Another variation on the common subexpression problem

Abstract

In their work from 1980 (to which our title makes reference) Downey et al. (1980) study various cases of the problem of constructing the ‘congruence closure’ of a relation R on a graph (which is the unique extension R' of R on the vertex set V such that R' is an equivalence relation and closed under congruences; the problem is viewed by Kozen (1977) as the word problem for finite algebras and by Nelson and Oppen (1980) as the decision problem for the quantifier-free theory of equality with uninterpreted function symbols). In this work they prove the upper bound O( n log n) for the general case ( n=| V|) and linear upper bounds for various special cases. We give an almost linear algorithm for yet another special case: the case of congruences of finite index (without any acyclicity restrictions), in the notation of Kozen (1977). It is there that the relationship to the finite tree automata of Thatcher and Wright (1968) was pointed out the first time. In fact, the problem reduces to the equivalence problem for finite tree automata. We are able to extend the algorithm by Hopcroft and Karp (1971) for the equivalence problem for deterministic finite automata (dfa's). The approach, the use of certain methods for monadic structures also for nonmonadic ones, is of further-reaching interest. Finally, by showing that Kozen's lower bound for the general problem, logspace-complete for P(not parallelizable), is also valid for the equivalence problem for tree automata, a nice gap between the theories on monadic and nonmonadic structures is established, because Kannelakis and Revesz (1988) show in a recent paper that the equivalence problem for dfa's lies in NC 2.