_add(k2)
    example_res = n1 + n2

    # try to find a better version of k3 with Z3
    solver = z3.Solver()
    solver.set("timeout", 8000)

    var1 = BitVec('v1')
    var2 = BitVec('v2')

    ones = BitVec('ones')
    unknowns = BitVec('unknowns')
    better_k3 = KnownBits(ones, unknowns)
    print(k1, k2, k3)

    # we're trying to find an example for a better k3, so we use check, without
    # negation:
    res = solver.check(z3.And(
        # better_k3 should be a valid knownbits instance
        better_k3.is_well_formed(),
        # it should be better than k3, ie there are known bits in better_k3
        # that we don't have in k3
        better_k3.knowns & ~k3.knowns != 0,
        # now encode the correctness condition for better_k3 with a ForAll:
        # for all concrete values var1 and var2, it must hold that if
        # var1 is in k1 and var2 is in k2 it follows that var1 + var2 is in
        # better_k3
        z3.ForAll(
        [var1, var2],
        z3.Implies(
            z3.And(k1.contains(var1), k2.contains(var2)),
            better_k3.contains(var1 + var2)))))
    # if this query is satisfiable, we have found a better result for the
    # abstract_add
    if res == z3.sat:
        model = solver.model()
        rk3 = KnownBits(model.eval(ones).as_signed_long(), model.eval(unknowns).as_signed_long())
        print("better", rk3)
        assert 0
    if res == z3.unknown:
        print("timeout")

It does not actually fail for abstract_add (nor the other abstract functions). To see the test failing we can add some imprecision to the implementation of abstract_add to see Hypothesis and Z3 find examples of values that are not optimally precise (for example by setting some bits of unknowns in the implementation of abstract_add unconditionally).

Using the Abstract Domain in the Toy Optimizer for Generalized Constant Folding

Now after all this work we can finally actually use the knownbits abstract domain in the toy optimizer. The code for this follows Max' intro post about abstract interpretation quite closely.

For completeness sake, in the fold there's the basic infrastructure classes that make up the IR again (they are identical or at least extremely close to the previous toy posts).

toy infrastructure

class Value:
    def find(self):
        raise NotImplementedError("abstract")


@dataclass(eq=False)
class Operation(Value):
    name : str
    args : list[Value]

    forwarded : Optional[Value] = None

    def find(self) -> Value:
        op = self
        while isinstance(op, Operation):
            next = op.forwarded
            if next is None:
                return op
            op = next
        return op

    def arg(self, index):
        return self.args[index].find()

    def make_equal_to(self, value : Value):
        self.find().forwarded = value


@dataclass(eq=False)
class Constant(Value):
    value : object

    def find(self):
        return self


class Block(list):
    def __getattr__(self, opname):
        def wraparg(arg):
            if not isinstance(arg, Value):
                arg = Constant(arg)
            return arg
        def make_op(*args):
            op = Operation(opname,
                [wraparg(arg) for arg in args])
            self.append(op)
            return op
        return make_op


def bb_to_str(l : Block, varprefix : str = "var"):
    def arg_to_str(arg : Value):
        if isinstance(arg, Constant):
            return str(arg.value)
        else:
            return varnames[arg]

    varnames = {}
    res = []
    for index, op in enumerate(l):
        # give the operation a name used while
        # printing:
        var =  f"{varprefix}{index}"
        varnames[op] = var
        arguments = ", ".join(
            arg_to_str(op.arg(i))
                for i in range(len(op.args))
        )
        strop = f"{var} = {op.name}({arguments})"
        res.append(strop)
    return "\n".join(res)

Now we can write some first tests, the first one simply checking constant folding:

def test_constfold_two_ops():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.int_add(5, 4)
    var2 = bb.int_add(var1, 10)
    var3 = bb.int_add(var2, var0)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = int_add(19, optvar0)"""

Calling the transfer functions on constant KnownBits produces a constant results, as we have seen. Therefore "regular" constant folding should hopefully be achieved by optimizing with the KnownBits abstract domain too.

The next two tests are slightly more complicated and can't be optimized by regular constant-folding. They follow the motivating examples from the start of this blog post, a hundred years ago:

def test_constfold_via_knownbits():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.int_or(var0, 1)
    var2 = bb.int_and(var1, 1)
    var3 = bb.dummy(var2)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = int_or(optvar0, 1)
optvar2 = dummy(1)"""

def test_constfold_alignment_check():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.int_invert(0b111)
    # mask off the lowest three bits, thus var2 is aligned
    var2 = bb.int_and(var0, var1)
    # add 16 to aligned quantity
    var3 = bb.int_add(var2, 16)
    # check alignment of result
    var4 = bb.int_and(var3, 0b111)
    var5 = bb.int_eq(var4, 0)
    # var5 should be const-folded to 1
    var6 = bb.dummy(var5)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = int_and(optvar0, -8)
optvar2 = int_add(optvar1, 16)
optvar3 = dummy(1)"""

Here is simplify to make these tests pass:

def unknown_transfer_functions(*abstract_args):
    return KnownBits.all_unknown()


def simplify(bb: Block) -> Block:
    abstract_values = {} # dict mapping Operation to KnownBits

    def knownbits_of(val : Value):
        if isinstance(val, Constant):
            return KnownBits.from_constant(val.value)
        return abstract_values[val]

    opt_bb = Block()
    for op in bb:
        # apply the transfer function on the abstract arguments
        name_without_prefix = op.name.removeprefix("int_")
        method_name = f"abstract_{name_without_prefix}"
        transfer_function = getattr(KnownBits, method_name, unknown_transfer_functions)
        abstract_args = [knownbits_of(arg.find()) for arg in op.args]
        abstract_res = abstract_values[op] = transfer_function(*abstract_args)
        # if the result is a constant, we optimize the operation away and make
        # it equal to the constant result
        if abstract_res.is_constant():
            op.make_equal_to(Constant(abstract_res.ones))
            continue
        # otherwise emit the op
        opt_bb.append(op)
    return opt_bb

The code follows the approach from the previous blog post very closely. The only difference is that we apply the transfer function first, to be able to detect whether the abstract domain can tell us that the result has to always be a constant. This code makes all three tests pass.

Using the `KnownBits` Domain for Conditional Peephole Rewrites

So far we are only using the KnownBits domain to find out that certain operations have to produce a constant. We can also use the KnownBits domain to check whether certain operation rewrites are correct. Let's use one of the examples from the Mining JIT traces for missing optimizations with Z3 post, where Z3 found the inefficiency (x << 4) & -0xf == x << 4 in PyPy JIT traces. We don't have shift operations, but we want to generalize this optimization anyway. The general form of this rewrite is that under some circumstances x & y == x, and we can use the KnownBits domain to detect situations where this must be true.

To understand when x & y == x is true, we can think about individual pairs of bits a and b. If a == 0, then a & b == 0 & b == 0 == a. If b == 1 then a & b == a & 1 == a. So if either a == 0 or b == 1 is true, a & b == a follows. And if either of these conditions is true for all the bits of x and y, we can know that x & y == x.

We can write a method on KnownBits to check for this condition:

class KnownBits:
    ...

    def is_and_identity(self, other):
        """ Return True if n1 & n2 == n1 for any n1 in self and n2 in other.
        (or, equivalently, return True if n1 | n2 == n2)"""
        return self.zeros | other.ones == -1

Since my reasoning about this feels ripe for errors, let's check that our understanding is correct with Z3:

def test_prove_is_and_identity():
    solver, k1, n1, k2, n2 = z3_setup_variables()
    prove(z3.Implies(k1.is_and_identity(k2), n1 & n2 == n1), solver)

Now let's use this in the toy optimizer. Here are two tests for this rewrite:

def test_remove_redundant_and():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.int_invert(0b1111)
    # mask off the lowest four bits
    var2 = bb.int_and(var0, var1)
    # applying the same mask is not redundant
    var3 = bb.int_and(var2, var1)
    var4 = bb.dummy(var3)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = int_and(optvar0, -16)
optvar2 = dummy(optvar1)"""

def test_remove_redundant_and_more_complex():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.getarg(1)
    # var2 has bit pattern ????
    var2 = bb.int_and(var0, 0b1111)
    # var3 has bit pattern ...?1111
    var3 = bb.int_or(var1, 0b1111)
    # var4 is just var2
    var4 = bb.int_and(var2, var3)
    var5 = bb.dummy(var4)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = getarg(1)
optvar2 = int_and(optvar0, 15)
optvar3 = int_or(optvar1, 15)
optvar4 = dummy(optvar2)"""

The first test could also be made to pass by implementing a reassociation optimization that turns (x & c1) & c2 into x & (c1 & c2) and then constant-folds the second and. But here we want to use KnownBits and conditionally rewrite int_and to its first argument. So to make the tests pass, we can change simplify like this:

def simplify(bb: Block) -> Block:
    abstract_values = {} # dict mapping Operation to KnownBits

    def knownbits_of(val : Value):
        ...

    opt_bb = Block()
    for op in bb:
        # apply the transfer function on the abstract arguments
        name_without_prefix = op.name.removeprefix("int_")
        method_name = f"abstract_{name_without_prefix}"
        transfer_function = getattr(KnownBits, method_name, unknown_transfer_functions)
        abstract_args = [knownbits_of(arg.find()) for arg in op.args]
        abstract_res = abstract_values[op] = transfer_function(*abstract_args)
        # if the result is a constant, we optimize the operation away and make
        # it equal to the constant result
        if abstract_res.is_constant():
            op.make_equal_to(Constant(abstract_res.ones))
            continue
        # <<<< new code
        # conditionally rewrite int_and(x, y) to x
        if op.name == "int_and":
            k1, k2 = abstract_args
            if k1.is_and_identity(k2):
                op.make_equal_to(op.arg(0))
                continue
        # >>>> end changes
        opt_bb.append(op)
    return opt_bb

And with that, the new tests pass as well. A real implementation would also check the other argument order, but we leave that out for the sake of brevity.

