Vol. 2 (4:3) 2006, pp. 1–44 www.lmcs-online.org

We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs. Keywords: Elementary Linear Logic, Geometry of interaction, Complexity, Semantics.

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper

We present a novel way of using proof nets for the multimodal Lambek calculus, which provides a general treatment of both the unary and binary connectives. We also introduce a correctness criterion which is valid for a large class of structural rules and prove basic soundness, completeness and cut elimination results. Finally, we will present a correctness criterion for the original Lambek calculus L as an instance of our general correctness criterion. 1 Introduction One of the most important proof theoretic innovations of linear logic has been the introduction of proof nets as a redundancy-free and elegant way to represent proofs. Proof nets are usually presented as members of a larger class of structures, called proof structures. Proof nets are those proof structures which satisfy some condition, a correctness criterion. In the theory of proof nets, correctness criteria usually fall into one of two categories. On the one hand, there are the graph theoretic or geometric correctness c...

We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that the simply-typed -calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the -autonomous type theory characterise -autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any -autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that the standard simply-typed -calculus with surjective pairing is...

We consider the following decision problems: PROOFNET: Given a multiplicative linear logic (MLL) proof structure, is it a proof net? ESSNET: Given an essential net (of an intuitionistic MLL sequent), is it correct? In this paper we show that linear-time algorithms for ESS-NET can be obtained by constructing the dominator tree of the input essential net. As a corollary, by showing that PROOFNET is linear-time reducible to ESSNET (by the trip translation), we obtain a linear-time algorithm for PROOFNET. We show further that these linear-time algorithms can be optimized to simple one-pass algorithms – each node of the input structure is visited at most once. As another application of dominator trees, we obtain lineartime algorithms for sequentializing proof nets (i.e. given a proof net, find a derivation for the underlying MLL sequent) and essential nets.

Abstract. We introduce a new class of multiplicative proof nets, J-proof nets, which are a typed version of Faggian and Maurel’s multiplicative L-nets. In J-proof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct, we obtain a simple proof of the sequentialisation theorem. 1

We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.

Proofs Without Syntax [37] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.

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