Deciding Properties of Nonregular Programs

Abstract

Extensions of propositional dynamic logic (PDL) with nonregular programs are considered. Three classes of nonregular languages are defined, and for each of them it is shown that for any language L in the class, PDL, with L added to the set of regular programs as a new program, is decidable. The first class consists of the languages accepted by pushdown automata that act only on the basis of their input symbol, except when determining whether they reject or continue. The second class (which contains even noncontext-free languages) consists of the languages accepted by deterministic stack machines, but which have a unique new symbol prefixing each word. The third class represents a certain delicate combination of these, and, in particular, it serves to prove the 1983 conjecture that PDL with the addition of the language $\{ {a^i b^i c^i |i \geqslant 0} \}$ is decidable.