This rewrite also generalizes the rewrites int_and(0, x) -> 0 and int_and(-1, x) -> x, let's add a test for those:

def test_remove_and_simple():
    bb = Block()
    var0 = bb.getarg(0)
    var1 = bb.getarg(1)
    var2 = bb.int_and(0, var0) # == 0
    var3 = bb.int_invert(var2) # == -1
    var4 = bb.int_and(var1, var3) # == var1
    var5 = bb.dummy(var4)

    opt_bb = simplify(bb)
    assert bb_to_str(opt_bb, "optvar") == """\
optvar0 = getarg(0)
optvar1 = getarg(1)
optvar2 = dummy(optvar1)"""

This test just passes. And that's it for this post!

Conclusion

In this post we've seen the implementation, testing and proofs about a 'known bits' abstract domain, as well as its use in the toy optimizer to generalize constant folding, and to implement conditional peephole rewrites.

In the next posts I'll write about the real implementation of a knownbits domain in PyPy's JIT, its combination with the existing interval abstract domain, how to deal with gaining information from conditions in the program, and some lose ends.

Sources:

Known bits in LLVM
Tristate numbers for known bits in Linux eBPF
Sound, Precise, and Fast Abstract Interpretation with Tristate Numbers
Verifying the Veriﬁer: eBPF Range Analysis Veriﬁcation
Bit-Twiddling: Addition with Unknown Bits is a super readable blog post by Dougall J. I've taken the ones and unknowns naming from this post, which I find significantly clearer than value and mask, which the Linux kernel uses.
Bits, Math and Performance(?), a fantastic blog by Harold Aptroot. There are a lot of relevant posts about known bits, range analysis etc. Harold is also the author of Haroldbot, a website that can be used for bitvector calculations, and also checks bitvector identities.
Sharpening Constraint Programming approaches for Bit-Vector Theory
Deriving Abstract Transfer Functions for Analyzing Embedded Software
Synthesizing Abstract Transformers

There's a subtletly about the Z3 proofs that I'm sort of glossing over here. Python integers are of arbitrary width, and the KnownBits code is actually carefully written to work for integers of any size. This property is tested by the Hypothesis tests, which don't limit the sizes of the generated random integers. However, the Z3 proofs only check bitvectors of a fixed bitwidth of 64. There are various ways to deal with this situation. For most "real" compilers, the bitwidth of integers would be fixed anyway. Then the components ones and unknowns of the KnownBits class would use the number of bits the corresponding integer variable has, and the Z3 proofs would use the same width. This is what we do in the PyPy JIT. ↩
The less close connection between implementation and proof for abstract_eq is one of the reasons why it makes sense to do unit-testing in addition to proofs. For a more detailed explanation of why both tests and proofs are good to have, see Jeremy Siek's blog post, as well as the Knuth quote. ↩

Mining JIT traces for missing optimizations with Z3

CF Bolz-Tereick — Fri, 19 Jul 2024 17:01:09 GMT

In my last post I've described how to use Z3 to find simple local peephole optimization patterns for the integer operations in PyPy's JIT. An example is int_and(x, 0) -> 0. In this post I want to scale up the problem of identifying possible optimizations to much bigger instruction sequences, also using Z3. For that, I am starting with the JIT traces of real benchmarks, after they have been optimized by the optimizer of PyPy's JIT. Then we can ask Z3 to find inefficient integer operations in those traces.

Starting from the optimized traces of real programs has some big advantages over the "classical" superoptimization approach of generating and then trying all possible sequences of instructions. It avoids the combinatorial explosion that happens with the latter approach. Also, starting from the traces of benchmarks or (even better) actual programs makes sure that we actually care about the missing optimizations that are found in this way. And because the traces are analyzed after they have been optimized by PyPy's optimizer, we only get reports for missing optimizations, that the JIT isn't able to do (yet).

The techniques and experiments I describe in this post are again the result of a bunch of discussions with John Regehr at a conference a few weeks ago, as well as reading his blog posts and papers. Thanks John! Also thanks to Max Bernstein for super helpful feedback on the drafts of this blog post (and for poking me to write things in general).

High-Level Approach

The approach that I took works as follows:

Run benchmarks or other interesting programs and then dump the IR of the JIT traces into a file. The traces have at that point been already optimized by the PyPy JIT's optimizer.
For every trace, ignore all the operations on non-integer variables.
Translate every integer operation into a Z3 formula.
For every operation, use Z3 to find out whether the operation is redundant (how that is done is described below).
If the operation is redundant, the trace is less efficient than it could have been, because the optimizer could also have removed the operation. Report the inefficiency.
Minimize the inefficient programs by removing as many operations as possible to make the problem easier to understand.

In the post I will describe the details and show some pseudocode of the approach. I'll also make the proper code public eventually (but it needs a healthy dose of cleanups first).

Dumping PyPy Traces

PyPy will write its JIT traces into the file out if the environment variable PYPYLOG is set as follows:

PYPYLOG=jit-log-opt:out pypy <program.py>

This environment variable works for PyPy, but also for other virtual machines built with RPython.

(This is really a side point for the rest of the blog post, but since the question came up I wanted to clarify it: Operations on integers in the Python program that the JIT is running don't all correspond 1-to-1 with the int_... operations in the traces. The int_... trace operations always operate on machine words. The Python int type supports arbitrarily large integers. PyPy will optimistically try to lower the operations on Python integers into machine word operations, but adds the necessary guards into the trace to make sure that overflow outside of the range of machine words is caught. In case one of these guards fails the interpreter switches to a big integer heap-allocated representation.)

Encoding Traces as Z3 formulas

The last blog post already contained the code to encode the results of individual trace operations into Z3 formulas, so we don't need to repeat that here. To encode traces of operations we introduce a Z3 variable for every operation in the trace and then call the z3_expression function for every single one of the operations in the trace.

For example, for the following trace:

[i1]
i2 = uint_rshift(i1, 32)
i3 = int_and(i2, 65535)
i4 = uint_rshift(i1, 48)
i5 = int_lshift(i4, 16)
i6 = int_or(i5, i3)
jump(i6, i2) # equal

We would get the Z3 formula:

z3.And(i2 == LShR(i1, 32),
       i3 == i2 & 65535,
       i4 == LShR(i1, 48),
       i5 == i4 << 16)

Usually we won't ask for the formula of the whole trace at once. Instead we go through the trace operation by operation and try to find inefficiencies in the current one we are looking at. Roughly like this (pseudo-)code:

def newvar(name):
    return z3.BitVec(name, INTEGER_WIDTH)

def find_inefficiencies(trace):
    solver = z3.Solver()
    var_to_z3var = {}
    for input_argument in trace.inputargs:
        var_to_z3var[input_argument] = newz3var(input_argument)
    for op in trace:
        var_to_z3var[op] = z3resultvar = newz3var(op.resultvarname)
        arg0 = op.args[0]
        z3arg0 = var_to_z3var[arg0]
        if len(op.args) == 2:
            arg1 = op.args[1]
            z3arg1 = var_to_z3var[arg1]
        else:
            z3arg1 = None
        res, valid_if = z3_expression(op.name, z3arg0, z3arg1)
        # checking for inefficiencies, see the next sections
        ...
        if ...:
            return "inefficient", op

        # not inefficient, assert op into the solver and continue with the next op
        solver.add(z3resultvar == res)
    return None # no inefficiency found

Identifying constant booleans with Z3

To get started finding inefficiencies in a trace, we can first focus on boolean variables. For every operation in the trace that returns a bool we can ask Z3 to prove that this variable must be always True or always False. Most of the time, neither of these proofs will succeed. But if Z3 manages to prove one of them, we know have found an ineffiency: instead of computing the boolean result (eg by executing a comparison) the JIT's optimizer could have replaced the operation with the corresponding boolean constant.

Here's an example of an inefficiency found that way: if x < y and y < z are both true, PyPy's JIT could conclude that x < z must also be true. However, currently the JIT cannot make that conclusion because it only reasons about the concrete ranges (lower and upper bounds) for every integer variable, but it has no way to remember anything about relationships between different variables. This kind of reasoning would quite often be useful to remove list/string bounds checks. Here's a talk about how LLVM does this (but it might be too heavyweight for a JIT setting).

Here are some more examples found that way:

x - 1 == x is always False
x - (x == -1) == -1 is always False. The pattern x - (x == -1) happens a lot in PyPy's hash computations: To be compatible with the CPython hashes we need to make sure that no object's hash is -1 (CPython uses -1 as an error value on the C level).

