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.     

On the relationship between annotated logic programs and nonmonotonic formalisms

Abstract

In the past we developed a semantics for a restricted annotated logic language for inheritance reasoning. Here we generalize it to annotated Horn logic programs. We first provide a formal account of the language, describe its semantics, and provide an interpreter written in Prolog for it. We then investigate its relationship to Belnap's 4-valued logic, Gelfond and Lifschitz's semantics for logic programs with negation, Brewka's prioritized default logics and other annotated logics due to Kifer et al.     

Autoepistemic logic of first order and its expressive power

We study the expressive power of first-order autoepistemic logic. We argue that full introspection of rational agents should be carried out by minimizing positive introspection and maximizing negative introspection. Based on full introspection, we propose the maximal well-founded semantics that characterizes autoepistemic reasoning processes of rational agents, and show that breadth of the semantics covers all theories in autoepistemic logic of first order, Moore's AE logic, and Reiter's default logic. Our study demonstrates that the autoepistemic logic of first order is a very powerful framework for nonmonotonic reasoning, logic programming, deductive databases, and knowledge representation.This research is partially supported by NSERC grant OGP42193.     

Modal logic for default reasoning

In the paper we introduce a variant of autoepistemic logic that is especially suitable for expressing default reasonings. It is based on the notion of iterative expansion. We show a new way of translating default theories into the language of modal logic under which default extensions correspond exactly to iterative expansions. Iterative expansions have some attractive properties. They are more restrictive than autoepistemic expansions, and, for some classes of theories, than moderately grounded expansions. At the same time iterative expansions avoid several undesirable properties of strongly grounded expansions, for example, they are grounded in the whole set of the agent's initial assumptions and do not depend on their syntactic representation.Iterative expansions are defined syntactically. We define a semantics which leads to yet another notion of expansion — weak iterative expansion — and we show that there is an important class of theories, that we call -programs, for which iterative and weak iterative expansions coincide. Thus, for -programs, iterative expansions can be equivalently defined by semantic means.This work was partially supported by Army Research Office under grant DAAL03-89-K-0124, and by National Science Foundation and the Commonwealth of Kentucky EPSCoR program under grant RII 8610671.     

On computing logic programs

In this paper we present and compare some classical problem-solving methods for computing the stable models of logic programs with negation. Using a graph theoretic representation of logic programs and their stable models, we discuss and compare linear programming, propositional satisfiability, constraint satisfaction, and graph methods.     

A logic programming system for nonmonotonic reasoning

The evolution of logic programming semantics has included the introduction of a new explicit form of negation, beside the older implicit (or default) negation typical of logic programming. The richer language has been shown adequate for a spate of knowledge representation and reasoning forms.The widespread use of such extended programs requires the definition of a correct top-down querying mechanism, much as for Prolog wrt. normal programs. One purpose of this paper is to present and exploit a SLDNF-like derivation procedure, SLX, for programs with explicit negation under well-founded semantics (WFSX) and prove its soundness and completeness. (Its soundness wrt. the answer-sets semantics is also shown.) Our choice ofWFSX as the base semantics is justi-fied by the structural properties it enjoys, which are paramount for top-down query evaluation.Of course, introducing explicit negation requires dealing with contradiction. Consequently, we allow for contradiction to appear, and show moreover how it can be removed by freely changing the truth-values of some subset of a set of predefined revisable literals. To achieve this, we introduce a paraconsistent version ofWFSX, WFSX _p , that allows contradictions and for which our SLX top-down procedure is proven correct as well.This procedure can be used to detect the existence of pairs of complementary literals inWESX _p simply by detecting the violation of integrity rulesf L, -L introduced for eachL in the language of the program. Furthermore, integrity constraints of a more general form are allowed, whose violation can likewise be detected by SLX.Removal of contradiction or integrity violation is accomplished by a variant of the SLX procedure that collects, in a formula, the alternative combinations of revisable literals' truth-values that ensure the said removal. The formulas, after simplification, can then be satisfied by a number of truth-values changes in the revisable, among true, false, and undefined. A notion of minimal change is defined as well that establishes a closeness relation between a program and its revisions. Forthwith, the changes can be enforced by introducing or deleting program rules for the revisable literals.To illustrate the usefulness and originality of our framework, we applied it to obtain a novel logic programming approach, and results, in declarative debugging and model-based diagnosis problems.     

SSL协议的扩展Rubin逻辑形式化分析

SSL协议是一个用于因特网上进行保密通信的实用安全协议,由于它的复杂性,很多形式化分析方法都不适合分析它[1].而适用于分析非单调密码协议的Rubin逻辑,不同于大多数采用"知识"和"信念"的逻辑分析发现安全缺陷的逻辑分析方法,它完整地分析协议过程中的出现所有"动作",不但能清晰地看到SSL协议的不足,还可指出进一步完善SSL协议的方法.     

