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McCarthy's Ambiguous Operator (2005) (randomhacks.net)
157 points by kristianp 7 days ago | hide | past | web | favorite | 42 comments

Author here. It's fun to see this post 14 years later!

If you like this, you might also enjoy:

- Prolog: This is an entire language built around choice and backtracking. https://en.wikipedia.org/wiki/Prolog

- The Reasoned Schemer, which builds a Prolog-in-Scheme from first principles. This book is excellent, or even mind-blowing if you're never seen this before. https://mitpress.mit.edu/books/reasoned-schemer

- Oz/Mozart: This language takes the basic idea even further, adding concurrency and constraint solving. There's an older book about this that's nice: https://www.info.ucl.ac.be/~pvr/book.html It feels like Oz was an evolutionary dead end, but an interesting one.

- Probability monads. You can extend this general idea to include probability distributions over values. I have an older series of blog posts at http://www.randomhacks.net/probability-monads/ that explains this idea piece-by-piece, but there's been a ton of research in the past decade on probabilistic languages: http://www.probabilistic-programming.org/wiki/Home

It's definitely a fun topic!

There are also icon and txr. I always liked the icon approach where backtracking is fairly explicit, but still tightly integrated into the language. Txr is more for text analysis.

is there any new wild idea in this field ? so far all I could find was FD/CLP over prolog like search.

The McCarthy '63 paper referred to seems to be here: http://www-formal.stanford.edu/jmc/basis1/basis1.html

A brilliant paper. IMHO one of the cornerstones of computer science. I can warmly recommend it. 'pg has a good explanation of it here: http://www.paulgraham.com/rootsoflisp.html?viewfullsite=1 (Windows users might want to use an online converter to convert it from Postscript to PDF.)

I suppose this is a good time to point out the toy Lisp interpreter I wrote a while ago: http://patient0.github.io/FirstClassLisp/

It supports first class macros, continuations, and has the amb "operator" (implemented using a macro of course!): https://github.com/Patient0/FirstClassLisp/blob/master/Lisp/...

amb without side-effects can also be expressed in terms of the List monad, e.g.:

    do x <- [1, 2, 3]
       y <- [4, 5, 6]
       if x * y == 8 then pure (x, y)
                     else []
which can be equivalently reworded as the list comprehension:

    [(x, y) | x <- [1, 2, 3], y <- [4, 5, 6], x * y == 8]
i.e., the underlying structure of amb without side-effects can also be understood as just a search of the input space for values that fulfill some criteria, rather than as backtracking via continuations.

This analogy misses something fundamental. See how the assignments to x and y are of very different character than the predicate? Notice the return path (`else []`). `amb` doesn't need this.

`amb` merges the assignments and predicate - this is important because you may not know your argument count statically, i.e. if your arguments are a list. But beyond that, `amb` may backtrack an arbitrarily deep call stack, so the the error return does not need to be expressed explicitly. You may think that is good or bad but it is not the same as the Haskell solution.

We can implement it with a single `amb` function, too (I took some shortcuts that might have hidden some of the nature of the implementation)!

    -- Lifts a list into Amb.
    amb :: [a] -> Amb a
If we assume Amb is just List, then:

    amb = id
If we write the example in the original article in desugared style, we get:

    amb [1, 2, 3] >>= \x ->
    amb [4, 5, 6] >>= \y ->
    if x * y /= 8 then amb [] else pure () >>
    pure (x, y)
(we are forced to use an awkward condition with `pure ()` on the else branch when calling `amb` because Haskell requires us to return values on all branches. We can rewrite it equivalently in terms of `when` to hide that detail:

    amb [1, 2, 3] >>= \x ->
    amb [4, 5, 6] >>= \y ->
    when (x * y /= 8) (amb []) >>
    pure (x, y)
It now looks more similar to the original example.)

It ends up looking like continuation-passing style: conceptually, if we encounter `amb []` in the nested function, the nested computation ends and we start examining the next value in the list values being assigned to `x`: the implementation given in the original post does end up using callcc, so with some imagination you might be able to derive some kind of equivalence here :)

What is magical and mind-bending about `amb` is that it rewinds through arbitrary complicated call stacks, similar to an exception. This is why it's necessarily a special form.

The Haskell version requires each callee to be annotated with the `Amb` return type. It cannot be used to escape unless the caller is prepared for it to escape. That's a significant limitation (but all we're really saying is that Haskell doesn't support call/cc).

That's not an amb function. That's a builtin type that's inherently amb-y.

You still get credit for solving the basic problem but let's not be misleading.

Also don't forget to call head.

In Haskell monads, you define a monadic type and then "bind" is the function that does something specific for that type. "It's a type" because Haskell uses static types and polymorphism to factor out common logic used by different functions.

Of course Haskell list is a built-in type, but it's not magical except for the special bracket syntax. You can make a sugar-free user-defined type

    data List a = Nil | Cons a (List a)
if you want.

Yes, I know all of that. I wasn't saying it was magical, I was saying that in that particular code "amb expressed in terms of the List monad" is an accurate description, but "amb function" isn't really true. The monad is doing 100% of the ambiguous operation and the "amb function" does 0%.

