On combinations of propositional dynamic logic and doxastic modal logics

We prove completeness and decidability results for a family of combinations of propositional dynamic logic and unimodal doxastic logics in which the modalities may interact. The kind of interactions we consider include three forms of commuting axioms, namely, axioms similar to the axiom of perfect recall and the axiom of no learning from temporal logic, and a Church–Rosser axiom. We investigate the influence of the substitution rule on the properties of these logics and propose a new semantics for the test operator to avoid unwanted side effects caused by the interaction of the classic test operator with the extra interaction axioms. This paper is a revised and extended version of Schmidt and Tishkovsky (2003).     

Uniform semantic treatment of default and autoepistemic logics

We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise one-to-one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.     

Epistemic logic and logical omniscience: A survey

This survey brings together a collection of epistemic logics and discusses their approaches in alleviating the logical omniscience problem. Of particular note is the logic of implicit and explicit belief. Explicit belief refers to information actively held by an agent, while implicit belief refers to the logical consequence of explicit belief. Ramifications of Levesque's logic include nonstandard epistemic logic and the logics of awareness and local reasoning. Models of nonstandard epistemic logic are defined with respect to nonstandard proportional logic to weaken its semantics. In the logic of awareness, an agent can only believe a concept that it is aware of. Closely related to awareness are S-1 and S-3 epistemic operators which can be used to model skeptical and credulous agents. The logic of local reasoning provides a semantics for representing the fact that agents can have different clusters of beliefs which may contradict each other. Other variations include epistemic structures which are generalizations of the logic of local reasoning and fusion epistemic models which provide an account that agents can combine information conjunctively or disjunctively. Another closely related approach is the logic of explicit propostions which captures the insight that agents can hold beliefs independently without putting them together. © 1997 John Wiley & Sons, Inc.     

Logics of communication and change

Current dynamic epistemic logics for analyzing effects of informational events often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions involving common knowledge are essential to successful multi-agent communication. We propose new systems that extend the epistemic base language with a new notion of ‘relativized common knowledge’, in such a way that the resulting full dynamic logic of information flow allows for a compositional analysis of all epistemic postconditions via perspicuous ‘reduction axioms’. We also show how such systems can deal with factual alteration, rather than just information change, making them cover a much wider range of realistic events. After a warm-up stage of analyzing logics for public announcements, our main technical results are expressivity and completeness theorems for a much richer logic that we call LCC. This is a dynamic epistemic logic whose static base is propositional dynamic logic (PDL), interpreted epistemically. This system is capable of expressing all model-shifting operations with finite action models, while providing a compositional analysis for a wide range of informational events. This makes LCC a serious candidate for a standard in dynamic epistemic logic, as we illustrate by analyzing some complex communication scenarios, including sending successive emails with both ‘cc’ and ‘bcc’ lines, and other private announcements to subgroups. Our proofs involve standard modal techniques, combined with a new application of Kleene’s Theorem on finite automata, as well as new Ehrenfeucht games of model comparison.     

基于动态描述逻辑DDL的动作理论

基于一阶谓词逻辑或高阶逻辑的动作理论与采用命题语言的动作理论之间存在一个关于描述和推理能力的鸿沟;作为描述逻辑的动态扩展,动态描述逻辑DDL为基于描述逻辑的动作刻画和推理提供了一种途径.系统地研究了基于DDL的动作表示和推理问题.首先,在应用描述逻辑对静态领域知识进行刻画的基础上,引入带参数的原子动作定义式和带参数的复...     

动态描述逻辑的Tableau判定算法

动态描述逻辑在描述逻辑的基础上引入了动态维,用于描述和推理动态领域的知识,但目前缺少有效的判定算法作为支撑.文中以描述逻辑ALCO的动态扩展为例,构建出动态描述逻辑D-ALCO.以D-ALCO的构建过程为基础,将ALCO的Tableau算法、命题动态逻辑的Tableau算法以及对可能模型途径的处理有机地结合起来,给出了D-ALCO的Tableau判定算法,证明了算法的可终止性、可靠性和完备性.应用该算法,可以在采用开世界假设的情况下对D-ALCO中公式的可满足性进行判定.对于D-ALCQO、D-ALCQIO等具有更强描述能力的动态描述逻辑,可以对该算法扩展后得到相应的Tableau判定算法.     

A concurrent dynamic logic of knowledge,belief and certainty for multi-agent systems

This paper extends the logic of knowledge, belief and certainty from one agent to multi-agent systems, and gives a good combination between logic of knowledge, belief, certainty in multi-agent systems and actions that have concurrent and dynamic properties. Based on it, we present a concurrent dynamic logic of knowledge, belief and certainty for MAS, which is called CDKBC logic. Furthermore, a CDKBC model is given for interpreting this logic. We construct a CDKBC proof system for the logic and show that the proof system is sound and complete, and prove that the validity problem for the system is EXPTIME-complete.     

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.     

Justification logics, logics of knowledge, and conservativity

Several justification logics have been created, starting with the logic LP, (Artemov, Bull Symbolic Logic 7(1):1–36, 2001). These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.     

The dynamics of default reasoning

In this paper we study default reasoning from a dynamic, agent-oriented, semantics-based point of view. In a formal framework used to specify and to reason about rational agents, we introduce actions that model the (attempted) jumping to conclusions that is a fundamental part of reasoning by default. Application of such an action consists of three parts. First it is checked whether the formula that the agent tries to jump to is a default, thereafter it is checked whether the default formula can consistently be incorporated by the agent, and if this is the case the formula is included in the agent's beliefs. As for all actions in our framework, we define the ability and opportunity of agents to apply these actions, and the states of affairs following application. To formalise formulae being defaults, we introduce the modality of common possibility. This modality is related to, but not reducible to, the notions of common knowledge and ‘everybody knows’-knowledge. To model the qualitative difference that exists between hard, factual knowledge and beliefs derived by default, we employ different modalities to represent these concepts, thus combining knowledge, beliefs, and defaults in one framework. Based on the concepts used to model the default reasoning of agents, we look into the dynamics of the supernormal fragment of default logic. We show in particular that by sequences of jumps to conclusions agents can end up with extensions in the sense of default logic of their belief.     

