Formalisms For Nonmonotonic Reasoning Research Articles

Space Efficiency of Propositional Knowledge Representation Formalisms

We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency of a formalism F in representing a certain piece of knowledge A, is the size of the shortest formula of F that represents A. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space efficiency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms with the same time complexity do not necessarily belong to the same space efficiency class.

An abstract machine for efficiently computing queries to well-founded models

The well-founded semantics has gained wide acceptance partly because it is a skeptical semantics. That is, the well-founded model posits as unknown atoms which are deemed true or false in other formalisms such as stable models. This skepticism makes the well-founded model not only useful in itself, but also suitable as a basis for other forms of non-monotonic reasoning. For instance, since algorithms to compute stable models are intractable, the atoms relevant to such algorithms can be limited to those undefined in the well-founded model. Thus, an engine that efficiently evaluates programs according to the well-founded semantics can be seen as a prerequisite to practical systems for non-monotonic reasoning. This paper describes the architecture of the Warren Abstract Machine (WAM)-based abstract machine underlying the XSB system. This abstract machine, called the SLG-WAM, uses tabling to efficiently compute the well-founded semantics of non-ground normal logic programs in a goal-directed way. To do so, the SLG-WAM requires sophisticated extensions to its core tabling engine for fixed-order stratified programs. A mechanism must be implemented to represent answers that are neither true nor false, and the delay and simplification operations – which serve to break and to resolve cycles through negation, must be implemented. We describe fully these extensions to our tabling engine, and demonstrate the efficiency of our implementation in two ways. First, we present a theorem that bounds the need for delay to those literals which are not dynamically stratified for a fixed-order computation. Second, we present performance results that indicate that the overhead of delay and simplification to Prolog – or tabled – evaluations is minimal.

On the decidability and complexity of reasoning about only knowing

We study reasoning in Levesque's logic of only knowing. In particular, we first prove that extending a decidable subset of first-order logic with the ability of reasoning about only knowing preserves decidability of reasoning, as long as quantifying-in is not allowed in the language, and define a general method for reasoning about only knowing in such a case. Then, we show that the problem of reasoning about only knowing in the propositional case lies at the second level of the polynomial hierarchy. Thus, it is as hard as reasoning in the majority of propositional formalisms for nonmonotonic reasoning, like default logic, circumscription, and autoepistemic logic, and it is easier than reasoning in propositional formalisms based on the minimal knowledge paradigm, which is strictly related to the notion of only knowing. Finally, we identify a syntactic restriction in which reasoning about only knowing is easier than in the general propositional case, and provide a specialized deduction method for such a restricted setting.

Reasoning about Minimal Belief and Negation as Failure

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic.

Reasoning about Minimal Knowledge in Nonmonotonic Modal Logics

We study the problem of embedding Halpern and Mosess modal logic of minimal knowledge states into two families of modal formalism for nonmonotonic reasoning, McDermott and Doyles nonmonotonic modal...

On first-order conditional logics

Conditional logics have been developed as a basis from which to investigate logical properties of “weak” conditionals representing, for example, counterfactual and default assertions. This work has largely centred on propositional approaches. However, it is clear that for a full account a first-order logic is required. Existing or obvious approaches to first-order conditional logics are inadequate; in particular, various representational issues in default reasoning are not addressed by extant approaches. Further, these problems are not unique to conditional logic, but arise in other nonmonotonic reasoning formalisms. I argue that an adequate first-order approach to conditional logic must admit domains that vary across possible worlds; as well the most natural expression of the conditional operator binds variables (although this binding may be eliminated by definition). A possible worlds approach based on Kripke structures is developed, and it is shown that this approach resolves various problems that arise in a first-order setting, including specificity arising from nested quantifiers in a formula and an analogue of the lottery paradox that arises in reasoning about default properties.

