953 research outputs found
Focusing in Asynchronous Games
Game semantics provides an interactive point of view on proofs, which enables
one to describe precisely their dynamical behavior during cut elimination, by
considering formulas as games on which proofs induce strategies. We are
specifically interested here in relating two such semantics of linear logic, of
very different flavor, which both take in account concurrent features of the
proofs: asynchronous games and concurrent games. Interestingly, we show that
associating a concurrent strategy to an asynchronous strategy can be seen as a
semantical counterpart of the focusing property of linear logic
A Systematic Approach to Canonicity in the Classical Sequent Calculus
International audienceThe sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means of abstracting away the details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that make the foci as parallel as possible are canonical. Moreover, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique is a systematic way to recover the desired essence of any sequent proof without abandoning the sequent calculus
Multiplicative-Additive Focusing for Parsing as Deduction
Spurious ambiguity is the phenomenon whereby distinct derivations in grammar
may assign the same structural reading, resulting in redundancy in the parse
search space and inefficiency in parsing. Understanding the problem depends on
identifying the essential mathematical structure of derivations. This is
trivial in the case of context free grammar, where the parse structures are
ordered trees; in the case of categorial grammar, the parse structures are
proof nets. However, with respect to multiplicatives intrinsic proof nets have
not yet been given for displacement calculus, and proof nets for additives,
which have applications to polymorphism, are involved. Here we approach
multiplicative-additive spurious ambiguity by means of the proof-theoretic
technique of focalisation.Comment: In Proceedings WoF'15, arXiv:1511.0252
Multi-Focusing on Extensional Rewriting with Sums
International audienceWe propose a logical justification for the rewriting-based equivalence procedure for simply-typed lambda-terms with sums of Lindley [Lin07]. It relies on maximally multi-focused proofs, a notion of canonical derivations introduced for linear logic. Lindley's rewriting closely corresponds to preemptive rewriting [CMS08], a technical device used in the meta-theory of maximal multi-focus
Polarizing Double Negation Translations
Double-negation translations are used to encode and decode classical proofs
in intuitionistic logic. We show that, in the cut-free fragment, we can
simplify the translations and introduce fewer negations. To achieve this, we
consider the polarization of the formul{\ae}{} and adapt those translation to
the different connectives and quantifiers. We show that the embedding results
still hold, using a customized version of the focused classical sequent
calculus. We also prove the latter equivalent to more usual versions of the
sequent calculus. This polarization process allows lighter embeddings, and
sheds some light on the relationship between intuitionistic and classical
connectives
A Hybrid Linear Logic for Constrained Transition Systems
Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. The logic has a focused cut-free sequent calculus that can be used to internalize the rules of particular constrained transition systems; we illustrate this with an adequate encoding of the synchronous stochastic pi-calculus
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