Measuring nondeterminism in pushdown automata

Abstract

The amount of nondeterminism that a pushdown automaton requires to recognize an input string can be measured by the minimum number of guesses that it must make to accept the string, where guesses are measured in bits of information. When this quantity is unbounded, the rate at which it grows as the length of the string increases serves as a measure of the pushdown automaton's "rate of consumption" of nondeterminism. We show that this measure is similar to other complexity measures in that it gives rise to an infinite hierarchy of complexity classes of context-free languages differing in the amount of the resource (in this case, nondeterminism) that they require. In addition, we show that there are context-free languages that can only be recognized by a pushdown automaton whose nondeterminism grows linearly, resolving an open problem in the literature. In particular, the set of palindromes is such a language.

Context-dependent nondeterminism for pushdown automata

“…This measure of nondeterminism has been introduced for finite automata in [5]. In [6,8] it is studied in connection with pushdown automata. In order to be more precise, we continue with the stepwise formalization of pushdown automata with bounded branching.…”

“…We next show that the inclusions are proper. We first consider the language L 0 = ({a n b n c | n ≥ 0} ∪ {a n b 2n c | n ≥ 0}) * which is not in Γ ∪ (DCFL) due to [6] and Theorem 3. Obviously, L 0 ∈ L * which shows the first proper inclusion.…”

“…In contrast to the assertion assume that L is accepted by a (∞, Z 0 )-nPDA M = Q, {a, b, c}, Γ , δ, q 0 , Z 0 , F . For each state q ∈ Q we define the set W q of prefixes of accepted words which reinitialize the pushdown store in state q: W q = {w ∈ {a, b, c} + | (q 0 , w, Z 0 ) + (q, λ, Z 0 ) and there exists v ∈ {a, b, c} * such that (q, v, Z 0 ) * (q f , λ, γ ), for some q f ∈ F, γ ∈ Γ * } In [6,7] it is shown that the language L w = {ww R | w ∈ {a, b} * } does not belong to Γ ∪ (DCFL). By Theorem 3 we obtain that L w is not accepted by any (fin, Z 0 )-nPDA.…”

“…In [6,8] it is studied in connection with pushdown automata. In [8] infinite hierarchies in between the deterministic context-free (DCFL) and context-free languages (CFL) depending on the amount of nondeterminism or on the amount of ambiguity are shown.…”

“…In [8] infinite hierarchies in between the deterministic context-free (DCFL) and context-free languages (CFL) depending on the amount of nondeterminism or on the amount of ambiguity are shown. In [6] lower bounds for the minimum amount of nondeterminism to accept certain context-free languages are established.…”