As an analogy let's say I define "(" to do write to stdout, and "print" as id. Even though "print(4)" works as expected, my "print function" is a lie.

Or in C source code where adjacent strings get merged, I could [#define CONCAT ] so that ["ab" CONCAT "cd"] gets preprocessed to ["ab" "cd"] and parsed as ["abcd"]. But I didn't actually write a concatenation operator. The concatenation happens because of something else, it would happen even if you didn't use CONCAT, and you can sprinkle CONCAT all over your source code with no effect.

    foo = do
        x <- [1,2 3]
        y <- [1,2,3]
        guard (x * y == 8)
        return (x, y)

Hmmm, I think there is some subtlety here in terms of the way an empty list is used in Haskell. This paper by Wadler from the 80s (predating monads) discusses the functional lore of the combinator approach to parsing, and it significantly leverages the properties of [] to represent failure. It's a "semipredicate" type of situation, where the return value is either a list or failure; conveniently, the empty list isn't one of the possible success values. So if I understand the Haskell idiom correctly, it's a bit of a hack which is inherent in the heritage of the list monad. (NB there's nothing about raw lists per se that means they form a nondeterministic choice monad—that's a culturally contingent Haskell thing.)


It's an arbitrary choice in the Prelude, yes. If you don't like it you can definitely your own Cons type (or wrap a list in a new type) and define any monad instance (semantics) you choose.

Cons/lists don't have to be involved at all: a Set type could be made to work for many applications.

I've observed several smart people struggling at the outset with the connection between recursively defined lists and nondeterminism in Haskell. It's not uncommon for people to assume that there is some essential connection between the two, because that's what "list monad" seems to suggest. In fact any collection type would do.

The comments on the article include two more nice Haskell solutions (with much better speed).

This actually sounds very Prologesque.

"Here, have some inputs. From those, find me the ones that satisfy this goal."

AMB is a very handy primitive to implement prolog.

Hmmm... AMB probably is sufficient for a toy version of Prolog.

I feel like anyone actually implementing prolog will need to fully understand horn logic. AMB is a useful operator when the specifics of the backtracking search can be abstracted away.

But an "actual" prolog implementation needs to have a well-defined depth-first search over the horn-logic clauses. There are a lot of optimizations to make horn-logic solvers way faster.

I envy the languages with continuations. That makes the implementation of backtracking so much easier.

Not only backtracking but also other control flow structures: https://curiosity-driven.org/continuations#flow-structures

Continuations also allowed me to implement python’s yield/send in 16 lines of Scheme code :-)


You can also implement scheme's call/cc with python's yield/send, right?

Not in full generality. (Well, obviously you can write a whole Scheme interpreter in Python! But you can't write something that does in Python just what call/cc does in Scheme.)

Suppose you do the following:

(define a-continuation (call-with-current-continuation (lambda (c) c)))

The value of a-continuation at this point is a continuation object that, when called with argument x, sets a-continuation to x and returns to toplevel.

So if you now do, say,

(define (showfibs a b) (print a) (if (> a 1000) (a-continuation a) 1) (showfibs b (+ a b)))


(showfibs 0 1)

then you'll get 011235813213455891442333776109871597 as output, the new value of a-continuation will be 1597, and you'll be back at your top-level REPL prompt.

I don't think anything you can do with Python generators has such a pervasive ability to mess with control flow.

Incorrect. In c terms, you can think of callcc as a procedure which memcpy s the stack onto the heap for later application. And when it is later applied, the heap allocated stack is memcpy ed back into the stack region of memory, destroying the current stack.

Python and c#’s yield are nothing like that. It’s just syntactic sugar for iterators.

Python yield actually supports coroutines with bidirectional communication, not just unidirectional iterators; while not as full featured as callcc, these cover quite a lot of the use cases.

Agreed regarding use cases. My scheme implementation does both yield and send, cleanly imho, because of call/cc.

that said oleg kiselyov said that generators were .. as powerful as calcc. Or maybe close enough. I forgot the details

SICP spends many pages on this.

right, actually one of the topics were i stopped on my first readings.


Contextualizing this with Hoare's work on pre and post conditions helped me to understand this chapter. Predication is fundamental to advanced computer science.

So it's a base operator for prolog unification?

No. From an implementation POV, Prolog adds two pieces to ordinary programming languages: backtracking plus unification. This post talks about backtracking, but doesn't touch upon unification at all.

I had been wondering what the `amb` operator in Rx is named after. It takes any number of reactive streams and evaluates to the one that first emits a value, discarding the others. I presume the analogy here is that because it doesn’t block, it figuratively has to ”look into the future” to figure out which stream to return. But is there a deeper theoretical connection there?

For me it looks like this in JavaScript: const [a, b] = permutations(array1, array2).filter(([a, b]) => a*b === 8).slice(0,2);

Regarding the ruby example, thinking about the performance of such a (non-)feature makes my head spin. Not to mention the possibility of side-effects wrecking the program state and the complexity of debugging code like that without making Rubys error handling aware of such a construct.

All in all, this seems like a fun hack, but a catastrophically bad idea in practice.

This is an NP-complete problem, so performance will be an issue anyway.

There's some discussion about removing callcc from the language here:


It looks like it's still included at the moment, though:


It sounds a lot like constraint programming.

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