Theoretical Computer Science
lo.logic type-theory proof-theory continuations curry-howard
Updated Thu, 23 Jun 2022 08:25:03 GMT

How do continuations represent negations (under the Curry–Howard correspondence)?


Under the CurryHoward correspondence, types can be thought of as propositions, and values inhabiting a type can be thought of as proofs that the corresponding proposition is true. (E.g., the Cartesian product is logical conjunction.)

Negation is represented as $A \to \bot$, where $\bot$ is the empty type. This works because if a type is inhabited, there can't be any function $A \to \bot$ (if you do you have such a function and the type is inhabited, feed the value into it and you get an element of the empty type, hence a contradiction), and vice versa (if $A$ is empty, the empty function $A \to \bot$ exists).

But once you add continuations to your language, an $A$-continuation can also be given the type $A \to \bot$. (see e.g. the symmetric lambda calculus). This apparently provides computational meaning to classical theorems; for example, the $\mathcal{C}$-operator (a basic operator for manipulating continuations), has type $\neg\neg A \to A$.

But can't you have continuations for non-empty types? If you add continuations, does $\neg A$ no longer express "$A$ is-empty/is-false/has-no-proofs"? It can't, right? This formulation makes an inhabitant of $\mathbb{N} \to \bot$ axiomatic, and $\mathbb{N}$ certainly isn't uninhabited. So what does "$\neg A$" actually express, if not "$A$ is-empty/is-false/has-no-proof"? And how do you actually express "$A$ is-empty/is-false/has-no-proof" when continuations exist?

Solution

When one associates negation with continuations, it is probably not ideal to think of it in terms of an 'empty' type. Continuation passing can be done with respect to any result type, and if that type is held abstract, it works a lot like 'false' in that there is no introduction rule for it.

With respect to the paper you linked (I'm by no means an expert on it, and just skimmed it, really), I think it might help to study the reduction rules. It appears to me that the goal of the logic is to construct an expression $e : +\mathsf{int}$ and reduce that expression to a numeral $n$. This means that the continuation $\bullet : int$ is not something that would be valid as a program in itself. Rather it is a clue that the abstract 'result type' is actually $\mathsf{int}$, because we are interested in using continuation-based computations that eventually result in integers. $\bullet$ is something that immediately completes the computation, hence the rules:

$$(begin)\ \ \ \ e : +int \leadsto \langle e | \bullet \rangle$$ $$(end)\ \ \ \ \langle n | \bullet \rangle \leadsto n$$

I don't know what happens if you have $\bullet$ involved in other situations, or if you can, but I might expect it to abort other continuations and end the computation. It seems more like it can only be the initial continuation based on the rules, though.

Now, one answer to your question is that this system is not even trying to reason about uninhabited types. It is using a correspondence between the logical rules and control flow, without the specific semantic meaning of negation. One thing you might want to check out is linear logic, because it actually has two sorts of 'false' or zero, one which is about control flow, and one which is about emptiness. The linearity is important for keeping them separate.

In that setting, $$ is the 'false' that has to do with control flow, and it gets things like double negation elimination and excluded middle (with a corresponding 'or' that is more related to representing functions via continuations than it is disjoint union), while there is also $0$, which gets the principle of explosion (which is arguably the distinct part of the empty type1). An interesting thing about this setting is that even though the logic seems classical, it is actually constructive because of the more restricted rules. Double negation by the 'control flow' bottom $$ is just a different (possibly more convenient) way of structuring a program that computes the doubly-negated type (and 'excluded middle' is a decomposed identity function).

Fillinski's 'Linear Continuations' might be a good place to start reading, although there are probably other references as well.

  1. In the continuation-with-abstract-result-type setting, you can also interpret explosion, but it has sort of a different meaning. Effectively such interpretations are always 'delimited' somewhere, even if it's only the end of your program. Explosion means that if you somehow get a value of the abstract result type, you can throw away your current continuation and jump to the delimiter. So it is an 'abort' operation. Linear logic doesn't allow this; all continuations must be used (and used exactly once), so explosion must be justified by involving an 'impossible' situation, like having a value of an empty type.

Comments (2)

  • +0 – While the second part of the answer looks speculative, there is indeed an interpretation in the same area as the first linked paper based on linear (involutive) negation. See Girard, A new constructive logic: classical logic (1991). Its continuation semantics is given by attributing to continuations an entirely different role than representing negation. — Jun 18, 2019 at 22:27  
  • +0 – Oh, another paper to look at is Linear Logic for Constructive Mathematics, which gives an interpretation of classical linear logic in terms of intuitionistic logic that is (I think) different than what you'll find in Girard's paper. Contrary to the title, the semantics are not exactly linear, but it still has some similar distinctions. — Jun 19, 2019 at 00:52  

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