Abstract
AbstractIn this paper, we investigate the complexity of the emptiness problem for Parikh automata equipped with a pushdown stack. Pushdown Parikh automata extend pushdown automata with counters which can only be incremented and an acceptance condition given as a semi-linear set, which we represent as an existential Presburger formula over the final values of the counters. We show that the non-emptiness problem both in the deterministic and non-deterministic cases is . If the input head can move in a two-way fashion, emptiness gets undecidable, even if the pushdown stack is visibly and the automaton deterministic. We define a restriction, called the single-use restriction, to recover decidability in the presence of two-wayness, when the stack is visibly. This syntactic restriction enforces that any transition which increments at least one dimension is triggered only a bounded number of times per input position. Our main contribution is to show that non-emptiness of two-way visibly Parikh automata which are single-use is NExpTime-c. We finally give applications to decision problems for expressive transducer models from nested words to words, including the equivalence problem.
Highlights
Motivated by applications in transducer theory for well-nested words, we investigate in this article extensions of Parikh automata with a pushdown stack
We study the complexity of the emptiness problem for Parikh automata with a pushdown store
We show that adding a stack can be done for free with respect to the emptiness problem, which remains, as for stackfree Parikh automata, NP-c
Summary
It is possible to define a model of single-use reversal-bounded two-way visibly pushdown counter machines, where the single-useness is put on transitions that modify the counters This model is expressively equivalent to 2VPPAsu in the non-determinstic case, and thanks to our result, has a decidable emptiness problem. A 2VPPA is single-use if for every input w and every accepting run ρ over w, there do not exist two different configurations (p, i, d, σ) and (p, i, d, σ ) with p a producing state, meaning that ρ does not reach any position in the same direction twice in any given state of P This property is a syntaxic restriction of the model. This equivalence implies by Parikh’s Theorem [24], semi-linearity of Val (P ) for any 2VPPAsu P
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