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On First-Order Logic and CPDA Graphs

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Abstract

Higher-order pushdown automata (n-PDA) are abstract machines equipped with a nested ‘stack of stacks of stacks’. Collapsible pushdown automata (n-CPDA) extend these devices by adding ‘links’ to the stack and are equi-expressive for tree generation with simply typed λY terms. Whilst the configuration graphs of HOPDA are well understood, relatively little is known about the CPDA graphs. The order-2 CPDA graphs already have undecidable MSO theories but it was only recently shown by Kartzow (Log. Methods Comput. Sci. 9(1), 2013) that first-order logic is decidable at the second level. In this paper we show the surprising result that first-order logic ceases to be decidable at order-3 and above. We delimit the fragments of the decision problem to which our undecidability result applies in terms of quantifer alternation and the orders of CPDA links used. Additionally we exhibit a natural sub-hierarchy enjoying limited decidability.

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Notes

  1. We are very grateful to Arnaud Carayol for suggesting this reduction.

  2. We will only use reachability properties rather than the full μ-calculus. However, the reader unfamiliar with this logic might wish to consult a survey such as [2].

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Acknowledgements

We are grateful for the helpful feedback from the anonymous reviewers of the related STACS submission [4] as well as the very helpful and thorough reviewers of the present submission. We would also like to thank Arnaud Carayol for his suggestion of a construction for showing that CPDA allowing unconstructible stacks have undecidable first-order theories.

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  1. Institut für Informatik Institution, Technische Universität München, Boltzmannstr. 3, 85748, Munich, Germany

    Christopher H. Broadbent

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Correspondence to Christopher H. Broadbent.

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Work supported by La Fondation Sciences Mathématiques de Paris.

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Broadbent, C.H. On First-Order Logic and CPDA Graphs. Theory Comput Syst 55, 771–832 (2014). https://doi.org/10.1007/s00224-014-9533-0

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