Jumbo Connectives In Type Theory And Logic

We make an argument that, for any study involving computational effects such as divergence or continuations, the traditional syntax of simply typed lambda-calculus cannot be regarded as canonical, because standard arguments for canonicity rely on isomorphisms that may not exist in an effectful setting. To remedy this, we define a “jumbo lambda-calculus” that fuses the traditional connectives together into more general ones, so-called “jumbo connectives”. We provide two pieces of evidence for our thesis that the jumbo formulation is advantageous. Firstly, we show that the jumbo lambda-calculus provides a “complete” range of connectives, in the sense of including every possible connective that, within the beta-eta theory, possesses a reversible rule. Secondly, in the presence of effects, we see that there is no decomposition of jumbo connectives into non-jumbo ones that is valid in both call-byvalue and call-by-name. Finally, we apply the concept of jumbo connectives to systems with isorecursive types (Jumbo FPC) and multiple conclusions (Jumbo LK). At each stage, we see that various connectives proposed in the literature are special cases of the jumbo connectives. 1 Canonicity and Connectives According to many authors [GLT88,LS86,Pit00], the “canonical” simply typed λ-calculus possesses the following types: A ::= 0 | A+A | 1 | A×A | A → A (1) There are two variants of this calculus. In some texts [GLT88,LS86] the × connective (type constructor) is a projection product, with elimination rules Γ ⊢ M : A×B Γ ⊢ πM : A Γ ⊢ M : A×B Γ ⊢ πM : B In other texts [Pit00], × is a pattern-match product, with elimination rule Γ ⊢ M : A×B Γ, x : A, y : B ⊢ N : C Γ ⊢ pm M as 〈x, y〉. N : C This choice of five connectives 0,+, 1,×,→ raises some questions. 1. Why not include a ternary sum type +(A,B,C)? 2. Why not include a type (A,B) → C of functions that take two arguments? 3. Why not include both a pattern-match product A × B and a projection product A Π B? In the purely functional setting, these can be answered using Ockham’s razor: 1. unnecessary—it would be isomorphic to (A+B) + C 2. unnecessary—it would be isomorphic to (A×B) → C, and to A → (B → C) 3. unnecessary—they would be isomorphic, so either one suffices. But these answers are not valid in the presence of effectful constructs, such as recursion or control operators. For example, in a call-by-name language with recursion, +(A,B,C) 6 ∼= (A+B)+C (a point made in [McC96b]), and A×B 6∼= A Π B. To see this, consider standard semantics that interprets each type by a pointed cpo. Then A + B denotes ([[A]] + [[B]])⊥, and A Π B denotes [[A]] × [[B]] whereas A×B denotes ([[A]]× [[B]])⊥. This suggests that, to obtain a canonical formulation of simply typed λcalculus (suitable for subsequent extension with effects), we should—at least a priori—replace Ockham’s minimalist philosophy with a maximalist one, treating many combinations of the above connectives as primitive. These combinations are called jumbo connectives. But how many connectives must we include to obtain a “complete” range? A first suggestion might be to include every possible combination of the original five as primitive, e.g. a ternary connective γ mapping A,B,C to (A → B) → C. But this seems unwieldy. We need some criterion of reasonableness that excludes γ but includes all the connectives mentioned above. We obtain this by noting that each of the above connectives possesses, within the βη equational theory, a reversible rule. For example: Γ, A ⊢ B ========= Γ ⊢ A → B Γ, A ⊢ C Γ, B ⊢ C =============== Γ, A+B ⊢ C The rule for A → B means that we can turn each inhabitant of Γ, A ⊢ B into an inhabitant of Γ ⊢ A → B, and vice versa, and these two operations are inverse (up to βη-equality). The rule for A + B is understood similarly. Note also that, in these rules, every part of the conclusion other than the type being introduced appears in each premise. Informally, we shall say that a connective is {0,+, 1,×,→}-like when, in the presence of βη, it possesses such a reversible rule. In this paper, we introduce a calculus called “jumbo λ-calculus”, and show that it contains every {0,+, 1,×,→}-like connective. As stated above, our main argument for the necessity of jumbo connectives in the effectful setting is that suggested decompositions are not a priori valid. But in Sect. 4 we take this further by showing that, a posteriori, they do not have a decomposition that is valid in both CBV and CBN. In the last part of the paper, we show how the concept of jumbo connectives can be applied to a system with iso-recursive types (Jumbo FPC) or with multiple conclusions (Jumbo LK). The latter example demonstrates how the form of the jumbo connectives is closely tied to the form of a sequent. Since a multipleconclusion sequent is more general than a single-conclusion one, the form of the jumbo connectives is likewise more general. Related work Both our arguments for jumbo connectives (invalidity of decompositions, possession of a reversible rule) have arisen in ludics [Gir01]. 1.1 Infinitely Wide Variant Frequently, in semantics, one wishes to study infinitely wide calculi with countable sum types and countable product types. (The latter are necessarily projection products.) We therefore say that a connective is {0,+, ∑ i∈N, 1,×, ∏ i∈N,→ }-like when it possesses a reversible rule with countably many premises. By contrast, a {0,+, 1,×,→}-like connective is required to have a finitary reversible rule i.e. one with finitely many premises. We therefore define two versions of the jumbo λ-calculus: – the finitary version, containing every {0,+, 1,×,→}-like connective – the infinitely wide version, containing every {0,+, ∑ i∈N, 1,×, ∏ i∈N,→}-like connective.