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History-Deterministic Vector Addition Systems

Authors Sougata Bose , David Purser , Patrick Totzke

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  • Theory of computation → Formal languages and automata theory
Keywords
  • Vector Addition Systems
  • History-determinism
  • Good-for Games

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Abstract

We consider history-determinism, a restricted form of non-determinism, for Vector Addition Systems with States (VASS) when used as acceptors to recognise languages of finite words. History-determinism requires that the non-deterministic choices can be resolved on-the-fly; based on the past and without jeopardising acceptance of any possible continuation of the input word.
Our results show that the history-deterministic (HD) VASS sit strictly between deterministic and non-deterministic VASS regardless of the number of counters. We compare the relative expressiveness of HD systems, and closure-properties of the induced language classes, with coverability and reachability semantics, and with and without ε-labelled transitions.
Whereas in dimension 1, inclusion and regularity remain decidable, from dimension two onwards, HD-VASS with suitable resolver strategies, are essentially able to simulate 2-counter Minsky machines, leading to several undecidability results: It is undecidable whether a VASS is history-deterministic, or if a language equivalent history-deterministic VASS exists. Checking language inclusion between history-deterministic 2-VASS is also undecidable.

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Sougata Bose, David Purser, and Patrick Totzke. History-Deterministic Vector Addition Systems. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CONCUR.2023.18

Author Details

Sougata Bose
  • University of Liverpool, UK
David Purser
  • University of Liverpool, UK
Patrick Totzke
  • University of Liverpool, UK

Funding

We gratefully acknowledge support from the EPSRC, project number EP/V025848/1 and the Royal Society, IES\R3\203109.

Acknowledgements

The authors thank Ismaël Jecker, Piotr Hofman and Filip Mazowiecki for fruitful discussions.

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