Probability Aggregates in Probability Answer Set Programming

Probability answer set programming [Saad and Pontelli, 2006; Saad, 2006; Saad, 2007a] is a declarative programming that has been shown effective for representing and reasoning about a variety of probability reasoning tasks [Saad, 2008a; Saad, 2011; Saad, 2007b; Saad, 2009; Saad, 2008b]. However, the lack of probability aggregates, e.g. expected values, in the language of disjunctive hybrid probability logic programs (DHPP) [Saad, 2007a] disallows the natural and concise representation of many interesting problems. In this paper, we extend DHPP to allow arbitrary probability aggregates. We introduce two types of probability aggregates; a type that computes the expected value of a classical aggregate, e.g., the expected value of the minimum, and a type that computes the probability of a classical aggregate, e.g, the probability of sum of values. In addition, we define a probability answer set semantics for DHPP with arbitrary probability aggregates including monotone, antimonotone, and nonmonotone probability aggregates. We show that the proposed probability answer set semantics of DHPP subsumes both the original probability answer set semantics of DHPP [Saad, 2007a] and the classical answer set semantics of classical disjunctive logic programs with classical aggregates [Faber et al., 2010], and consequently subsumes the classical answer set semantics of the original disjunctive logic programs [Gelfond and Lifschitz, 1991]. We show that the proposed probability answer sets of DHPP with probability aggregates are minimal probability models and hence incomparable, which is an important property for nonmonotonic probability reasoning.