Here's pseudo-code for how to implement checking boolean operations for inefficiencies:

def find_inefficiencies(trace):
    ...
    for op in trace:
        ...
        res, valid_if = z3_expression(op.name, z3arg0, z3arg1)
        # check for boolean constant result
        if op.has_boolean_result():
            if prove(solver, res == 0):
                return "inefficient", op, 0
            if prove(solver, res == 1):
                return "inefficient", op, 1
        # checking for other inefficiencies, see the next sections
        ...

        # not inefficient, add op to the solver and continue with the next op
        solver.add(z3resultvar == res)
    return None # no inefficiency found

Identifying redundant operations

A more interesting class of redundancy is to try to find two operations in a trace that compute the same result. We can do that by asking Z3 to prove for each pair of different operations in the trace to prove that the result is always the same. If a previous operation returns the same result, the JIT could have reused that result instead of re-computing it, saving time. Doing this search for equivalent operations with Z3 is quadratic in the number of operations, but since traces have a maximum length it is not too bad in practice.

This is the real workhorse of my script so far, it's what finds most of the inefficiencies. Here's a few examples:

The very first and super useful example the script found is int_eq(b, 1) == b if b is known to be a boolean (ie and integer 0 or 1). I have already implemented this optimization in the JIT.
Similarly, int_and(b, 1) == b for booleans.
(x << 4) & -0xf == x << 4
((x >> 63) << 1) << 2) >> 3 == x >> 63. In general the JIT is quite bad at optimizing repeated shifts (the infrastructure for doing better with that is already in place, so this will be a relatively easy fix).
(x & 0xffffffff) | ((x >> 32) << 32) == x. Having the JIT optimize this would maybe require first recognizing that (x >> 32) << 32 can be expressed as a mask: (x & 0xffffffff00000000), and then using (x & c1) | (x & c2) == x & (c1 | c2)
A commonly occurring pattern is variations of this one: ((x & 1345) ^ 2048) - 2048 == x & 1345 (with different constants, of course). xor is add without carry, and x & 1345 does not have the bit 2048 set. Therefore the ^ 2048 is equivalent to + 2048, which the - 2048 cancels. More generally, if a & b == 0, then a + b == a | b == a ^ b. I don't understand at all why this appears so often in the traces, but I see variations of it a lot. LLVM can optimize this, but GCC can't, thanks to Andrew Pinski for filing the bug!

And here's some implementation pseudo-code again:

def find_inefficiencies(trace):
    ...
    for op in trace:
        ...
        res, valid_if = z3_expression(op.name, z3arg0, z3arg1)
        # check for boolean constant result
        ...
        # searching for redundant operations
        for previous_op in trace:
            if previous_op is op:
                break # done, reached the current op
            previous_op_z3var = var_to_z3var[previous_op]
            if prove(solver, previous_op_z3var == res):
                return "inefficient", op, previous_op
        ...
        # more code here later
        ...

        # not inefficient, add op to the solver and continue with the next op
        solver.add(z3resultvar == res)
    return None # no inefficiency found

Synthesizing more complicated constants with exists-forall

To find out whether some integer operations always return a constant result, we can't simply use the same trick as for those operations that return boolean results, because enumerating 2⁶⁴ possible constants and checking them all would take too long. Like in the last post, we can use z3.ForAll to find out whether Z3 can synthesize a constant for the result of an operation for us. If such a constant exists, the JIT could have removed the operation, and replaced it with the constant that Z3 provides.

Here a few examples of inefficiencies found this way:

(x ^ 1) ^ x == 1 (or, more generally: (x ^ y) ^ x == y)
if x | y == 0, it follows that x == 0 and y == 0
if x != MAXINT, then x + 1 > x

Implementing this is actually slightly annoying. The solver.add calls for non-inefficient ops add assertions to the solver, which are now confusing the z3.ForAll query. We could remove all assertion from the solver, then do the ForAll query, then add the assertions back. What I ended doing instead was instantiating a second solver object that I'm using for the ForAll queries, that remains empty the whole time.

def find_inefficiencies(trace):
    solver = z3.Solver()
    empty_solver = z3.Solver()
    var_to_z3var = {}
    ...
    for op in trace:
        ...
        res, valid_if = z3_expression(op.name, z3arg0, z3arg1)
        # check for boolean constant result
        ...
        # searching for redundant operations
        ...
        # checking for constant results
        constvar = z3.BitVec('find_const', INTEGER_WIDTH)
        condition = z3.ForAll(
            var_to_z3var.values(),
            z3.Implies(
                *solver.assertions(),
                expr == constvar
            )
        )
        if empty_solver.check(condition) == z3.sat:
            model = empty_solver.model()
            const = model[constvar].as_signed_long()
            return "inefficient", op, const

        # not inefficient, add op to the solver and continue with the next op
        solver.add(z3resultvar == res)
    return None # no inefficiency found

Minimization

Analyzing an inefficiency by hand in the context of a larger trace is quite tedious. Therefore I've implemented a (super inefficient) script to try to make the examples smaller. Here's how that works:

First throw out all the operations that occur after the inefficient operation in the trace.
Then we remove all "dead" operations, ie operations that don't have their results used (all the operations that we can analyze with Z3 are without side effects).
Now we try to remove every guard in the trace one by one and check afterwards, whether the resulting trace still has an inefficiency.
We also try to replace every single operation with a new argument to the trace, to see whether the inefficiency is still present.

The minimization process is sort of inefficient and I should probably be using shrinkray or C-Reduce instead. However, it seems to work well in practice and the runtime isn't too bad.

Results

So far I am using the JIT traces of three programs: 1) Booting Linux on the Pydrofoil RISC-V emulator, 2) booting Linux on the Pydrofoil ARM emulator, and 3) running the PyPy bootstrap process on top of PyPy.

I picked these programs because most Python programs don't contain interesting amounts of integer operations, and the traces of the emulators contain a lot of them. I also used the bootstrap process because I still wanted to try a big Python program and personally care about the runtime of this program a lot.

The script identifies 94 inefficiencies in the traces, a lot of them come from repeating patterns. My next steps will be to manually inspect them all, categorize them, and implement easy optimizations identified that way. I also want a way to sort the examples by execution count in the benchmarks, to get a feeling for which of them are most important.

I didn't investigate the full set of Python benchmarks that PyPy uses yet, because I don't expect them to contain interesting amounts of integer operations, but maybe I am wrong about that? Will have to try eventually.

Conclusion

This was again much easier to do than I would have expected! Given that I had the translation of trace ops to Z3 already in place, it was a matter of about a day's of programming to use this infrastructure to find the first problems and minimizing them.

Reusing the results of existing operations or replacing operations by constants can be seen as "zero-instruction superoptimization". I'll probably be rather busy for a while to add the missing optimizations identified by my simple script. But later extensions to actually synthesize one or several operations in the attempt to optimize the traces more and find more opportunities should be possible.

Finding inefficiencies in traces with Z3 is significantly less annoying and also less error-prone than just manually inspecting traces and trying to spot optimization opportunities.

Random Notes and Sources

Again, John's blog posts:

and papers:

I remembered recently that I had seen the approach of optimizing the traces of a tracing JIT with Z3 a long time ago, as part of the (now long dead, I think) SPUR project. There's a workshop paper from 2010 about this. SPUR was trying to use Z3 built into the actual JIT (as opposed to using Z3 only to find places where the regular optimizers could be improved). In addition to bitvectors, SPUR also used the Z3 support for arrays to model the C# heap and remove redundant stores. This is still another future extension for all the Z3 work I've been doing in the context of the PyPy JIT.

Finding Simple Rewrite Rules for the JIT with Z3

CF Bolz-Tereick — Fri, 12 Jul 2024 19:14:09 GMT

In June I was at the PLDI conference in Copenhagen to present a paper I co-authored with Max Bernstein. I also finally met John Regehr, who I'd been talking on social media for ages but had never met. John has been working on compiler correctness and better techniques for building compilers and optimizers since a very long time. The blog post Finding JIT Optimizer Bugs using SMT Solvers and Fuzzing was heavily inspired by this work. We talked a lot about his and his groups work on using Z3 for superoptimization and for finding missing optimizations. I have applied some of the things John told me about to the traces of PyPy's JIT, and wanted to blog about that. However, my draft felt quite hard to understand. Therefore I have now written this current post, to at least try to provide a somewhat gentler on-ramp to the topic.

In this post we will use the Python-API to Z3 to find local peephole rewrite rules for the operations in the intermediate representation of PyPy's tracing JIT. The code for this is simple enough that we can go through all of it.

The PyPy JIT produces traces of machine level instructions, which are optimized and then turned into machine code. The optimizer uses a number of approaches to make the traces more efficient. For integer operations it applies a number of arithmetic simplification rules rules, for example int_add(x, 0) -> x. When implementing these rules in the JIT there are two problems: How do we know that the rules are correct? And how do we know that we haven't forgotten any rules? We'll try to answer both of these, but the first one in particular.

We'll be using Z3, a satisfiability module theories (SMT) solver which has good bitvector support and most importantly an excellent Python API. We can use the solver to reason about bitvectors, which are how we will model machine integers.

