Guest Post: Final Encoding in RPython Interpreters

Introduction¶

This post started as a quick note summarizing a recent experiment I carried out upon a small RPython interpreter by rewriting it in an uncommon style. It is written for folks who have already written some RPython and want to take a deeper look at interpreter architecture.

Some experiments are about finding solutions to problems. This experiment is about taking a solution which is already well-understood and applying it in the context of RPython to find a new approach. As we will see, there is no real change in functionality or the number of clauses in the interpreter; it's more like a comparison between endo- and exoskeletons, a different arrangement of equivalent bones and plates.

Overview¶

An RPython interpreter for a programming language generally does three or four things, in order:

1. Read and parse input programs
2. Encode concrete syntax as abstract syntax
3. Optionally, optimize or reduce the abstract syntax
4. Evaluate the abstract syntax: read input data, compute, print output data, etc.

Today we'll look at abstract syntax. Most programming languages admit a concrete parse tree which is readily abstracted to provide an abstract syntax tree (AST). The AST is usually encoded with the initial style of encoding. An initial encoding can be transformed into any other encoding for the same AST, looks like a hierarchy of classes, and is implemented as a static structure on the heap.

In contrast, there is also a final encoding. A final encoding can be transformed into by any other encoding, looks like an interface for the actions of the interpreter, and is implemented as an unwinding structure on the stack. From the RPython perspective, Python builtin modules like os or sys are final encodings for features of the operating system; the underlying implementation is different when translated or untranslated, but the interface used to access those features does not change.

In RPython, an initial encoding is built from a hierarchy of classes. Each class represents a type of tree nodes, corresponding to a parser production in the concrete parse tree. Each class instance therefore represents an individual tree node. The fields of a class, particularly those filled during .__init__(), store pre-computed properties of each node; methods can be used to compute node properties on demand. This seems like an obvious and simple approach; what other approaches could there be? We need an example.

Final Encoding of Brainfuck¶

We will consider Brainfuck, a simple Turing-complete programming language. An example Brainfuck program might be:

[-]

This program is built from a loop and a decrement, and sets a cell to zero. In an initial encoding which follows the algebraic semantics of Brainfuck, the program could be expressed by applying class constructors to build a structure on the heap:

Loop(Plus(-1))

A final encoding is similar, except that class constructors are replaced by methods, the structure is built on the stack, and we are parameterized over the choice of class:

lambda cls: cls.loop(cls.plus(-1))

In ordinary Python, transforming between these would be trivial, and mostly is a matter of passing around the appropriate class. Indeed, initial and final encodings are equivalent; we'll return to that fact later. However, in RPython, all of the types must line up, and classes must be determined before translation. We'll need to monomorphize our final encodings, using some RPython tricks later on. Before that, let's see what an actual Brainfuck interface looks like, so that we can cover all of the difficulties with final encoding.

Before we embark, please keep in mind that local code doesn't know what cls is. There's no type-safe way to inspect an arbitrary semantic domain. In the initial-encoded version, we can ask isinstance(bf, Loop) to see whether an AST node is a loop, but there simply isn't an equivalent for final-encoded ASTs. So, there is an implicit challenge to think about: how do we evaluate a program in an arbitrary semantic domain? For bonus points, how do we optimize a program without inspecting the types of its AST nodes?

What follows is a dissection of this module at the given revision. Readers may find it satisfying to read the entire interpreter top to bottom first; it is less than 300 lines.

Core Functionality¶

Final encoding is given as methods on an interface. These five methods correspond precisely to the summands of the algebra of Brainfuck.

class BF(object):
    # Other methods elided
    def plus(self, i): pass
    def right(self, i): pass
    def input(self): pass
    def output(self): pass
    def loop(self, bfs): pass

Note that the .loop() method takes another program as an argument. Initial-encoded ASTs have other initial-encoded ASTs as fields on class instances; final-encoded ASTs have other final-encoded ASTs as parameters to interface methods. RPython infers all of the types, so the reader has to know that i is usually an integer while bfs is a sequence of Brainfuck operations.

We're using a class to implement this functionality. Later, we'll treat it as a mixin, rather than a superclass, to avoid typing problems.

