Cut-free proof systems for logics of weak excluded middle

 In this work we perform a proof-theoretical investigation of some logical systems in the neighborhood of substructural, intermediate and many-valued logics. The common feature of the logics we consider is that they satisfy some weak forms of the excluded-middle principle. We first propose a cut-free hypersequent calculus for the intermediate logic LQ, obtained by adding the axiom *A∨**A to intuitionistic logic. We then propose cut-free calculi for systems W _n , obtained by adding the axioms *A∨(A ⊕ ⋯ ⊕ A) (n−1 times) to affine linear logic (without exponential connectives). For n=3, the system W _n coincides with 3-valued Łukasiewicz logic. For n>3, W _n is a proper subsystem of n-valued Łukasiewicz logic. Our calculi can be seen as a first step towards the development of uniform cut-free Gentzen calculi for finite-valued Łukasiewicz logics.     

Theory of truth degrees of propositions in the logic system Ln*

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

A spatial equational logic for the applied π-calculus

Spatial logics have been proposed to reason locally and modularly on algebraic models of distributed systems. In this paper we define the spatial equational logic A π L whose models are processes of the applied π-calculus. This extension of the π-calculus allows term manipulation and records communications as aliases in a frame, thus augmenting the predefined underlying equational theory. Our logic allows one to reason locally either on frames or on processes, thanks to static and dynamic spatial operators. We study the logical equivalences induced by various relevant fragments of A π L, and show in particular that the whole logic induces a coarser equivalence than structural congruence. We give characteristic formulae for some of these equivalences and for static equivalence. Going further into the exploration of A π L’s expressivity, we also show that it can eliminate standard term quantification.     

A Büchi-Like Theorem for Weighted Tree Automata over Multioperator Monoids

A multioperator monoid A\mathcal{A} is a commutative monoid with additional operations on its carrier set. A weighted tree automaton over A\mathcal{A} is a finite state tree automaton of which each transition is equipped with an operation of A\mathcal{A}. We define M-expressions over A\mathcal{A} in the spirit of formulas of weighted monadic second-order logics and, as our main result, we prove that if A\mathcal{A} is absorptive, then the class of tree series recognizable by weighted tree automata over A\mathcal{A} coincides with the class of tree series definable by M-expressions over A\mathcal{A}. This result implies the known fact that for the series over semirings recognizability by weighted tree automata is equivalent with definability in syntactically restricted weighted monadic second-order logic. We prove this implication by providing two purely syntactical transformations, from M-expressions into formulas of syntactically restricted weighted monadic second-order logic, and vice versa.     

Theory of truth degrees of formulas in Łukasiewicz<Emphasis Type="Italic">n</Emphasis>-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Łukasiewiczn-valued propositional logicL _n is proposed. A limit theorem is obtained, which says that the truth functionτ _n induced by truth degrees converges to the integrated truth functionτ whenn converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Łukasiewicz logic and the continuous valued Łukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.     

Infinitary logics and 0–1 laws

We investigate the infinitary logic L_∞ω^ω, in which sentences may have arbitrary disjunctions and conjunctions, but they involve only finite numbers of distinct variables. We show that various fixpoint logics can be viewed as fragments of L_∞ω^ω, and we describe a game-theoretic characterization of the expressive power of the logic. Finally, we study asymptotic probabilities of properties expressible in L_∞ω^ω on finite structures. We show that the 0–1 law holds for L_∞ω^ω, i.e., the asymptotic probability of every sentence in this logic exists and is equal to either 0 or 1. This result subsumes earlier work on asymptotic probabilities for various fixpoint logics and reveals the boundary of 0–1 laws for infinitary logics.     

A Restricted Second Order Logic for Finite Structures

We introduce a restricted version of second order logic SO^ωin which the second order quantifiers range over relations that are closed under the equivalence relation ≡^kofkvariable equivalence, for somek. This restricted second order logic is an effective fragment of the infinitary logicL^ω_∞ω, but it differs from other such fragments in that it is not based on a fixed point logic. We explore the relationship of SO^ωwith fixed point logics, showing that its inclusion relations with these logics are equivalent to problems in complexity theory. We also look at the expressibility of NP-complete problems in this logic.     