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.     

Temporal Languages for Epistemic Programs

This paper adds temporal logic to public announcement logic (PAL) and dynamic epistemic logic (DEL). By adding a previous-time operator to PAL, we express in the language statements concerning the muddy children puzzle and sum and product. We also express a true statement that an agent’s beliefs about another agent’s knowledge flipped twice, and use a sound proof system to prove this statement. Adding a next-time operator to PAL, we provide formulas that express that belief revision does not take place in PAL. We also discuss relationships between announcements and the new knowledge agents thus acquire; such relationships are related to learning and to Fitch’s paradox. We also show how inverse programs and hybrid logic each can be used to help determine whether or not an arbitrary structure represents the play of a game. We then add a past-time operator to DEL, and discuss the importance of adding yet another component to the language in order to prove completeness.     

Dynamic Epistemic Logic for Implicit and Explicit Beliefs

Epistemic logic with its possible worlds semantic model is a powerful framework that allows us to represent an agent’s information not only about propositional facts, but also about her own information. Nevertheless, agents represented in this framework are logically omniscient: their information is closed under logical consequence. This property, useful in some applications, is an unrealistic idealisation in some others. Many proposals to solve this problem focus on weakening the properties of the agent’s information, but some authors have argued that solutions of this kind are not completely adequate because they do not look at the heart of the matter: the actions that allow the agent to reach such omniscient state. Recent works have explored how acts of observation, inference, consideration and forgetting affect an agent’s implicit and explicit knowledge; the present work focuses on acts that affect an agent’s implicit and explicit beliefs. It starts by proposing a framework in which these two notions can be represented, and then it looks into their dynamics, first by reviewing the existing notion of belief revision, and then by introducing a rich framework for representing diverse forms of inference that involve both knowledge and beliefs.     

Combining and automating classical and non-classical logics in classical higher-order logics

Numerous classical and non-classical logics can be elegantly embedded in Church??s simple type theory, also known as classical higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.     

Reasoning about knowledge and belief: a survey

We examine a number of logics of knowledge and belief from the perspective of knowledge-based systems. We are concerned with the beliefs of a knowledge-based system, including both the system's base set of beliefs–those garnered directly from the world–and beliefs that follow from the base set. Three things to consider with such logics are the expressive power of the language of the logic, the correctness and completeness of the inferences sanctioned, and the speed with which it is possible to determine whether a given sentence is believed. The influential possible worlds approach to representing belief has the property of logical omniscience, which makes for inferences that are unacceptable in the context of belief and may take too much time to make. We examine a number of weak logics which attempt to deal with these problems. These logics divide into three categories: those that admit incomplete or inconsistent situations into their semantics, those that posit a number of distinct states for a believer which correspond roughly to frames of mind, and those that incorporate axioms or other syntactic entities directly into the semantics. As to expressive power, we consider whether belief should be represented by a predicate or a sentential operator and examine the boundary between self-referential and inconsistent systems. Finally, we consider logics of believing only , which add the assumption that a system's base set of beliefs are, in a certain sense, all that it believes.     

Strong Negation and Equivalence in the Safe Belief Semantics

The safe belief semantics uses intermediate logics to definean extension of answer sets to all propositional formulas, butonly considering one kind of negation. In this work we extendsafe beliefs adding the strong negation connective. The mainfeature of our extension is that strong negation can occur beforeany formula, and not only at the atomic level. We give resultsconcerning the relation between strong negation extensions ofintermediate logics and safe beliefs and consider the way inwhich strong negation can be eliminated from any formula whilepreserving its semantics. We also propose two new notions ofequivalence: substitution equivalence and contextualized equivalence.We prove that they are both more general than strong equivalenceand, for propositional formulas where strong negation may occurat the non-atomic level, substitution equivalence captures anotion of equivalence that cannot be captured by strong equivalencealone.     

Reasoning about knowledge,belief and certainty in hierarchical multi-agent systems

Multi-agent systems (MAS) have received extensive studies in the last decade. However, little attention is paid to investigation on reasoning about logics in MAS with hierarchical structures. This paper proposes a complete quantified temporal KBC (knowledge, belief and certainty) logic and corresponding reasoning in hierarchical multi-agent systems (HMAS). The key point is that internal beliefs and certainty, and external belief and certainty are considered in our logic. The internal beliefs and certainty show every agent is autonomous, while the external belief and certainty indicate the mutual influence of mental attitudes between two different agents on different layers in HMAS. To interpret this logic, we propose four classes of corresponding quantified interpreted systems, and define first-order KBC axiomatisations over HMAS, which are sound and complete with respect to the corresponding semantical classes. Finally, we give a case study to show the advantages in terms of expressiveness of our logic.     

AGM Contraction and Revision of Rules

In this paper we study AGM contraction and revision of rules using input/output logical theories. We replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of theories, and we replace the classical consequence operator Cn by an input/output logic. The results in this paper suggest that, in general, results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic. First, we generalise belief base contraction to rule base contraction, and show that two representation results of Hansson still hold for rule base contraction. Second, we show that the six so-called basic postulates of AGM contraction are consistent only for some input/output logics, but not for others. In particular, we show that the notorious recovery postulate can be satisfied only by basic output, but not by simple-minded output. Third, we show how AGM rule revision can be defined in terms of AGM rule contraction using the Levi identity. We highlight various topics for further research.