Vladimir Lifschitz

Classical negation in logic programs and disjunctive databases

An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter.

Representing action and change by logic programs

Abstract We represent properties of actions in a logic programming language that uses both classical negation and negation as failure. The method is applicable to temporal projection problems with incomplete information, as well as to reasoning about the past. It is proved to be sound relative to a semantics of action based on states and transition functions.

Strongly equivalent logic programs

A logic program <inline-equation><f><g>P</g><subscrpt>1</subscrpt></f> </inline-equation> is said to be equivalent to a logic program <inline-equation><f><g>P</g><subscrpt>2</subscrpt></f></inline-equation> in the sense of the answer set semantics if <inline-equation><f><g>P</g><subscrpt>1</subscrpt></f></inline-equation> and <inline-equation><f><g>P</g><subscrpt>2</subscrpt></f></inline-equation> have the same answer sets. We are interested in the following stronger condition: for every logic program, <inline-equation><f><g>P</g>, <g>P</g><subscrpt>1</subscrpt>, ∪ <g>P</g></f></inline-equation> has the same answer sets as <inline-equation><f><g>P</g><subscrpt>2</subscrpt> ∪ <g>P</g></f></inline-equation>. The study of strong equivalence is important, because we learn from it how one can simplify a part of a logic program without looking at the rest of it. The main theorem shows that the verification of strong equivalence can be accomplished by cheching the equivalence of formulas in a monotonic logic, called the logic of here-and-there, which is intermediate between classical logic and intuitionistic logic.

Nonmonotonic causal theories

The nonmonotonic causal logic defined in this paper can be used to represent properties of actions, including actions with conditional and indirect effects, nondeterministic actions, and concurrently executed actions. It has been applied to several challenge problems in the theory of commonsense knowledge. We study the relationship between this formalism and other work on nonmonotonic reasoning and knowledge representation, and discuss its implementation, called the Causal Calculator.

Answer set planning

In “answer set programming”[5,7] solutions to a problem are represented by answer sets (known also as stable models), and not by answer substitutions produced in response to a query, as in conventional logic programming. Instead of Prolog, answer set programming uses software systems capable of computing answer sets. Four such systems were demonstrated at the Workshop on Logic- Based AI held in June of 1999 in Washington, DC: dlv1, smodels,2, DeReS3 and ccalc4.

Nested expressions in logic programs

We extend the answer set semantics to a class of logic programs with nested expressions permitted in the bodies and heads of rules. These expressions are formed from literals using negation as failure, conjunction (,) and disjunction (;) that can be nested arbitrarily. Conditional expressions are introduced as abbreviations. The study of equivalent transformations of programs with nested expressions shows that any such program is equivalent to a set of disjunctive rules, possibly with negation as failure in the heads. The generalized answer set semantics is related to the Lloyd–Topor generalization of Clark’s completion and to the logic of minimal belief and negation as failure.

Formal theories of action

ABSTRACT We apply circumscription to formalizing reasoning about the effects of actions in the framework of situation calculus. An axiomatic description of causal connections between actions and changes allows us to solve the qualification problem and the frame problem using only simple forms of circumscription. The method is applied to the Hanks—McDermott shooting problem and to a blocks world in which blocks can be moved and painted.

Stable models and circumscription

The concept of a stable model provided a declarative semantics for Prolog programs with negation as failure and became a starting point for the development of answer set programming. In this paper we propose a new definition of that concept, which covers many constructs used in answer set programming and, unlike the original definition, refers neither to grounding nor to fixpoints. It is based on a syntactic transformation similar to parallel circumscription.

Weight constraints as nested expressions

We compare two recent extensions of the answer set (stable model) semantics of logic programs. One of them, due to Lifschitz, Tang and Turner, allows the bodies and heads of rules to contain nested expressions. The other, due to Niemela and Simons, uses weight constraints. We show that there is a simple, modular translation from the language of weight constraints into the language of nested expressions that preserves the programs answer sets. Nested expressions can be eliminated from the result of this translation in favor of additional atoms. The translation makes it possible to compute answer sets for some programs with weight constraints using satisfiability solvers, and to prove the strong equivalence of programs with weight constraints using the logic of here-and-there.

Minimal belief and negation as failure

Abstract Fangzhen Lin and Yoav Shoham defined a propositional nonmonotonic logic which uses two independent modal operators. One of them represents minimal knowledge, the other is related to the ideas of justification (as understood in default logic) and of negation as failure. We describe a simplified version of that system, show how quantifiers can be included in it, and study its relation to circumscription and default logic, to logic programming, and to the theory of epistemic queries developed by Hector Levesque and Ray Reiter.