Logic programming and reasoning with incomplete information

The purpose of this paper is to expand the syntax and semantics of logic programs and disjunctive databases to allow for the correct representation of incomplete information in the presence of multiple extensions. The language of logic programs with classical negation, epistemic disjunction, and negation by failure is further expanded by new modal operators K and M (where for the set of rulesT and formulaF, KF stands for F is known to be true by a reasoner with a set of premisesT and MF means F may be believed to be true by the same reasoner). Sets of rules in the extended language will be called epistemic specifications. We will define the semantics of epistemic specifications (which expands the semantics of disjunctive databases from) and demonstrate their applicability to formalization of various forms of commonsense reasoning. In particular, we suggest a new formalization of the closed world assumption which seems to better correspond to the assumption's intuitive meaning.     

Sequent calculi for skeptical reasoning in predicate default logic and other nonmonotonic logics

Sequent calculi for skeptical consequence in predicate default logic, predicate stable model logic programming, and infinite autoepistemic theories are presented and proved sound and complete. While skeptical consequence is decidable in the finite propositional case of all three formalisms, the move to predicate or infinite theories increases the complexity of skeptical reasoning to being ₁¹-complete. This implies the need for sequent rules with countably many premises, and such rules are employed. AMS subject classification 03B42, 68N17, 68T27This paper grew directly out of the authors dissertation, written under the direction of Anil Nerode.     

析取封闭世界假设的一种过程语义

析取信息的表示是一个重要的研究问题.DCWA(析取封闭假设)为一般演绎数据库提供了一种谨慎语义,并且扩充了标准的良基语义.同时DCWA支持争论推理,为广义封闭世界假设提供了一种逼近.基于此,提出了DCWA的过程语义,并证明了它的可靠性和完备性.     

The Complexity of Read-Once Resolution

We investigate the complexity of deciding whether a propositional formula has a read-once resolution proof. We give a new and general proof of Iwama–Miynano's theorem which states that the problem whether a formula has a read-once resolution proof is NP-complete. Moreover, we show for fixed k2 that the additional restriction that in each resolution step one of the parent clauses is a k-clause preserves the NP-completeness. If we demand that the formulas are minimal unsatisfiable and read-once refutable then the problem remains NP-complete. For the subclasses MU(k) of minimal unsatisfiable formulas we present a pol-time algorithm deciding whether a MU(k)-formula has a read-once resolution proof. Furthermore, we show that the problems whether a formula contains a MU(k)-subformula or a read-once refutable MU(k)-subformula are NP-complete.     

Completeness of a combination of neighbourhood logic and temporal logic

This paper presents a completeness result, with respect to a possible world semantics, for a combination of a first-order temporal logic and neighbourhood logic. This logic was considered by Qiu and Zhou (1998, Proceedings of the PROCOMET 98, pp 444–461) to define semantics of a real-time OCCAM-like programming language.Received June 1999Accepted in revised form September 2003 by M. R. Hansen and C. B. Jones     

Closed world assumption for disjunctive reasoning

In this paper,the relationship between argumentation and closed world reasoning for disjunctive information is studied.In particular,the authors propose a simple and intuitive generalization of the closed world assumption(CWA) for general disjunctive deductive databases(with default negation).This semantics,called DCWA,allows a natural argumentation-based interpretation and can be used to represent reasoning for disjunctive information.We compare DCWA with GCWA and prove that DCWA extends Minker‘s GCWA to the class of disjunctive databases with defacult negation.Also we compare our semantics with some related approaches.In addition,the computational complexity of DCWA is investigated.     

Stable semantics for disjunctive programs

We introduce the stable model semantics fordisjunctive logic programs and deductive databases, which generalizes the stable model semantics, defined earlier for normal (i.e., non-disjunctive) programs. Depending on whether only total (2-valued) or all partial (3-valued) models are used we obtain thedisjunctive stable semantics or thepartial disjunctive stable semantics, respectively. The proposed semantics are shown to have the following properties:

? For normal programs, the disjunctive (respectively, partial disjunctive) stable semantics coincides with thestable (respectively,partial stable) semantics.

? For normal programs, the partial disjunctive stable semantics also coincides with thewell-founded semantics.

? For locally stratified disjunctive programs both (total and partial) disjunctive stable semantics coincide with theperfect model semantics.

? The partial disjunctive stable semantics can be generalized to the class ofall disjunctive logic programs.

? Both (total and partial) disjunctive stable semantics can be naturally extended to a broader class of disjunctive programs that permit the use ofclassical negation.

? After translation of the programP into a suitable autoepistemic theory \( \hat P \) the disjunctive (respectively, partial disjunctive) stable semantics ofP coincides with the autoepistemic (respectively, 3-valued autoepistemic) semantics of \( \hat P \) .

     

Disjunctive LP+integrity constraints= stable model semantics

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.     

