Tabling is a powerful resolution mechanism for logic programs that captures their least fixed point semantics more faithfully than plain Prolog. In many tabling applications, we are not interested in the set of all answers to a goal, but only require an aggregation of those answers. Several works have studied efficient techniques, such as lattice-based answer subsumption and mode-directed tabling, to do so for various forms of aggregation.

While much attention has been paid to expressivity and efficient implementation of the different approaches, soundness has not been considered. This paper shows that the different implementations indeed fail to produce least fixed points for some programs. As a remedy, we provide a formal framework that generalises the existing approaches and we establish a soundness criterion that explains for which programs the approach is sound.

Distribution semantics is one of the most prominent approaches for the combination of logic programming and probability theory. Many languages follow this semantics, such as Independent Choice Logic, PRISM, pD, Logic Programs with Annotated Disjunctions (LPADs), and ProbLog. When a program contains functions symbols, the distribution semantics is well–defined only if the set of explanations for a query is finite and so is each explanation. Well–definedness is usually either explicitly imposed or is achieved by severely limiting the class of allowed programs. In this paper, we identify a larger class of programs for which the semantics is well–defined together with an efficient procedure for computing the probability of queries. Since Logic Programs with Annotated Disjunctions offer the most general syntax, we present our results for them, but our results are applicable to all languages under the distribution semantics. We present the algorithm “Probabilistic Inference with Tabling and Answer subsumption” (PITA) that computes the probability of queries by transforming a probabilistic program into a normal program and then applying SLG resolution with answer subsumption. PITA has been implemented in XSB and tested on six domains: two with function symbols and four without. The execution times are compared with those of ProbLog, cplint, and CVE. PITA was almost always able to solve larger problems in a shorter time, on domains with and without function symbols.

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. We show that, by adapting its transformation and library functions, PITA can be parameterized to PITA(IND, EXC) which supports the restricted PLP of PRISM, including optimizations that reduce non-discriminating arguments and the computation of Viterbi paths. Furthermore, we show PITA to be competitive with PRISM for complex queries to Hidden Markov Model examples, and sometimes much faster. We further show how PITA can be parameterized to PITA(COUNT) which computes the number of different explanations for a subgoal, and to PITA(POSS) which scalably implements Possibilistic Logic Programming. PITA is a supported package in version 3.3 of XSB.