Logic of knowledge and discovery via interacting agents - Decision algorithm for true and satisfiable statements

In this paper we study logical properties of the operation chance discovery (CD) via structures based on special Kripke/Hintikka models. These models use as bases partially ordered sets of indexes (indexes of steps in a computation, or ones indicating time points in a time flow), and clusters of states associated to each index. The language chosen to build the logical formulas includes modal/temporal operations, operations for the agents’ knowledge, local and global operations for CD, operation of local common knowledge, and an operation for chance of discovery via agents’ interactions. We introduce and study a logic (of knowledge and discovery via interaction of agents), LDK_a, which is defined by semantics, as the set of all formulas that are valid in all suggested models. The paper provides an algorithm to recognize logical laws (and satisfiable formulas) of LDK_a. The algorithm replaces a formula with a rule in a special, so-called reduced normal form, and, then it verifies the validity of this rule in specific models of exponential size in the size of the rule. We show that the problem of computing the true logical laws of LDK_a is decidable.     

Decidable metric logics

The common metric temporal logic for continuous time were shown to be insufficient, when it was proved that they cannot express a modality suggested by Pnueli. Moreover no finite temporal logic can express all the natural generalizations of this modality. It followed that if we look for an optimal decidable metric logic we must accept infinitely many modalities, or adopt a different formalism.Here we identify a fragment of the second order monadic logic of order with the “+1” function, that expresses all the Pnueli modalities and much more. Its main advantage over the temporal logics is that it enables us to say not just that within prescribed time there is a point where some punctual event will occur, but also that within prescribed time some process that starts now (or that started before, or that will start soon) will terminate. We prove that this logic is decidable with respect to satisfiability and validity, over continuous time. The proof depends heavily on the theory of compositionality. In particular every temporal logic that has truth tables in this logic is automatically decidable. We extend this result by proving that any temporal logic, that has all its modalities defined by means more general than truth tables, in a logic stronger than the one just described, has a decidable satisfiability problem. We suggest that this monadic logic can be the framework in which temporal logics can be safely defined, with the guarantee that their satisfiability problem is decidable.     

Reasoning in Medical and Tutoring Systems: A Decidable First-Order Temporal Paraconsistent Non-Commutative Logic

Reasoning in medical and tutoring systems requires expressions relating not only to time-dependency, paraconsistency, constructiveness, and resource-sensitivity, but also order-sensitivity. Our objective in this study is to construct a decidable rst-order logic for appropriately expressing this reasoning. To meet this objective, we introduce a rst-order temporal paraconsistent non-commutative logic as a Gentzen-type sequent calculus. This logic has no structural rules but has some bounded temporal operators and a paraconsistent negation connective. The main result of this study is to show this logic to be decidable. Based on this logic, we present some illustrative examples for reasoning in medical and tutoring systems.     

A Tableau for Bundled CTL

We present a sound, complete and relatively straightforwardtableau method for deciding valid formulas in the propositionalversion of the bundled (or suffix and fusion closed) computationtree logic BCTL*. This proves that BCTL* is decidable. It isalso moderately useful to have a tableau available for a reasonablyexpressive branching-time temporal logic. However, the maininterest in this should be that it leads us closer to beingable to devise a tableau-based technique for theorem-provingin the important full computational tree logic CTL*.     

时态描述逻辑ALC-LTL的Tableau判定算法

时态描述逻辑ALC-LTL将描述逻辑ALC的描述能力与线性时态逻辑LTL的刻画能力结合起来,在具有较强描述能力的同时还使得可满足性问题保持在EXPTIME-完全这个级别。针对ALC-LTL缺少有效的判定算法的现状,将LTL的Tableau判定算法与描述逻辑ALC的推理机制有机地结合起来,给出了ALC-LTL的Tableau判定算法并证明了算法的可终止性、可靠性和完备性。该算法具有很好的可扩展性。当ALC-工`I'I、中的描述逻辑从ALC改变为任何一个具有可判定性特征的描述逻辑X时,只需要对算法进行简单修改,就可以得到相应的时态描述逻辑X-LTL的Tableau判定算法。     

