Type Inference of Joy Expressions

Two kinds of type inference are provided, a simple inferencer that can handle functions that have a single stack effect (aka “type signature”) and that can generate Python code for a limited subset of those functions, and a more complex inferencer/interpreter hybrid that can infer the stack effects of most Joy expressions, including multiple stack effects, unbounded sequences of values, and combinators (if enough information is available.)

joy.utils.types

Curently (asterix after name indicates a function that can be auto-compiled to Python):

_Tree_add_Ee = ([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1]) *
_Tree_delete_R0 = ([a2 ...1] a1 -- [a2 ...1] a2 a1 a1) *
_Tree_delete_clear_stuff = (a3 a2 [a1 ...1] -- [...1]) *
_Tree_get_E = ([a3 a4 ...1] a2 a1 -- a4) *
add = (n1 n2 -- n3)
and = (b1 b2 -- b3)
bool = (a1 -- b1)
ccons = (a2 a1 [...1] -- [a2 a1 ...1]) *
cons = (a1 [...0] -- [a1 ...0]) *
div = (n1 n2 -- n3)
divmod = (n2 n1 -- n4 n3)
dup = (a1 -- a1 a1) *
dupd = (a2 a1 -- a2 a2 a1) *
dupdd = (a3 a2 a1 -- a3 a3 a2 a1) *
eq = (n1 n2 -- b1)
first = ([a1 ...1] -- a1) *
first_two = ([a1 a2 ...1] -- a1 a2) *
floordiv = (n1 n2 -- n3)
fourth = ([a1 a2 a3 a4 ...1] -- a4) *
ge = (n1 n2 -- b1)
gt = (n1 n2 -- b1)
le = (n1 n2 -- b1)
lshift = (n1 n2 -- n3)
lt = (n1 n2 -- b1)
modulus = (n1 n2 -- n3)
mul = (n1 n2 -- n3)
ne = (n1 n2 -- b1)
neg = (n1 -- n2)
not = (a1 -- b1)
over = (a2 a1 -- a2 a1 a2) *
pm = (n2 n1 -- n4 n3)
pop = (a1 --) *
popd = (a2 a1 -- a1) *
popdd = (a3 a2 a1 -- a2 a1) *
popop = (a2 a1 --) *
popopd = (a3 a2 a1 -- a1) *
popopdd = (a4 a3 a2 a1 -- a2 a1) *
pow = (n1 n2 -- n3)
pred = (n1 -- n2)
rest = ([a1 ...0] -- [...0]) *
rolldown = (a1 a2 a3 -- a2 a3 a1) *
rollup = (a1 a2 a3 -- a3 a1 a2) *
rrest = ([a1 a2 ...1] -- [...1]) *
rshift = (n1 n2 -- n3)
second = ([a1 a2 ...1] -- a2) *
sqrt = (n1 -- n2)
stack = (... -- ... [...]) *
stuncons = (... a1 -- ... a1 a1 [...]) *
stununcons = (... a2 a1 -- ... a2 a1 a1 a2 [...]) *
sub = (n1 n2 -- n3)
succ = (n1 -- n2)
swaack = ([...1] -- [...0]) *
swap = (a1 a2 -- a2 a1) *
swons = ([...1] a1 -- [a1 ...1]) *
third = ([a1 a2 a3 ...1] -- a3) *
truediv = (n1 n2 -- n3)
tuck = (a2 a1 -- a1 a2 a1) *
uncons = ([a1 ...0] -- a1 [...0]) *
unit = (a1 -- [a1 ]) *
unswons = ([a1 ...1] -- [...1] a1) *
class joy.utils.types.AnyJoyType(number)[source]

Joy type variable. Represents any Joy value.

class joy.utils.types.BooleanJoyType(number)[source]
accept

alias of __builtin__.bool

class joy.utils.types.FloatJoyType(number)[source]
accept

alias of __builtin__.float

class joy.utils.types.IntJoyType(number)[source]
accept

alias of __builtin__.int

exception joy.utils.types.JoyTypeError[source]
class joy.utils.types.NumberJoyType(number)[source]
class joy.utils.types.StackJoyType(number)[source]
accept

alias of __builtin__.tuple

joy.utils.types.compilable(f)[source]

Return True if a stack effect represents a function that can be automatically compiled (to Python), False otherwise.

joy.utils.types.compile_(name, f, doc=None)[source]

Return a string of Python code implementing the function described by the stack effect. If no doc string is passed doc_from_stack_effect() is used to generate one.

joy.utils.types.compose(*functions)[source]

