Semënov Arithmetic, Affine {VASS}, and String Constraints

Authors Andrei Draghici , Christoph Haase , Florin Manea

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Andrei Draghici
  • Department of Computer Science, University of Oxford, UK
Christoph Haase
  • Department of Computer Science, University of Oxford, UK
Florin Manea
  • Computer Science Department and Campus-Institut Data Science, Göttingen University, Germany

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Andrei Draghici, Christoph Haase, and Florin Manea. Semënov Arithmetic, Affine {VASS}, and String Constraints. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study extensions of Semënov arithmetic, the first-order theory of the structure ⟨ℕ,+,2^x⟩. It is well-known that this theory becomes undecidable when extended with regular predicates over tuples of number strings, such as the Büchi V₂-predicate. We therefore restrict ourselves to the existential theory of Semënov arithmetic and show that this theory is decidable in EXPSPACE when extended with arbitrary regular predicates over tuples of number strings. Our approach relies on a reduction to the language emptiness problem for a restricted class of affine vector addition systems with states, which we show decidable in EXPSPACE. As an application of our result, we settle an open problem from the literature and show decidability of a class of string constraints involving length constraints.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • arithmetic theories
  • Büchi arithmetic
  • exponentiation
  • vector addition systems with states
  • string constraints


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