Generalized Connectives for Multiplicative Linear Logic

In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ? and disjunction ?, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms

Ludics and its Applications to natural Language Semantics

Proofs, in Ludics, have an interpretation provided by their counter-proofs, that is the objects they interact with. We follow the same idea by proposing that sentence meanings are given by the counter-meanings they are opposed to in a dialectical interaction. The conception is at the intersection of a proof-theoretic and a game-theoretic accounts of semantics, but it enlarges them by allowing to deal with possibly infinite processes

An Analytic Propositional Proof System on Graphs

In this paper we present a proof system that operates on graphs instead of formulas. Starting from the well-known relationship between formulas and cographs, we drop the cograph-conditions and look at arbitrary undirected) graphs. This means that we lose the tree structure of the formulas corresponding to the cographs, and we can no longer use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic with mix, and we argue that our graphs form a notion of generalized connective

When are prime formulae characteristic?

In the setting of the modal logic that characterizes modal refinement over modal transition systems, Boudol and Larsen showed that the formulae for which model checking can be reduced to preorder checking, that is, the characteristic formulae, are exactly the consistent and prime ones. This paper presents general, sufficient conditions guaranteeing that characteristic formulae are exactly the consistent and prime ones. It is shown that the given conditions apply to the logics characterizing all the semantics in van Glabbeek's branching-time spectrum

On natural deduction in fixpoint logics

In the current paper we present a powerful technique of obtaining natural deduction (or, in other words, Gentzen-like) proof systems for first-order fixpoint logics. The term "fixpoint logics" refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with e.g. dynamic logic, temporal logic and, of course, mu-calculus among them. Fixpoint logics were intensively studied during the last decade. In this paper we are going to present some results concerning deductive systems for first-order fixpoint logics. In particular we shall present some powerful and general technique for obtaining natural deduction (Gentzen-like) systems for fixpoint logics. As those logics are usually totally undecidable, we show how to obtain complete (but infinitary) proof systems as well as relatively complete (finitistic) ones. More precisely, given fixpoint equations on formulas defining nonclassical connectives of a logic, we automatically derive Gentzen-like proof systems for the logic. The discussion of implementation problems is also provided

Querying the Guarded Fragment

Evaluating a Boolean conjunctive query Q against a guarded first-order theory F is equivalent to checking whether "F and not Q" is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since Q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories F and (unions of) conjunctive queries Q that (i) Q is true in each model of F iff Q is true in each finite model of F and (ii) determining whether F implies Q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded bisimulation invariant PTIME.Comment: This is an improved and extended version of the paper of the same title presented at LICS 201

Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries

Nonstandard utilities for lexicographically decomposable orderings

Using a basic theorem from mathematical logic, I show that there are field-extensions of R on which a class of orderings that do not admit any real-valued utility functions can be represented by uncountably large families of utility functions. These are the lexicographically decomposable orderings studied in Beardon et al. (2002a). A corollary to this result yields an uncountably large family of very simple utility functions for the lexicographic ordering of the real Cartesian plane. I generalise these results to the lexicographic ordering of R^n, for every n > 2, and to lexicographic products of lexicographically decomposable chains. I conclude by showing how almost all of these results may be obtained without any appeal to the Axiom of Choice