LOGIC PROGRAMMING AND THEOREM PROVING LOGIC PROGRAMMING AND THEOREM PROVING LOGIC PROGRAMMING A-System: Problem Solving through Abduction Antonis C. Kakas Bert Van Nuffelen and Marc Denecker Dep. of Computer Science Dep. of Computer Science University of Cyprus K.U.Leuven B.O.Box 20537 Celestijnenlaan 200A CY-1678 Nicosia, Cyprus B-3000 Leuven, Belgium g [email protected] fbertv, marcd @cs.kuleuven.ac.be Abstract represented in a high-level declarative way with logic pro- gramming rules and classical first-order sentences (integrity This paper presents a new system, called the A- constraints). Our work aims to examine the possibility of de- System, performing abductive reasoning within the veloping ALP into a declarative problem solving framework, framework of Abductive Logic Programming. It is suitable for a variety of AI problems, that is computationally based on a hybrid computational model that imple- viable for problems of practical scale. ments the abductive search in terms of two tightly A principled implementation of the A-System is developed coupled processes: a reduction process of the high- based on a hybrid computational model that formalizes the level logical representation to a lower-level con- abductive search for a solution in terms of two interleaving straint store and a lower-level constraint solving processes: the logical reduction of the high-level representa- process. A set of initial ”proof of principle” ex- tion of the problem and lower-level constraint solving. The periments demonstrate the versatility of the ap- abductive search is linked tightly to the construction of an as- proach stemming from its declarative representa- sociated constraint store. tion of problems and the good underlying compu- To validate our approach we have carried out a set of ”proof tational behaviour of the system. The approach of- of principle” experiments with an initial implementation of fers a general methodology of declarative problem the system that aim to test (a) the general underlying compu- solving in AI where an incremental and modular re- tational behaviour of the A-System on different domains and finement of the high-level representation with extra (b) the flexibility of the framework to incorporate additional domain knowledge can improve and scale the com- problem specific knowledge in a modular and computation- putational performance of the framework. ally enhancing way. In particular, some of these experiments are designed to test the extend to which the high-level rep- 1 Introduction resentation of problems can be computationally improved via Over the last two decades it has become clear that abduction extra domain knowledge added to it. The experiments con- can play a central role in addressing a variety of problems sidered include constraint satisfaction problems and standard in Artificial Intelligence. These problems include diagnosis AI Planning Systems Competition problems. [Poole et al., 1987; Console et al., 1996], planning [Missiaen The A-System has been developed as a follow up of two [ ] et al., 1995; Kakas et al., 2000; Shanahan, 2000] knowledge earlier ALP systems, the ACLP system Kakas et al., 2000 [ ] assimilation and belief revision, [Inoue and Sakama, 1995; and SLDNFAC Denecker and Schreye, 1998 , bringing to- Pagnucco, 1996] multi-agent coordination,[Ciampolini et al., gether features from these two systems. The main devel- 2000; Kowalski and Sadri, 1999] and knowledge intensive opment in the design of the A-System over these previous learning [Muggleton, 2000; Mooney, 2000]. systems is the fact that the non-determinism in the abduct- The essential feature of this abductive approach to problem ive computation is now made explicit allowing the possibil- solving is the fact that it allows the application problems to ity of implementing this as a form of parameterized heuristic be formalized directly in their high-level declarative repres- search coupled with a process of deterministic propagation entation. A close link therefore emerges between declarative of the state of computation. The A-System is available from problem solving in AI and the logical reasoning of abduction. www.cs.kuleuven.ac.be/dtai/kt. However, despite this variety of applications for abduction and its potential benefits it has not been easy to develop gen- 2 Declarative Programming with Abduction eral systems for abduction that are computationally effective Declarative Problem Solving with a high-level representation for problems of practical size and complexity. of the problem at hand is closely related to abduction.The This paper presents a new system, called the A-System, reason for this stems from the fact that in a declarative rep- that supports abductive reasoning within the framework of resentation where one describes the expert knowledge on the Abductive Logic Programming (ALP) [Kakas et al., 1998; problem