Symmetric Neural Networks and Propositional Logic Satisfiability

Abstract

St. LOUJS, MO 63230 U S AConnectionist networks with symmetric weights (like Hopfield networks and Boltzmann Machines) use gradient descent to find a minimum for quadratic 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 the set of solutions that minimizes the function is equal to the set of 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 WFF into an energy function that describes it and vice versa.High-order models that use sigma-pi units are shown to be equivalent to the standard quadratic models with additional hidden units. An algorithm to convert high-order networks to low-order ones is used 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 capabilities of symmetric connectionist models. The techniques developed for the satisfiability problem may be applied to a wide range of other problems, such as associative memories, finding maximal consistent subsets, automatic deduction, and even nonmonotonic reasoning.

Discrete Hopfield Neural Network in Restricted Maximum k-Satisfiability Logic Programming

“…Thus, the energy value is vital to separate local minimum and global minimum solution. Global minimum energy supposed to be can be pre-calculated because the total magnitude of the energy that corresponds to MAX-kSAT clauses is always constant (Pinkas 1991;Wan Abdullah 1993). The retrieval power of HNN always depends on how the synaptic weights are computed.…”

“…Thus, the logic programming can be interpreted as a problem in combinatorial optimization standpoint. Pinkas (1991) expanded the idea of logic program by integrating the competent propositional knowledge or logical mapping system via symmetric connectionist network. Hence, the proposed symmetric connectionist network (SCN) has attracted researchers to revive many domains of artificial neural network such as Hopfield Neural Network, Boltzmann Machine, Harmony Theory and Mean Field Theory.…”

Performance analysis of activation function in higher order logic programming

“…We use neurons to store the truth value of atoms to write a cost function for minimization when all the clauses are satisfied. In addition, a bi-directional mapping between propositional logic formulas and energy functions of symmetric neural networks had defined by Gadi Pinkas [6,7] and Wan Abdullah [4]. We show below how logic programming can be interpreted as a combinatorial optimization problem and implemented on a neural network.…”

“…In addition, the optimization of logical inconsistency is carried out by the network after the connection strengths are defined from the higher order logic program [5]. Consequently, Pinkas [6,7] emphasized on a bi-directional mapping between the prepositional logic formulas and Hopfield network by integrating the energy minimization scheme to achieve global convergence. In this paper, we will ponder and analyze the performance of doing higher order logic programming in Hopfield network.…”

Activation function comparison in neural-symbolic integration

“…Basically, logic programming can be treated as a problem in combinatorial optimization standpoint [3,7]. Therefore, it can be carried out in a neural network to obtain desired solutions.…”