Duality for modal μ-logics

Completeness for μ-calculi: A coalgebraic approach

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

A classical method for model-checking timed properties—such as those expressed using timed extensions of temporal logic—is to rely on the use of observers. In this context, a major problem is to prove the correctness of observers. Essentially, this boils down to proving that: (1) every trace that contradicts a property can be detected by the observer; but also that (2) the observer is innocuous, meaning that it cannot interfere with the system under observation. In this paper, we describe a method for automatically testing the correctness of realtime observers. This method is obtained by automating an approach often referred to as visual verification, in which the correctness of a system is performed by inspecting a graphical representation of its state space. Our approach has been implemented on the tool Tina, a model-checking toolbox for Time Petri Net.

Functional Reactive Programming (FRP) models reactive systems with events and signals, which have previously been observed to correspond to the "eventually" and "always" modalities of linear temporal logic (LTL). In this paper, we define a constructive variant of LTL with least fixed point and greatest fixed point operators in the spirit of the modal mu-calculus, and give it a proofs-as-programs interpretation as a foundational calculus for reactive programs. Previous work emphasized the propositions-as-types part of the correspondence between LTL and FRP; here we emphasize the proofs-as-programs part by employing structural proof theory. We show that the type system is expressive enough to enforce liveness properties such as the fairness of schedulers and the eventual delivery of results. We illustrate programming in this calculus using (co)iteration operators. We prove type preservation of our operational semantics, which guarantees that our programs are causal. We give also a proof of strong normalization which provides justification that our programs are productive and that they satisfy liveness properties derived from their types.

The author characterizes Rabin definability (see M.O. Rabin, 1969) of properties of infinite trees of fixed-point definitions based on the basic operations of a standard powerset algebra of trees and involving the least and greatest fixed-point operators as well as the finite union operator and functional composition. A strict connection is established between a hierarchy resulting from alternating the least and greatest fixed-point operators and the hierarchy induced by Rabin indices of automata. The characterization result is actually proved on a more general level, namely, for arbitrary powerset algebra, where the concept of Rabin automaton is replaced by the more general concept of infinite grammar.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Constructive linear-time temporal logic: Proof systems and Kripke semantics

A rough set logic based on Heyting-Brouwer algebras HBRSL is proposed as a basis for reasoning about rough information. It is an extension of Duntsch’s logic with intuitionistic implication, and is seen as a variant of Heyting- Brouwer logic. A Kripke semantics and natural deduction for the logic are presented and the completeness theorem is proved.

Two cut‐free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic (FPL) are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei‐Ardeshir's calculi for Visser's Basic Propositional Logic.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt , and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose.

The linear time μ-calculus extends LTL with arbitrary least and greatest fixpoint operators. This gives it the power to express all ω-regular languages, i.e. strictly more than LTL. The validity problem is PSPACE-complete for both LTL and the linear time μ-calculus. In practice it is more difficult for the latter because of nestings of fixpoint operators and variables with several occurrences. We present a simple sound and complete infinitary proof system for the linear time μ-calculus and then present two decision procedures for provability in the system, hence validity of formulas. One uses nondeterministic Büchi automata, the other one a generalisation of size-change termination analysis (SCT) known from functional programming. The main novelties of this paper are the connection with SCT and the fact that both decision procedures have a better asymptotic complexity than earlier ones and have been implemented.

Dynamic epistemic logic (\(\mathbf {DEL}\)) is known as a large family of logics that extend standard epistemic logic with dynamic operators. Such dynamic operators can be regarded as epistemic actions over Kripke semantics (or its variant). Therefore, \(\mathbf {DEL}\) is often used to model changes of agents’ knowledge, belief or preference over Kripke semantics in terms of dynamic operators in many literatures. As a variant of \(\mathbf {DEL}\), (van Benthem and Liu, J Appl Non-Classical Logics 17(2):157–18 2007; Liu, Reasoning about preference dynamics, Springer Science & Business Media, Berlin 2011) proposed dynamic logic of relation changers (\(\mathbf {DLRC}\)). They provided a general framework to capture many dynamic operators in terms of relation changing operation written in programs of propositional dynamic logic, and they also provided a sound and complete Hilbert-style axiomatization for \(\mathbf {DLRC}\). While \(\mathbf {DLRC}\) can cover many dynamic operators in a uniform manner, proof theory for \(\mathbf {DLRC}\) is not well-studied except the Hilbert-style axiomatization. Therefore, we propose a cut-free labelled sequent calculus for \(\mathbf {DLRC}\). We show that our sequent calculus is equipollent with the Hilbert-style axiomatization.

Since this lecture is called the Beth lecture, It is natural to ask oneself what connection is there between Beth and intuitionism? and, to my mind, the most obvious answer is: the Beth models for intuitionistic propositional and predicate logic. I would like to remind you of a certain picture that we all use when thinking in terms of possible worlds, namely the picture in which we are always at a certain stage, considering a number of possible future alternatives, and then one of those alternatives is realized and we get to a new stage, at which we are faced with some other alternatives, one of those alternatives materializes, and then the whole process continues in the same way. That means that, at each moment, we are at the end of a certain path, which I will draw by a thick line, symbolizing that which has already materialized, and we are considering all possible future alternatives, one of those alternatives materializes, as a result of which the thick path gets extended, and then the pattern repeats itself: Open image in new window We all think in terms of this picture: it underlies possible world semantics, and it is natural to ask: where does it really come from? I mean: who had this picture for the first time? Most of us have probably met it in connection with Kripke semantics (Kripke 1965), but actually Kripke semantics was predated by Beth semantics by a couple of years: Beth’ papers were published in 1956 and 1959 (Beth 1956a, 1959), and Beth in turn drew on Brouwer here, because he took the nodes, or stages, to form a spread in Brouwer’s sense (Brouwer 1918B).

We study a framework for the specification of architecture styles as families of architectures involving a common set of types of components and coordination mechanisms. The framework combines two logics: 1 interaction logics for the specification of architectures as generic coordination schemes involving a configuration of interactions between typed components; 2 configuration logics for the specification of architecture styles as sets of interaction configurations. The presented results build on previous work on architecture modelling in BIP. We show how propositional interaction logic can be extended into a corresponding configuration logic by adding new operators on sets of interaction configurations. We provide a complete axiomatisation of the propositional configuration logic, as well as a decision procedure for checking that an architecture satisfies given logical specifications. To allow genericity of specifications, we study first-order and second-order extensions of the propositional logic. We provide examples illustrating the application of the results to the characterization of architecture styles. Finally, we provide an experimental evaluation using the Maude rewriting system to implement the decision procedure for the propositional logic.