基于四值语义的缺省逻辑

基于公式变换,给出一组缺省理论的变换方法,将命题语言L中的缺省理论变换到对应的命题语言L^-＋中,保证了所得到的缺省理论的所有扩张均不平凡,并通过一种弱变换可同时保证缺省扩张的存在性．为缺省理论定义了各种四值模型,使得缺省逻辑具有非单调超协调推理能力,并证明了L^-＋中的缺省扩张与L中缺省理论的四值模型之间具有一一对应关系．四值模型描述了公式变换的语义,基于四值语义的缺省推理通过缺省理论的变换技术能在标准的缺省逻辑中实现．     

On nonmonotonic reasoning with the method of sweeping presumptions

Reasoning almost always occurs in the face of incomplete information. Such reasoning is nonmonotonic in the sense that conclusions drawn may later be withdrawn when additional information is obtained. There is an active literature on the problem of modeling such nonmonotonic reasoning, yet no category of method-let alone a single method-has been broadly accepted as the right approach. This paper introduces a new method, called sweeping presumptions, for modeling nonmonotonic reasoning. The main goal of the paper is to provide an example-driven, nontechnical introduction to the method of sweeping presumptions, and thereby to make it plausible that sweeping presumptions can usefully be applied to the problems of nonmonotonic reasoning. The paper discusses a representative sample of examples that have appeared in the literature on nonmonotonic reasoning, and discusses them from the point of view of sweeping presumptions.     

Logic programming with infinite sets

Using the ideas from current investigations in Knowledge Representation we study the use of a class of logic programs for reasoning about infinite sets. Our programs reason about the codes for various infinite sets. Depending on the form of atoms allowed in the bodies of clauses we obtain a variety of completeness results for various classes of arithmetic sets of integers.AMS subject classification 68T27, 03B70     

概率逻辑、不确定逻辑和模糊逻辑之比较

通过一个实例分析比较了概率逻辑、主观概率逻辑、不确定逻辑和模糊逻辑的思想方法。提出了自己的观点：基于数据统计的概率逻辑是最科学的。不确定逻辑比主观概率逻辑更科学。当具有不确定性的原子命题具有独立性时,不确定逻辑和模糊逻辑的观点是一致的。而对于处理带有不确定性的相关性命题,不确定逻辑比模糊逻辑更科学。但是模糊逻辑在建立推理理论方面见长。     

The Nature of Nonmonotonic Reasoning

Conclusions reached using common sense reasoning from a set of premises are often subsequently revised when additional premises are added. Because we do not always accept previous conclusions in light of subsequent information, common sense reasoning is said to be nonmonotonic. But in the standard formal systems usually studied by logicians, if a conclusion follows from a set of premises, that same conclusion still follows no matter how the premise set is augmented; that is, the consequence relations of standard logics are monotonic. Much recent research in AI has been devoted to the attempt to develop nonmonotonic logics. After some motivational material, we give four formal proofs that there can be no nonmonotonic consequence relation that is characterized by universal constraints on rational belief structures. In other words, a nonmonotonic consequence relation that corresponds to universal principles of rational belief is impossible. We show that the nonmonotonicity of common sense reasoning is a function of the way we use logic, not a function of the logic we use. We give several examples of how nonmonotonic reasoning systems may be based on monotonic logics.     

A theory of legal reasoning and a logic to match

This paper describes a model of legal reasoning and a logic for reasoning with rules, principles and goals that is especially suited to this model of legal reasoning. The paper consists of three parts. The first part describes a model of legal reasoning based on a two-layered view of the law. The first layer consists of principles and goals that express fundamental ideas of a legal system. The second layer contains legal rules which in a sense summarise the outcome of the interaction of the principles and goals for a number of case types. Both principles, goals and rules can be used in legal arguments, but their logical roles are different. One characteristic of the model of legal reasoning described in the first part of the paper is that it takes these logical differences into account. Another characteristic is that it pays serious attention to the phenomena of reasoning about the validity and acceptance of rules, respectively principles and goals, and about the application of legal rules, and the implications of these arguments for the use of rules, principles and goals in deriving legal conclusions for concrete cases.The second part of the paper first describes a logic (Reason-Based Logic) that is especially suited to deal with legal arguments as described in terms of the previously discussed model. The facilities of the logic are illustrated by means of examples that correspond to the several aspects of the model.The third part of the paper deals with a number of logico-philosophical reflections on Reason-Based Logic. The occasion is also used to compare these presuppositions with theories of defeasible reasoning based on the comparison of arguments.The ideas developed in this paper are based on the draft of my book Reasoning with rules which will be published by Kluwer Academic Publishers in the Law and Philosophy Series. The book offers not only more elaborate and sometimes different treatments of the topics of this paper, but also pays more attention to the philosophical background of this work.     

