Decidability of the Clark's Completion Semantics for Monadic Programs and Queries

There are many different semantics for general logic programs (i.e. programs that use negation in the bodies of clauses). Most of these semantics are Turing complete (in a sense that can be made precise), implying that they are undecidable. To obtain decidability one needs to put additional restrictions on programs and queries. In logic programming it is natural to put restrictions on the underlying first-order language. In this note we show the decidability of the Clark's completion semantics for monadic general programs and queries. To appear in Theory and Practice of Logic Programming (TPLP

An encompassing framework for Paraconsistent Logic Programs

AbstractWe propose a framework which extends Antitonic Logic Programs [Damásio and Pereira, in: Proc. 6th Int. Conf. on Logic Programming and Nonmonotonic Reasoning, Springer, 2001, p. 748] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting's bilattice approaches, this framework allows a precise definition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [Pereira and Alferes, in: European Conference on Artificial Intelligence, 1992, p. 102], according to which explicit negation entails default negation. We then define Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalizing many paraconsistent semantics for logic programs. In particular, Paraconsistent Well-Founded Semantics with eXplicit negation (WFSXp) [Alferes et al., J. Automated Reas. 14 (1) (1995) 93–147; Damásio, PhD thesis, 1996]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program

PDL with Negation of Atomic Programs

Propositional dynamic logic (PDL) is one of the most succesful variants of modal logic. To make it even more useful for applications, many extensions of PDL have been considered in the literature. A very natural and useful such extension is with negation of programs. Unfortunately, it is long-known that reasoning with the resulting logic is undecidable. In this paper, we consider the extension of PDL with negation of atomic programs, only. We argue that this logic is still useful, e.g. in the context of description logics, and prove that satisfiability is decidable and EXPTIME-complete using an approach based on Büchi tree automata

A Parameterised Hierarchy of Argumentation Semantics for Extended Logic Programming and its Application to the Well-founded Semantics

Argumentation has proved a useful tool in defining formal semantics for assumption-based reasoning by viewing a proof as a process in which proponents and opponents attack each others arguments by undercuts (attack to an argument's premise) and rebuts (attack to an argument's conclusion). In this paper, we formulate a variety of notions of attack for extended logic programs from combinations of undercuts and rebuts and define a general hierarchy of argumentation semantics parameterised by the notions of attack chosen by proponent and opponent. We prove the equivalence and subset relationships between the semantics and examine some essential properties concerning consistency and the coherence principle, which relates default negation and explicit negation. Most significantly, we place existing semantics put forward in the literature in our hierarchy and identify a particular argumentation semantics for which we prove equivalence to the paraconsistent well-founded semantics with explicit negation, WFSX$_p$. Finally, we present a general proof theory, based on dialogue trees, and show that it is sound and complete with respect to the argumentation semantics.Comment: To appear in Theory and Practice of Logic Programmin

Three-valued completion for abductive logic programs

AbstractIn this paper, we propose a three-valued completion semantics for abductive logic programs, which solves some problems associated with the Console et al. two-valued completion semantics. The semantics is a generalization of Kunen's completion semantics for general logic programs, which is known to correspond very well to a class of effective proof procedures for general logic programs. Secondly, we propose a proof procedure for abductive logic programs, which is a generalization of a proof procedure for general logic programs based on constructive negation. This proof procedure is sound and complete with respect to the proposed semantics. By generalizing a number of results on general logic programs to the class of abductive logic programs, we present further evidence for the idea that limited forms of abduction can be added quite naturally to general logic programs

Learning logic programs with negation as failure

Normal logic programs are usually shorter and easier to write and understand than definite logic programs. As a consequence, it is worth investigating their learnability, if Inductive Logic Program- ming is to be proposed as an alternative tool for software development and Software Engineering at large. In this paper we present an exten- sion of the ILP system TRACY, called TRACY-not, able to learn normal logic programs. The method is proved to be sound, in the sense that it outputs a program which is complete and consistent w.r.t.the ex- amples, and complete, in the sense that it does find a solution when it exists. Compared to learning systems based on extensionality,TRACY and TRACY not are less dependent on the kind and number of training examples, which is due to the intensional evaluation of the hypothe- ses and, for TRACY-not, to the possibility to have restricted hypothesis spaces through the use of negation

Abduction in Well-Founded Semantics and Generalized Stable Models

Abductive logic programming offers a formalism to declaratively express and solve problems in areas such as diagnosis, planning, belief revision and hypothetical reasoning. Tabled logic programming offers a computational mechanism that provides a level of declarativity superior to that of Prolog, and which has supported successful applications in fields such as parsing, program analysis, and model checking. In this paper we show how to use tabled logic programming to evaluate queries to abductive frameworks with integrity constraints when these frameworks contain both default and explicit negation. The result is the ability to compute abduction over well-founded semantics with explicit negation and answer sets. Our approach consists of a transformation and an evaluation method. The transformation adjoins to each objective literal $O$ in a program, an objective literal $not(O)$ along with rules that ensure that $not(O)$ will be true if and only if $O$ is false. We call the resulting program a {\em dual} program. The evaluation method, \wfsmeth, then operates on the dual program. \wfsmeth{} is sound and complete for evaluating queries to abductive frameworks whose entailment method is based on either the well-founded semantics with explicit negation, or on answer sets. Further, \wfsmeth{} is asymptotically as efficient as any known method for either class of problems. In addition, when abduction is not desired, \wfsmeth{} operating on a dual program provides a novel tabling method for evaluating queries to ground extended programs whose complexity and termination properties are similar to those of the best tabling methods for the well-founded semantics. A publicly available meta-interpreter has been developed for \wfsmeth{} using the XSB system.Comment: 48 pages; To appear in Theory and Practice in Logic Programmin