-
-# Type Inference
+# The Blissful Elegance of Typing Joy
This notebook presents a simple type inferencer for Joy code. It can infer the stack effect of most Joy expressions. It's built largely by means of existing ideas and research. (A great overview of the existing knowledge is a talk ["Type Inference in Stack-Based Programming Languages"](http://prl.ccs.neu.edu/blog/2017/03/10/type-inference-in-stack-based-programming-languages/) given by Rob Kleffner on or about 2017-03-10 as part of a course on the history of programming languages.)
s[u] = v
elif isinstance(v, int):
s[v] = u
+ else:
+ s = False
return s
```
print e
```
+ Cannot unify (1, 2) and (1001, 1002).
+
+
#### `unify()` version 2
The problem is that the `unify()` function as written doesn't handle the case when both terms are tuples. We just have to add a clause to deal with this recursively:
s = unify(a, c, s)
if s != False:
s = unify(b, d, s)
+ else:
+ s = False
return s
```
return (n2, stack)
-(Eventually I should come back around to this becuase it's not tooo difficult to exend this code to be able to compile e.g. `n3 = mul(n1, n2)` for `mul` and insert it in the right place with the right variable names. It requires a little more support from the library functions, in that we need to know to call `mul()` the Python function for `mul` the Joy function, but since *most* of the math functions (at least) are already wrappers it should be straightforward.)
+(Eventually I should come back around to this becuase it's not tooo difficult to exend this code to be able to compile e.g. `n2 = mul(n1, n1)` for `mul` with the right variable names and insert it in the right place. It requires a little more support from the library functions, in that we need to know to call `mul()` the Python function for `mul` the Joy function, but since *most* of the math functions (at least) are already wrappers it should be straightforward.)
#### `compilable()`
The functions that *can* be compiled are the ones that have only `AnyJoyType` and `StackJoyType` labels in their stack effect comments. We can write a function to check that:
```python
+
+```
+
+
+```python
+
+```
+
+
+```python
+
+```
+
+
+```python
+
+```
+
+
+```python
+
+```
+
+
+```python
unstack = (S[1], S[0]), S[1]
enstacken = S[0], (S[0], S[1])
```
+
+```python
+
+```
+
## Part VI: Multiple Stack Effects
...
```python
+
+```
+
+
+```python
def meta_compose(F, G):
for f, g in product(F, G):
try:
## Part VII: Typing Combinators
-In order to compute the stack effect of combinators you kinda have to have the quoted programs they expect available. In the most general case, the `i` combinator, you can't say anything about it's stack effect other than it expects one quote:
+In order to compute the stack effect of combinators you kinda have to have the quoted programs they expect available. In the most general case, the `i` combinator, you can't say anything about its stack effect other than it expects one quote:
i (... [.1.] -- ... .1.)
Without any information about the contents of the quote we can't say much about the result.
-I think there's a way forward. If we convert our list of terms we are composing into a stack structure we can use it as a *Joy expression*, then we can treat the *output half* of a function's stack effect comment as a Joy interpreter stack, and just execute combinators directly. We can hybridize the compostition function with an interpreter to evaluate combinators, compose non-combinator functions, and put type variables on the stack. For combinators like `branch` that can have more than one stack effect we have to "split universes" again and return both.
+### Hybrid Inferencer/Interpreter
+I think there's a way forward. If we convert our list (of terms we are composing) into a stack structure we can use it as a *Joy expression*, then we can treat the *output half* of a function's stack effect comment as a Joy interpreter stack, and just execute combinators directly. We can hybridize the compostition function with an interpreter to evaluate combinators, compose non-combinator functions, and put type variables on the stack. For combinators like `branch` that can have more than one stack effect we have to "split universes" again and return both.
+
+#### Joy Types for Functions
+We need a type variable for Joy functions that can go in our expressions and be used by the hybrid inferencer/interpreter. They have to store a name and a list of stack effects.
```python
def __repr__(self):
return self.name
-
-
-class SymbolJoyType(FunctionJoyType): prefix = 'F'
-class CombinatorJoyType(FunctionJoyType): prefix = 'C'
-
-
-
```
-
-```python
-def flatten(g):
- return list(chain.from_iterable(g))
-
-
-ID = S[0], S[0] # Identity function.