To find rewrite rules, we will consider the binary operations (i.e. those taking two arguments) in PyPy traces that take and produce integers. The completely general form op(x, y) is not simplifiable on its own. But if either x == y or if one of the arguments is a constant, we can potentially simplify the operation into a simpler form. The results are either the variable x, or a (potentially different) constant. We'll ignore constant-folding where both arguments of the binary operation are constants. The possible results for a simplifiable binary operation are the variable x or another constant. This leaves the following patterns as possibilities:

op(x, x) == x
op(x, x) == c1
op(x, c1) == x
op(c1, x) == x
op(x, c1) == c2
op(c1, x) == c2

Our approach will be to take every single supported binary integer operation, instantiate all of these patterns, and try to ask Z3 whether the resulting simplification is valid for all values of x.

Quick intro to the Z3 Python-API

Here's a terminal session showing the use of the Z3 Python API:

>>>> import z3
>>>> # construct a Z3 bitvector variable of width 8, with name x:
>>>> x = z3.BitVec('x', 8)
>>>> # construct a more complicated formula by using operator overloading:
>>>> x + x
x + x
>>>> x + 1
x + 1

Z3 checks the "satisfiability" of a formula. This means that it tries to find an example set of concrete values for the variables that occur in a formula, such that the formula becomes true. Examples:

>>>> solver = z3.Solver()
>>>> solver.check(x * x == 3)
unsat
>>>> # meaning no x fulfils this property
>>>>
>>>> solver.check(x * x == 9)
sat
>>>> model = solver.model()
>>>> model
[x = 253]
>>>> model[x].as_signed_long()
-3
>>>> # 253 is the same as -3 in two's complement arithmetic with 8 bits

In order to use Z3 to prove something, we can ask Z3 to find counterexamples for the statement, meaning concrete values that would make the negation of the statement true:

>>>> solver.check(z3.Not(x ^ -1 == ~x))
unsat

The result unsat means that we just proved that x ^ -1 == ~x is true for all x, because there is no value for x that makes not (x ^ -1 == ~x) true (this works because -1 has all the bits set).

If we try to prove something incorrect in this way, the following happens:

>>>> solver.check(z3.Not(x ^ -1 == x))
sat

sat shows that x ^ -1 == x is (unsurprisingly) not always true, and we can ask for a counterexample:

>>>> solver.model()
[x = 0]

This way of proving this works because the check calls try to solve an (implicit) "exists" quantifier, over all the Z3 variables used in the formula. check will either return z3.unsat, which means that no concrete values make the formula true; or z3.sat, which means that you can get some concrete values that make the formula true by calling solver.model().

In math terms we prove things using check by de-Morgan's rules for quantifiers:

$$ \lnot \exists x: \lnot f(x) \implies \forall x: f(x) $$

Now that we've seen the basics of using the Z3 API on a few small examples, we'll use it in a bigger program.

Encoding the integer operations of RPython's JIT into Z3 formulas

Now we'll use the API to reason about the integer operations of the PyPy JIT intermediate representation (IR). The binary integer operations are:

opnames2 = [
"int_add",
"int_sub",
"int_mul",
"int_and",
"int_or",
"int_xor",
"int_eq",
"int_ne",
"int_lt",
"int_le",
"int_gt",
"int_ge",
"uint_lt",
"uint_le",
"uint_gt",
"uint_ge",
"int_lshift",
"int_rshift",
"uint_rshift",
"uint_mul_high",
"int_pydiv",
"int_pymod",
]

There's not much special about the integer operations. Like in LLVM, most of them are signedness-independent: int_add, int_sub, int_mul, ... work correctly for unsigned integers but also for two's-complement signed integers. Exceptions for that are order comparisons like int_lt etc. for which we have unsigned variants uint_lt etc. All operations that produce a boolean result return a full-width integer 0 or 1 (the PyPy JIT supports only word-sized integers in its intermediate representation)

In order to reason about the IR operations, some ground work:

import z3

INTEGER_WIDTH = 64
solver = z3.Solver()
solver.set("timeout", 10000) # milliseconds, ie 10s
xvar = z3.BitVec('x', INTEGER_WIDTH)
constvar = z3.BitVec('const', INTEGER_WIDTH)
constvar2 = z3.BitVec('const2', INTEGER_WIDTH)
TRUEBV = z3.BitVecVal(1, INTEGER_WIDTH)
FALSEBV = z3.BitVecVal(0, INTEGER_WIDTH)

And here's the a function to turn an integer IR operation of PyPy's JIT into Z3 formulas:

def z3_expression(opname, arg0, arg1=None):
    """ computes a tuple of (result, valid_if) of Z3 formulas. `result` is the
    formula representing the result of the operation, given argument formulas
    arg0 and arg1. `valid_if` is a pre-condition that must be true for the
    result to be meaningful. """
    result = None
    valid_if = True # the precondition is mostly True, with few exceptions
    if opname == "int_add":
        result = arg0 + arg1
    elif opname == "int_sub":
        result = arg0 - arg1
    elif opname == "int_mul":
        result = arg0 * arg1
    elif opname == "int_and":
        result = arg0 & arg1
    elif opname == "int_or":
        result = arg0 | arg1
    elif opname == "int_xor":
        result = arg0 ^ arg1
    elif opname == "int_eq":
        result = cond(arg0 == arg1)
    elif opname == "int_ne":
        result = cond(arg0 != arg1)
    elif opname == "int_lt":
        result = cond(arg0 < arg1)
    elif opname == "int_le":
        result = cond(arg0 <= arg1)
    elif opname == "int_gt":
        result = cond(arg0 > arg1)
    elif opname == "int_ge":
        result = cond(arg0 >= arg1)
    elif opname == "uint_lt":
        result = cond(z3.ULT(arg0, arg1))
    elif opname == "uint_le":
        result = cond(z3.ULE(arg0, arg1))
    elif opname == "uint_gt":
        result = cond(z3.UGT(arg0, arg1))
    elif opname == "uint_ge":
        result = cond(z3.UGE(arg0, arg1))
    elif opname == "int_lshift":
        result = arg0 << arg1
        valid_if = z3.And(arg1 >= 0, arg1 < INTEGER_WIDTH)
    elif opname == "int_rshift":
        result = arg0 << arg1
        valid_if = z3.And(arg1 >= 0, arg1 < INTEGER_WIDTH)
    elif opname == "uint_rshift":
        result = z3.LShR(arg0, arg1)
        valid_if = z3.And(arg1 >= 0, arg1 < INTEGER_WIDTH)
    elif opname == "uint_mul_high":
        # zero-extend args to 2*INTEGER_WIDTH bit, then multiply and extract
        # highest INTEGER_WIDTH bits
        zarg0 = z3.ZeroExt(INTEGER_WIDTH, arg0)
        zarg1 = z3.ZeroExt(INTEGER_WIDTH, arg1)
        result = z3.Extract(INTEGER_WIDTH * 2 - 1, INTEGER_WIDTH, zarg0 * zarg1)
    elif opname == "int_pydiv":
        valid_if = arg1 != 0
        r = arg0 / arg1
        psubx = r * arg1 - arg0
        result = r + (z3.If(arg1 < 0, psubx, -psubx) >> (INTEGER_WIDTH - 1))
    elif opname == "int_pymod":
        valid_if = arg1 != 0
        r = arg0 % arg1
        result = r + (arg1 & z3.If(arg1 < 0, -r, r) >> (INTEGER_WIDTH - 1))
    elif opname == "int_is_true":
        result = cond(arg0 != FALSEBV)
    elif opname == "int_is_zero":
        result = cond(arg0 == FALSEBV)
    elif opname == "int_neg":
        result = -arg0
    elif opname == "int_invert":
        result = ~arg0
    else:
        assert 0, "unknown operation " + opname
    return result, valid_if

def cond(z3expr):
    """ helper function to turn a Z3 boolean result z3expr into a 1 or 0
    bitvector, using z3.If """
    return z3.If(z3expr, TRUEBV, FALSEBV)

We map the semantics of a PyPy JIT operation to Z3 with the z3_expression function. It takes the name of a JIT operation and its two (or one) arguments into a pair of Z3 formulas, result and valid_if. The resulting formulas are constructed with the operator overloading of Z3 variables/formulas.

The first element result of the result of z3_expression represents the result of performing the operation. valid_if is a bool that represents a condition that needs to be True in order for the result of the operation to be defined. E.g. int_pydiv(a, b) is only valid if b != 0. Most operations are always valid, so they return True as that condition (we'll ignore valid_if for a bit, but it will become more relevant further down in the post).

We can define a helper function to prove things by finding counterexamples:

def prove(cond):
    """ Try to prove a condition cond by searching for counterexamples of its negation. """
    z3res = solver.check(z3.Not(cond))
    if z3res == z3.unsat:
        return True
    elif z3res == z3.unknown: # eg on timeout
        return False
    elif z3res == z3.sat:
        return False
    assert 0, "should be unreachable"

Finding rewrite rules

Now we can start finding our first rewrite rules, following the first pattern op(x, x) -> x. We do this by iterating over all the supported binary operation names, getting the z3 expression for op(x, x) and then asking Z3 to prove op(x, x) == x.

for opname in opnames2:
    result, valid_if = z3_expression(opname, xvar, xvar)
    if prove(result == xvar):
        print(f"{opname}(x, x) -> x, {result}")

This yields the simplifications:

int_and(x, x) -> x
int_or(x, x) -> x

Synthesizing constants

Supporting the next patterns is harder: op(x, x) == c1, op(x, c1) == x, and op(c1, x) == x. We don't know which constants to pick to try to get Z3 to prove the equality. We could iterate over common constants like 0, 1, MAXINT, etc, or even over all the 256 values for a bitvector of length 8. However, we will instead ask Z3 to find the constants for us too.