Return the stack effect of the composition of some of stack effects.

joy.utils.types.defs()[source]

Return a dict of named stack effects.

joy.utils.types.delabel(f, seen=None, c=None)[source]

Fix up type variable numbers after relabel().

joy.utils.types.doc_from_stack_effect(inputs, outputs)[source]

Return a crude string representation of a stack effect.

joy.utils.types.reify(meaning, name, seen=None)[source]

Apply substitution dict to term, returning new term.

joy.utils.types.relabel(left, right)[source]

Re-number type variables to avoid collisions between stack effects.

joy.utils.types.unify(u, v, s=None)[source]

Return a substitution dict representing a unifier for u and v.

joy.utils.polytypes

Example output of the infer() function. The first number on each line is the depth of the Python stack. It goes down when the function backtracks. The next thing on each line is the currently-computed stack effect so far. It starts with the empty “identity function” and proceeds through the expression, which is the rest of each line. The function acts like an interpreter but instead of executing the terms of the expression it composes them, but for combinators it does execute them, using the output side of the stack effect as the stack. This seems to work fine. With proper definitions for the behavior of the combinators that can have more than one effect (like branch or loop) the infer() function seems to be able to handle anything I throw at it so far.

  7 (--) ∘ pop swap rolldown rest rest cons cons
 10 (a1 --) ∘ swap rolldown rest rest cons cons
 13 (a3 a2 a1 -- a2 a3) ∘ rolldown rest rest cons cons
 16 (a4 a3 a2 a1 -- a2 a3 a4) ∘ rest rest cons cons
 19 ([a4 ...1] a3 a2 a1 -- a2 a3 [...1]) ∘ rest cons cons
 22 ([a4 a5 ...1] a3 a2 a1 -- a2 a3 [...1]) ∘ cons cons
 25 ([a4 a5 ...1] a3 a2 a1 -- a2 [a3 ...1]) ∘ cons
 28 ([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1]) ∘
----------------------------------------
([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1])

Here’s another example (implementing ifte) using some combinators:

  7 (--) ∘ [pred] [mul] [div] [nullary bool] dipd branch
  8 (-- [pred ...2]) ∘ [mul] [div] [nullary bool] dipd branch
  9 (-- [pred ...2] [mul ...3]) ∘ [div] [nullary bool] dipd branch
 10 (-- [pred ...2] [mul ...3] [div ...4]) ∘ [nullary bool] dipd branch
 11 (-- [pred ...2] [mul ...3] [div ...4] [nullary bool ...5]) ∘ dipd branch
 15 (-- [pred ...5]) ∘ nullary bool [mul] [div] branch
 19 (-- [pred ...2]) ∘ [stack] dinfrirst bool [mul] [div] branch
 20 (-- [pred ...2] [stack ]) ∘ dinfrirst bool [mul] [div] branch
 22 (-- [pred ...2] [stack ]) ∘ dip infra first bool [mul] [div] branch
 26 (--) ∘ stack [pred] infra first bool [mul] [div] branch
 29 (... -- ... [...]) ∘ [pred] infra first bool [mul] [div] branch
 30 (... -- ... [...] [pred ...1]) ∘ infra first bool [mul] [div] branch
 34 (--) ∘ pred s1 swaack first bool [mul] [div] branch
 37 (n1 -- n2) ∘ [n1] swaack first bool [mul] [div] branch
 38 (... n1 -- ... n2 [n1 ...]) ∘ swaack first bool [mul] [div] branch
 41 (... n1 -- ... n1 [n2 ...]) ∘ first bool [mul] [div] branch
 44 (n1 -- n1 n2) ∘ bool [mul] [div] branch
 47 (n1 -- n1 b1) ∘ [mul] [div] branch
 48 (n1 -- n1 b1 [mul ...1]) ∘ [div] branch
 49 (n1 -- n1 b1 [mul ...1] [div ...2]) ∘ branch
 53 (n1 -- n1) ∘ div
 56 (f2 f1 -- f3) ∘
 56 (i1 f1 -- f2) ∘
 56 (f1 i1 -- f2) ∘
 56 (i2 i1 -- f1) ∘
 53 (n1 -- n1) ∘ mul
 56 (f2 f1 -- f3) ∘
 56 (i1 f1 -- f2) ∘
 56 (f1 i1 -- f2) ∘
 56 (i2 i1 -- i3) ∘
----------------------------------------
(f2 f1 -- f3)
(i1 f1 -- f2)
(f1 i1 -- f2)
(i2 i1 -- f1)
(i2 i1 -- i3)

Multiple Stack Effects

By adjusting the machinery in types.py to handles lists of stack effect comments we can capture more information about the type signatures of some functions, and we can introduce a kind of Kleene Star or sequence type that can stand for an unbounded sequence of other types.

class joy.utils.polytypes.CombinatorJoyType(name, sec, number, expect=None)[source]

Represent combinators.