Theory of truth degrees of propositions in the logic system L n *

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

一类扩展的动态描述逻辑

作为描述逻辑的扩展,动态描述逻辑为语义Web服务的建模和推理提供了一种有效途径.在将语义Web服务建模为动作之后,动态描述逻辑从动作执行结果的角度提供了丰富的推理机制,但对于动作的执行过程却不能加以处理.借鉴Pratt关于命题动态逻辑的相关研究,一方面,对动态描述逻辑中动作的语义重新进行定义,将每个动作解释为由关于可能世界的序列组成的集合;另一方面,在动态描述逻辑中引入动作过程断言,用来对动作的执行过程加以刻画.在此基础上提出一类扩展的动态描述逻辑EDDL(X),其中的X表示从ALC(attributive language with complements)到SHOIN(D)等具有不同描述能力的描述逻辑.以X为描述逻辑ALCQO(attributive language with complements,qualified number restrictions and nominals)的情况为例,给出了EDDL(ALCQO)的表判定算法,并证明了算法的可终止性、可靠性和完备性.EDDL(X)可以从动作执行过程和动作执行结果两个方面对动作进行全面的刻画和推理,为语义Web服务的建模和推理提供了进一步的逻辑支持.     

The Expressive Power of Finitely Many Generalized Quantifiers

We consider extensions of first order logic (FO) and fixed point logic (FP) by means of generalized quantifiers in the sense of Lindström. We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature, strengthening earlier results. We also prove a stronger version of this result for PSPACE, which enables us to establish that PSPACE ≠ FO on any infinite class of ordered structures, a weak version of the ordered conjecture formulated by Ph. G. Kolaitis and M. Y. Vardi (Fixpoint logic vs. infinitary logic in finite-model theory, in "Proceedings, 7th IEEE Symposium on Logic in Computer Science, 1992," pp. 46-57). These results are obtained by defining a notion of element type for bounded variable logics with finitely many generalized quantifiers. Using these, we characterize the classes of finite structures over which the infinitary logic L^ω_∞ω extended by a finite aw set of generalized quantifiers Q is no more expressive than first order logic extended by the quantifiers in Q.     

Functional dependencies,supervenience, and consequence relations

An analogy between functional dependencies and implicational formulas of sentential logic has been discussed in the literature. We feel that a somewhat different connexion between dependency theory and sentential logic is suggested by the similarity between Armstrong's axioms for functional dependencies and Tarski's defining conditions for consequence relations, and we pursue aspects of this other analogy here for their theoretical interest. The analogy suggests, for example, a different semantic interpretation of consequence relations: instead of thinking ofB as a consequence of a set of formulas {A₁,...,A _n} whenB is true on every assignment of truth-values on which eachA _i is true, we can think of this relation as obtaining when every pair of truth-value assignments which give the same truth-values toA ₁, the same truth-values toA ₂,..., and the same truth-values toA _n, also make the same assignment in respect ofB. We describe the former as the consequence relation inference-determined by the class of truth-value assignments (valuations) under consideration, and the latter as the consequence relation supervenience-determined by that class of assignments. Some comparisons will be made between these two notions.     

A logic for the specification and proof of regular controllable processes of CCS

Summary This work has been motivated by the following general problem: find logics for the specification and proof of programs, described by terms of some algebra with given congruence relation. This relation is supposed to define a satisfactory concept for the behavioural comparison of programs. We require these logics to be adequate with respect to the term language, in the sense that two programs, behaviourally equivalent satisfy the same formulas and conversely. The term language considered is the subset of controllable, regular terms of CCS, on a vocabulary of actions A, with observational congruence. A term is said to be controllable if it is congruent to some term without occurrence of . We obtain an adequate logic whose language of formulas is obtained from constants true, false and ¦Nil¦ by using operators , , fixpoint operators, + and a for aA; the latter can be considered as extensions of the operators + and a for aA of CCS. As a result, controllable CCS terms can be considered as formulas of this logic and the problem of program verification is reduced to the proof of the validity of a formula.     