-
-
-def infer(e, F=ID):
- if not e:
- return [F]
-
- n, e = e
-
- if isinstance(n, SymbolJoyType):
- res = flatten(infer(e, Fn) for Fn in MC([F], n.stack_effects))
-
- elif isinstance(n, CombinatorJoyType):
- res = []
- for combinator in n.stack_effects:
- fi, fo = F
- new_fo, ee, _ = combinator(fo, e, {})
- ee = update(FUNCTIONS, ee) # Fix Symbols.
- new_F = fi, new_fo
- res.extend(infer(ee, new_F))
- else:
- lit = s9, (n, s9)
- res = flatten(infer(e, Fn) for Fn in MC([F], [lit]))
-
- return res
-
-
-```
+#### Specialized for Simple Functions and Combinators
+For non-combinator functions the stack effects list contains stack effect comments (represented by pairs of cons-lists as described above.)
```python
-f0, f1, f2, f3, f4, f5, f6, f7, f8, f9 = F = map(FloatJoyType, _R)
-i0, i1, i2, i3, i4, i5, i6, i7, i8, i9 = I = map(IntJoyType, _R)
-n0, n1, n2, n3, n4, n5, n6, n7, n8, n9 = N
-s0, s1, s2, s3, s4, s5, s6, s7, s8, s9 = S
+class SymbolJoyType(FunctionJoyType):
+ prefix = 'F'
```
+For combinators the list contains Python functions.
-```python
-import joy.library
-
-FNs = '''ccons cons divmod_ dup dupd first
- over pm pop popd popdd popop pred
- rest rolldown rollup rrest second
- sqrt stack succ swaack swap swons
- third tuck uncons'''
-
-FUNCTIONS = {
- name: SymbolJoyType(name, [NEW_DEFS[name]], i)
- for i, name in enumerate(FNs.strip().split())
- }
-FUNCTIONS['sum'] = SymbolJoyType('sum', [(((Ns[1], s1), s0), (n0, s0))], 100)
-FUNCTIONS['mul'] = SymbolJoyType('mul', [
- ((i2, (i1, s0)), (i3, s0)),
- ((f2, (i1, s0)), (f3, s0)),
- ((i2, (f1, s0)), (f3, s0)),
- ((f2, (f1, s0)), (f3, s0)),
-], 101)
-FUNCTIONS.update({
- combo.__name__: CombinatorJoyType(combo.__name__, [combo], i)
- for i, combo in enumerate((
- joy.library.i,
- joy.library.dip,
- joy.library.dipd,
- joy.library.dipdd,
- joy.library.dupdip,
- joy.library.b,
- joy.library.x,
- joy.library.infra,
- ))
- })
-
-def branch_true(stack, expression, dictionary):
- (then, (else_, (flag, stack))) = stack
- return stack, CONCAT(then, expression), dictionary
-def branch_false(stack, expression, dictionary):
- (then, (else_, (flag, stack))) = stack
- return stack, CONCAT(else_, expression), dictionary
+```python
+class CombinatorJoyType(FunctionJoyType):
-FUNCTIONS['branch'] = CombinatorJoyType('branch', [branch_true, branch_false], 100)
-```
+ prefix = 'C'
+ def __init__(self, name, sec, number, expect=None):
+ super(CombinatorJoyType, self).__init__(name, sec, number)
+ self.expect = expect
-```python
-globals().update(FUNCTIONS)
+ def enter_guard(self, f):
+ if self.expect is None:
+ return f
+ g = self.expect, self.expect
+ new_f = list(compose(f, g, ()))
+ assert len(new_f) == 1, repr(new_f)
+ return new_f[0][1]
```
-
-```python
-from itertools import chain
-from joy.utils.stack import list_to_stack as l2s
-```
+For simple combinators that have only one effect (like ``dip``) you only need one function and it can be the combinator itself.
```python
-expression = l2s([n1, n2, (mul, s2), (stack, s3), dip, infra, first])
-```
-
+import joy.library
-```python
-expression
+dip = CombinatorJoyType('dip', [joy.library.dip], 23)
```
-
-
-
- (n1, (n2, ((mul, s2), ((stack, s3), (dip, (infra, (first, ())))))))
-
-
+For combinators that can have more than one effect (like ``branch``) you have to write functions that each implement the action of one of the effects.
