Energy Minimization and the Satisfiability of Propositional Logic

Abstract

Interactive connectionist networks with symmetric weights (like Hopfield nets and Boltzman Machines), use gradient descent to find a minimum for quadratic binary functions called energy functions. We show an equivalence between the problem of satisfiability in propositional calculus and the problem of minimizing those energy functions. The equivalence is in the sense that for any satisfiable Well Formed Formula (WFF) we can find a quadratic function that describes it, such that if “true” and “false” are mapped to “one” and “zero”, respectively, then the set of solutions that minimize the function is equal to the set of models (truth assignments) that satisfy the WFF. We also show that in the same sense every quadratic energy function describes some satisfiable WFF. Algorithms are given to transform any propositional Well Formed Formula into an energy function that describes it and vice versa. Using Sigma-Pi units we define high order energy minimization models that are stable, and we show equivalence between those high order models and quadratic ones. In fact we show that Sigma-Pi units are not needed if additional simple hidden units are added into the network. An algorithm to convert high order energy functions into low order functions is used as a powerful tool to implement a satisfiability problem solver on a connectionist network. The results give better understanding of the role of hidden units and of the limitations and the capabilities of these interactive connectionist models. The techniques developed here for the satisfiability problem may be applied as well to a wide range of other problems.