Programming in temporal-nonmonotonic reasoning

Nonmonotonic reasoning has been proposed as an extension to classical first-order logic. Now people are interested in temporal reasoning with nonmonotonic logic [6]. We combine the monotonic logic [7] with a temporal logic to get a more general reasoning language. We discuss a monotonic logic TML which has predicate formulas, temporal formulas and a special modal formula, and give a completeness theorem of it. We use TH() to designate the set of theorems of a temporal-nonmonotonic theory which has the same language with TML. The completeness theorem of the temporal-nonmonotonic logic naturally arises. Like the relationship between predicate logic with a practical logic programming language PROLOG, we propose a useful temporal-nonmonotonic reasoning language TN for the temporal-nonmonotonic logic. As an appendix we supply an algorithm for the programming language TN.     

一类具有3种否定的模糊模态命题逻辑

对不同否定知识的认知、区分、表达、推理及计算是模糊知识研究处理的一个基础。具有矛盾否定、对立否定和中介否定的模糊命题逻辑形式系统FLCOM是一种能够完整描述模糊知识中的不同否定及其关系与规律的理论。基于FLCOM和中介模态命题逻辑MK,提出一类具有3种否定的模糊模态命题逻辑MKCOM及其扩充系统MTCOM,MS₄COM和MS₅COM;讨论了MKCOM的语义和语法解释,并证明了MKCOM的可靠性定理和完备性定理。     

Naming Worlds in Modal and Temporal Logic

In this paper we suggest adding to predicate modal and temporal logic a locality predicate W which gives names to worlds (or time points). We also study an equal time predicate D(x, y)which states that two time points are at the same distance from the root. We provide the systems studied with complete axiomatizations and illustrate the expressive power gained for modal logic by simulating other logics. The completeness proofs rely on the fairly intuitive notion of a configuration in order to use a proof technique similar to a Henkin completion mixed with a tableau construction. The main elements of the completeness proofs are given for each case, while purely technical results are grouped in the appendix. This revised version was published online in August 2006 with corrections to the Cover Date.     

Complexity of some problems in positive and related calculi

A problem of recognizing important properties of propositional calculi is considered, and complexity bounds for some decidable properties are found. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L+Ax has the property P or not. In Maksimova and Voronkov (Bull. Symbol. Logic 6 (2000) 118) the complexity of tabularity, pretabularity, and interpolation problems over the intuitionistic logic (Int) and over modal logic S4 was studied.In the present paper, positive and positively axiomatizable calculi are investigated. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation and projective Beth's property over the positive fragment Int⁺ of the intuitionistic logic. Some complexity bounds for properties of propositional calculi over the intuitionistic or the minimal logic are found.     

Embedding defaults into terminological knowledge representation formalisms

We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic. Semantically, one has the unpleasant effect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only finitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are applied only to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a finite terminological default theory, which means that this type of default reasoning is decidable. We describe an algorithm for computing extensions and show how the inference procedures of terminological systems can be modified to give optimal support to this algorithm.This is a revised and extended version of a paper presented at the3rd International Conference on Principles of Knowledge Representation and Reasoning, October 1992, Cambridge, MA.     

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. Supported by the National Natural Science Foundation of China (Grant Nos. 10331010 and 10771129), and the Foundation of 211 Construction of Shaanxi Normal University     

A finite model construction for coalgebraic modal logic

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatisation in rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy–Milner logic, graded modal logic and probabilistic modal logic.     

Tableau-based characterization and theorem proving for default logic

In this paper, we present a new method for computing extensions and for deriving formulae from a default theory. It is based on the semantic tableaux method and works for default theories with a finite set of defaults that are formulated over a decidable subset of first-order logic. We prove that all extensions (if any) of a default theory can be produced by constructing the semantic tableau ofone formula built from the general laws and the default consequences. This result allows us to describe an algorithm that provides extensions if there are any, and to decide if there are none. Moreover, the method gives a necessary and sufficient criterion for the existence of extensions of default theories with finitely many defaults provided they are formulated on a decidable subset of FOL.This work was completed while the author was at CNRS, Marseille.     

