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

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Summary

Introduction

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

Basic definitions
Distinguishability
Reachability is EXPTIME-solvable
Reachability is EXPTIME-hard
Global reachability
Higher-order pushdown systems
Simulation of k-PA without constants
Simulation of k-PA with constants
Conclusion
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