This can be done by using quantifiers, in this case z3.ForAll. The query we pose to Z3 is "does there exist a constant c1 such that for all x the following is true: op(x, c1) == x? Note that the constant c1 is not necessarily unique, there could be many of them. We generate several matching constant, and add that they must be different to the condition of the second and further queries.

We can express this in a helper function:

def find_constant(z3expr, number_of_results=5):
    condition = z3.ForAll(
        [xvar],
        z3expr
    )
    for i in range(number_of_results):
        checkres = solver.check(condition)
        if checkres == z3.sat:
            # if a solver check succeeds, we can ask for a model, which is
            # concrete values for the variables constvar
            model = solver.model()
            const = model[constvar].as_signed_long()
            yield const
            # make sure we don't generate the same constant again on the
            # next call
            condition = z3.And(constvar != const, condition)
        else:
            # no (more) constants found
            break

We can use this new function for the three mentioned patterns:

# try to find constants for op(x, x) == c
for opname in opnames2:
    result, valid_if = z3_expression(opname, xvar, xvar)
    for const in find_constant(result == constvar):
        print(f"{opname}(x, x) -> {const}")
# try to find constants for op(x, c) == x and op(c, x) == x
for opname in opnames2:
    result, valid_if = z3_expression(opname, xvar, constvar)
    for const in find_constant(result == xvar):
        print(f"{opname}(x, {const}) -> x")
    result, valid_if = z3_expression(opname, constvar, xvar)
    for const in find_constant(result == xvar):
        print(f"{opname}({const}, x) -> x")
# this code is not quite correct, we'll correct it later

Together this yields the following new simplifications:

# careful, these are not all correct!
int_sub(x, x) -> 0
int_xor(x, x) -> 0
int_eq(x, x) -> 1
int_ne(x, x) -> 0
int_lt(x, x) -> 0
int_le(x, x) -> 1
int_gt(x, x) -> 0
int_ge(x, x) -> 1
uint_lt(x, x) -> 0
uint_le(x, x) -> 1
uint_gt(x, x) -> 0
uint_ge(x, x) -> 1
uint_rshift(x, x) -> 0
int_pymod(x, x) -> 0
int_add(x, 0) -> x
int_add(0, x) -> x
int_sub(x, 0) -> x
int_mul(x, 1) -> x
int_mul(1, x) -> x
int_and(x, -1) -> x
int_and(-1, x) -> x
int_or(x, 0) -> x
int_or(0, x) -> x
int_xor(x, 0) -> x
int_xor(0, x) -> x
int_lshift(x, 0) -> x
int_rshift(x, 0) -> x
uint_rshift(x, 0) -> x
int_pydiv(x, 1) -> x
int_pymod(x, 0) -> x

Most of these look good at first glance, but the last one reveals a problem: we've been ignoring the valid_if expression up to now. We can stop doing that by changing the code like this, which adds z3.And(valid_if, ...) to the argument of the calls to find_constant:

# try to find constants for op(x, x) == c, op(x, c) == x and op(c, x) == x
for opname in opnames2:
    result, valid_if = z3_expression(opname, xvar, xvar)
    for const in find_constant(z3.And(valid_if, result == constvar)):
        print(f"{opname}(x, x) -> {const}")
# try to find constants for op(x, c) == x and op(c, x) == x
for opname in opnames2:
    result, valid_if = z3_expression(opname, xvar, constvar)
    for const in find_constant(z3.And(result == xvar, valid_if)):
        print(f"{opname}(x, {const}) -> x")
    result, valid_if = z3_expression(opname, constvar, xvar)
    for const in find_constant(z3.And(result == xvar, valid_if)):
        print(f"{opname}({const}, x) -> x")

And we get this list instead:

int_sub(x, x) -> 0
int_xor(x, x) -> 0
int_eq(x, x) -> 1
int_ne(x, x) -> 0
int_lt(x, x) -> 0
int_le(x, x) -> 1
int_gt(x, x) -> 0
int_ge(x, x) -> 1
uint_lt(x, x) -> 0
uint_le(x, x) -> 1
uint_gt(x, x) -> 0
uint_ge(x, x) -> 1
int_add(x, 0) -> x
int_add(0, x) -> x
int_sub(x, 0) -> x
int_mul(x, 1) -> x
int_mul(1, x) -> x
int_and(x, -1) -> x
int_and(-1, x) -> x
int_or(x, 0) -> x
int_or(0, x) -> x
int_xor(x, 0) -> x
int_xor(0, x) -> x
int_lshift(x, 0) -> x
int_rshift(x, 0) -> x
uint_rshift(x, 0) -> x
int_pydiv(x, 1) -> x

Synthesizing two constants

For the patterns op(x, c1) == c2 and op(c1, x) == c2 we need to synthesize two constants. We can again write a helper method for that:

def find_2consts(z3expr, number_of_results=5):
    condition = z3.ForAll(
        [xvar],
        z3expr
    )
    for i in range(number_of_results):
        checkres = solver.check(condition)
        if checkres == z3.sat:
            model = solver.model()
            const = model[constvar].as_signed_long()
            const2 = model[constvar2].as_signed_long()
            yield const, const2
            condition = z3.And(z3.Or(constvar != const, constvar2 != const2), condition)
        else:
            return

And then use it like this:

for opname in opnames2:
    # try to find constants c1, c2 such that op(c1, x) -> c2
    result, valid_if = z3_expression(opname, constvar, xvar)
    consts = find_2consts(z3.And(valid_if, result == constvar2))
    for const, const2 in consts:
        print(f"{opname}({const}, x) -> {const2}")
    # try to find constants c1, c2 such that op(x, c1) -> c2
    result, valid_if = z3_expression(opname, xvar, constvar)
    consts = find_2consts(z3.And(valid_if, result == constvar2))
    for const, const2 in consts:
        print("%s(x, %s) -> %s" % (opname, const, const2))

Which yields some straightforward simplifications:

int_mul(0, x) -> 0
int_mul(x, 0) -> 0
int_and(0, x) -> 0
int_and(x, 0) -> 0
uint_lt(x, 0) -> 0
uint_le(0, x) -> 1
uint_gt(0, x) -> 0
uint_ge(x, 0) -> 1
int_lshift(0, x) -> 0
int_rshift(0, x) -> 0
uint_rshift(0, x) -> 0
uint_mul_high(0, x) -> 0
uint_mul_high(1, x) -> 0
uint_mul_high(x, 0) -> 0
uint_mul_high(x, 1) -> 0
int_pymod(x, 1) -> 0
int_pymod(x, -1) -> 0

A few require a bit more thinking:

int_or(-1, x) -> -1
int_or(x, -1) -> -1

The are true because in two's complement, -1 has all bits set.

The following ones require recognizing that -9223372036854775808 == -2**63 is the most negative signed 64-bit integer, and 9223372036854775807 == 2 ** 63 - 1 is the most positive one:

int_lt(9223372036854775807, x) -> 0
int_lt(x, -9223372036854775808) -> 0
int_le(-9223372036854775808, x) -> 1
int_le(x, 9223372036854775807) -> 1
int_gt(-9223372036854775808, x) -> 0
int_gt(x, 9223372036854775807) -> 0
int_ge(9223372036854775807, x) -> 1
int_ge(x, -9223372036854775808) -> 1

The following ones are true because the bitpattern for -1 is the largest unsigned number:

uint_lt(-1, x) -> 0
uint_le(x, -1) -> 1
uint_gt(x, -1) -> 0
uint_ge(-1, x) -> 1

Strength Reductions

All the patterns so far only had a variable or a constant on the target of the rewrite. We can also use the machinery to do strengh-reductions where we generate a single-argument operation op1(x) for input operations op(x, c1) or op(c1, x). To achieve this, we try all combinations of binary and unary operations. (We won't consider strength reductions where a binary operation gets turned into a "cheaper" other binary operation here.)

opnames1 = [
"int_is_true",
"int_is_zero",
"int_neg",
"int_invert",
]

for opname in opnames2:
    for opname1 in opnames1:
        result, valid_if = z3_expression(opname, xvar, constvar)
        # try to find a constant op(x, c) == g(x)
        result1, valid_if1 = z3_expression(opname1, xvar)
        consts = find_constant(z3.And(valid_if, valid_if1, result == result1))
        for const in consts:
            print(f"{opname}(x, {const}) -> {opname1}(x)")

        # try to find a constant op(c, x) == g(x)
        result, valid_if = z3_expression(opname, constvar, xvar)
        result1, valid_if1 = z3_expression(opname1, xvar)
        consts = find_constant(z3.And(valid_if, valid_if1, result == result1))
        for const in consts:
            print(f"{opname}({const}, x) -> {opname1}(x)")

Which yields the following new simplifications:

int_sub(0, x) -> int_neg(x)
int_sub(-1, x) -> int_invert(x)
int_mul(x, -1) -> int_neg(x)
int_mul(-1, x) -> int_neg(x)
int_xor(x, -1) -> int_invert(x)
int_xor(-1, x) -> int_invert(x)
int_eq(x, 0) -> int_is_zero(x)
int_eq(0, x) -> int_is_zero(x)
int_ne(x, 0) -> int_is_true(x)
int_ne(0, x) -> int_is_true(x)
uint_lt(0, x) -> int_is_true(x)
uint_lt(x, 1) -> int_is_zero(x)
uint_le(1, x) -> int_is_true(x)
uint_le(x, 0) -> int_is_zero(x)
uint_gt(x, 0) -> int_is_true(x)
uint_gt(1, x) -> int_is_zero(x)
uint_ge(x, 1) -> int_is_true(x)
uint_ge(0, x) -> int_is_zero(x)
int_pydiv(x, -1) -> int_neg(x)

Conclusions

With not very little code we managed to generate a whole lot of local simplifications for integer operations in the IR of PyPy's JIT. The rules discovered that way are "simple", in the sense that they only require looking at a single instruction, and not where the arguments of that instruction came from. They also don't require any knowledge about the properties of the arguments of the instructions (e.g. that they are positive).

