On solving cycle-free context-free grammar equivalence problem using numerical analysis

Abstract

Abstract In this paper we consider the problem of cycle-free context-free grammars equivalence. To every context-free grammar there corresponds a system of formal equations. Formally applying the iteration method to this system we obtain the grammar axiom in the form of a formal power series composed of the words generated by the grammar ”multiplied” by the respective ambiguities. We define a transform that attributes a matrix meaning to the system of formal equations and to formal power series: terminal symbols are substituted by matrices and formal sum and product are substituted by the matrix ones. In order to effectively compute the sum of a matrix series we numerically solve the system of matrix equations. We prove distinguishability theorems showing that if two formal power series generated by cycle-free context-free grammars are different, then there exists a matrix substitution such that the sums of the respective matrix series are different. Based on this result, we suggest a procedure that can resolve the problem of equivalence of cycle-free context-free grammars in many practical cases. The results obtained in this paper form a theoretical basis for algorithms oriented to automatic assessment of students’ answers in computer science. We present the respective algorithms. Then we compare our approach with a simple heuristic method based on CYK algorithm and discuss the limitations of our method.