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

Taking I/O seriously: resolution reconsidered for disk

Journal ArticleModern compilation techniques can give Prolog programs, in the best cases, a speed comparable to C. However, Prolog has proven to be unacceptable for data-oriented queries for two major reasons: its poor termination and complexity properties for Datalog, and its tuple-at-a-time strategy. A number of tabling frameworks and systems have addressed the first problem, including the XSB system which has achieved Prolog speeds for tabled programs. Yet tabling systems such as XSB continue to use the tuple-at-a-time paradigm. As a result, these systems are not amenable to a tight interconnection with disk-resident data. However, in a tabling framework the difference between tuple-at-a-time behavior and set-at-a-time can be viewed as one of scheduling. Accordingly, we define a breadth-first set-at-a-time tabling strategy and prove it iteration equivalent to a form of semi-naive magic evaluation. That is, we extend the well-known asymptotic results of Seki [10] by proving that each iteration of the tabling strategy produces the same information as semi-naive magic. Further, this set-at-a-time scheduling is amenable to implementation in an engine that uses Prolog compilation. We describe both the engine and its performance, which is comparable with the tuple-at-a-time strategy even for in-memory Datalog queries. Because of its performance and its fine level of integration of Prolog with a database-style search, the set-at-a-time engine appears as an important key to linking logic programming and deductive databases

Beyond depth-first strategies: improving tabled logic programs through alternative scheduling

Journal ArticleTabled evaluation ensures termination for programs with finite models by keeping track of which subgoals have been called. Given several variant subgoals in an evaluation, only the fi rst one encountered will use program-clause resolution; the rest will resolve with the answers generated by the first subgoal. This use of answer resolution prevents infi nite looping that sometimes happens in SLD. Because answers that are produced in one path of the computation may be consumed, asynchronously, in others, tabling systems face an important scheduling choice not present in traditional top-down evaluation: when to schedule answer resolution. This paper investigates alternate scheduling strategies for tabling in a WAM implementation, the SLG-WAM. The original SLG-WAM had a simple mechanism for scheduling answer resolution that was expensive in terms of trailing and choice-point creation. We propose here a more sophisticated scheduling strategy, batched scheduling, which reduces the overheads of these operations and provides dramatic space reduction as well as speedups for many programs. We also propose a second strategy, local scheduling, which has applications to nonmonotonic reasoning, and when combined with answer subsumption, can arbitrarily improve the performance of some programs

Beyond depth-first: improving tabled logic programs through alternative scheduling strategies

Journal ArticleTabled evaluations ensure termination of logic programs with fi nite models by keeping track of which subgoals have been called Given several variant subgoals in an evaluation, only the fi rst one encountered will use program clause resolution the rest uses answer resolution This use of answer resolution prevents infi nite looping which happens in SLD Given the asynchronicity of answer generation and answer return, tabling systems face an important scheduling choice not present in traditional top-down evaluation How does the order of returning answers to consuming subgoals affect program efficiency This paper investigates alternate scheduling strategies for tabling in a WAM implementation, the SLG-WAM. The original SLG-WAM had a simple mechanism of scheduling answers to be returned to callers which was expensive in terms of trailing and choice point creation We propose here a more sophisticated scheduling strategy, Batched Scheduling, which reduces the overheads of these operations and provides dramatic space reduction as well as speedups for many programs We also propose a second strategy, Local Scheduling, which has applications to non-monotonic reasoning and when combined with answer subsumption can improve the performance of some programs by arbitrary amounts

The PITA System: Tabling and Answer Subsumption for Reasoning under Uncertainty

Many real world domains require the representation of a measure of uncertainty. The most common such representation is probability, and the combination of probability with logic programs has given rise to the field of Probabilistic Logic Programming (PLP), leading to languages such as the Independent Choice Logic, Logic Programs with Annotated Disjunctions (LPADs), Problog, PRISM and others. These languages share a similar distribution semantics, and methods have been devised to translate programs between these languages. The complexity of computing the probability of queries to these general PLP programs is very high due to the need to combine the probabilities of explanations that may not be exclusive. As one alternative, the PRISM system reduces the complexity of query answering by restricting the form of programs it can evaluate. As an entirely different alternative, Possibilistic Logic Programs adopt a simpler metric of uncertainty than probability. Each of these approaches -- general PLP, restricted PLP, and Possibilistic Logic Programming -- can be useful in different domains depending on the form of uncertainty to be represented, on the form of programs needed to model problems, and on the scale of the problems to be solved. In this paper, we show how the PITA system, which originally supported the general PLP language of LPADs, can also efficiently support restricted PLP and Possibilistic Logic Programs. PITA relies on tabling with answer subsumption and consists of a transformation along with an API for library functions that interface with answer subsumption

Tabling and Answer Subsumption for Reasoning on Logic Programs with Annotated Disjunctions

Abstract Probabilistic Logic Programming is an active field of research, with many proposals for languages, semantics and reasoning algorithms. One such proposal, Logic Programming with Annotated Disjunctions (LPADs) represents probabilistic information in a sound and simple way. This paper presents the algorithm "Probabilistic Inference with Tabling and Answer subsumption" (PITA) for computing the probability of queries. Answer subsumption is a feature of tabling that allows the combination of different answers for the same subgoal in the case in which a partial order can be defined over them. We have applied it in our case since probabilistic explanations (stored as BDDs in PITA) possess a natural lattice structure. PITA has been implemented in XSB and compared with ProbLog, cplint and CVE. The results show that, in almost all cases, PITA is able to solve larger problems and is faster than competing algorithms