The rewrites in this post have mostly been in PyPy's JIT already. But now we mechanically confirmed that they are correct. I've also added the remaining useful looking ones, in particular int_eq(x, 0) -> int_is_zero(x) etc.

If we wanted to scale this approach up, we would have to work much harder! There are a bunch of problems that come with generalizing the approach to looking at sequences of instructions:

Combinatorial explosion: if we look at sequences of instructions, we very quickly get a combinatorial explosion and it becomes untractable to try all combinations.
Finding non-minimal patterns: Some complicated simplifications can be instances of simpler ones. For example, because int_add(x, 0) -> x, it's also true that int_add(int_sub(x, y), 0) -> int_sub(x, y). If we simply generate all possible sequences, we will find the latter simplification rule, which we would usually not care about.
Unclear usefulness: if we simply generate all rewrites up to a certain number of instructions, we will get a lot of patterns that are useless in the sense that they typically aren't found in realistic programs. It would be much better to somehow focus on the patterns that real benchmarks are using.

In the next blog post I'll discuss an alternative approach to simply generating all possible sequences of instructions, that tries to address these problems. This works by analyzing the real traces of benchmarks and mining those for inefficiencies, which only shows problems that occur in actual programs.

Sources

I've been re-reading a lot of blog posts from John's blog:

but also papers:

Another of my favorite blogs has been Philipp Zucker's blog in the last year or two, lots of excellent posts about/using Z3 on there.

Finding JIT Optimizer Bugs using SMT Solvers and Fuzzing

CF Bolz-Tereick — Sun, 11 Dec 2022 18:00:00 GMT

In this blog post I want to describe a recent bug finding technique that I've added to the PyPy JIT testing infrastructure. This technique uses the Z3 theorem prover to find bugs in the optimizer of PyPy's JIT, in particular its integer operation optimizations. The approach is based on things I have learned from John Regehr's blog (this post is a good first one to read), Twitter, and on his (et al) paper Alive2: Bounded Translation Validation for LLVM. The work was triggered by a recent miscompilation bug my current bachelor student Nico Rittinghaus found.

Background: Python Integers in the PyPy JIT

The optimizer of PyPy's JITs operates on traces, which are linear sequences of instructions with guards. The instructions in the traces operate on different machine-level data types, machine integers, doubles, pointers, bools, etc. In this post we'll be mostly concerned with machine integers.

To given some wider context I'll explain a bit how Python ints in the user code relate to the types that are used in traces when the PyPy Python implementation is used. When PyPy turns a regular Python 3 function into a trace, there is a lot of work happening in the JIT frontend to try to observe and infer the types that the Python function concretely uses at runtime. The traces are generated under these typing assumptions. Therefore, code that uses ints in the Python code can typically be translated into traces that operate on machine integers. In order to make sure that the Python integer semantics are upheld, many of the operations in the traces need to check that the integer results of some operations still fit into a machine integer. If that is not the case (a rare situation for most programs), the trace is left via a guard, execution falls back to the interpreter, and there a big integer representation is chosen for the too big value (the big integer representation is done via a pointer and some storage on the heap).

All of this machinery is not going to be too relevant for the rest of the post. For the post it's important to know that trace instructions operate on machine integers and other low-level types, and some of the operations can optionally check whether the results still fit into a machine integer. These trace operations are improved by the optimizer, which tries to transform the trace into one that behaves the same, but is less costly to execute.

Background: Bounds Analysis in PyPy's JIT

The optimizer of PyPy's JIT has an analysis based on abstract interpretation that tries to find out whether the integer values stored in a variable are actually not using the full 64 bit (or 32 bit) range, but instead fit into some smaller range. This means that for every integer variable x in a trace, the JIT compiler tracks upper and lower bounds of the runtime value of that variable: a range [a, b] such that for every concrete runtime value v that gets stored in variable x, a <= v <= b must be true. a and b start out as the most general MININT and MAXINT, but sometimes there is extra information that makes it possible to improve these known bounds, and that is often useful to optimize the code.

A typical example is that the JIT knows that the length of a string is non-negative, so for this kind of code: x = len(s) where s is a string, x gets a range [0, MAXINT] assigned. With this information we could for example remove a check x + 10 < 0 completely, because it can never be true.

The bounds information is useful for optimization, but the analysis of the bounds is also a source of bugs in the JIT, because the reasoning is often subtle and easy to get wrong in corner cases. We already use a number of testing techniques to try to make sure that it is correct. A simple one is property-based testing using Hypothesis on the operations on bounds. Even though Hypothesis is fantastic, it unfortunately does not catch absolutely all the bugs even if we'd like it too, as we'll see in the next section.

Motivation: A JIT Miscompilation

I am currently supervising a Bachelor thesis by Nico Rittinghaus, who is extending the integer analysis in the JIT. He'll probably write a separate blog post about that soon. In the process of his work, the current bounds analysis code got a lot of scrutiny, and we found out that one of the unit tests of the bounds analysis was actually incorrect, and the example code in that unit test was optimized incorrectly. This case of incorrect optimization is not a big deal for regular Python code, because it involved a "wrapping integer addition operation", i.e. one where overflowing results just wrap around to negative values. All the additions and other arithmetic operations that the PyPy Python frontend generates actually have overflow checks (to be able to switch to a big integer representation if needed). However, it's still possible to trigger the problem with the __pypy__.intop.int_add API which is a function that exposes wraparound arithmetic on Python ints.

Here's the miscompilation. The JIT optimizes the following function:

import __pypy__

def wrong(x):
    a = __pypy__.intop.int_add(x, 10)
    if a < 15:
        if x < 6:
            return 0
        return 1
    return 2

Into the following code:

import __pypy__

def wrong(x):
    a = __pypy__.intop.int_add(x, 10)
    if a < 15:
        return 0
    return 2

Basically the faulty reasoning of the JIT looks like this: if int_add(x, 10) < 15 then it must follow that x < 5, which is stronger than x < 6, so the second if is always true. This sounds good, but is actually wrong if the addition + 10 wrapped around. So if x == MAXINT, then int_add(x, 10) == MININT + 9 < 15. But MAXINT < 5 is not correct.

Note how the same reasoning with overflow-checking addition is correct! If x + 10 < 15 and the + didn't overflow, then indeed x < 6. And if your mind bends starting to think about all this, you understand some of the difficulty of getting the JIT correct in this area.

How could we have avoided this bug?

One exercise I try to do after finding bugs is to reflect on ways that the bug could have been avoided. I think this is particularly important in the JIT, where bugs are potentially really annoying to find and can cause very strange behaviour in basically arbitrary Python code.

It's easy to always answer this question with "try to think more carefully when working", but that approach cannot be relied on in complicated situations, because humans don't concentrate perfectly for long stretches of time.

A situation-specific problem I identified was the bad design of the range analysis API. A range is not just represented by two numbers, instead it's two numbers and two bools that are supposed to represent that some operation did or did not underflow/overflow. The meaning of these bools was quite hard to grasp and easy to get wrong, so probably they should never have been introduced in the first place (and my bugfix indeed removed them).

But in the rest of this blog post I want to talk about another, systematic approach that can be applied to the problem of mis-optimizations of integer operations, and that is done by applying an SMT solver to the problem.

An SMT solver (Satisfyability Modulo Theories) is a tool that can be used to find out whether mathematical formulas are "satisfiable", i.e. whether some chosen set of inputs exists that will make the formulas evaluate to true. SMT solvers are commonly used in a wide range of CS applications including program correctness proofs, program synthesis, etc. The most widely known one is probably Z3 by Microsoft Research which has the nice advantage of coming with an easy-to-use Python binding.

Going into this I basically knew next to nothing about SMT solvers (despite having been embedded in a formal methods research group for years!) so it was an interesting new world to learn about.

As briefly mentioned in the introduction, the approach I took followed a similar (but much more properly executed) one applied to LLVM operations, called Alive2. Krister Waldfridsson has done similar work for GCC recently, described on his blog.

Z3 Proof of Concept

The first thing I did was to try to get Z3 find the above bug, by encoding the input program into an SMT formula by hand and trying to get Z3 to prove the condition that the JIT thinks is always true. The Z3 code for this looks as follows:

from z3 import BitVec, Implies, prove
x = BitVec('x', 64)
a = x + 10
cond1 = a < 15
cond2 = x < 6
prove(Implies(cond1, cond2))

Here, x is defined to be a bit vector variable of width 64, which is a datatype that can be used to represent bounded machine integers. Addition on bit vectors performs wraparound arithmetic, like the __pypy__.intop.int_add call in the original code. The JIT optimized the second condition away, so essentially it was convinced that the first condition implies the second one. The above snippet tries to get Z3 to confirm this.

