Autoepistemic Logic Research Articles

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.

Autoepistemic logics as a unifying framework for the semantics of logic programs

In this paper, it is shown that a three-valued autoepistemic logic provides an elegant unifying framework for some of the major semantics of normal and disjunctive logic programs and logic programs with classical negation, namely, the stable semantics, the well-founded semantics, supported models, Fitting's semantics, Kunen's semantics, the stationary semantics, and answer sets. For the first time, so many semantics are embedded into one logic. The framework extends previous results—by Gelfond, Lifschitz, Marek, Subrahmanian, and Truszczynski —on the relationships between logic programming and Moore's autoepistemic logic. The framework suggests several new semantics for negation-as-failure. In particular, we will introduce the epistemic semantics for disjunctive logic programs. In order to motivate the epistemic semantics, an interesting class of applications called ignorance tests will be formalized; it will be proved that ignorance tests can be defined by means of the epistemic semantics, but not by means of the old semantics for disjunctive programs. The autoepistemic framework provides a formal foundation for an environment that integrates different forms of negation. The role of classical negation and various forms of negation-by-failure in logic programming will be briefly discussed.

A decision method for nonmonotonic reasoning based on autoepistemic reasoning

A novel decision method for autoepistemic reasoning is developed and proved correct. The method is applicable in a general setting, i.e., for an autoepistemic logic based on a given classical logic. It provides a decision procedure for a tightly grounded from of autoepistemic reasoning based on L-hierarchic expansions as well as for autoepistemic reasoning based on Moorestyle expansions and N-expansions. Prominent formalizations of nonmonotonic reasoning, such as default logic and circumscription, can be embedded into autoepistemic logic based on L-hierarchic expansions using simple local translations. Hence, the method can serve as a unified reasoning tool for a wide range of forms of nonmonotonic reasoning. The method is conceptually simple, and the inherent sources of complexity and targets for optimization are clearly identifiable. As an example of exploiting optimization possibilities, a new decision method for Reiter's default logic is developed where ideas from autoepistemic reasoning are used to prune the search space for applicable default rules when constructing extensions of a default theory.

Belief, provability, and logic programs

ABSTRACT The main goal of this paper is to establish a nonmonotonic epistemic logic ϵβ with two modalities—provability and belief—capable of expressing and comparing a variety of known semantics for extended logic programs, and clarify their meaning. In particular we present here, for the first time, embeddings into epistemic logic of logic programs extended with a second kind of negation under the well-founded semantics, and contrast them to the recent embeddings into autoepistemic logics of such programs under stable models based semantics. Because of the newly established relationship between our epistemic logic ϵβ and extended program semantics, the former benefits from the procedures and implementations of the latter, and can be applied to at least the same class of AI problems that the latter can. Moreover, one issue of epistemic logic introduced here, belief revision, can profit from adapting techniques employed by the latter for contradiction removal. Furthermore, the language of the epistemic log...

Possible world semantics and autoepistemic reasoning

The possible world semantics of modal logic has been employed to study the nonmonotonic reasoning formalism, Moore's autoepistemic logic. Only Kripke structure, which is also known as a standard model, has been used to represent the autoepistemic expansions of modal theories. Here we argue that the mode of autoepistemic reasoning, which is formalized by autoepistemic logic, is not necessarily associated with a standard model. Another model of the possible world semantics, the minimal model, is helpful in knowledge representation and belief reasoning. In particular, the minimal model can be used as a structure to represent all the autoepistemic expansions of a modal theory. We extend the minimal model with standard model to deal with both knowledge and belief and introduce a null value called unknown to provide a refined model for knowledge and belief. We also present a notion of epistemic base of basic sentences to identify the possible worlds in a minimal model. By reformulating Moore's autoepistemic logic with respect to epistemic base, some reasonable modal theories that do not have any autoepistemic expansion will have generalized autoepistemic expansions.

Towards a theory of belief reconstruction: Application to communication

We present a theory of belief reconstruction to be embedded in an agent's communication model, which accounts for both belief persistence and revision. We analyse Cohen and Levesque (1990a)'s theories, highlight problems which arise, and show that our theory does not have these problems. The starting point of our theory is called the observation principle. It accounts for a distinction between what an agent observes from another agent, and the action the latter has really performed. The theory is couched in an autoepistemic logic used objectively, along the same lines as in (Levesque, 1990). When applied to a communication context, it is shown that it correctly predicts the changes in an observer's beliefs in test cases such as sincere assertion and (detected or non-detected) lie. Such test cases highlight the ability of the theory to handle not only normal dialogue situations but also those where problems arise due to erroneous perception, such as misrecognition in spoken communication.