Complexity and undecidability results for logic programming

This paper surveys complexity, degree of uncomputability, and expressive power results for logic programming. Some major decision problem complexity results and other results for logic programming are also covered. It also proves several new results filling in previous gaps in the literature. The paper considers seven logic programming semantics: the van Emden-Kowalski semantics for definite (Horn) logic programs; the perfect model semantics for stratified and for locally stratified logic programs; and the two- and three-valued program completion semantics, the well-founded semantics, and the stable semantics, all for normal logic programs, under skeptical inference. The main results concern expressibility and query complexity/uncomputability in five contexts: for propositional logic programs, for first order logic programs with infinite Herbrand universes on their Herbrand universes (a closed domain assumption), for first order logic programs with infinite Herbrand universes on those universes extended with infinitely many new elements (an open domain assumption), and for logic programs without function or constant symbols evaluated over varying extensional databases (DATALOG-type results, data complexity results only) under both closed and open domain assumptions. Several of the open domain assumption results are new to this paper. Other results surveyed are (1) results about the family of all stable models of a program and (2) decision questions about when a logic program has nice properties with respect to a semantics (e.g., a unique stable model). One decision result, for well-founded semantics, is new to this paper.Work supported in part by NSF grant IRI-8905166.     

On the complexity of 2D discrete fixed point problem

We study a computational complexity version of the 2D Sperner problem, which states that any three coloring of vertices of a triangulated triangle, satisfying some boundary conditions, will have a trichromatic triangle. In introducing a complexity class PPAD, Papadimitriou [C.H. Papadimitriou, On graph-theoretic lemmata and complexity classes, in: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 1990, 794–801] proved that its 3D analogue is PPAD-complete about fifteen years ago. The complexity of 2D-SPERNER itself has remained open since then.     

Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation

Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunctive logic programs, we provide two declarative characterizations of stable models in terms of unfounded sets. One shows that the set of stable models coincides with the family of unfounded-free models (i.e., a model is stable iff it contains no unfounded atoms). The other proves that stable models can be defined equivalently by a property of their false literals, as a model is stable iff the set of its false literals coincides with its greatest unfounded set. We then generalize the well-founded operator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of function-free programs. The algorithm's soundness and completeness are proved and some complexity issues are discussed.     

Set theory for verification. II: Induction and recursion

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs, and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski theorem over a suitable set.Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion.Recursive data structures are expressed by applying the Knaster-Tarski theorem to a set, such asV , that is closed under Cartesian product and disjoint sum.Worked examples include the transitive closure of a relation, lists, variable-branching trees, and mutually recursive trees and forests. The Schröder-Bernstein theorem and the soundness of propositional logic are proved in Isabelle sessions.     

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.     

The equivalence of Horn and network complexity for Boolean functions

Summary We propose in this paper a logical complexity measure — Horn complexity — for Boolean functions which measures the minimal length of quasi-Horn definitions of such functions by propositional formulae. The interest for this complexity measure comes on the one hand from the observation that the satisfiability problem for Horn formulae is in P, on the other hand from a strong connection to Cook's problem. We show the proposed Horn complexity to be polynomially equivalent to network complexity and therefore to Turing complexity for Boolean functions.A preliminary version of this work has appeared in the Proceedings of the 5^th Scandinavian Logic Symposium, edited by B. Mayoh and F. Jensen, Aalborg University Press, Aalborg 1979     

Timing Analysis of Combinational Circuits in Intuitionistic Propositional Logic

Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof-theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way.We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic (Information and Computation, Vol. 137, No. 1, 1–33, 1997), in which validity is validity up to stabilisation, and implication comes out as boundedly gives rise to. We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers. Following this second point of view an intensional semantics for proofs is presented which allows us effectively to compute quantitative stabilisation bounds.We discuss the application of the theory to the timing analysis of combinational circuits. To test our ideas we have implemented an experimental prototype tool and run several examples.     

An alternative approach to the semantics of disjunctive logic programs and deductive databases

In this paper, we study a new semantics of logic programming and deductive databases. Thepossible model semantics is introduced as a declarative semantics of disjunctive logic programs. The possible model semantics is an alternative theoretical framework to the classical minimal model semantics and provides a flexible inference mechanism for inferring negation in disjunctive logic programs. We also present a proof procedure for the possible model semantics and show that the possible model semantics has an advantage from the computational complexity point of view.This is a revised and extended version of the paper [36] which was presented at the Tenth International Conference on Logic Programming, Budapest, 21–25 June 1993.     

New problems complete for nondeterministic log space

It is shown that a variety of problems have computational complexity equivalent to that of finding a path through a directed graph. These results, which parallel those of Karp at a lower complexity level, concern satisfiability of propositional formulas with two literals per clause, generation of elements by an associative binary operation, solution of linear equations with two variables per equation, equivalence of generalized sequential machines with final states, and deciding theLL(k) andLR(k) conditions for context-free grammars. Finally, several problems are shown equivalent to reachability in undirected graphs, including bipartiteness and satisfiability of formulas with the exclusive or connective.The work of this author was partly supported by Kansas General Research Grant 3756-5038. A preliminary version appears in the Proceedings of the Eighth Princeton Conference on Information Sciences and Systems, Princeton University, pp. 184–188, March 1974.