A Continuous Valued Neural Network with a New Evaluation Function of Degree of Unsatisfaction for Solving CSP

We have proposed a neural network called the Lagrange programming neural network with polarized high-order connections (LPPH) for solving the satisfiability problem (SAT) of propositional calculus. The LPPH has gradient descent dynamics for variables and gradient ascent dynamics for Lagrange multipliers, which represent the weights of the clauses of the SAT. Each weight wr increases according to the degree of unsatisfaction of clauseCr. This causes changes in the energy landscape of the Lagrangian function, on which the values of the variables change in the gradient descent direction. It was proved that the LPPH is not trapped by any point that is not a solution of the SAT. Experimental results showed that the LPPH can find solutions faster than existing methods. In the LPPH dynamics, a function hr(x) calculates the degree of unsatisfaction of clause Cr via multiplication. However, this definition of hr(x) has a disadvantage when the number of literals in a clause is large. In the present paper, we propose a new definition of hr(x) in order to overcome this disadvantage using the “min” operator. In addition, we extend the LPPH to solve the constraint satisfaction problem (CSP). Our neural network can update all neurons simultaneously to solve the CSP. In contrast, conventional discrete methods for solving the CSP must update variables sequentially. This is advantageous for VLSI implementation. key words: neural network, constraint satisfaction problem, Lagrangian method