```python
-expression = l2s([n1, n2, mul])
-```
+def branch_true(stack, expression, dictionary):
+ (then, (else_, (flag, stack))) = stack
+ return stack, concat(then, expression), dictionary
+def branch_false(stack, expression, dictionary):
+ (then, (else_, (flag, stack))) = stack
+ return stack, concat(else_, expression), dictionary
-```python
-expression
+branch = CombinatorJoyType('branch', [branch_true, branch_false], 100)
```
+You can also provide an optional stack effect, input-side only, that will then be used as an identity function (that accepts and returns stacks that match the "guard" stack effect) which will be used to guard against type mismatches going into the evaluation of the combinator.
+#### `infer()`
+With those in place, we can define a function that accepts a sequence of Joy type variables, including ones representing functions (not just values), and attempts to grind out all the possible stack effects of that expression.
-
- (n1, (n2, (mul, ())))
-
-
+One tricky thing is that type variables *in the expression* have to be updated along with the stack effects after doing unification or we risk losing useful information. This was a straightforward, if awkward, modification to the call structure of `meta_compose()` et. al.
```python
-infer(expression)
-```
-
-
+ID = S[0], S[0] # Identity function.
- [(s1, (f1, s1)), (s1, (i1, s1))]
+def infer(*expression):
+ return sorted(set(_infer(list_to_stack(expression))))
+def _infer(e, F=ID):
+ _log_it(e, F)
+ if not e:
+ return [F]
+ n, e = e
-```python
-infer(expression)
-```
+ if isinstance(n, SymbolJoyType):
+ eFG = meta_compose([F], n.stack_effects, e)
+ res = flatten(_infer(e, Fn) for e, Fn in eFG)
+ elif isinstance(n, CombinatorJoyType):
+ fi, fo = n.enter_guard(F)
+ res = flatten(_interpret(f, fi, fo, e) for f in n.stack_effects)
+ elif isinstance(n, Symbol):
+ assert n not in FUNCTIONS, repr(n)
+ func = joy.library._dictionary[n]
+ res = _interpret(func, F[0], F[1], e)
+ else:
+ fi, fo = F
+ res = _infer(e, (fi, (n, fo)))
- [(s1, (f1, s1)), (s1, (i1, s1))]
+ return res
+def _interpret(f, fi, fo, e):
+ new_fo, ee, _ = f(fo, e, {})
+ ee = update(FUNCTIONS, ee) # Fix Symbols.
+ new_F = fi, new_fo
+ return _infer(ee, new_F)
-```python
-for stack_effect_comment in infer(expression):
- print doc_from_stack_effect(*stack_effect_comment)
+def _log_it(e, F):
+ _log.info(
+ u'%3i %s ∘ %s',
+ len(inspect_stack()),
+ doc_from_stack_effect(*F),
+ expression_to_string(e),
+ )
```
- (-- f1)
- (-- i1)
-
+#### Work in Progress
+And that brings us to current Work-In-Progress. The mixed-mode inferencer/interpreter `infer()` function seems to work well. There are details I should document, and the rest of the code in the `types` module (FIXME link to its docs here!) should be explained... There is cruft to convert the definitions in `DEFS` to the new `SymbolJoyType` objects, and some combinators. Here is an example of output from the current code :
```python
-expression
-```
-
-
+1/0 # (Don't try to run this cell! It's not going to work. This is "read only" code heh..)
+logging.basicConfig(format='%(message)s', stream=sys.stdout, level=logging.INFO)
- (n1, (n2, (mul, ())))
-
+globals().update(FUNCTIONS)
+h = infer((pred, s2), (mul, s3), (div, s4), (nullary, (bool, s5)), dipd, branch)
+print '-' * 40
-```python
-infer(expression)
+for fi, fo in h:
+ print doc_from_stack_effect(fi, fo)
```
+ ---------------------------------------------------------------------------
-
- [(s1, (f1, s1)), (s1, (i1, s1))]
-
-
-
-And that brings us to current Work-In-Progress. I'm pretty hopeful that the mixed-mode inferencer/interpreter `infer()` function along with the ability to specify multiple implementations for the combinators will permit modelling of the stack effects of e.g. `ifte`. If I can keep up the pace I should be able to verify that conjecture by the end of June.
+ ZeroDivisionError Traceback (most recent call last)
+
+ <ipython-input-1-9a9d60354c35> in <module>()
+ ----> 1 1/0 # (Don't try to run this cell! It's not going to work. This is "read only" code heh..)