When run, the above program prints:

counterexample
[x = 9223372036854775803]

Which shows the bug. As a small side-note, I thought it was cool that the process of "proving" something in Z3 basically means trying to find an example for the negation of the formula. If no counterexample can be found for the negation, the original formula is true. If the original formula turns out to be false (like here) we get a nice example that shows the problem to go with it.

It's not realistic to hand-translate all the hundreds of unit-tests into Z3 formulas and then ask Z3 to prove the optimizations. Instead, we want to have a program that does this for us.

SMT Checking of the JIT Optimizer

What we want from this program is the following: given an unoptimized trace and its optimized version, we want to use Z3 to check whether the optimized trace behaves identically to the unoptimized one. One question is what "behaves identically" means. What we care about is the outputs of the trace being the same values, no matter how they are computed. Also, for every guard we want to make sure that it fails in identical ways in the optimized and unoptimized versions. A guard is only allowed to be optimized away if it can never fail. The code that comes after a guard can assume that the guard has not failed, because otherwise execution would have left the trace. All of this should be true regardless for the values of the input variables of the trace.

So in order to check that the two traces are behaving identically, we do the following:

We create Z3 variables for every input variable. We use the same input variables both for the unoptimized as well as the optimized trace.
We align the two traces at the corresponding guards. Thankfully the optimizer keeps track of which optimized guard corresponds to which unoptimized input guard.
All the operations before a guard are translated into Z3 formulas, for both versions of the trace.
For two corresponding guards, we ask Z3 to prove that the guard conditions are identical.
For a guard that was optimized away we ask Z3 to prove that the condition is always true.
After a guard, we tell Z3 that from now on it can assume that the guard condition is true.
We repeat this, guard for guard, until we reach the end of the trace. There, we ask Z3 to prove that the output variables in the unoptimized trace and the optimized trace are identical (every trace can return one or many values).

I implemented this, it's not a lot of code, basically a couple of hundred lines of (somewhat hacky) Python code. So far I only support integer operations. Here are some parts of the code to give you a flavor of what this looks like.

This is the code that translates operations into Z3 formulas:

def add_to_solver(self, ops, state):
    for op in ops:
        if op.type != 'v': # is it an operation with a result
            res = self.newvar(op)
        else: # or does it return void
            res = None

       # ...

        # convert arguments
        if op.numargs() == 1:
            arg0 = self.convertarg(op, 0)
        elif op.numargs() == 2:
            arg0 = self.convertarg(op, 0)
            arg1 = self.convertarg(op, 1)

        # compute results
        if opname == "int_add":
            expr = arg0 + arg1
        elif opname == "int_sub":
            expr = arg0 - arg1
        elif opname == "int_mul":
            expr = arg0 * arg1
        elif opname == "int_and":
            expr = arg0 & arg1
        elif opname == "int_or":
            expr = arg0 | arg1
        elif opname == "int_xor":
            expr = arg0 ^ arg1

        # ...  more operations, some shown below

        self.solver.add(res == expr)

New Z3 variables are defined by the helper function newvar, which adds the operation to a dictionary box_to_z3 mapping boxes (=variables) to Z3 variables. Due to the SSA property that traces have, a variable must be defined before its first use.

Here's what newvar looks like (LONG_BIT is a constant that is either 64 or 32, depending on the target architecture):

def newvar(self, box, repr=None):
    # ... some logic around making the string representation
    # somewhat nicer omitted
    result = z3.BitVec(repr, LONG_BIT)
    self.box_to_z3[box] = result
    return result

The convert method turns an operation argument (either a constant or a variable) into a Z3 formula (either a constant bit vector or an already defined Z3 variable). convertarg is a helper function that takes an operation, reads its nth argument and converts it.

def convert(self, box):
    if isinstance(box, ConstInt):
        return z3.BitVecVal(box.getint(), LONG_BIT)
    return self.box_to_z3[box]

def convertarg(self, box, arg):
    return self.convert(box.getarg(arg))

The lookup of variables in box_to_z3 that convert does cannot fail, because the variable must have been defined before use.

Comparisons return the bit vector 0 or bit vector 1, we use a helper function cond to turn the Z3 truth value of the comparison into a bit vector:

def cond(self, z3expr):
    return z3.If(z3expr, TRUEBV, FALSEBV)


def add_to_solver(self, ops, state):
        # ... start as above

        # more cases
        elif opname == "int_eq":
            expr = self.cond(arg0 == arg1)
        elif opname == "int_ne":
            expr = self.cond(arg0 != arg1)
        elif opname == "int_lt":
            expr = self.cond(arg0 < arg1)
        elif opname == "int_le":
            expr = self.cond(arg0 <= arg1)
        elif opname == "int_gt":
            expr = self.cond(arg0 > arg1)
        elif opname == "int_ge":
            expr = self.cond(arg0 >= arg1)
        elif opname == "int_is_true":
            expr = self.cond(arg0 != FALSEBV)
        elif opname == "uint_lt":
            expr = self.cond(z3.ULT(arg0, arg1))
        elif opname == "uint_le":
            expr = self.cond(z3.ULE(arg0, arg1))
        elif opname == "uint_gt":
            expr = self.cond(z3.UGT(arg0, arg1))
        elif opname == "uint_ge":
            expr = self.cond(z3.UGE(arg0, arg1))
        elif opname == "int_is_zero":
            expr = self.cond(arg0 == FALSEBV)

        # ... rest as above

So basically for every trace operation that operates on integers I had to give a translation into Z3 formulas, which is mostly straightforward.

Guard operations get converted into a Z3 boolean by their own helper function, which looks like this:

def guard_to_condition(self, guard, state):
    opname = guard.getopname()
    if opname == "guard_true":
        return self.convertarg(guard, 0) == TRUEBV
    elif opname == "guard_false":
        return self.convertarg(guard, 0) == FALSEBV
    elif opname == "guard_value":
        return self.convertarg(guard, 0) == self.convertarg(guard, 1)

    # ... some more exist, shown below

Some operations are a bit trickier. An important example in the context of this blog post are integer operations that check for overflow. The overflow operations return a result, but also a boolean whether the operation overflowed or not.

def add_to_solver(self, ops, state):

        # ... more cases

        elif opname == "int_add_ovf":
            expr = arg0 + arg1
            m = z3.SignExt(LONG_BIT, arg0) + z3.SignExt(LONG_BIT, arg1)
            state.no_ovf = m == z3.SignExt(LONG_BIT, expr)
        elif opname == "int_sub_ovf":
            expr = arg0 - arg1
            m = z3.SignExt(LONG_BIT, arg0) - z3.SignExt(LONG_BIT, arg1)
            state.no_ovf = m == z3.SignExt(LONG_BIT, expr)
        elif opname == "int_mul_ovf":
            expr = arg0 * arg1
            m = z3.SignExt(LONG_BIT, arg0) * z3.SignExt(LONG_BIT, arg1)
            state.no_ovf = m == z3.SignExt(LONG_BIT, expr)

        # ...

The boolean is computed by comparing the result of the bit vector operation with the result of converting the input bit vectors into an abstract (arbitrary precision) integer and the result back to bit vectors. Let's go through the addition case step by step, the other cases work analogously.

The addition in the first elif that computes expr is an addition on bit vectors, therefore it is performing wraparound arithmetic. z3.SignExt(LONG_BIT, arg0) sign-extends arg0 from a bit vector of LONG_BIT bits to an abstract, arbitrary precision integer. The addition in the second line is therefore an addition between abstract integers, so it will never overflow and just compute the correct result as an integer.

The condition to check for overflow is now: if the results of the two different ways to do the addition are the same, then overflow did not occur. So in order to compute state.no_ovf in the addition case the code converts the result of the bit vector wraparound addition to an abstract integer (using SignExt again), and then compares that to the integer result.

This boolean can then be checked by the guard operations guard_no_overflow and guard_overflow.

def guard_to_condition(self, guard, state):

    # ... more cases

    elif opname == "guard_no_overflow":
        assert state.no_ovf is not None
        return state.no_ovf
    elif opname == "guard_overflow":
        assert state.no_ovf is not None
        return z3.Not(state.no_ovf)

    # ... more cases

Finding the Bug, Again

Let's actually make all of this more concrete by applying it to the trace of our original bug. The input trace and the incorrectly optimized trace for that look like this (differences highlighted):

# input                       # optimized
[i0]                          [i0]
i1 = int_add(i0, 10)          i1 = int_add(i0, 10)
i2 = int_lt(i1, 15)           i2 = int_lt(i1, 15)
guard_true(i2)                guard_true(i2)
i3 = int_lt(i0, 6)            jump(0)
guard_true(i3)
jump(0)

Note that the trace represents just one of the paths through the control flow graph of the original function, which is typical for tracing JITs (the other paths could incrementally get added later).

The first guards in both these traces correspond to each other, so the first chunks to check are the first three operations (lines 1-4). Those operations don't get changed by the optimizer at all.

