Probabilistic Logic Programming

We present a new approach to probabilistic logic pro- grams with a possible worlds semantics. Classical program clauses are extended by a subinterval of that describes the range for the conditional probability of the head of a clause given its body. We show that deduction in the defined probabilistic logic pro grams is computationally more complex than deduction in classical logic programs. More precisely, restricted deduction problems t hat are P- complete for classical logic programs are already NP-hard f or proba- bilistic logic programs. We then elaborate a linear program ming ap- proach to probabilistic deduction that is efficient in inter esting spe- cial cases. In the best case, the generated linear programs h ave a num- ber of variables that is linear in the number of ground instan ces of purely probabilistic clauses in a probabilistic logic prog ram. As a second contribution, we show that deduction in probabilistic logic programs is computationally more complex than deduction in classical logic programs: restricted deduction problems t hat are P- complete for classical logic programs are already NP-hard f or prob- abilistic logic programs. Hence, any attempt towards effici ent deduc- tion in probabilistic logic programs should be guided by looking for efficient special-case, average-case, or approximation te chniques. As a third contribution, by generalizing own work from (18), we elaborate a linear programming approach to deduction in probabilis- tic logic programs, which is efficient in interesting specia l cases. In our framework, probabilistic deduction problems can easil y be rep- resented by linear programs. However, these initial linear programs have a number of variables that is exponential in the cardina lity of the Herbrand base. Moreover, also the Herbrand base of a probabilistic logic program is generally quite large. Motivated by this ob servation, we elaborate a technique that, in the best case, yields linea r programs with a number of variables that is linear in the number of ground in- stances of purely probabilistic clauses. This result is ver y promising. The work in (22), in contrast, is based on solving an exponential number of linear programs over an exponential number of variables (both in the cardinality of the Herbrand base) in each fixpoin t itera- tion step and in each compilation step for SLDp-refutation. The rest of this paper is organized as follows. In Section 2, w e present the syntax and the semantics of probabilistic logic programs and of queries addressed to them. Section 3 gives an illustra tive ex- ample. Section 4 concentrates on the computational complexity of probabilistic deduction in our framework. In Sections 5 and 6, we present and discuss an optimized linear programming approach to probabilistic deduction. Section 7 summarizes the main results.