Deontic logic in the representation of law: Towards a methodology

There seems to be no clear consensus in the existing literature about the role of deontic logic in legal knowledge representation — in large part, we argue, because of an apparent misunderstanding of what deontic logic is, and a misplaced preoccupation with the surface formulation of legislative texts. Our aim in this paper is to indicate, first, which aspects of legal reasoning are addressed by deontic logic, and then to sketch out the beginnings of a methodology for its use in the analysis and representation of law.The essential point for which we argue is that deontic logic — in some form or other —needs to be taken seriously whenever it is necessary to make explicit, and then reason about, the distinction between what ought to be the case and what is the case, or as we also say, between the ideal and the actual. We take the library regulations at Imperial College as the main illustration, and small examples from genuinely legal domains to introduce specific points. In conclusion, we touch on the role of deontic logic in the development of the theory of normative positions.Deontic logic and the theory of normative positions are of relevance to legal knowledge representation, but also to the analysis and. representation of normative systems generally. The emphasis of the paper is on legal knowledge representation, but we seek to place the discussion within the context of a broader range of issues concerning the role of deontic logic in Computer Science.     

Dualities between alternative semantics for logic programming and nonmonotonic reasoning

The Gelfond-Lifschitz operator associated with a logic program (and likewise the operator associated with default theories by Reiter) exhibits oscillating behavior. In the case of logic programs, there is always at least one finite, nonempty collection of Herbrand interpretations around which the Gelfond-Lifschitz operator bounces around. The same phenomenon occurs with default logic when Reiter's operator is considered. Based on this, a stable class semantics and extension class semantics has been proposed. The main advantage of this semantics was that it was defined for all logic programs (and default theories), and that this definition was modelled using the standard operators existing in the literature such as Reiter's operator. In this paper our primary aim is to prove that there is a very interestingduality between stable class theory and the well-founded semantics for logic programming. In the stable class semantics, classes that were minimal with respect to Smyth's power-domain ordering were selected. We show that the well-founded semantics precisely corresponds to a class that is minimal w.r.t. Hoare's power domain ordering: the well-known dual of Smyth's ordering. Besides this elegant duality, this immediately suggests how to define a well-founded semantics for default logic in such a way that the dualities that hold for logic programming continue to hold for default theories. We show how the same technique may be applied to strong autoepistemic logic: the logic of strong expansions proposed by Marek and Truszczynski.     

逻辑系统中的拟对偶性与拟排中律

在经典逻辑和t-模基础逻辑中提出了排中律与拟排中律的新概念,说明经典逻辑(带对偶非的MTL)满足排中律(拟排中律),证明了Go?del模糊逻辑(Lukasiewicz(简称Luk)模糊逻辑)关于最小算子ù与最大算子(关于t-模与t-余模对补算子c)既不满足拟对偶性也不满足拟排中律,检验了Luk模糊逻辑关于Lukt-模与Lukt-余模对?算子满足拟对偶性和排中律。     

Displacement logic for anaphora

The displacement calculus of Morrill, Valentín and Fadda (2011) [25] aspires to replace the calculus of Lambek (1958) [13] as the foundation of categorial grammar by accommodating intercalation as well as concatenation while remaining free of structural rules and enjoying Cut-elimination and its good corollaries. Jäger (2005) [11] proposes a type logical treatment of anaphora with syntactic duplication using limited contraction. Morrill and Valentín (2010) [24] apply (modal) displacement calculus to anaphora with lexical duplication and propose extension with a negation as failure in conjunction with additives to capture binding conditions. In this paper we present an account of anaphora developing characteristics and employing machinery from both of these proposals.     

A Critical Reexamination of Default Logic, Autoepistemic Logic, and Only Knowing

Fifteen years of work on nonmonotonic logic has certainly increased our understanding of the area. However, given a problem in which nonmonotonic reasoning is called for, it is far from clear how one should go about modeling the problem using the various approaches. We explore this issue in the context on two of the best–known approaches, Reiter's default logic and Moore's autoepistemic logic, as well as two related notions of "only knowing," due to Halpern and Moses and to Levesque. In particular, we return to the original technical definitions given in these papers and examine the extent to which they capture the intuitions they were designed to capture.     

基于信度语义的算子模糊逻辑

本文在算子集与值域不同的算子格上，基于信度语义定义了一种归结推理更自然、具有结合性的算子模糊逻辑ＢＡＯＦＬ及一种语义有层次性的，非单调的算子模糊逻辑ＮＭＯＦＬ。