Deterministic Pushdown Automata Research Articles

On the recursion-theoretic complexity of relative succinctness of representations of languages

In Hartmanis (1980) a simple proof is given of the fact (originally proved in Valiant (1976)) that the relative succinctness of representing deterministic context-free languages by deterministic vs. nondeterministic pushdown automata is not recursively bounded, in the following sense: there is no recursive function which, for deterministic context-free languages L, can bound the size of the minimal deterministic pushdown automaton accepting L as a function of the size of a nondeterministic pushdown automaton accepting L. It is then stated that ...even if we would know (or be given) which pushdown automata describe deterministic languages, we still could not effectively write down the corresponding deterministic pushdown automata because of their enormous size which grows nonrecursively in the size of the nondeterministic pushdown This does not, however, rule out a priori the possibility that a partial recursive bound might exist, as a function of the description of the nondeterministic pushdown automaton rather than of its size; indeed, the proof in Hartmanis (1980) uses the fact that the bounding function is total. It is the purpose of this note to make a case for using partial bounding functions in questions of relative succinctness. It will be shown that for the examples considered in Hartmanis (1980) (deterministic context-free languages, unambiguous languages, regular languages), the best possible partial bound as a function of the description of the unrestricted automata, while still nonrecursive, has lower recursiontheoretic complexity than the best possible bound as a functions of the size of the unrestricted automata. An example wi l l be given of a class of languages for which there exists a partial recursive bound on the size of the restricted automata, again as a function of the description of the unrestricted automata, while no (total) recursive bound exists as a function of the size of the unrestricted automata.

An effective simulation of deterministic pushdown automata with many two-way and one-way heads

An effective simulation of deterministic pushdown automata with many two-way and one-way heads

A simplicity test for deterministic pushdown automata

A context-free language is said to be simple if it is accepted by a single-state deterministic pushdown automaton with empty stack acceptance. This paper proves that it is decidable whether a deterministic pushdown automaton (dpda) accepts a simple language. For this purpose, we prove that it is decidable whether a real-time dpda with empty stack acceptance accepts a simple language. To prove this, we present two conditions called nonsingularity and separability conditions and show that for a real-time dpda M with empty stack acceptance, (i) the language accepted by M is simple if and only if M satisfies the two conditions, and (ii) it is decidable whether M satisfies the two conditions.

A real-time strictness test for deterministic pushdown automata

A context-free language is said to be real-time strict if it is accepted by a real-time deterministic pushdown automaton with empty stack acceptance. This paper proves that there exists an effective procedure for deciding whether the language accepted by a deterministic pushdown automaton (dpda) is real-time strict. The result implies the following: For any subclass ℰ of real-time dpda's with empty stack acceptance, the problem for deciding whether a dpda is equivalent to some machine in ℰ reduces to that for deciding whether a real-time dpda with empty stack acceptance is equivalent to some machine in ℰ.

Two decidability results for deterministic pushdown automata

A real-time deterministic pushdown automaton is said to be stack uniform if for each input letter a, every a-tule has the same effect on the length of the pushdown store, i.e., if ( s 1 , A) → a ( S 2, w) and ( s 1′ , A′) → a ( s 2′ , w′) are two a-rules, then the lengths of w and w′ are equal. It is shown that the equivalence problem for stack uniform automata and the inclusion problem for stack uniform automata with empty store acceptance are decidable.

A note on non-singular deterministic pushdown automata

The non-singular deterministic pushdown automata were first defined by Valiant as an example of a class of machines with a decidable equivalence problem [3]. No algorithm currently exist for deciding whether or not a deterministic pushdown automation is non-singular, so the applicability of Valiant's equivalence decision procedure cannot be readily (if ever) determined. In this paper, it is shown that the equivalence problem for non-singular automata is reducible to the problem of deciding whether or not a deterministic pushdown automaton is non-singular.

Polynomial Algorithms for Deterministic Pushdown Automata

An algorithm is presented for converting a deterministic pushdown automaton (dpda) of size n into an equivalent dpda that always halts. The dpda produced is of size $O(n)$. The algorithm operates in linear time on a random access machine (but may require the allocation of $O(n^2 )$ storage), and in time $O(n^2 )$ on a multi-tape Turing machine. Related results on polynomial time algorithms for dpda equivalence problems and for two-way pushdown automata language recognition problems are discussed.

Some open problems in the theory of computation as questions about two-way deterministic pushdown automaton languages

We considered some of the important unsolved problems in the theory of computation concerning the relationship between deterministic and nondeterministic computations, and between tape and time bounded computations. For each such problem we find an equivalent problem concerning two-way deterministic pushdown automaton languages.

A note on the succinctness of descriptions of deterministic languages

It is shown that the relative succinctness that may be achieved by describing deterministic context-free languages by general unambiguous grammars rather than by deterministic pushdown automata is not bounded by any recursive function.

A result on the equivalence problem for deterministicpushdown automata

This paper shows that it is decidable whether or not two deterministic pushdownautomata, one of which is nonsingular (introduced by L. G. Valiant), are equivalent (in the sense that they accept the same languages by empty stack). This is an extension of Valiant's result that the equivalence of two nonsingular deterministic pushdown automata is decidable.

Syntactic analysis of CF-languages by deterministic pushdown automata

Syntactic analysis of CF-languages by deterministic pushdown automata

A multidimensional approach to syntactic pattern recognition

This paper describes a syntactic method for representing the primitive parts of a pattern as nodes of a type of directed graph. A linear representation of the digraph can then be presented to a regular unordered tree automaton for classification. Regular unordered tree automata can be simulated by deterministic pushdown automata, so this procedure can be implemented easily. Regular u-tree automata and the corresponding generative systems, regular u-tree grammars are formally defined. Several results are shown which are applicable to all syntactic pattern recognition schemes involving the use of primitives.

Real-Time Strict Deterministic Languages

The family of strict deterministic languages has been studied for its theoretical properties and applications to parsing. In particular, these languages have been shown to be precisely the prefix-free deterministic languages. Deterministic pushdown automata are called (quasi-)real time if they have no (only a bounded number of consecutive) null moves. It is shown that for strict deterministic languages, the quasi-real-time and real-time constraints are equivalent (except for $\{ \Lambda \} $). A grammatical characterization of these languages is also given. For quasi-real-time strict deterministic languages, an easy and elegant decision method is given for testing regularity. For all known methods of accepting deterministic languages, it is shown that the families of real-time languages are a proper subset of the full families. A relation is established among these sets, the simple deterministic languages, and some hierarchies.