On a theory of probabilistic deductive databases

Abstract

We propose a framework for modeling uncertainty where both belief and doubt can be given independent, first-class status. We adopt probability theory as the mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of a confidence level, consisting of a pair of intervals of probability, one for each of her belief and doubt. The space of confidence levels naturally leads to the notion of a trilattice, similar in spirit to Fitting's bilattices. Intuitively, the points in such a trilattice can be ordered according to truth, information, or precision. We develop a framework for probabilistic deductive databases by associating confidence levels with the facts and rules of a classical deductive database. While the trilattice structure offers a variety of choices for defining the semantics of probabilistic deductive databases, our choice of semantics is based on the truth-ordering, which we find to be closest to the classical framework for deductive databases. In addition to proposing a declarative semantics based on valuations and an equivalent semantics based on fixpoint theory, we also propose a proof procedure and prove it sound and complete. We show that while classical Datalog query programs have a polynomial time data complexity, certain query programs in the probabilistic deductive database framework do not even terminate on some input databases. We identify a large natural class of query programs of practical interest in our framework, and show that programs in this class possess polynomial time data complexity, i.e. not only do they terminate on every input database, they are guaranteed to do so in a number of steps polynomial in the input database size.

Deductive Algorithmic Knowledge

“…An interesting question is whether this approach can shed light on the epistemic content of probabilistic deductive systems, of the kind found in the recent probabilistic deductive database literature [Lukasiewicz 1999;Lakshmanan and Sadri 2001]. Presumably, the ideas developed by Halpern and Pucella [2005] may be applicable in this setting.…”

On the Relationship between Hybrid Probabilistic Logic Programs and Stochastic Satisfiability

“…A similar relationship between SSAT and other probabilistic logic programming frameworks, e.g., (Ng & Subrahmanian 1992;1993;1994;Dekhtyar & Subrahmanian 2000;Kern-Isberner & Lukasiewicz 2004;Lukasiewicz 1998;Baral, Gelfond, & Rushton 2004;Kersting & Raedt 2000;Lakshmanan & Sadri 2001;Poole 1997;Vennekens, Verbaeten, & Bruynooghe 2004), has not been studied. However, the relationship between the probabilistic logic programming frameworks (Ng & Subrahmanian 1992;1993;1994;Dekhtyar & Subrahmanian 2000;Kern-Isberner & Lukasiewicz 2004;Lukasiewicz 1998) and a different extension to SAT, namely, Probabilistic SAT (PSAT) (Boole 1854) has been studied.…”

“…It was shown that the HPP (Saad & Pontelli 2006) framework is more suitable for reasoning and decision making tasks, including those arising from planning under probabilistic uncertainty (Saad 2007). In addition, it subsumes Lakshmanan and Sadri's (Lakshmanan & Sadri 2001) probabilistic implication-based framework as well as it is a natural extension of classical logic programming with answer set semantics. As a step towards enhancing its reasoning capabilities, the framework of HPP was extended to cope with non-monotonic negation (Saad & Pontelli 2005) by introducing the notion of Normal Hybrid Probabilistic Logic Programs (NHPP) and providing two different semantics namely; stable probabilistic model semantics and wellfounded probabilistic model semantics.…”

Multi-lattices as a Basis for Generalized Fuzzy Logic Programming

“…There are approaches which are based either on the structure of lattice (residuated lattice [4,13] or multi-adjoint lattice [9]), or more restrictive structures, such as bilattices or trilattices [7], or even more general structures such as algebraic domains [11]. One can also find some attempts aiming at weakening the restrictions imposed on a (complete) lattice, namely, the "existence of least upper bounds and greatest lower bounds" is relaxed to the "existence of minimal upper bounds and maximal lower bounds".…”