On the Relationship between Probabilistic Logic and CMS P.HANSEN GERAD and Dept. Methodes Quantitatives de Gestion Ecole des Hautes Etudes Commercials pierreh @crt.umontreal.ca Abstract We discuss the relationship between probabilis- tic logic and CMS. Given a set of logical sen- tences and their probabilities of being true, the outcome of a probabilistic logic system con- sists of lower and upper bounds on the prob- ability of an additional sentence to be true. These bounds are computed using a linear pro- gramming formulation. In -CMS systems, the outcome is defined by the probabilities of the support and the plausibility (with the assump- tions being independent) after a first phase which consists of computing the prime impli- cants depending only on the variables of the assumptions. We propose to reformulate a CMS system without independence conditions on the assumptions, using the linear program- ming framework of probabilistic logic, and show how to exploit its particular structure to solve it efficiently. When an independence condition is imposed on the assumptions the two systems give different results. Comparisons are made on small problems using the assumption-based ev- idential language program (ABEL) of [Anrig et a/., 1998] and the PSAT program of [Jaumard et a/., 1991]. 1 Introduction Many different models have been proposed for reasoning under uncertainty in expert systems. Among those based on the use of logic and probability theory which require a moderate amount of input data, two important families are the probabilistic Clause Maintenance Systems (TT- CMS) [Reiter and de Kleer, 1987; deKleer and Williams, 1987], and the probabilistic logic models (Nilsson, 1986; Jaumard et ol., 1991; Hansen et a/., 1995]. While these families have been developed separately, they have much in common, with however the difference that models of the former family usually suppose independence of their assumptions, while those of the latter do not. In this paper, we explore -CMS systems, with and without in- dependence of the assumptions, from the probabilistic logic viewpoint. The main results are the following: (i) when relaxing the independence condition, two formula- tions can be proposed for -CMS systems; (ii) the sec- ond formulation suggests an algorithm which exploits the particular structure of -CMS and is, on large instances, 7000 times faster than the column generation algorithm for probabilistic logic of (Jaumard et a/., 1991] applied to the first formulation; (iii) with the independence con- dition, probability intervals of an additional proposition obtained by CMS may overlap with those of the corre- sponding probabilistic logic model, which suggests that CMS systems do not exploit the available information to its fullest degree. The paper is organized as follows: probabilistic logic models and CMS systems are briefly reviewed in the next two sections. The two probabilistic logic models for CMS systems are presented in Section 4. Algorithms are discussed in Section 5 and computational results in Section 6. Brief conclusions are drawn in the last section. 2 Probabilistic Logic Nilsson [Nilsson, 1986] has presented a semantical gen- eralization of propositional logic in which the truth val- ues of sentences are probability values (probabilistic logic, also called probabilistic satisfiability or PSAT for short). Let X denote a set of propositional variables and S a set of propositional sentences over X defined with the usual operators (dis- junction), (conjunction), and -. (negation). A literal is a propositional variable or its negation Propositional sentences are assumed to be written using a DNF (Disjunctive Normal Form) expres- sion in this paragraph. Let be a valuation for 5, where is equal to 1 if has value true, and to 0 otherwise, is a possible world if there exists a truth assignment over X which leads to w over S and it is an impossible world otherwise. Observe that two different truth assignments on X may lead to the same possible world. Let W denote the set of possible worlds and set to | W\ (note that AUTOMATED REASONING B. JAUMARD GERAD and Dept. Math, and hid. Eng. Ecole Polytechnique de Montreal brigittOcrt.umontreal.ca A. D. PARHEIRA Dept. Math, and Ind. Eng. Ecole Polytechnique de Montreal anderson@crt.umontreal.ca