These type variables carry Joy functions that implement the behaviour of Joy combinators and they can appear in expressions. For simple combinators the implementation functions can be the combinators themselves.

These types can also specify a stack effect (input side only) to guard against being used on invalid types.

joy.utils.polytypes.FUNCTIONS = {'!=': ne, '&': and, '*': mul, '+': add, '++': succ, '-': sub, '--': pred, '/': truediv, '<': lt, '<<': lshift, '<=': le, '<>': ne, '=': eq, '>': gt, '>=': ge, '>>': rshift, '_Tree_add_Ee': _Tree_add_Ee, '_Tree_delete_R0': _Tree_delete_R0, '_Tree_delete_clear_stuff': _Tree_delete_clear_stuff, '_Tree_get_E': _Tree_get_E, 'add': add, 'and': and, 'b': b, 'bool': bool, 'branch': branch, 'ccons': ccons, 'clear': clear, 'concat_': concat_, 'cons': cons, 'dip': dip, 'dipd': dipd, 'dipdd': dipdd, 'div': div, 'divmod': divmod, 'dup': dup, 'dupd': dupd, 'dupdd': dupdd, 'dupdip': dupdip, 'eq': eq, 'first': first, 'first_two': first_two, 'floordiv': floordiv, 'fourth': fourth, 'ge': ge, 'gt': gt, 'i': i, 'infra': infra, 'le': le, 'loop': loop, 'lshift': lshift, 'lt': lt, 'modulus': modulus, 'mul': mul, 'ne': ne, 'neg': neg, 'not': not, 'nullary': nullary, 'over': over, 'pop': pop, 'popd': popd, 'popdd': popdd, 'popop': popop, 'popopd': popopd, 'popopdd': popopdd, 'pow': pow, 'pred': pred, 'product': product, 'rest': rest, 'roll<': rolldown, 'roll>': rollup, 'rolldown': rolldown, 'rollup': rollup, 'rrest': rrest, 'rshift': rshift, 'second': second, 'sqrt': sqrt, 'stack': stack, 'stuncons': stuncons, 'stununcons': stununcons, 'sub': sub, 'succ': succ, 'sum': sum, 'swaack': swaack, 'swap': swap, 'swons': swons, 'third': third, 'truediv': truediv, 'truthy': bool, 'tuck': tuck, 'uncons': uncons, 'unit': unit, 'unswons': unswons, 'x': x}

Docstring for functions in Sphinx?

class joy.utils.polytypes.FunctionJoyType(name, sec, number)[source]
class joy.utils.polytypes.IntJoyType(number)[source]
class joy.utils.polytypes.KleeneStar(number)[source]

A sequence of zero or more AnyJoyType variables would be:

A*

The A* works by splitting the universe into two alternate histories:

A* → ∅

A* → A A*

The Kleene star variable disappears in one universe, and in the other it turns into an AnyJoyType variable followed by itself again.

We have to return all universes (represented by their substitution dicts, the “unifiers”) that don’t lead to type conflicts.

kind

alias of joy.utils.types.AnyJoyType

class joy.utils.polytypes.SymbolJoyType(name, sec, number)[source]

Represent non-combinator functions.

These type variables carry the stack effect comments and can appear in expressions (as in quoted programs.)

joy.utils.polytypes.compose(f, g, e)[source]

Yield the stack effects of the composition of two stack effects. An expression is carried along and updated and yielded.

joy.utils.polytypes.defs()[source]

Return a dict of FunctionJoyType instances to be used with infer().

joy.utils.polytypes.infer(*expression)[source]

Return a list of stack effects for a Joy expression.

For example:

h = infer(pop, swap, rolldown, rest, rest, cons, cons)
for fi, fo in h:
    print doc_from_stack_effect(fi, fo)

Prints:

([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1])
joy.utils.polytypes.meta_compose(F, G, e)[source]

Yield the stack effects of the composition of two lists of stack effects. An expression is carried along and updated and yielded.

joy.utils.polytypes.type_check(name, stack)[source]

Trinary predicate. True if named function type-checks, False if it fails, None if it’s indeterminate (because I haven’t entered it into the FUNCTIONS dict yet.)