Relating only knowing to minimal belief and negation as failure

Abstract In this paper, we relate Lifschitz's logic of minimal belief and negation as failure (MBNF) to Levesque's logic of only knowing (OL). MBNF can be viewed as an integration of Lin and Shoham's logic of grounded knowledge and the theory of epistemic queries by Levesque and Reiter. Lifschitz showed that MBNF can be used as a general framework to compare different non-monotonic formalisms such as default logic and circumscription, as well as several forms of logic programs, including disjunctive logic programs with classical negation. Levesque's logic of only knowing is an epistemic formalism which can be used to model agent's knowledge and belief and to answer epistemic queries. The OL logic allows one to express the statement φ is all that is known'. Levesque showed that autoepistemic logic of Moore can be embedded in OL and he also gave a proof theory for OL. We show that a substantial subset of propositional MBNF can be embedded in OL (and hence in autoepistemic logic). In particular, the class of...

Unifying default reasoning and belief revision in a modal framework

We present a logic for reasoning about belief revision in which the process of revising a knowledge base by some sentence is represented with a conditional connective. The conditional is not primitive however; it is defined in terms of two unary modal operators. We show that our notion of revision is equivalent to that determined by the classic AGM postulates. Furthermore, unlike current models of revision, our approach does not require the Limit Assumption. We also present a model for subjunctive query answering that allows the expression of subjunctive or factual premises, integrity constraints, and notions of entrenchment and plausibility. The modal framework we adopt is sufficiently general to allow the expression of other forms of defeasible reasoning, and facilitates the demonstration of some interesting connections between revision, default reasoning and autoepistemic logic. In particular, we show that the normative conditional for default reasoning (developed in a companion paper) and our subjunctive conditional are identical. Default reasoning can thus be viewed as the revision of a theory of expectations in manner that naturally relates priorities of default rules to the entrenchment of expectations.

THE LOGIC OF ONLY KNOWING AS A UNIFIED FRAMEWORK FOR NON-MONOTONIC REASONING

We propose to use the logic of only knowing (OL) by Levesque [10] as a unified framework that encompasses various non-monotonic formalisms and logic programming. OL is a modal logic which can be used to formalize an agent's introspective reasoning and to answer epistemic queries to databases. The OL logic allows one to formally express the statement “α is all I know” (in symbols, Oα) and to perform inferencing based on only-knowing, which is very useful for commonsense reasoning. Another nice thing about the OL logic is that it has a clear model-theoretic semantics and a simple proof theory, which is sound for the quantificational case, and both sound and complete for the prepositional case. We establish the relations between OL and various non-monotonic logics (such as default logic, circumscription) and logic programming, thus extending the existing works relating the OL logic with other non-monotonic reasoning formalisms (e.g., Levesque showed that autoepistemic logic can be embeded in OL). This is accomplished by finding the connection between OL and MBNF, the logic of Minimal Belief and Negation as Failure proposed by Lifschitz [12, 13], which is known to have close relationship with logic programming and other non-monotonic logics. Our results show that OL can be used as a unified framework to compare different non-monotonic formalisms based on the same domain.

Autoepistemic logic programming

An autoepistemic logic programming language is derived from a subset of a three-valued autoepistemic logic, called 3AEL. Autoepistemic programs generalize several ideas underlying logic programming: stable, supported, and well-founded models, Fitting's semantics, Kunen's semantics, and abductive frameworks can all be captured through simple autoepistemic translations; moreover, SLDNF-resolution and a generate-and-test method for stable semantics are generalized to provide sound and complete proof methods for autoepistemic programs. These methods extend existing proof methods for 3AEL. Thus autoepistemic logic programming, besides contributing to the understanding of 3AEL, can be seen as a unifying framework for the theory of logic programs. It should also be regarded as a first step toward a flexible environment where different forms of inference can be formally integrated.

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.

