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.

💡 Research Summary

The paper presents the PITA system (Probabilistic Inference with Tabling and Answer subsumption), a unified framework for reasoning under uncertainty that can handle three major paradigms: general probabilistic logic programming (PLP) based on Logic Programs with Annotated Disjunctions (LPADs), restricted PLP as exemplified by PRISM, and possibilistic logic programming. PITA works by transforming a given LPAD program into a normal Prolog program enriched with an extra argument that carries an auxiliary data structure representing explanations for each subgoal. These explanations are stored as Boolean formulas and efficiently manipulated using Binary Decision Diagrams (BDDs) or, in specialized modes, as simple numeric aggregates.

The transformed program is executed on the XSB Prolog engine, which provides tabling (memoisation of subgoal calls and answers) and a powerful answer‑subsumption mechanism. While ordinary tabling only avoids duplicate answers, answer subsumption allows the combination of multiple answers according to a user‑defined semi‑lattice operation (join). In the probabilistic setting the join corresponds to logical disjunction of explanations (implemented as BDD OR); in the restricted PLP mode (PITA(IND,EXC)) it reduces to ordinary addition or max, and in the possibilistic mode (PITA(POSS)) the join is the maximum of necessity degrees.

PITA is parameterised into three main configurations:

1. PITA(IND,EXC) – mimics PRISM’s independence and exclusivity assumptions. It removes non‑discriminating arguments, aggregates explanations by simple addition, and supports Viterbi‑style most‑probable‑explanation computation. Experiments on Hidden Markov Model (HMM) benchmarks show that PITA(IND,EXC) matches or outperforms PRISM, especially on repeated‑sequence queries where it is significantly faster.

2. PITA(COUNT) – counts the number of distinct explanations for a subgoal, providing a “explanation‑count” query mode useful when the exact probability is not required but the combinatorial richness of explanations matters.

3. PITA(POSS) – implements possibilistic logic programming (CPL1). Each clause carries a lower bound on its necessity degree; answer subsumption takes the maximum of necessity values, and inference follows the sound rules of possibilistic reasoning (minimum for conjunction, maximum for fusion). The implementation scales well to large possibilistic knowledge bases, delivering fast necessity‑degree queries.

The authors detail the formal semantics of LPADs, the construction of composite choices, selections, and worlds, and show how the probability of a query reduces to the probability of a DNF formula—a known #P‑hard problem. By representing the DNF with BDDs, PITA achieves dynamic‑programming‑style evaluation in polynomial time with respect to the BDD size.

The paper also reviews tabling and answer subsumption in XSB, explaining how stratified programs can safely use semi‑lattice joins, and how mode‑directed tabling in B‑Prolog can be seen as a special case. For function‑free LPADs, the left‑to‑right dynamic stratification guarantees termination of the tabling process.

Experimental evaluation covers three axes: (a) comparison with PRISM on HMM queries, demonstrating comparable or superior runtime; (b) Viterbi path extraction, showing that PITA’s Viterbi extension incurs negligible overhead; (c) possibilistic reasoning benchmarks, where PITA(POSS) remains linear in the number of clauses and outperforms existing possibilistic engines.

Implementation-wise, PITA is shipped as a package in XSB version 3.3, requiring only a declarative table/answer‑subsumption directive to activate. The transformation step runs in linear time relative to program size, and the auxiliary BDD structures grow proportionally to the number of distinct explanations, which is manageable for many real‑world datasets.

In conclusion, the paper demonstrates that by leveraging tabling with answer subsumption, a single Prolog‑based engine can efficiently support a spectrum of uncertainty formalisms—from full distribution‑semantics PLP to restricted PRISM‑style models and to possibilistic logic—offering a versatile, scalable solution for researchers and practitioners dealing with probabilistic and possibilistic knowledge representation.