Abstract
We investigate reachability in pushdown automata over infinite alphabets. We show that, in terms of reachability/emptiness, these machines can be faithfully represented using only 3r elements of the alphabet, where r is the number of registers. We settle the complexity of associated reachability/emptiness problems. In contrast to register automata, the emptiness problem for pushdown register automata is EXPTIME-complete, independent of the register storage policy used. We also solve the global reachability problem by representing pushdown configurations with a special register automaton. Finally, we examine extensions of pushdown storage to higher orders and show that reachability is undecidable at order 2.
Highlights
Recent years have seen lively interest in automata over infinite alphabets
To ensure the algorithm is a smooth analogy of the saturation algorithm of [5], it manipulates a succinct variant of register automata which we call a register manipulating register automaton (RMRA), but we show that such machines can always be transformed to a machine of the usual kind for at most an exponential increase in size
We show that the reachability problem for higher-order register pushdown automata is undecidable, already at order 2 and with one register
Summary
Recent years have seen lively interest in automata over infinite alphabets. This has largely been driven by applications which, diverse, had a common thread in dealing with alphabets of potentially unbounded size for which finitedomain abstractions were deemed unsatisfactory. Other examples include array-accessing programs [1], which are allowed to use the array to store and compare elements of an unbounded domain, as well as programs with restricted integer parameters [7] Such applications call for a robust theory of automata over infinite alphabets which can lead to an understanding comparable to that of the finite-alphabet setting. We first observe that one can no longer establish a uniform bound on the number of symbols of the infinite alphabet that suffice to represent arbitrary runs The existence of such a bound would imply decidability of the associated reachability problems, but the lack of a bound is not sufficient for establishing undecidability: the decidable class of data automata from [4] contains an automaton that can recognise all words consisting of distinct letters. We show that the reachability problem for higher-order register pushdown automata is undecidable, already at order 2 and with one register
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