Disproving Termination of O and S-like Combinators by Tree Automata

Abstract

Smullyan introduced many combinators, including the combinators M, O, and P. Recently, we showed that P is not terminating by a tree automaton using our SAT-based implementation. From observations on the results, we found similarities in the form of the M, O, and P rules, as well as similarities of the terms that disprove their termination. In addition, Curry and Feys introduced a family of combinators S_n. In this paper, we disprove the termination of a family of combinators that, generalizes the combinators M, O, and P, by a family of tree automata. Furthermore, we disprove the termination of the family of the combinators S_n by family of tree automata.