Frame definability in finitely valued modal logics

In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics.     

Rudimentary Languages and Second-Order Logic

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing a direct logical characterization of rudimentary languages and a representation result of second-order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity.     

Measures on finite concrete logics

We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets.

     

Z-自动机与分划逻辑

本文分解Ｚ 自动机成为ω 自动机偶的复合；建立了Ｚ的Ｌ（ＭＰ）理论，利用Ｚ 自动机的技巧证明了Ｚ的Ｌ（ＭＰ）理论的可判定性．作为应用，证明了有限状态过程的正确性问题是可判定的．     

Arithmetic height functions over finitely generated fields

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). Oblatum 7-VI-1999 & 21-IX-1999 / Published online: 24 January 2000     

Approximating max-cut under graph-MSO constraints

We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.     

Monadic second-order model-checking on decomposable matroids

A notion of branch-width, which generalizes the one known for graphs, can be defined for matroids. We first give a proof of the polynomial time model-checking of monadic second-order formulas on representable matroids of bounded branch-width, by reduction to monadic second-order formulas on trees. This proof is much simpler than the one previously known. We also provide a link between our logical approach and a grammar that allows one to build matroids of bounded branch-width. Finally, we introduce a new class of non-necessarily representable matroids, described by a grammar and on which monadic second-order formulas can be checked in linear time.     

Prefinitely axiomatizable modal and intermediate logics

A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.     

Subintuitionistic logics and the implications they prove

This is an investigation of the implications of IPC which remain provable when one weakens intuitionistic logic in various ways. The research is concerned with logics with Kripke models as introduced by G. Corsi in 1987, and others like G. Restall, Do?en, Visser. This leads to conservativity results for IPC with regard to classes of implications in some of these logics. Moreover, similar results are reached for some weaker subintuitionistic systems with neighborhood models introduced by the authors in 2016. In addition, the relationship between two types of neighborhood models introduced in that work is clarified. This clarification leads also to modal companions for weaker logics.     

Highly nonlinear functions over finite fields

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from ${\mathbb{F}}_q^n$ to ${\mathbb{F}}_q$ to the set of affine functions from ${\mathbb{F}}_q^n$ to ${\mathbb{F}}_q$. We prove the conjecture for each q such that the characteristic of ${\mathbb{F}}_q$ lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the $[q^n,n+1]$ Reed-Muller code over ${\mathbb{F}}_q$ and so answers a question raised by Leducq in 2013. Our results extend the case $q=2$, which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.     

On intermediate inquisitive and dependence logics: An algebraic study

This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain representation theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics.     

Permutation polynomials from trace functions over finite fields

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields.     

Entropy of formulas

A probability distribution can be given to the set of isomorphism classes of models with universe {1, ..., n} of a sentence in first-order logic. We study the entropy of this distribution and derive a result from the 0–1 law for first-order sentences.      

Separation logic and logics with team semantics

Separation logic is a successful logical system for formal reasoning about programs that mutate their data structures. Team semantics, on the other side, is the basis of modern logics of dependence and independence. Separation logic and team semantics have been introduced with quite different motivations, and are investigated by research communities with rather different backgrounds and objectives. Nevertheless, there are obvious similarities between these formalisms. Both separation logic and logics with team semantics involve the manipulation of second-order objects, such as heaps and teams, by first-order syntax without reference to second-order variables. Moreover, these semantical objects are closely related; it is for instance obvious that a heap can be seen as a team, and the separating conjunction of separation logic is (essentially) the same as the team-semantical disjunction. Based on such similarities, the possible connections between separation logic and team semantics have been raised as a question at several occasions, and lead to informal discussions between these research communities. The objective of this paper is to make this connection precise, and to study its potential but also its obstacles and limitations.     

The Beth property and interpolation in lattice-based algebras and logics

We deal with logics based on lattices with an additional unary operation. Interrelations of different versions of interpolation, the Beth property, and amalgamation, as they bear on modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of implicative lattices, have been studied in many works. Sometimes these relations can and sometimes cannot be extended to the logics without implication considered in the paper. Supported by INTAS (grant No. 04-77-7080) and by RFBR (grant No. 06-01-00358). Supported by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 307–334, May–June, 2008.     

Unifying hidden-variable problems from quantum mechanics by logics of dependence and independence

We study hidden-variable models from quantum mechanics and their abstractions in purely probabilistic and relational frameworks by means of logics of dependence and independence, which are based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties and their simultaneous realisability can thus be formulated and proven on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the ‘no-go’ theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property.