This paper introduces an abductive framework for updating knowledge bases represented by extended disjunctive programs. We first provide a simple transformation from abductive programs to update programs which are logic programs specifying changes on abductive hypotheses. Then, extended abduction, which was introduced by the same authors as a generalization of traditional abduction, is computed by the answer sets of update programs. Next, different types of updates, view updates and theory updates are characterized by abductive programs and computed by update programs. The task of consistency restoration is also realized as special cases of these updates. Each update problem is comparatively assessed from the computational complexity viewpoint. The result of this paper provides a uniform framework for different types of knowledge base updates, and each update is computed using existing procedures of logic programming.
Inoue characterizes inconsistency resolution in an ELP @math by the abductive program @math . Then, he considers a maximal consistent subset of the hypotheses @math , which is computed using a program transformation from the abductive program to an ELP. We characterized the same problem by the abductive program @math in , but the result is the same as @cite_38 for ELPs. The problem is also characterized by the abductive program @math in @cite_27 . This formulation, however, produces different results in general. For instance, given the inconsistent program @math , @math has the minimal explanation @math which produces the updated program @math . On the other hand, @math has the minimal explanation @math and the result of update is @math . Thus, @math permits the introduction of new facts as well as the deletion of facts to resolve inconsistency. Generally, permitting introduction of sentences increases the number of possible solutions. Nevertheless, this type of inconsistency resolution is also realized by computing consistent U-minimal answer sets of the update program of @math .
@cite_38 @cite_27
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Horn clause logic programming can be extended to include abduction with integrity constraints. In the resulting extension of logic programming, negation by failure can be simulated by making negative conditions abducible and by imposing appropriate denials and disjunctions as integrity constraints. This gives an alternative semantics for negation by failure, which generalises the stable model semantics of negation by failure. The abductive extension of logic programming extends negation by failure in three ways: (1) computation can be perfonned in alternative minimal models, (2) positive as well as negative conditions can be made abducible, and (3) other integrity constraints can also be accommodated. * This paper was written while the first author was at Imperial College. 235 Introduction The tenn "abduction" was introduced by the philosopher Charles Peirce [1931] to refer to a particular kind of hypothetical reasoning. In the simplest case, it has the fonn: From A and A fB infer B as a possible "explanation" of A. Abduction has been given prominence in Charniak and McDennot's [1985] "Introduction to Artificial Intelligence", where it has been applied to expert systems and story comprehension. Independently, several authors have developed deductive techniques to drive the generation of abductive hypotheses. Cox and Pietrzykowski [1986] construct hypotheses from the "dead ends" of linear resolution proofs. Finger and Genesereth [1985] generate "deductive solutions to design problems" using the "residue" left behind in resolution proofs. Poole, Goebel and Aleliunas [1987] also use linear resolution to generate hypotheses. All impose the restriction that hypotheses should be consistent with the "knowledge base". Abduction is a fonn of non-monotonic reasoning, because hypotheses which are consistent with one state of a knowledge base may become inconSistent when new knowledge is added. Poole [1988] argues that abduction is preferable to noh-monotonic logics for default reasoning. In this view, defaults are hypotheses fonnulated within classical logic rather than conclusions derived withln some fonn of non-monotonic logic. The similarity between abduction and default reasoning was also pointed out in [Kowalski, 1979]. In this paper we show how abduction can be integrated with logic programming, and we concentrate on the use of abduction to generalise negation by failure. Conditional Answers Compared with Abduction In the simplest case, a logic program consists of a set of Horn Clauses, which are used backward to_reduce goals to sub goals. The initial goal is solved when there are no subgollls left; To explain observations from nonmonotonic background theories, one often needs removal of some hypotheses as well as addition of other hypotheses. Moreover, some observations should not be explained, while some are to be explained. In order to formalize these situations, extended abduction was introduced by Inoue and Sakama (1995) to generalize traditional abduction in the sense that it can compute negative explanations by removing hypotheses and antidexplanations to unexplain negative observations. In this paper, we propose a computational mechanism for extended abduction. When a background theory is written in a normal logic program, we introduce its transaction program for computing extended abduction. A transaction program is a set of nonddeterministic production rules that declaratively specify addition and deletion of abductive hypotheses. Abductive explanations are then computed by the fixpoint of a transaction program using a bottomdup model generation procedure. The correctness of the proposed procedure is shown for the class of acyclic covered abductive logic programs. In the context of deductive databases, a transaction program provides a declarative specification of database update. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. 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 @math in a program, an objective literal @math along with rules that ensure that @math will be true if and only if @math is false. We call the resulting program a dual program. The evaluation method, ABDUAL, then operates on the dual program. ABDUAL 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, ABDUAL is asymptotically as efficient as any known method for either class of problems. In addition, when abduction is not desired, ABDUAL 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 ABDUAL using the XSB system.