+ 2
+ 3 logging.basicConfig(format='%(message)s', stream=sys.stdout, level=logging.INFO)
+ 4
+ 5 globals().update(FUNCTIONS)
+
+
+ ZeroDivisionError: integer division or modulo by zero
+
+
+The numbers at the start of the lines are the current depth of the Python call stack. They're followed by the current computed stack effect (initialized to `ID`) then the pending expression (the inference of the stack effect of which is the whole object of the current example.)
+
+In this example we are implementing (and inferring) `ifte` as `[nullary bool] dipd branch` which shows off a lot of the current implementation in action.
+
+ 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)
## Conclusion
-(for now...)
+We built a simple type inferencer, and a kind of crude "compiler" for a subset of Joy functions. Then we built a more powerful inferencer that actually does some evaluation and explores branching code paths
Work remains to be done:
- docstrings all around
- improve this notebook (it kinda falls apart at the end narratively. I went off and just started writing code to see if it would work. It does, but now I have to come back and describe here what I did.
-I'm starting to realize that, with the inferencer/checker/compiler coming along, and with the UI ready to be rewritten in Joy, I'm close to a time when my ephasis is going to have to shift from crunchy code stuff to squishy human stuff. I'm going to have to put normal people in front of this and see if, in fact, they *can* learn the basics of programming with it.
+## Appendix: Joy in the Logical Paradigm
-The rest of this stuff is junk and/or unfinished material.
+For *type checking* to work the type label classes have to be modified to let `T >= t` succeed, where e.g. `T` is `IntJoyType` and `t` is `int`. If you do that you can take advantage of the *logical relational* nature of the stack effect comments to "compute in reverse" as it were. There's a working demo of this at the end of the `types` module. But if you're interested in all that you should just use Prolog!
-## Appendix: Joy in the Logical Paradigm
-For this to work the type label classes have to be modified to let `T >= t` succeed, where e.g. `T` is `IntJoyType` and `t` is `int`
+Anyhow, type *checking* is a few easy steps away.
```python
AnyJoyType.accept = tuple, int, float, long, str, unicode, bool, Symbol
StackJoyType.accept = tuple
```
-
-
-```python
-F = infer(l2s((pop, swap, rolldown, rest, rest, cons, cons)))
-
-for f in F:
- print doc_from_stack_effect(*f)
-```
-
- ([a4 a5 .1.] a3 a2 a1 -- [a2 a3 .1.])
-
-
-
-```python
-from joy.parser import text_to_expression
-```
-
-
-```python
-F = infer(l2s((pop, pop, pop)))
-
-for f in F:
- print doc_from_stack_effect(*f)
-```
-
- (a3 a2 a1 --)
-
-
-
-```python
-s = text_to_expression('0 1 2')
-s
-```
-
-
-
-
- (0, (1, (2, ())))
-
-
-
-
-```python
-F[0][0]
-```
-
-
-
-
- (a1, (a2, (a3, s1)))
-
-
-
-
-```python
-L = unify(s, F[0][0])
-L
-```
-
-
-
-
- ()
-
-
-
-
-```python
-s = text_to_expression('0 1 2 [3 4]')
-s
-```
-
-
-
-
- (0, (1, (2, ((3, (4, ())), ()))))
-
-
-
-
-```python
-F[0][0]
-```
-
-
-
-
- (a1, (a2, (a3, s1)))
-
-
-
-
-```python
-L = unify(s, F[0][0])
-L
-```
-
-
-
-
- ()
-
-
-
-
-```python
-L = unify(F[0][0], s)
-L
-```
-
-
-
-
- ()
-
-
-
-
-```python
-F[1][0]
-```
-
-
- ---------------------------------------------------------------------------
-
- IndexError Traceback (most recent call last)
-
- <ipython-input-133-58a8e44e9cba> in <module>()
- ----> 1 F[1][0]
-
-
- IndexError: list index out of range
-
-
-
-```python
-s[0]
-```
-
-
-```python
-A[1] >= 23
-```
-
-## [Abstract Interpretation](https://en.wikipedia.org/wiki/Abstract_interpretation)
-I *think* this might be sorta what I'm doing above with the `kav()` function...
- In any event "mixed-mode" interpreters that include values and type variables and can track constraints, etc. will be, uh, super-useful. And Abstract Interpretation should be a rich source of ideas.
