# 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)

  Joy type variable. Represents any Joy value.

class `joy.utils.types.``BooleanJoyType`(number)

class `joy.utils.types.``FloatJoyType`(number)

class `joy.utils.types.``IntJoyType`(number)

exception `joy.utils.types.``JoyTypeError`

class `joy.utils.types.``NumberJoyType`(number)

class `joy.utils.types.``StackJoyType`(number)

`joy.utils.types.``compilable`(f)

  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)

  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)

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

`joy.utils.types.``defs`()

  Return a dict of named stack effects.

`joy.utils.types.``delabel`(f, seen=None, c=None)

  Fix up type variable numbers after relabel().

`joy.utils.types.``doc_from_stack_effect`(inputs, outputs)

  Return a crude string representation of a stack effect.

`joy.utils.types.``relabel`(left, right)

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

`joy.utils.types.``unify`(u, v, s=None)

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

`joy.utils.types.``update`(s, term)

  Apply substitution dict to term, returning new term.

## `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.``AnyStarJoyType`(number)
  `kind`

    alias of `joy.utils.types.AnyJoyType`

class `joy.utils.polytypes.``CombinatorJoyType`(name, sec, number, expect=None)

  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.

class `joy.utils.polytypes.``FunctionJoyType`(name, sec, number)

class `joy.utils.polytypes.``IntJoyType`(number)

class `joy.utils.polytypes.``KleeneStar`(number)

  A sequence of zero or more AnyJoyType variables would be:

  > A*

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

  > A* -> 0

  > 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.``NumberStarJoyType`(number)
  `kind`

    alias of `joy.utils.types.NumberJoyType`

`joy.utils.polytypes.``Ss` = [s1*, s2*, s3*, s4*, s5*, s6*, s7*, s8*, s9*, s10*]

  Docstring for functions in Sphinx?

class `joy.utils.polytypes.``StackStarJoyType`(number)
  `kind`

    alias of `joy.utils.types.StackJoyType`

class `joy.utils.polytypes.``SymbolJoyType`(name, sec, number)

  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)

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

`joy.utils.polytypes.``defs`()

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

`joy.utils.polytypes.``infer`(*expression)

  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)

  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.``unify`(u, v, s=None)

  Return a tuple of substitution dicts representing unifiers for u and v.