A System of Interaction and Structure IV: The Exponentials and Decomposition

arXiv:0903.5259v2 [cs.LO] 27 Jul 2010 A System of Interaction and Structure IV: The Exponentials and Decomposition Lutz Straßburger INRIA Saclay–ˆIle-de-France and ´Ecole Polytechnique, France and Alessio Guglielmi University of Bath, UK and INRIA Nancy–Grand Est, France We study a system, called NEL, which is the mixed commutative/non-commutative linear logic BV augmented with linear logic’s exponentials. Equivalently, NEL is MELL augmented with the non-commutative self-dual connective seq. In this paper, we show a basic compositionality property of NEL, which we call decomposition. This result leads to a cut-elimination theorem, which is proved in the next paper of this series. To control the induction measure for the theorem, we rely on a novel technique that extracts from NEL proofs the structure of exponentials, into what we call !-?-Flow-Graphs. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—proof theory General Terms: Deep Inference, Calculus of Structures, Linear Logic, Noncommutativity Additional Key Words and Phrases: Decomposition, Cut Elimination, !-?-Flow-Graphs 1. INTRODUCTION This is the fourth in a series of papers dedicated to the proof theory of a self-dual non-commutative operator, called seq, in the context of linear logic. The first paper “A System of Interaction and Structure” [Guglielmi 2007] intro- duced seq in the context of multiplicative linear logic. The resulting logic is called BV. The proof system for BV is presented in the formalism called the calculus of structures, which is the simplest formalism in the methodology of deep inference. In fact, deep inference was born precisely for giving BV a normalization theory. In the second paper “A System of Interaction and Structure II: The Need for Deep Inference” [Tiu 2006], Alwen Tiu shows that deep inference is necessary to obtain analyticity for BV. In other words, traditional Gentzen proof theory is not sufficient to deal with seq. The third paper, currently being elaborated, explores the connection between BV and pomset logic [Retor´e 1997]. This fourth paper, and the fifth paper “A System of Interaction and Structure V: The Exponentials and Splitting” [Guglielmi and Straßburger 2009] are devoted to the proof theory of system BV when it is enriched with linear logic’s exponentials. We call NEL (non-commutative exponential linear logic) the resulting system. We can also consider NEL as MELL (multiplicative exponential linear logic [Girard 1987]) plus seq. NEL, which was first presented in [Guglielmi and Straßburger 2002], is conservative over BV and over MELL augmented by the mix and nullary mix rules [Fleury and Retor´e 1994; Retor´e 1993; Abramsky and Jagadeesan 1994]. Note that, ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY, Pages 1–43. 2 · Lutz Straßburger and Alessio Guglielmi like BV, NEL cannot be analytically expressed outside deep inference. System NEL can be immediately understood by anybody acquainted with the sequent calculus, and is aimed at the same range of applications as MELL, but it offers, of course, explicit sequential composition. NEL is especially interesting because it is Turing-complete [Straßburger 2003c]. The complexity of MELL is currently unknown, but MELL is widely conjectured to be decidable. If that was the case, then the line towards Turing-completeness would clearly be crossed by seq, which, in fact, has been interpreted already as an effective mechanism to structure a Turing machine tape. This is something that MELL, which is fully commutative, apparently cannot do. This paper is devoted to the decomposition theorem. Together with the splitting theorem in [Guglielmi and Straßburger 2009] it immediately yields cut-elimination, which will be claimed in [Guglielmi and Straßburger 2009]. Decomposition (which was first pioneered in [Guglielmi and Straßburger 2001; Straßburger 2003b] for BV and MELL) is as follows: we can transform every NEL derivation into an equivalent one, composed of eleven derivations carried into eleven disjoint subsystems of NEL. This means that we can study small subsystems of NEL in isolation and then compose them together with considerable more freedom than in the sequent calculus, where, for example, contraction can not be isolated in a derivation. Decomposition is made available in the calculus of structures by exploiting the top-down symmetry of derivations that is typical of deep inference. Such a result is unthinkable in formalisms lacking locality, like Gentzen systems. The technique by which we prove the result is an evolution and simplification of a technique that was first developed in [Straßburger 2003b] for MELL, but that would not work unmodified in the presence of seq. In fact, seq makes matters more complicated, due to similar phenomena to those unveiled by Tiu [Tiu 2006], and that make seq intractable for Gentzen methods. Some of the main results of this paper have already been presented, with

…(Full text truncated)…

This content is AI-processed based on ArXiv data.