Monoid¶

In order to optimize input programs, we'll need to represent the underlying monoid of Brainfuck programs. To do this, we add the signature for a monoid:

class BF(object):
    # Other methods elided
    def unit(self): pass
    def join(self, l, r): pass

This is technically a unital magma, since RPython doesn't support algebraic laws, but we will enforce the algebraic laws later on during optimization. We also want to make use of the folklore that free monoids are lists, allowing callers to pass a list of actions which we'll reduce with recursion:

class BF(object):
    # Other methods elided
    def joinList(self, bfs):
        if not bfs: return self.unit()
        elif len(bfs) == 1: return bfs[0]
        elif len(bfs) == 2: return self.join(bfs[0], bfs[1])
        else:
            i = len(bfs) >> 1
            return self.join(self.joinList(bfs[:i]), self.joinList(bfs[i:]))

.joinList() is a little bulky to implement, but Wirth's principle applies: the interpreter is shorter with it than without it.

Idioms¶

Finally, our interface includes a few high-level idioms, like the zero program shown earlier, which are defined in terms of low-level behaviors. In an initial encoding, these could be defined as module-level functions; here, we define them on the mixin class BF.

class BF(object):
    # Other methods elided
    def zero(self): return self.loop(self.plus(-1))
    def move(self, i): return self.scalemove(i, 1)
    def move2(self, i, j): return self.scalemove2(i, 1, j, 1)
    def scalemove(self, i, s):
        return self.loop(self.joinList([
            self.plus(-1), self.right(i), self.plus(s), self.right(-i)]))
    def scalemove2(self, i, s, j, t):
        return self.loop(self.joinList([
                self.plus(-1), self.right(i), self.plus(s), self.right(j - i),
                self.plus(t), self.right(-j)]))

Interface-oriented Architecture¶

Applying Interfaces¶

Now, we hack at RPython's object model until everything translates. First, consider the task of pretty-printing. For Brainfuck, we'll simply regurgitate the input program as a Python string:

class AsStr(object):
    import_from_mixin(BF)
    def unit(self): return ""
    def join(self, l, r): return l + r
    def plus(self, i): return '+' * i if i > 0 else '-' * -i
    def right(self, i): return '>' * i if i > 0 else '<' * -i
    def loop(self, bfs): return '[' + bfs + ']'
    def input(self): return ','
    def output(self): return '.'

Via rlib.objectmodel.import_from_mixin, no stressing with covariance of return types is required. Instead, we shift from a Java-esque view of classes and objects, to an OCaml-ish view of prebuilt classes and constructors. AsStr is monomorphic, and any caller of it will have to create their own covariance somehow. For example, here are the first few lines of the parsing function:

@specialize.argtype(1)
def parse(s, domain):
    ops = [domain.unit()]
    # Parser elided to preserve the reader's attention

By invoking rlib.objectmodel.specialize.argtype, we make copies of the parsing function, up to one per call site, based on our choice of semantic domain. Oleg calls these "symantics" but I prefer "domain" in code. Also, note how the parsing stack starts with the unit of the monoid, which corresponds to the empty input string; the parser will repeatedly use the monoidal join to build up a parsed expression without inspecting it. Here's a small taste of that:

while i < len(s):
    char = s[i]
    if char == '+': ops[-1] = domain.join(ops[-1], domain.plus(1))
    elif char == '-': ops[-1] = domain.join(ops[-1], domain.plus(-1))
    # and so on

The reader may feel justifiably mystified; what breaks if we don't add these magic annotations? Well, the translator will throw UnionError because the low-level types don't match. RPython only wants to make one copy of functions like parse() in its low-level representation, and each copy of parse() will be compiled to monomorphic machine code. In this interpreter, in order to support parsing to an optimized string and also parsing to an evaluator, we need two copies of parse(). It is okay to not fully understand this at first.