A survey of complexity results for non-monotonic logics

This paper surveys the main results appearing in the literature on the computational complexity of non-monotonic inference tasks. We not only give results about the tractability/intractability of the individual problems but we also analyze sources of complexity and explain intuitively the nature of easy/hard cases. We focus mainly on non-monotonic formalisms, like default logic, autoepistemic logic, circumscription, closed-world reasoning, and abduction, whose relations with logic programming are clear and well studied. Complexity as well as recursion-theoretic results are surveyed.

Modal nonmonotonic logics

Many nonmonotonic formalism, including default logic, logic programming with stable models, and autoepistemic logic, can be represented faithfully by means of modal nonmonotonic logics in the family proposed by McDermott and Doyle. In this paper properties of logics in this family are thoroughly investigated. We present several results on characterization of expansions. These results are applicable to a wide class of nonmonotonic modal logics. Using these characterization results, algorithms for computing expansions for finite theories are developed. Perhaps the most important finding of this paper is that the structure of the family of modal nonmonotonic logics is much simpler than that of the family of underlying modal (monotonic) logics. Namely, it is often the case that different monotonic modal logics collapse to the same nonmonotonic system. We exhibit four families of logics whose nonmonotonic variants coincide: 5-KD45, TW5-SW5, N-WK , and W5-D4WB . These nonmonotonic logics naturally represent logics related to commonsense reasoning and knowledge representation such as autoepistemic logic, reflexive autoepistemic logic, default logic, and truth maintenance with negation.

On the autoepistemic reconstruction of logic programming

Current semantics of logic programs normally ignore thesyntactical aspects of the programs. As a result, only the meanings ofsome well-behaved programs can be captured by these semantics. In this paper however, we propose a new semantics of logic programs that can reflectsome of the syntactical behaviours of the programs. The central notion of the semantics is the concept of aneutral clause p ← A which does not affect the behaviour of p in a program. The logic that underlies the semantics is based on anintensional extension of Levesque’s autoepistemicpredicate logic. It differs from existing autoepistemic logics in that it isquantificational andconstructive. We will also compare and contrast our semantics with some well-known semantics. In particular, we will show how to capture the undefined value of a logic program without resorting to a three-valued nonmonotonic formalism. This is achieved by translating an incoherent AE logic program to a program with multiple AE extensions whose intersection can then be used to characterize the undefined value of a logic program.

Autoepistemic logic revisited

Autoepistemic logic revisited

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 ГΔ 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 ГΔ 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.

Autoepistemic circumscription and logic programming

We propose a framework of autoepistemic reasoning in which the underlying semantics is determined by the choice of a nonmonotonic inference mechanism and by specifying abelief constraint. While the latter makes the approach flexible in meeting possibly different applications, the former links the resulting semantics to a nonmonotonic reasoning formalism and thus allows adoption of existing techniques. In this paper we choosecircumscription as the underlying inference mechanism and use two different belief constraints to define two semantics,the stable circumscriptive semantics andthe well-founded circumscriptive semantics, for autoepistemic theories. The former coincides with Moore's autoepistemic logic for logic programs and is arguably more desirable in handling disjunctive autoepistemic theories. The latter is a reconstruction and extension of Przymusinski's iterative method for computing the leastAEL(circ) expansions for logic programs. We show that for logic programs the two construction methods coincide. However, while Przymusinski's construction method is restricted to logic programs only, the well-founded circumscriptive semantics is applicable to more general autoepistemic theories.

A logic of knowledge and justified assumptions

In this paper we define the logic GK of knowledge and justified assumptions. GK is best understood as a formalization of autoepistemic reasoning processes that are more general than those in Moore's autoepistemic logic, and is formally defined via a modification of Shoham's preference semantics. We show that GK includes not only Moore's autoepistemic logic, but also Reiter's default logic. To our knowledge GK is the first complete semantic unification of the two logics. Similarly to circumscription, GK is based on the notion of logical minimization, and thus provides a bridge between circumscription and fixed-point nonmonotonic logics, an outstanding problem in nonmonotonic logics. As an application of this bridge, we propose a formalization of logic programs with negation-as-failure in circumscription.

Reflexive Autoepistemic Logic

We propose a new variant of autoepistemic logic which, intuitively, corresponds to understanding a belief operator L as “is known”, in contrast to the interpretation of L as “is believed” in Moore’s autoepistemic logic. Formal properties of the new logic and relationship to Moore’s logic are studied in detail.

More on Modal Aspects of Default Logic1

Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.