Probabilistic Logic of k-Valued Formulaic Events Without Combinatorial Explosion

In a finite-valued k-valued propositional calculus, the truth table of a formula containing n variables consists of kⁿ rows, which leads to an exponential growth in the number of possible interpretations and gives rise to the problem of combinatorial explosion. In works on probabilistic inference and artificial intelligence (Pearl; Russell and Norvig), it is shown that as the number of variables and dependencies increases, the computational complexity of logical and probabilistic inference grows rapidly. For many-valued logical systems, this results in an exponential increase in the number of interpretations and hinders the application of traditional truth-table methods in intelligent systems. This paper proposes a probabilistic logic of events intended for knowledge representation and probabilistic inference in artificial intelligence systems. The proposed approach is based on event algebra and probability theory and makes it possible to significantly reduce the computational complexity associated with combinatorial explosion. The 2-, 3-, and 4-valued logics of formulaic events are constructed without combinatorial explosion, and the general k-valued case is given: for n variables, the table consists of (k-1) (n+k-1) rows instead of kⁿ.