Composing Interfaces¶

Earlier, we noted that an interpreter can optionally optimize input programs after parsing. To support this, we'll precompose a peephole optimizer onto an arbitrary domain. We could also postcompose with a parser instead, but that sounds more difficult. Here are the relevant parts:

def makePeephole(cls):
    domain = cls()
    def stripDomain(bfs): return domain.joinList([t[0] for t in bfs])
    class Peephole(object):
        import_from_mixin(BF)
        def unit(self): return []
        def join(self, l, r): return l + r
        # Actual definition elided... for now...
    return Peephole, stripDomain

Don't worry about the actual optimization yet. What's important here is the pattern of initialization of semantic domains. makePeephole is an SML-style functor on semantic domains: given a final encoding of Brainfuck, it produces another final encoding of Brainfuck which incorporates optimizations. The helper stripDomain is a finalizer which performs the extraction from the optimizer's domain to the underlying cls that was passed in at translation time. For example, let's optimize pretty-printing:

AsStr, finishStr = makePeephole(AsStr)

Now, it only takes one line to parse and print an optimized AST without ever building it on the heap. To be pedantic, fragments of the output string will be heap-allocated, but the AST's node structure will only ever be stack-allocated. Further, to be shallow, the parser is written to prevent malicious input from causing a stack overflow, and this forces it to maintain a heap-allocated RPython list of intermediate operations inside loops.

print finishStr(parse(text, AsStr()))

Performance¶

But is it fast? Yes. It's faster than the prior version, which was initial-encoded, and also faster than Andrew Brown's classic version (part 1, part 2). Since Brown's interpreter does not perform much optimization, we will focus on how final encoding can outperform initial encoding.

JIT¶

First, why is it faster than the same interpreter with initial encoding? Well, it still has initial encoding from the JIT's perspective! There is an Op class with a hierarchy of subclasses implementing individual behaviors. A sincere tagless-final student, or those who remember Stop Writing Classes (2012, Pycon US), will recognize that the following classes could be plain functions, and should think of the classes as a concession to RPython's lack of support for lambdas with closures rather than an initial encoding. We aren't ever going to directly typecheck any Op, but the JIT will generate typechecking guards anyway, so we effectively get a fully-promoted AST inlined into each JIT trace. First, some simple behaviors:

class Op(object): _immutable_ = True

class _Input(Op):
    _immutable_ = True
    def runOn(self, tape, position):
        tape[position] = ord(os.read(0, 1)[0])
        return position
Input = _Input()

class _Output(Op):
    _immutable_ = True
    def runOn(self, tape, position):
        os.write(1, chr(tape[position]))
        return position
Output = _Output()

class Add(Op):
    _immutable_ = True
    _immutable_fields_ = "imm",
    def __init__(self, imm): self.imm = imm
    def runOn(self, tape, position):
        tape[position] += self.imm
        return position

The JIT does technically have less information than before; it no longer knows that a sequence of immutable operations is immutable enough to be worth unrolling, but a bit of rlib.jit.unroll_safe fixes that:

class Seq(Op):
    _immutable_ = True
    _immutable_fields_ = "ops[*]",
    def __init__(self, ops): self.ops = ops
    @unroll_safe
    def runOn(self, tape, position):
        for op in self.ops: position = op.runOn(tape, position)
        return position

Finally, the JIT entry point is at the head of each loop, just like with prior interpreters. Since Brainfuck doesn't support mid-loop jumps, there's no penalty for only allowing merge points at the head of the loop.

class Loop(Op):
    _immutable_ = True
    _immutable_fields_ = "op",
    def __init__(self, op): self.op = op
    def runOn(self, tape, position):
        op = self.op
        while tape[position]:
            jitdriver.jit_merge_point(op=op, position=position, tape=tape)
            position = op.runOn(tape, position)
        return position

That's the end of the implicit challenge. There's no secret to it; just evaluate the AST. Here's part of the semantic domain for evaluation, as well as the "functor" to optimize it. In AsOps.join() are the only isinstance() calls in the entire interpreter! This is acceptable because Seq is effectively a type wrapper for an RPython list, so that a list of operations is also an operation; its list is initial-encoded and available for inspection.

class AsOps(object):
    import_from_mixin(BF)
    def unit(self): return Shift(0)
    def join(self, l, r):
        if isinstance(l, Seq) and isinstance(r, Seq):
            return Seq(l.ops + r.ops)
        elif isinstance(l, Seq): return Seq(l.ops + [r])
        elif isinstance(r, Seq): return Seq([l] + r.ops)
        return Seq([l, r])
    # Other methods elided!

