is Computation = Proof search.

Abstract. The proof-theoretic approach to logic programming has benefited from the introduction of focused proof systems, through the non-determinism reduction and control they provide when searching for proofs in the sequent calculus. However, this technique was not available in the calculus

. Keywords-Proof theory; focused proof systems; linear logic I.

dale.miller at inria.fr saurin at lix.polytechnique.fr Abstract. Probably the most significant result concerning cut-free sequent cal-culus proofs in linear logic is the completeness of focused proofs. This com-pleteness theorem has a number of proof theoretic applications — e.g. in games semantics

We present a sequent calculus for non-commutative logic which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in the general case, we show that our focalized sequent calculus is equivalent to the original one by studying the permutabilities of rules for NL and showing that all LL permutabilities involved in focalization can be lifted to NL permutabilities. These results are based on a study of the partitions of partially ordered sets modulo entropy. 1 Introduction Non-commutative logic, NL for short, was introduced by Abrusci and the second author in [1, 15] (see also section 3). It unies commutative linear logic [6] and cyclic linear logic [16], a classical conservative extension of the Lambek calculus [10]. The present paper investigates the \focalization" property for non-commutative logic. 1.1 The property of focalization Rules of the sequent calculus for linear logic [6] are well-known to split into two c...

Abstract It is now well-established that the so-called focalization property plays a central role in the design of programming languages based on proof search, and more generally in the proof theory of linear logic. We present here a sequent calculus for non-commutative logic (NL) which enjoys

of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence

proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs—a well known, minimalistic

are formulated as a search of such sequence of steps that would minimize a loss function while one is seeking the most information. Examples are shown as the existence proof of the proposed theory on obtaining range from focus and sterolvergence on 2-D segmentation of an image and 3-0 shape parametrization.

Abstract. In previous work, the author gave a higher-order analysis of focusing proofs (in the sense of Andreoli’s search strategy), with a role for infinitary rules very similar in structure to Buchholz’s Ω-rule. Among other benefits, this “pattern-based ” description of focusing simplifies