The expressive power of circumscription

Circumscription is a form of nonmonotonic reasoning, introduced by McCarthy (1997) as a way of characterizing defaults using second order logic. The consequences of circumscription are those formulas true in the minimal models under a pre-order on models. In the case of domain circumscription the pre-order was the sub-model relation. Formula circumscription (McCarthy, 1980, 1986) is characterized by minimizing a set of formulas—one model is preferred to another model when the extensions of the minimized formulas in the first are subsets of the extensions in the second. We show that the propositional version of formula circumscription can capture all pre-orders on valuations of finite languages. We consider the question of infinite languages, and give the corresponding representation theorems. We further show that there are natural defaults (inertia in temporal projection), captured by inductive definitions, that cannot be captured by circumscription in the first order case. Finally, contrary to previous claims, we show that propositional formula circumscription can capture all preferential consequence relations over finite propositional languages, as defined by Kraus et al. (1990). Thus, in the finite propositional case, there is no restriction on the kinds of preferential defaults that circumscription can describe.

On the relation between default and modal nonmonotonic reasoning

The notion of a default consequence relation is introduced as a generalization of both default and modal formalizations of nonmonotonic reasoning. It is used to study a general problem of correspondence between these two formalisms.

Stereotypical reasoning: logical properties

Stereotypical reasoning assumes that the situation at hand is one of a kind and that it enjoys the properties generally associated with that kind of situation. It is one of the most basic forms of nonmonotonic reasoning. A formal model for stereotypical reasoning is proposed and the logical properties of this form of reasoning are studied. Stereotypical reasoning is shown to be cumulative under weak assumptions. Keywords: Prototypical Reasoning, Stereotypical Reasoning, Nonmonotonic Consequence Relations.

Autoepistemic logic of knowledge and beliefs

In recent years, various formalizations of nonmonotonic reasoning and different semantics for normal and disjunctive logic programs have been proposed, including autoepistemic logic, circumscription, CWA, GCWA, ECWA, epistemic specifications, stable, well-founded, stationary and static semantics of normal and disjunctive logic programs. In this paper we introduce a simple nonmonotonic knowledge representation framework which isomorphically contains all of the above-mentioned nonmonotonic formalisms and semantics as special cases and yet is significantly more expressive than each one of these formalisms considered individually. The new formalism, called the Autoepistemic Logic of Knowledge and Beliefs, AELB, is obtained by augmenting Moore's autoepistemic logic, AEL, already employing the knowledge operator, L , with an additional belief operator, B . As a result, we are able to reason not only about formulae F which are known to be true (i.e., those for which L F holds) but also about those which are only believed to be true (i.e., those for which B F holds). The proposed logic constitutes a powerful new formalism which can serve as a unifying frame-work for several major nonmonotonic formalisms. It allows us to better understand mutual relationships existing between different formalisms and semantics and enables us to provide them with simpler and more natural definitions. It also naturally leads to new, even more expressive, flexible and modular formalizations and semantics.

Efficient evaluation of a class of ordered logic programs

Ordered Logic ( OL ) enhances logic programming by the capability of modeling complex knowledge, by permitting a natural representation of several forms of non-monotonic reasoning. Both well-founded and stable model semantics for OL programs can be found in the literature [7,20,22]. This paper focuses on the problem of the efficient evaluation of the well-founded semantics of OL programs. First, a class of OL programs, called quasi-stratified programs, is presented. Then, an algorithm for the efficient bottom-up evaluation of the well-founded model of quasi-stratified programs is provided. Quasi-stratified programs evaluation has been implemented in KIWIS, an advanced environment for large database systems, whose query language is based on OL.

Extending Prolog with nonmonotonic reasoning

Nonmonotonic reasoning has been developed to capture common sense inferences. This paper considers nonmonotonic reasoning in logic programs with negation, specifically its implementation using Prolog and its integration with Prolog execution. Even within logic programming frameworks, a variety of inferencing methods, model-theoretic semantics, and language features have been proposed for different forms of nonmonotonic reasoning. A challenging problem is to incorporate them into an integrated system. This paper describes an implementation of such a system in Prolog and different reasoning capabilities that are supported. By combining Prolog technology with various mechanisms of nonmonotonic reasoning, the resulting system offers a more realistic testbed for applications of common sense reasoning.