AsOps, finishOps = makePeephole(AsOps)

And finally here is the actual top-level code to evaluate the input program. As before, once everything is composed, the actual invocation only takes one line.

tape = bytearray("\x00" * cells)
finishOps(parse(text, AsOps())).runOn(tape, 0)

Peephole Optimization¶

Our peephole optimizer is an abstract interpreter with one instruction of lookahead/rewrite buffer. It implements the aforementioned algebraic laws of the Brainfuck monoid. It also implements idiom recognition for loops. First, the abstract interpreter. The abstract domain has six elements:

class AbstractDomain(object): pass
meh, aLoop, aZero, theIdentity, anAdd, aRight = [AbstractDomain() for _ in range(6)]

We'll also tag everything with an integer, so that anAdd or aRight can be exact annotations. This is the actual Peephole.join() method:

def join(self, l, r):
    if not l: return r
    rv = l[:]
    bfHead, adHead, immHead = rv.pop()
    for bf, ad, imm in r:
        if ad is theIdentity: continue
        elif adHead is aLoop and ad is aLoop: continue
        elif adHead is theIdentity:
            bfHead, adHead, immHead = bf, ad, imm
        elif adHead is anAdd and ad is aZero:
            bfHead, adHead, immHead = bf, ad, imm
        elif adHead is anAdd and ad is anAdd:
            immHead += imm
            if immHead: bfHead = domain.plus(immHead)
            elif rv: bfHead, adHead, immHead = rv.pop()
            else:
                bfHead = domain.unit()
                adHead = theIdentity
        elif adHead is aRight and ad is aRight:
            immHead += imm
            if immHead: bfHead = domain.right(immHead)
            elif rv: bfHead, adHead, immHead = rv.pop()
            else:
                bfHead = domain.unit()
                adHead = theIdentity
        else:
            rv.append((bfHead, adHead, immHead))
            bfHead, adHead, immHead = bf, ad, imm
    rv.append((bfHead, adHead, immHead))
    return rv

If this were to get much longer, then implementing a DSL would be worth it, but this is a short-enough method to inline. The abstract interpretation is assumed by induction for the left-hand side of the join, save for the final instruction, which is loaded into a rewrite register. Each instruction on the right-hand side is inspected exactly once. The logic for anAdd followed by anAdd is exactly the same as for aRight followed by aRight because they both have underlying Abelian groups given by the integers. The rewrite register is carefully pushed onto and popped off from the left-hand side in order to cancel out theIdentity, which itself is merely a unifier for anAdd or aRight of 0.

Note that we generate a lot of garbage. For example, parsing a string of n '+' characters will cause the peephole optimizer to allocate n instances of the underlying domain.plus() action, from domain.plus(1) up to domain.plus(n). An older initial-encoded version of this interpreter used hash consing to avoid ever building an op more than once, even loops. It appears more efficient to generate lots of immutable garbage than to repeatedly hash inputs and search mutable hash tables, at least for optimizing Brainfuck incrementally during parsing.

Finally, let's look at idiom recognition. RPython lists are initial-coded, so we can dispatch based on the length of the list, and then inspect the abstract domains of each action.

def isConstAdd(bf, i): return bf[1] is anAdd and bf[2] == i

def oppositeShifts(bf1, bf2):
    return bf1[1] is bf2[1] is aRight and bf1[2] == -bf2[2]

def oppositeShifts2(bf1, bf2, bf3):
    return (bf1[1] is bf2[1] is bf3[1] is aRight and
            bf1[2] + bf2[2] + bf3[2] == 0)