domain rather than some method for its solution of- Denecker and Kakas, 2000]. In this framework problems are ten, the task of solving the problems consists in filling the LOGIC PROGRAMMING AND THEOREM PROVING 591 missing information from this representation pertaining to the The constraint predicates are defined, as in CLP, by an un- goal at hand. Typically, in a logical framework of represent- derlying constraint theory which is independent of any par- È; A; Á C µ ation the solution consists in finding the extension of some ticular abductive theory ´ . We will assume that the predicate(s) which are incompletely specified in the repres- constraint theory is a finite domain theory that also includes entation. The high-level theory describing the problem is then equality over logical terms. The formal details of this are extended to a new one such that the problem goal is satisfied. beyond the scope of this paper. Computing such extensions of the theory representing our In ALP the abductive problem task is defined as follows. problem is an abductive task. Indeed, abduction as a problem È ; A; Ì µ Definition 2.2 Given an abductive theory ´ and solving method assumes that the general data structure for the Ä some query É, consisting of a conjunction of literals over , solution to a problem (or solution carrier) is at the predicate ¡ A an abductive solution (or explanation) for É is a set of level and hence that a solution is described in the same terms ground abducible atoms together with an answer substitution and level as the problem itself. È [ ¡ such that is consistent and: A framework for problem solving with abduction therefore È [ ¡ j= 8´ ´Éµµ needs to be expressing enough to allow high-level declarative ¯ È [ ¡ j= ÁC: representations of complex problems with missing or incom- ¯ plete information. At the same time for such a framework to have a practical value it should provide ways of improving This definition is generic in that it defines the notion of an the computational effectiveness of this high-level represent- abductive solution in terms of any given semantics of standard ation for any particular problem. One such framework that (constraint) Logic Programming (LP). Each particular choice = combines representational expressiveness and computational of semantics defines its own entailment relation j and hence flexibility is that of Abductive Constraint Logic Programming its own notion of what is an abductive solution. In the con- (ACLP) a framework that integrates together Abductive Lo- text of ID-logic, its inductive definition semantics essentially [ gic Programming (ALP) with methods from Constraint Logic coincides with the well-founded model semantics Gelder et ] 1 Programming (CLP). The A-system is developed within this al., 1991 for LP when this is a two valued model .Intherest framework. Let us briefly review the underlying framework of this paper we will be adopting this well-founded model of ALP. semantics for LP for the development of our A-System. A recent way to formalize ALP is to view this as a special A computed abductive solution ¡ therefore gives a ground case of ID-logic [Denecker, 2000], a logic extending classical definition of the open predicates which in turn through the [ ¡ logic with inductive definitions. Let a language of predicates, logic program, È , extends this to the defined predicates. As an example of an abductive theory we give below a Ä, be given consisting of three disjoint types of predicates: part of the theory representing a planning domain. In this, È A (i) D the defined predicates, (ii) the open or abducible ÚehicÐe < 6= dÖ iÚ e , is the only open predicate, , , are con- predicates and (iii) C the constraint predicates. Then a theory in ALP is defined as follows. straint predicates and all others are defined predicates. The Ð ÓcaØiÓÒ ØiÑe predicates Ú ehicÐ e, and are simple predicates È; A; Á C µ Definition 2.1 An abductive theory is a triple ´ defining the finite domain of variables, e.g. using CLP nota- ´ÌÖÙckµ: ÌÖÙckiÒ ½::½¼ where: tion Ú ehicÐ e for a problem where we have ten trucks. È ¯ is a constraint logic program where only defined pre- aØ´Ì Ö Ùck ; ÄÓc; Ì µ: dicates appearing in the head of these rules. ÚehicÐe ÚehicÐe aØ´Ì Ö Ùck ; ÄÓcµ; iÒiØ A ¯ is a set of ground abducible atoms over open predic- 2 :cÐ iÔÔed aØ´Ì Ö Ùck ; ÄÓc; ¼;̵ : ates. ÚehicÐe aØ´Ì Ö Ùck ; ÄÓc; Ì µ: ÚehicÐe ÁC Ä ¯ is a set of first order formulae over , called integrity Ú ehicÐ e´Ì Ö Ùck ; ÄÓc; E µ; constraints. dÖ iÚ e E<Ì; ÚehicÐe aØ´Ì Ö Ùck ; ÄÓc; E ; Ì µ: cÐ iÔÔed The representation of an application problem in an abduct- : È; A; Á C µ ive theory ´ splits, in this view of ALP in terms cÐ iÔÔed Ú ehicÐ e aØ´ÌÖÙck;ÄÓc½;E;̵: of ID-logic, into two parts. The set of rules in È represents Ú ehicÐ e´Ì Ö Ùck ; ÄÓc¾;Cµ; the expert’s strong definitional knowledge of the problem, i.e.