Decision procedures for families of deterministic pushdown automata

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Abstract

The existence and complexity of decision procedures for
families of deterministic pushdown automata are investigated, with
special emphasis on positive decidability results for those questions,
such as equivalence, which are known to become undecidable when the
deterministic restriction is removed.
The equivalence problem is proved decidable for the following
three deterministic families, all of which are already extensive
enough to have undecidable inclusion problems:
(a) nonsingular automata - a realtime subfamily, which extends
the largest corresponding classes with previously known
equivalence tests,
(b) finite-turn automata - characterised by having a bound on
the number of times the direction of the stack movement can
change, and
(c) one-counter automata - defined by restricting the stack
alphabet to just one symbol.
The problem of whether a language defined by a machine in one
family, can be recognised by one in another, is a convenient
formulation of numerous decidable or potentially decidable questions.
We show that such questions as whether a deterministic context-free
language can be recognised by a machine in any one of the above
named classes, must be, if decidable at all, at least as difficult
to decide as whether such a language is regular. We re-examine the
regularity test of Stearns, and obtain an improved algorithm. We do this by reducing by an exponential order the upper bound on the
possible state complexity of regular sets recognised by deterministic
pushdown automata of a given size, to a level close to one known
to be achievable.
We pursue an application of this analysis to a schema theoretic
problem. We consider the succinctness with which certain functional
schemas can be used to express equivalent large flowchart schemas, and
obtain closely matching upper and lower bounds for a measure of this.

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