The Duality of Computation under Focus

Electronic Proceedings in Theoretical Computer Science, 2014

We apply an idea originated in the theory of programming languages-monadic meta-language with a distinction between values and computations-in the design of a calculus of cut-elimination for classical logic. The cut-elimination calculus we obtain comprehends the call-by-name and call-byvalue fragments of Curien-Herbelin's λ µμ-calculus without losing confluence, and is based on a distinction of "modes" in the proof expressions and "mode" annotations in types. Modes resemble colors and polarities, but are quite different: we give meaning to them in terms of a monadic meta-language where the distinction between values and computations is fully explored. This metalanguage is a refinement of the classical monadic language previously introduced by the authors, and is also developed in the paper.

Mathematical Structures in Computer Science, 2008

X is an untyped continuation-style formal language with a typed subset which provides a Curry-Howard isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X , we will show how elaborate calculi can be embedded, like the λ-calculus, Bloo and Rose's calculus of explicit substitutions λx, Parigot's λµ and Curien and Herbelin's λµμ.

Catgorial Grammar, 2004

The purpose of this paper is to show that we can work in the spirit of minimalist grammars by means of a labelled commutative (and associative) calculus (Oehrle's Mon:LP), enriched with constraints on the use of assumptions. Lexical entries are considered proper axioms, some of which are coupled with (sequences of) hypotheses which must necessarily be introduced and discharged before the use of their associated axiom. Like similar proposals by P. de Groote or R. Muskens, our calculus has two interfaces. Each of them provides a homomorphism of types (from syntactic types to semantic ones, or -types, and from syntactic types to phonetic ones, or -types). Move is simply the link between a hypothesis A and its lifted type (A X) X when a (or )-term of the later type applies to a piece of proof the last step of which consists in precisely discharging A. The ability for a phrase to overtly move is governed by the form of its -term. Moreover, the elimination rule for ª is used to simulate head-movements. An enrichment with the exponential of Linear Logic: "!" is also proposed in order to treat binding phenomena in the spirit of Kayne (2003). We see that this enrichment opens the field to new insights concerning ellipsis and coordination. Proximity with other formalisms like Lambda grammars and Abstract Categorial Grammars is discussed.

Theory of Computing Systems / Mathematical Systems Theory, 2009

In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.

ArXiv, 2013

In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) b...