Operational characterization of extensions in some logics for default reasoning

Default reasoning is one of the most common forms of nonmonotonic reasoning, and is found in many application domains. Default logic and its variants have been proposed as knowledge representation methods capable of default reasoning. One problem of all these methods is that their concepts are given in terms of fixed-point equations, thus making them difficult to understand and use. In this paper, we present simple, constructive approaches for the semantics of some default logic variants, and show their correctness.

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.

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.

Preference logics: Towards a unified approach to nonmonotonicity in deductive reasoning

The notion ofpreference is central to most forms of nonmonotonic reasoning. Shoham, in his dissertation, used this notion to give a single semantical point of view from which most nonmonotonic reasoning systems could be studied. In this paper, we study the notion of preference closely and devise a class of logics of preference that extract the logicalcore of the notion of preference. Earlier attempts have been largely unsuccessful, because of adoption as matters of logic of certain theory-specific preference principles such asasymmetry andtransitivity. Soundness, completeness and decidability proofs for the logics are given. We define the notion of a preferential theory and reframe nonmonotonicity as a symbolic optimization problem where defaults are coded aspreference criteria which place preference orders on the models of a first-order theory. We study the relationship between normal default theories and show the correspondence between models of extensions and optimal worlds. We give a preferential account of some forms of circumscription. Thelocal nature of preference logic is contrasted with the global notion of normality and preference that is used by conditional logics of normality and cumulative inference operations. In related papers, we give a completely declarative semantics for the stable models of normal logic programs, a deontic logic based on preferences that is free of the anomalies of standard deontic logic, and extend Horn clause logic programming to impose partial orders on the bodies of clauses as declarative specification of the relaxation criteria for the truth-hood of the heads.

Logic programming and knowledge representation

In this paper, we review recent work aimed at the application of declarative logic programming to knowledge representation in artificial intelligence. We consider extensions of the language of definite logic programs by classical (strong) negation, disjunction, and some modal operators and show how each of the added features extends the representational power of the language. We also discuss extensions of logic programming allowing abductive reasoning, meta-reasoning and reasoning in open domains. We investigate the methodology of using these languages for representing various forms of nonmonotonic reasoning and for describing knowledge in specific domains. We also address recent work on properties of programs needed for successful applications of this methodology such as consistency, categoricity and complexity.

Non-monotonic reasoning with logic programming

Our purpose is to exhibit a modular systematic method of representing non-monotonic reasoning problems with the Well-Founded Semantics WFS of extended logic programs augmented with eXplicit negation (WFSX), augmented by its Contradiction Removal Semantics (CRSX) when needed. We apply this semantics, and its contradiction removal semantics counterpart, to represent non-monotonic reasoning problems. We show how to cast in the language of logic programs extended with explicit negation such forms of non-monotonic reasoning as defeasible reasoning, abductive reasoning, and hypothetical reasoning and apply them to such different domains of knowledge representation as hierarchies and reasoning about actions. We then abstract a modular systematic method of representing non-monotonic problems in a logic programming semantics comprising two forms of negation avoiding some drawbacks of other proposals, with which we relate our work.

Approaches to abductive reasoning: an overview

Abduction is a form of non-monotonic reasoning that has gained increasing interest in the last few years. The key idea behind it can be represented by the following inference rule $$O = \mathop C\limits_| - N = \mathop P\limits_|^| - O - \mathop C\limits_|^| - .$$ i.e., from an occurrence of ω and the rule “ϕ implies ω”, infer an occurrence of ϕ as aplausible hypothesis or explanation for ω. Thus, in contrast to deduction, abduction is as well as induction a form of “defeasible” inference, i.e., the formulae sanctioned are plausible and submitted to verification. In this paper, a formal description of current approaches is given. The underlying reasoning process is treated independently and divided into two parts. This includes a description ofmethods for hypotheses generation andmethods for finding the best explanations among a set of possible ones. Furthermore, the complexity of the abductive task is surveyed in connection with its relationship to default reasoning. We conclude with the presentation of applications of the discussed approaches focusing on plan recognition and plan generation.

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.