Some Decision Problems Concerning NPDAs, Palindromes, and Dyck Languages

Abstract

We address several types of decision questions related to context-free languages when an NPDA is given as input. First we consider the question of whether the NPDA makes a bounded number of stack reversals (over all accepting inputs) and show that this problem is undecidable even when the NPDA is only 2-ambiguous. We consider the same problem for counter machines (i.e., whether the counter makes a bounded number of reversals) and show that it is also undecidable. On the other hand, we show that the problem is decidable for unambiguous NPDAs even when augmented with reversal-bounded counters. Next, we look at problems of equivalence, containment and disjointness with fixed languages. With the fixed language L ₀ being one of the following: P = $\{ x \# x^r \ |$ x ∈ (0 + 1)^* }, P _u = $\{ x x^r \ |$ x ∈ (0 + 1)^* }, D _k = Dyck language with k-type of parentheses, or S _k = two-sided Dyck language with k types of parentheses, we consider problems such as: ‘Is L(M) ∩ L ₀ = ∅?’, ‘Is L(M) ⊆ L ₀?’, or ‘Is L(M) = L ₀?’, where M is an input NPDA (or a restricted form of it). For example, we show that the problem, ‘Is L(M) ∩ P?’, is undecidable when M is a deterministic one-counter acceptor, while the problem ‘Is L(M) ⊆ P?’ is decidable even for NPDAs augmented with reversal-bounded counters. Another result is that the problem ‘Is L(M) ⊆ P _u ?’ is decidable in polynomial time for M an NPDA. We also show several other related decidability and undecidability results.