-
-
-## Junk
-
-
-```python
-class SymbolJoyType(AnyJoyType): prefix = 'F'
-
-W = map(SymbolJoyType, _R)
-
-k = S[0], ((W[1], S[2]), S[0])
-Symbol('cons')
-print doc_from_stack_effect(*k)
-
-```
-
-
-```python
-dip_a = ((W[1], S[2]), (A[1], S[0]))
-```
-
-
-```python
-d = relabel(S[0], dip_a)
-print doc_from_stack_effect(*d)
-```
-
-
-```python
-s = list(unify(d[1], k[1]))[0]
-s
-```
-
-
-```python
-j = update(s, k)
-```
-
-
-```python
-print doc_from_stack_effect(*j)
-```
-
-
-```python
-j
-```
-
-
-```python
-cons
-```
-
-
-```python
-for f in MC([k], [dup]):
- print doc_from_stack_effect(*f)
-```
-
-
-```python
-l = S[0], ((cons, S[2]), (A[1], S[0]))
-```
-
-
-```python
-print doc_from_stack_effect(*l)
-```
-
-
-```python
-
-def dip_t(F):
- (quote, (a1, sec)) = F[1]
- G = F[0], sec
- P = S[3], (a1, S[3])
- a = [P]
- while isinstance(quote, tuple):
- term, quote = quote
- a.append(term)
- a.append(G)
- return a[::-1]
-
-
-
-
-```
-
-
-```python
-from joy.utils.stack import iter_stack
-```
-
-
-```python
-a, b, c = dip_t(l)
-```
-
-
-```python
-a
-```
-
-
-```python
-b
-```
-
-
-```python
-c
-```
-
-
-```python
-MC([a], [b])
-```
-
-
-```python
-kjs = MC(MC([a], [b]), [c])
-kjs
-```
-
-
-```python
-print doc_from_stack_effect(*kjs[0])
-```
-
- (a0 [.0.] -- [a0 .0.] a1)
-
- a0 [.0.] a1 [cons] dip
- ----------------------------
- [a0 .0.] a1
-
-### `concat`
-
-How to deal with `concat`?
-
- concat ([.0.] [.1.] -- [.0. .1.])
-
-We would like to represent this in Python somehow...
-
-
-```python
-concat = (S[0], S[1]), ((S[0], S[1]),)
-```
-
-But this is actually `cons` with the first argument restricted to be a stack:
-
- ([.0.] [.1.] -- [[.0.] .1.])
-
-What we have implemented so far would actually only permit:
-
- ([.0.] [.1.] -- [.2.])
-
-
-```python
-concat = (S[0], S[1]), (S[2],)
-```
-
-
-Which works but can lose information. Consider `cons concat`, this is how much information we *could* retain:
-
- (1 [.0.] [.1.] -- [1 .0. .1.])
-
-As opposed to just:
-
- (1 [.0.] [.1.] -- [.2.])
-
-### represent `concat`
-
- ([.0.] [.1.] -- [A*(.0.) .1.])
-
-Meaning that `A*` on the right-hand side should all the crap from `.0.`.
-
- ([ .0.] [.1.] -- [ A* .1.])
- ([a .0.] [.1.] -- [a A* .1.])
- ([a b .0.] [.1.] -- [a b A* .1.])
- ([a b c .0.] [.1.] -- [a b c A* .1.])
-
-
-
-or...
-
- ([ .0.] [.1.] -- [ .1.])
- ([a .0.] [.1.] -- [a .1.])
- ([a b .0.] [.1.] -- [a b .1.])
- ([a b c .0.] [.1.] -- [a b c .1.])
- ([a A* c .0.] [.1.] -- [a A* c .1.])
-
-
-
- (a, (b, S0)) . S1 = (a, (b, (A*, S1)))
-
-
-```python
-class Astar(object):
- def __repr__(self):
- return 'A*'
-
-
-def concat(s0, s1):
- a = []
- while isinstance(s0, tuple):
- term, s0 = s0
- a.append(term)
- assert isinstance(s0, StackJoyType), repr(s0)
- s1 = Astar(), s1
- for term in reversed(a):
- s1 = term, s1
- return s1
-```
-
-
-```python
-a, b = (A[1], S[0]), (A[2], S[1])
-```
-
-
-```python
-concat(a, b)
-```
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