Specifying coalgebras with modal logic

We propose to use modal logic as a logic for coalgebras and discuss it in view of the work done on coalgebras as a semantics of object-oriented programming. Two approaches are taken: First, standard concepts of modal logic are applied to coalgebras. For a certain kind of functor it is shown that the logic exactly captures the notion of bisimulation and a complete calculus is given. Examples of verifications of object properties are given. Second, we discuss the relationship of this approach with the coalgebraic logic of Moss (Coalgebraic logic, Ann Pure Appl. Logic 96 (1999) 277–317.).     

An Event-Based Fragment of First-Order Logic over Intervals

We consider a new fragment of first-order logic with two variables. This logic is defined over interval structures. It constitutes unary predicates, a binary predicate and a function symbol. Considering such a fragment of first-order logic is motivated by defining a general framework for event-based interval temporal logics. In this paper, we present a sound, complete and terminating decision procedure for this logic. We show that the logic is decidable, and provide a NEXPTIME complexity bound for satisfiability. This result shows that even a simple decidable fragment of first-order logic has NEXPTIME complexity.     

Strong equivalence of logic programs under the infinite-valued semantics

We consider the notion of strong equivalence [V. Lifschitz, D. Pearce, A. Valverde, Strongly equivalent logic programs, ACM Transactions on Computational Logic 2 (4) (2001) 526-541] of normal propositional logic programs under the infinite-valued semantics [P. Rondogiannis, W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 (2) (2005) 441-467] (which is a purely model-theoretic semantics that is compatible with the well-founded one). We demonstrate that two such programs are strongly equivalent under the infinite-valued semantics if and only if they are logically equivalent in the corresponding infinite-valued logic. In particular, we show that strong equivalence of normal propositional logic programs is decidable, and more specifically coNP-complete. Our results have a direct implication for the well-founded semantics since, as we demonstrate, if two programs are strongly equivalent under the infinite-valued semantics, then they are also strongly equivalent under the well-founded semantics.     

Distributed reasoning with ontologies and rules in order-sorted logic programming

Integrating ontologies and rules on the Semantic Web enables software agents to interoperate between them; however, this leads to two problems. First, reasoning services in SWRL (a combination of OWL and RuleML) are not decidable. Second, no studies have focused on distributed reasoning services for integrating ontologies and rules in multiple knowledge bases. In order to address these problems, we consider distributed reasoning services for ontologies and rules with decidable and effective computation. In this paper, we describe multiple order-sorted logic programming that transfers rigid properties from knowledge bases. Our order-sorted logic contains types (rigid sorts), non-rigid sorts, and unary predicates that distinctly express essential sorts, non-essential sorts, and non-sortal properties. We formalize the order-sorted Horn-clause calculus for such properties in a single knowledge base. This calculus is extended by embedding rigid-property derivation for multiple knowledge bases, each of which can transfer rigid-property information from other knowledge bases. In order to enable the reasoning to be effective and decidable, we design a query-answering system that combines order-sorted linear resolution and rigid-property resolution as top-down algorithms.     

Conceptual modeling in full computation‐tree logic with sequence modal operator

In this paper, we propose a method for modeling concepts in full computation‐tree logic with sequence modal operators. An extended full computation‐tree logic, CTLS*, is introduced as a Kripke semantics with a sequence modal operator. This logic can appropriately represent hierarchical tree structures in cases where sequence modal operators in CTLS* are applied to tree structures. We prove a theorem for embedding CTLS* into CTL*. The validity, satisfiability, and model‐checking problems of CTLS* are shown to be decidable. An illustrative example of biological taxonomy is presented using CTLS* formulas. © 2011 Wiley Periodicals, Inc.     

Expressivity of coalgebraic modal logic: The limits and beyond

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modelled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioural equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.