On the Hardness of Approximate Reasoning

Many AI problems, when formulated, reduce to evaluat ing the probabi l i ty that a preposit ional expression is t rue. In this paper we show tha t this problem is computat ional ly int ractable even in surpr is ingly restricted cases and even if we settle for an approx imat ion to this probabi l i ty . We consider various methods used in approximate reasoning such as comput ing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfact ion and knowledge compi la t ion , that use approx imat ion to avoid computat iona l diff iculties, and reduce them to model-enumerat ion problems over a proposi t ional domain We prove that count ing satisfying assignments of proposi t ional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely l imi ted. Th is should he contrasted wi th the case of deductive reasoning, where Horn theories and theories w i th binary clauses are distinguished by the existence of l inear t ime satisf iabi l i ty al gor i thms. Wha t is even more surprising is that , as we show, even approx imat ing the number of sat isfy ing assignments (i.e., "approx imat ing" approx imate reasoning), is intractable for most of these restricted theories. We also ident i fy some restricted classes oi proposi t ional formulae for which we develop efficient a lgor i thms for count ing satisfying as-