Computation-oriented reductions of predicate to propositional logic

摘要

We give a sequential reduction technique for transforming a formula of the pure predicate calculus to a seq́uence of ‘increasingly accurate’ propositional logic ‘approximations’ to the predicate formula. At each stage of approximation, one of the following three cases occurs: (1) the approximation shows the predicate formula to be satisfiable; or (2) the approximation shows the predicate formula to be unsatisfiable; or (3) neither (1) nor (2) occurs, and a ‘more accurate’ refinement of the current propositional ‘approximation’ is obtained. The overall reduction technique is finite for the ∀-, ∃-, and ∃∀ satisfiability fragments of pure logic. In fact, it achieves the proven complexity bounds of Lewis [38] for these fragments, where nondeterminism is replaced by exponentiation. Since discrete programming and mixed integer programming can be used to treat propositional logic, this sequential reduction technique opens further possibilities for applications of mathematical programming to logic-based methods of decision support. In this respect, there is particular interest in the combination of logic with the nonlogical side constraints typical in mathematical programming, which are not efficiently handled by logic alone (e.g., material balances, capacity restrictions, demand requirements, etc.).