These two identical traces get translated to the following Z3 formulas:

i1unoptimized == input_i0 + 10
i2unoptimized == If(i1unoptimized < 15, 1, 0)
i1optimized == input_i0 + 10
i2optimized == If(i1optimized < 15, 1, 0)

To check that the two corresponding guards are the same, the solver is asked to prove that (i2unoptimized == 1) == (i2optimized == 1). This is correct, because the formulas for i2unoptimized and i2optimized are completely identical.

After checking that the guards behave the same, we add the knowledge to the solver that the guards passed. So the Z3 formulas become:

i1unoptimized == input_i0 + 10
i2unoptimized == If(i1unoptimized < 15, 1, 0)
i1optimized == input_i0 + 10
i2optimized == If(i1optimized < 15, 1, 0)
i1optimized == 1
i2optimized == 1

Now we continue with the remaining operations of the two traces (lines 6-8).

We start by adding the int_lt operation in the unoptimized trace to the Z3 formulas:

...
i3unoptimized == If(input_i0 < 6, 1, 0)

Because the second guard was optimized away, we need to ask Z3 to prove that i3unoptimized == 1 is always true, which fails and gives the following counterexample:

input_i0 = 9223372036854775800
i1unoptimized = 9223372036854775810
i2unoptimized = 0
i1optimized = 9223372036854775810
i2optimized = 1
i3unoptimized = 0

Thus demonstrating the bug. The fact that the Z3-based equivalence check also managed to find the original motivating bug without manually translating it to a formula is a good confirmation that the approach works.

Second bug

So with this code I applied the Z3-based equivalence check to all our optimizer unit tests. In addition to the bug we've been discussing the whole post, it also found another buggy test! I had found it too by hand by staring at all the tests in the process of writing all the Z3 infrastructure, but it was still a good confirmation that the process worked. This bug was in the range analysis for int_neg, integer negation. It failed to account that -MININT == MININT and therefore did a mis-optimization along the following lines:

import __pypy__

def wrong(x):
    a = __pypy__.intop.int_sub(0, x)
    if a < 0:
        if x > 0:
            return 0
        return 1
    return 2

Which was wrongly optimized into:

import __pypy__

def wrong(x):
    a = __pypy__.intop.int_sub(0, x)
    if a < 0:
        return 0
    return 2

This is wrong precisely for x == MININT.

Generating Random Traces

These two bugs were the only two that the Z3 checker found for existing unit tests. To try to find some more bugs I combined PyPy's existing random trace generator with the Z3 optimization checker. The random trace generator has so far been mostly used to find bugs in the machine code backends, particularly also in the register allocator. So far we haven't used it with our optimizer, but my experiments show that we should have!

I'm going to describe a little bit how the random trace generator works. It's actually not that complicated, but there's one neat trick to it.

The basic idea is straightforward, it starts out with an empty trace with a random number of input variables. Then it adds some number of operations to the trace, either regular operations or guards. Every operation takes already existing variables as input.

The neat trick is that our random trace generator keeps a concrete random example value for every one of the input variables, and an example result for every operation. In this way, it is possible to generate guards that are consistent with the example values to ensure that running the trace to its end is possible with at least one set of values.

Here's an example random trace that is generated, together with the random example inputs and the results of every operation at the end of every line:

[i0, i1, i2, i3, i4, i5] # example values: 9, 11, -8, -95, 46, 57
i6 = int_add_ovf(i3, i0) # -86
guard_no_overflow()
i7 = int_sub(i2, -35/ci) # 27
i8 = uint_ge(i3, i5) # 1
guard_true(i8)
i9 = int_lt(i7, i8) # 0
i10 = int_mul_ovf(34/ci, i7) # 918
guard_no_overflow()
i11 = int_and(i10, 63/ci) # 22
i12 = int_rshift(i3, i11) # -1
i13 = int_is_zero(i7) # 0
i14 = int_is_true(i13) # 0
guard_false(i13)
i15 = int_lt(i8, i4) # 1
i16 = int_and(i6, i0) # 8
i17 = uint_ge(i6, -6/ci) # 0
finish()

Note how every guard generated is true for the example values.

I have been running this combination of random trace generation and Z3 checking for many nights and it has found some bugs, which I'll describe in the next section. It should probably be run for a lot longer, but still a useful exercise already.

In this mode, I'm giving every Z3 call a time limit to make sure that the random tests don't just take arbitrarily long. This means that asking Z3 to prove something can have three outcomes, either it's proved, or Z3 finds a counterexample, or Z3 times out.

Bugs Found

In addition to the two bugs I've already described, I'll briefly list the additional bugs that were found by optimizing random traces and then trying to prove the equivalence with Z3.

Most of the bugs were actually identified by optimizing random traces alone, not by the Z3 component. They manifested as assert failures in the JIT compiler.

The JIT concluded after 12 == int_mul(x, 12) that x == 1, which is incorrect if overflow occurred (a counterexample is 0x8000000000000001).
An amusing bug, where from 0 == int_lshift(0x1000000000000000, x) with x <= 0 <= 15, the JIT concluded that 0x1000000000000000 == 0, triggering an assert. This wrong conclusion was again caused by not taking the possibility of overflow into account.
A corner case in an optimization for chained integer additions with a constant, where in complex enough expressions, the wrong IR API was used (which works correctly in simple cases). Again, this triggered an assert.

This shows that we should have been fuzzing our JIT optimizer already (not a surprising observation in hindsight, fuzz all the things!).

Thankfully, there was also one further bug that really failed in the Z3 verifier. It's a bug in common subexpression elimination / arithmetic simplification, which again does not take overflow correctly into account.

The buggy trace looks like this (unfortunately it's not easily possible to show this bug in Python code).

[a, b]
c = int_add(a, b)
r = int_sub_ovf(c, b)
guard_no_ovf()
finish(r)

This was optimized to:

[a, b]
finish(a)

Which is incorrect, because the guard can fail given the right inputs. But the optimizer concluded that the subtraction is safe, because its the inverse of an earlier addition, not taking into account that this earlier addition can have overflowed.

Note that a related optimization is actually correct. Given this code:

[a, b]
c = int_add_ovf(a, b)
guard_no_ovf()
r = int_sub(c, b)
finish(r)

It can be optimized to:

[a, b]
c = int_add_ovf(a, b)
guard_no_ovf()
finish(a)

Future Work and Conclusion

In the current form the Z3 checker is only a start, even though it has already been concretely useful. There are various directions into which we could extend it. In addition to generate random tests completely from scratch, we could also start from the existing manually written unit-tests and randomly mutate those.

I also want to extend the Z3 checker with support more operations, heap operations in particular (but it's not quite clear to me how to model garbage collection).

I also want to try to switch the code away from the Z3 API and use the more general smtlib interface directly, in order to be able to use other SMT checkers than Z3, eg CVC4.

But all in all this was a fun and not too hard way to find a bunch of bugs in our optimizer! And the infrastructure is now in place, which means that we run some random test cases every time we execute our tests. This is going to be particularly useful when we do further work on the integer reasoning of the JIT (like Nico is doing, for example). As of time of writing of this post, all the bugs mentioned have been fixed and the Z3 code has landed on the default branch and runs as part of PyPy's CI infrastructure.

Acknowledgements

Thanks to Saam Barati, Max Bernstein, Joshua Schmidt and Martin Berger, for great feedback on drafts of this post!

PyPy (Posts about z3)

A DSL for Peephole Transformation Rules of Integer Operations in the PyPy JIT

Motivation

The Peephole Rule DSL

Simple transformation rules

Computing constants and other intermediate results

Checks

Rule Ordering and Liveness

Checking rule coverage

Printing rule statistics

Termination and Confluence

Proofs

Checking for satisfiability

Implementation Notes

Conclusion

Acknowledgements

A Knownbits Abstract Domain for the Toy Optimizer, Correctly

Motivation

The Knownbits Abstract Domain

Transfer Functions

Property-based Tests with Hypothesis

When are Transfer Functions Correct? How do we test them?

Implementing Binary Transfer Functions

Addition and Subtraction

Proving correctness of the transfer functions with Z3

Cases where this style of Z3 proof doesn't work

Making Statements about Precision

Using the Abstract Domain in the Toy Optimizer for Generalized Constant Folding

Using the KnownBits Domain for Conditional Peephole Rewrites

Conclusion

Mining JIT traces for missing optimizations with Z3

High-Level Approach

Dumping PyPy Traces

Encoding Traces as Z3 formulas

Identifying constant booleans with Z3

Identifying redundant operations

Synthesizing more complicated constants with exists-forall

Minimization

Results

Conclusion

Random Notes and Sources

Finding Simple Rewrite Rules for the JIT with Z3

Quick intro to the Z3 Python-API

Encoding the integer operations of RPython's JIT into Z3 formulas

Finding rewrite rules

Synthesizing constants

Synthesizing two constants

Strength Reductions

Conclusions

Sources

Finding JIT Optimizer Bugs using SMT Solvers and Fuzzing

Background: Python Integers in the PyPy JIT

Background: Bounds Analysis in PyPy's JIT

Motivation: A JIT Miscompilation

How could we have avoided this bug?

Z3 Proof of Concept

SMT Checking of the JIT Optimizer

Finding the Bug, Again

Second bug

Generating Random Traces

Bugs Found

Future Work and Conclusion

Acknowledgements

Using the `KnownBits` Domain for Conditional Peephole Rewrites