Formulas In Modal Logic Research Articles

Polynomial-time checking of generalized Sahlqvist syntactic shape

The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.

Attack Graphs & Subset Sabotage Games

We consider an extended version of sabotage games played over Attack Graphs. Such games are two-player zero-sum reachability games between an Attacker and a Defender. This latter player can erase particular subsets of edges of the Attack Graph. To reason about such games we introduce a variant of Sabotage Modal Logic (that we call Subset Sabotage Modal Logic) in which one modality quantifies over non-empty subset of edges. We show that we can characterize the existence of winning Attacker strategies by formulas of Subset Sabotage Modal Logic.

Sahlqvist correspondence theory for second-order propositional modal logic

AbstractModal logic with propositional quantifiers (i.e. second-order propositional modal logic ($\textsf {SOPML}$)) has been considered since the early time of modal logic. Its expressive power and complexity are high, and its van Benthem–Rosen theorem and Goldblatt–Thomason theorem have been proved by ten Cate (2006, J. Philos. Logic, 35, 209–223). However, the Sahlqvist theory of $\textsf {SOPML}$ has not been considered in the literature. In the present paper, we fill in this gap. We develop the Sahlqvist correspondence theory for $\textsf {SOPML}$, which covers and properly extends existing Sahlqvist formulas in basic modal logic. We define the class of Sahlqvist formulas for $\textsf {SOMPL}$ step by step in a hierarchical way, each formula of which is shown to have a first-order correspondent over Kripke frames effectively computable by an algorithm $\textsf {ALBA}^{\textsf {SOMPL}}$. In addition, we show that certain $\varPi _2$-rules correspond to $\varPi _2$-Sahlqvist formulas in $\textsf {SOMPL}$, which further correspond to first-order conditions, and that even for very simple $\textsf {SOMPL}$ Sahlqvist formulas, they could already be non-canonical.

Remarks about the unification types of some locally tabular normal modal logics

AbstractIt is already known that unifiable formulas in normal modal logic $\textbf {K}+\square ^{2}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{2}\bot $ are unitary. In this paper, we prove that for all $d{\geq }3$, unifiable formulas in normal modal logic $\textbf {K}+\square ^{d}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{d}\bot $ are unitary.

Modelling and Verification Analysis of Cooperative and Non-Cooperative Games via a Modal Logic Approach

In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players (coalitions) due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing (e.g. through credible threats). Cooperative games are often analyzed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players’ actions and payoffs and analyzing Nash equilibriums. In this work, the cooperative and non-cooperative game problem is modeled by means of a modal logic formula. Then, using the concept of logic implication, and transforming this logical implication relation into a set of clauses, a modal resolution qualitative method for verification (satisfiability) as well as performance issues, for some queries is applied.

Logical Time and Space of the Network Intrusion

Nowadays, one of the biggest threats for modern computer networks are the cyber attacks. One of the possible ways how to increase the level of computer networks security is a deployment of a network intrusion detection system. This paper deals with the behavior of the network intrusion detection system during specific network intrusion. We formally describe this network intrusion by the modal linear logic formula. Based on this formula, logical space and logical time is expressed from the attacker, and the network environment point of view in the usage of the Ludics theory.

Dual characterizations for finite lattices via correspondence theory for monotone modal logic

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature.

Does Treewidth Help in Modal Satisfiability?

Many tractable algorithms for solving the Constraint Satisfaction Problem ( Csp ) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulas in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric, and transitive models, we show either that modal satisfiability is Fixed Parameter Tractable ( Fpt ), or that it is W[1]-hard. In particular, modal satisfiability in general models is Fpt , while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is Fpt .

De Jure and De Facto Validity in the Logic of Time and Modality

What formulas are tense-logically valid depends on the structure of time, for example on whether it has a beginning. Logicians have investigatedwhat formulas correspond towhat physical hypotheses about time. Analogously, we can investigate what formulas of modal logic correspond to what metaphysical hypotheses about necessity. It is widely held that physical hypotheses about time may be contingent. If so, tense-logical validity may be contingent. In contrast, validity in modal logic is typically taken to be non-contingent, as reflected by the general acceptance of the so-called ‘‘rule of necessitation.’’ But as has been argued by various authors in recent years, metaphysical hypotheses may likewise be contingent. If, in particular, hypotheses about the extent of possibility are contingent, we should expect modal-logical validity to be contingent too. Let ‘‘contingentism’’ be the view that everything that is not ruled out by logic is possible. I shall investigate what the right system of modal logic is, if contingentism is true. Given plausible assumptions, the system contains the McKinsey principle, and is thus not even contained in S5. It also contains simple and elegant iteration principles for the contingency operator: something is contingent if and only if it is contingently contingent.

