Multiplicative Linear Logic

this paper we concentrate on the so-called multiplicative fragment, i.e. the connectives\Omeg

Proof Nets with Explicit Negation for Multiplicative Linear Logic

Multiplicative linear logic (MLL) was introduced in [Gi87] as a one-sided sequent calculus: linear negation is a notion that is defined, via De Morgan identities. One obtains proof nets for MLL by identifying derivations in the one-sided calculus that are equal up to a permutation of inference rules. In this paper we consider a similar quotient for the formulation of MLL as a two-sided sequent calculus: to the usual set of links we add links also for the left rules. As a consequence, negation need no longer be defined, but can be treated as a basic connective. We develop the fundamental theory (substructures, empires and sequentialization) for this variation on the notion of proof net, and show how to obtain Girard's sequentialization theorem for the standard proof nets in one-sided sequent calculus as a corollary. [email protected] URL: http://www.math.uu.nl/people/puite/ Contents 1 Introduction 1 2 Proof structures of MLL 3 3 Proof nets of MLL 11 4 The proof net of a derivati..

An algebraical proof of the Contraction Criterion for Proof Nets

Abstract proof structures in multiplicative linear logic are graphs with some additional structure, and the class of proof nets is an inductively defined subclass. F. M'etayer established a correctness criterion by defining homology groups for proof structures, which characterize the proof nets among these. Using this result, we will present a completely algebraical proof of the traditional Contraction Criterion, due to Danos and Regnier. 1 Preliminaries 1.1 MLL The set of formulas of multiplicative linear logic (MLL) is defined as the smallest set containing atoms p 1 ; p 2 ; : : : and their formal negations p ? 1 ; p ? 2 ; : : : , and closed under the binary operations\Omega (tensor) and & (par ). Unary negation is defined by the (commutative) De Morgan laws, i.e. (a\Omega b) ? := a ? & b ? , etc. Derivable objects are sequents ) X, where X is a multiset of formulas. The rules of MLL read: axiom ) a ? ; a cut-rule ) X; a ) a ? ; Y ) X;Y tensor-rule ) X; a ) b; ..

Correctness Criteria based on a Homology of Proof Structures in Multiplicative Linear Logic

Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3 Graphs 15 3.1 Paired directed multigraphs : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 3.2 Constructions on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Proof nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 4 Homology 27 4.1 General theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.2 Application on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

An algebraical proof of the Danos-Regnier correctness criterion for proof nets

Abstract proof structures in multiplicative linear logic are graphs with some extra structure, and the class of proof nets is an inductively defined subclass. F. M'etayer established a correctness criterion by defining homology groups for proof structures, which characterize the proof nets among these ([Meta94a]). Using this result, we will present a completely algebraical proof of the traditional "acyclic and connected"-criterion, due to Danos and Regnier (see [DaRe89]). Main ingredient is a lemma stating that each proof structure satisfying the tree condition has an initial (as opposed to splitting) pair. 1 Preliminaries 1.1 MLL The set of formulas of multiplicative linear logic (MLL) is defined as the smallest set containing atoms p 1 ; p 2 ; : : : and their formal negations p ? 1 ; p ? 2 ; : : : , and closed under the binary operations\Omega (tensor) and & (par ). Unary negation is defined by the (commutative) De Morgan laws, i.e. (a\Omega b) ? := a ? & b ? , etc. The ..

Book review: Solving Mathematical Problems: A Personal Perspective

NO ABSTRAC

Proof Nets for the Multimodal Lambek Calculus

ion of \Gamma) 1. Let us call a sequence \Sigma of formulas a C-sequence if, for some \Gamma such that hh\Gammaii = \Sigma, the sequent \Gamma ` C is derivable. Then \Sigma is a C-sequence iff there is a proof structure from jj\Sigmajj to fCg, on which all contractions can be applied (together with the necessary structural conversions) such that we end with a hypothesis tree in which the order of the hypotheses equals \Sigma. 2. Let us call a multiset \Sigma of formulas a C-multiset if, for some \Gamma such that jj\Gammajj = \Sigma, the sequent \Gamma ` C is derivable. Then \Sigma is a C-multiset iff there is a proof structure from \Sigma to fCg, on which all contractions can be applied (together with the necessary structural conversions) such that we end with a hypothesis tree, i.e. iff there is a proof net from \Sigma to fCg. \Upsilon Example 3.10 The following proof structure A ffl 0 B Lffl 0 1 2 AA BB B n 0 (A n 0 C) Ln 0 1 2 A n 0 C Ln 0 1 2 CC 1 2 (A ffl 0 B) n 0 C Rn 0 has..

Van Bijsterveldt lanceert IMO 2011 ... en Euclides lanceert IMO-rubriek

No abstract

A bilateral-free notion of modules for Non commutative logic

manuscritA module is a piece of a proof structure. An adequate notion of type describes its behaviour, e.g. as a set of relations on the border, not making any reference to the original formulas or links of the module. Two modules connected along their border constitute a proof net precisely if their types are ``orthogonal''. For MLL the longtrip criterion of [Girard87] leads to a definition of type as (the biorthogonal of) the set of permutations on the border induced by the switchings. The correctness criterion for MNL, being based on bilateral longtrips ([AbRu00]), leads to a notion of type for MNL modules as a set of tuples of partial permutations and some relations on the border taking care of bilaterality and inclusion of the conclusions ([Abrusci99]). We will show that the bilaterality condition for MNL can be replaced by the requirement that the underlying commutative structure be a net. In other words it suffices to take into account commutative switchings as well: sequential-3-free switchings which may contain next-left-switches. This yields a definition of type as a pair of sets of decorated partial permutations on the border. The separation into pairs prevents composition of partial permutations being induced by two ``non-compatible'' switchings. The decoration is needed because for MNL it no longer holds that the longtrip meets all formulas, notably the border formulas and the conclusions. Finally, we generalize the composition of modules into an operation which yields modules again. This corresponds to a theory of types (without any reference to proof structures). Then all subsets of conclusions can be regarded as border formulas, while we compose two types only on the intersection of the not necessarily coinciding border