def loop(self, bfs):
    if len(bfs) == 1:
        bf, ad, imm = bfs[0]
        if ad is anAdd and imm in (1, -1):
            return [(domain.zero(), aZero, 0)]
    elif len(bfs) == 4:
        if (isConstAdd(bfs[0], -1) and
            bfs[2][1] is anAdd and
            oppositeShifts(bfs[1], bfs[3])):
            return [(domain.scalemove(bfs[1][2], bfs[2][2]), aLoop, 0)]
        if (isConstAdd(bfs[3], -1) and
            bfs[1][1] is anAdd and
            oppositeShifts(bfs[0], bfs[2])):
            return [(domain.scalemove(bfs[0][2], bfs[1][2]), aLoop, 0)]
    elif len(bfs) == 6:
        if (isConstAdd(bfs[0], -1) and
            bfs[2][1] is bfs[4][1] is anAdd and
            oppositeShifts2(bfs[1], bfs[3], bfs[5])):
            return [(domain.scalemove2(bfs[1][2], bfs[2][2],
                                       bfs[1][2] + bfs[3][2],
                                       bfs[4][2]), aLoop, 0)]
        if (isConstAdd(bfs[5], -1) and
            bfs[1][1] is bfs[3][1] is anAdd and
            oppositeShifts2(bfs[0], bfs[2], bfs[4])):
            return [(domain.scalemove2(bfs[0][2], bfs[1][2],
                                       bfs[0][2] + bfs[2][2],
                                       bfs[3][2]), aLoop, 0)]
    return [(domain.loop(stripDomain(bfs)), aLoop, 0)]

This ends the bonus question. How do we optimize an unknown semantic domain? We must maintain an abstract context which describes elements of the domain. In initial encoding, we ask an AST about itself. In final encoding, we already know everything relevant about the AST.

The careful reader will see that I didn't really answer that opening question in the JIT section. Because the JIT still ranges over the same operations as before, it can't really be slower; but why is it now faster? Because the optimizer is now slightly better in a few edge cases. It performs the same optimizations as before, but the rigor of abstract interpretation causes it to emit slightly better operations to the JIT backend.

Concretely, improving the optimizer can shorten pretty-printed programs. The Busy Beaver Gauge measures the length of programs which search for solutions to mathematical problems. After implementing and debugging the final-encoded interpreter, I found that two of my entries on the Busy Beaver Gauge for Brainfuck had become shorter by about 2%. (Most other entries are already hand-optimized according to the standard algebra and have no optimization opportunities.)

Discussion¶

Given that initial and final encodings are equivalent, and noting that RPython's toolchain is written to prefer initial encodings, what did we actually gain? Did we gain anything?

One obvious downside to final encoding in RPython is interpreter size. The example interpreter shown here is a rewrite of an initial-encoded interpreter which can be seen here for comparison. Final encoding adds about 20% more code in this case.

Final encoding is not necessarily more code than initial encoding, though. All AST encodings in interpreters are subject to the Expression Problem, which states that there is generally a quadratic amount of code required to implement multiple behaviors for an AST with multiple types of nodes; specifically, n behaviors for m types of nodes require n × m methods. Initial encodings improve the cost of adding new types of nodes; final encodings improve the cost of adding new behaviors. Final encoding may tend to win in large codebases for mature languages, where the language does not change often but new behaviors are added frequently and maintained for long periods.

Optimizations in final encoding require a bit of planning. The abstract-interpretation approach is solid but relies upon the monoid and its algebraic laws. In the worst case, an entire class hierarchy could be required to encode the abstraction.

It is remarkable to find a 2% improvement in residual program size merely by reimplementing an optimizer as an abstract interpreter respecting the algebraic laws. This could be the most important lesson for compiler engineers, if it happens to generalize.

Final encoding was popularized via the tagless-final movement in OCaml and Scala, including famously in a series of tutorials by Kiselyov et al. A "tag", in this jargon, is a runtime identifier for an object's type or class; a tagless encoding effectively doesn't allow isinstance() at all. In the above presentation, tags could be hacked in, but were not materially relevant to most steps. Tags were required for the final evaluation step, though, and the tagless-final insight is that certain type systems can express type-safe evaluation without those tags. We won't go further in this direction because tags also communicate valuable information to the JIT.

Summarizing Table¶

Credits¶

Thanks to folks in #pypy on Libera Chat: arigato for the idea, larstiq for pushing me to write it up, and cfbolz and mattip for reviewing and finding mistakes. The original IRC discussion leading to this blog post is available here.

This interpreter is part of the rpypkgs suite, a Nix flake for RPython interpreters. Readers with Nix installed can run this interpreter directly from the flake:

$ nix-prefetch-url https://github.com/MG-K/pypy-tutorial-ko/raw/refs/heads/master/mandel.b
$ nix run github:rpypkgs/rpypkgs#bf -- /nix/store/ngnphbap9ncvz41d0fkvdh61n7j2bg21-mandel.b