On the extension of classical propositional logic by means of a triangular norm

In this article, we introduce a generalized extension principle by substituting a more general triangular norm T for the min intersection operator in Zadeh's extension principle. We also introduce a family of propositional logics, sup- T extension logics, obtained by the extension of classical-logical functions. A few general properties of these sup-T extension logics are derived. It is also shown that classical binary logic and the Kleene ternary logic are special cases of these logics for any choice of T, obtained by a convenient restriction of the truth domain. the very practical decomposability property of classical logic is furthermore shown to hold for the sup-min extension logic, albeit in a somewhat more limited form.     

Cut-free sequent systems for temporal logic

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakening-free and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.     

On the finite degree of ambiguity of finite tree automata

Summary The degree of ambiguity of a finite tree automaton A, da(A), is the maximal number of different accepting computations of A for any possible input tree. We show: it can be decided in polynomial time whether or not da(A)<. We give two criteria characterizing an infinite degree of ambiguity and derive the following fundamental properties of an finite tree automaton A with n states and rank L>1 having a finite degree of ambiguity: for every input tree t there is a input tree t ₁ of depth less than 2²ⁿ·n! having the same number of accepting computations; the degree of ambiguity of A is bounded by 2^{2 2·log(L+1)·n}.     

Timed Modal Logics for Real-Time Systems

In this paper, a timed modal logic L _c is presented for the specification and verification of real-time systems. Several important results for L _c are discussed. First we address the model checking problem and we show that it is an EXPTIME-complete problem. Secondly we consider expressiveness and we explain how to express strong timed bisimilarity and how to build characteristic formulas for timed automata. We also propose a compositional algorithm for L _c model checking. Finally we consider several control problems for which L _c can be used to check controllability.     

Involutive monoidal t‐norm based logic and R0 logic

We investigate the relations between involutive monoidal t‐norm–based logics (IMTL) and R₀ logics. The main results are as follows: the logic IMTL and weak R₀ logic are equivalent; nilpotent minimum logic and R₀ logic are equivalent. © 2004 Wiley Periodicals, Inc.     

An Infinite Hierarchy of Temporal Logics over Branching Time

Many temporal logics have been suggested as branching time specification formalisms during the past 20 years. These logics were compared against each other for their expressive power, model checking complexity, and succinctness. Yet, unlike the case for linear time logics, no canonical temporal logic of branching time was agreed upon. We offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified yardstick to measure these logics. We define an infinite hierarchy BTL_k of temporal logics and prove its strictness. We examine the expressive power of commonly used branching time temporal logics. We show that CTL* has no finite base, and that almost all of its many sublogics suggested in the literature are inside the second level of our hierarchy. We introduce new Ehrenfeucht–Fraissé games on trees and use them as our main tool to prove inexpressibility.     

Generating Model Checkers from Algebraic Specifications

There is a great deal of research aimed toward the development of temporal logics and model checking algorithms which can be used to verify properties of systems. In this paper, we present a methodology and supporting tools which allow researchers and practitioners to automatically generate model checking algorithms for temporal logics from algebraic specifications. These tools are extensions of algebraic compiler generation tools and are used to specify model checkers as mappings of the form , where L _s is a temporal logic source language and L _t is a target language representing sets of states of a model M, such that . The algebraic specifications for a model checker define the logic source language, the target language representing sets of states in a model, and the embedding of the source language into the target language. Since users can modify and extend existing specifications or write original specifications, new model checking algorithms for new temporal logics can be easily and quickly developed; this allows the user more time to experiment with the logic and its model checking algorithm instead of developing its implementation. Here we show how this algebraic framework can be used to specify model checking algorithms for CTL, a real-time CTL, CTL*, and a custom extension called CTL ^e that makes use of propositions labeling the edges as well as the nodes of a model. We also show how the target language can be changed to a language of binary decision diagrams to generate symbolic model checkers from algebraic specifications.