A Logical Account of Formal Argumentation

In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment.

Theory of (n) truth degrees of formulas in modal logic and a consistency theorem

The theory of (n) truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the (n) truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of (n) truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of (n) similarity degrees among modal formulas is proposed and the (n) modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic.

Resolution in Modal, Description and Hybrid Logic

We provide a resolution‐based proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybrid machinery it avoids the complexities of earlier propositional resolution‐based methods for modal logic. It combines ideas from the method of prefixes used in tableaux, and resolution ideas in such a way that some of the heuristics and optimizations devised in either field are applicable.

Semantic analysis of concurrent ML by abstract model-checking

Abstract In this paper we present a new kind of semantics for Concurrent ML, apopular concurrent extension of the MLfunctional language equipped with message-passing inter-processcommunication along first-class channels. This semanticsis based on infinite domains of higher-dimensional transition systems that are able to model the asynchronous execution of concurrentoperations and is operational in nature. By dual abstract interpretation using folding of states and truncation oftransitions finite automata can be automatically derived that representa sound but imprecise semantics of a given program. They are used to compute staticproperties verified by the standardconcurrent execution of the program by means of abstractmodel-checking of modal logic formulae.

Program constructions that are safe for bisimulation

It has been known since the seventies that the formulas of modal logic are invariant for bisimulations between possible worlds models — while conversely, all bisimulation-invariant first-order formulas are modally definable. In this paper, we extend this semantic style of analysis from modal formulas to dynamic program operations. We show that the usual regular operations are safe for bisimulation, in the sense that the transition relations of their values respect any given bisimulation for their arguments. Our main result is a complete syntactic characterization of all first-order definable program operations that are safe for bisimulation. This is a semantic functional completeness result for programming, which may be contrasted with the more usual analysis in terms of computational power. The 'Safety Theorem' can be modulated in several ways. We conclude with a list of variants, extensions, and further developments.

Semantic analysis of concurrent ML by abstract model-checking

In this paper we present a new kind of semantics for Concurrent ML, apopular concurrent extension of the MLfunctional language equipped with message-passing inter-processcommunication along first-class channels. This semanticsis based on infinite domains of higher-dimensional transition systems that are able to model the asynchronous execution of concurrentoperations and is operational in nature.By dual abstract interpretation using folding of states and truncation oftransitions finite automata can be automatically derived that representa sound but imprecise semantics of a given program. They are used to compute staticproperties verified by the standardconcurrent execution of the program by means of abstractmodel-checking of modal logic formulae.

The global properties of valid formulas in modal logic K

Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated reasoning systems, thus the efficiency of the whole system is improved. This paper presents some global properties of valid formulas in modal logic K. Such properties are structure characters of formulas, so they are simple and easy to check. At the same time, some global properties of K unsatisfiable formula set are also given.

NP trees and Carnap's modal logic

In this paper we consider problems and complexity classes definable by interdependent queries to an oracle in NP. How the queries depend on each other is specified by a directed graph G. We first study the class of problems where G is a general dag and show that this class coincides with Δ p 2 . We then consider the class where G is a tree. Our main result states that this class is identical to P NP [O(log n)], the class of problems solvable in polynomial time with a logarithmic number of queries to an oracle in NP. This result has interesting applications in the fields of modal logic and artificial intelligence. In particular, we show that the following problems are all P NP [O(log n)] complete: validity-checking of formulas in Carnap's modal logic, checking whether a formula is almost surely valid over finite structures in modal logics K, T, and S4 (a problem recently considered by Halpern and Kapron [1992]), and checking whether a formula belongs to the stable set of beliefs generated by a propositional theory. We generalize the case of dags to the case where G is a general (possibly cyclic) directed graph of NP-oracle queries and show that this class corresponds to Π p 2 . We show that such graphs are easily expressible in autoepistemic logic. Finally, we generalize our complexity results to higher classes of the polynomial-time hierarchy.

Semantics-Based Translation Methods for Modal Logics

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic‘s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order function symbols are allowed. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the logic itself.

Modalities for Total Correctness

This paper is concerned with the relationship of termination problem for regular programs to the validity of certain formulas in modal logic.