Semënov Arithmetic, Affine VASS, and String Constraints (2306.14593v1)
Abstract: We study extensions of Sem\"enov arithmetic, the first-order theory of the structure $(\mathbb{N}, +, 2x)$. It is well-knonw that this theory becomes undecidable when extended with regular predicates over tuples of number strings, such as the B\"uchi $V_2$-predicate. We therefore restrict ourselves to the existential theory of Sem\"enov 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 results, we settle an open problem from the literature and show decidability of a class of string constraints involving length constraints.
- Norn: An SMT solver for string constraints. In Proc. Computer Aided Verification, CAV, volume 9206 of Lect. Notes Comp. Sci., pages 462–469, 2015. doi:10.1007/978-3-319-21690-4_29.
- The complexity of Presburger arithmetic with power or powers. In International Colloquium on Automata, Languages, and Programming, ICALP, 2023. To appear. URL: https://doi.org/10.48550/arXiv.2305.03037.
- Towards more efficient methods for solving regular-expression heavy string constraints. Theor. Comput. Sci., 943:50–72, 2023. doi:10.1016/j.tcs.2022.12.009.
- An SMT solver for regular expressions and linear arithmetic over string length. In Computer Aided Verification, CAV, volume 12760 of Lect. Notes Comp. Sci., pages 289–312, 2021. doi:10.1007/978-3-030-81688-9_14.
- Automatic structures. In Logic in Computer Science, LICS, pages 51–62. IEEE Computer Society, 2000. doi:10.1109/LICS.2000.855755.
- Logic and p𝑝pitalic_p-recognizable sets of integers. Bull. Belg. Math. Soc. Simon Stevin, 1(2):191–238, 1994. doi:10.36045/bbms/1103408547.
- On extensions of Presburger arithmetic. In Proc. 4th Easter Model Theory conference, Gross Köris, pages 17–34, 1986.
- Reachability in register machines with polynomial updates. In Proc. Mathematical Foundations of Computer Science, MFCS, volume 8087 of Lect. Notes Comp. Sci., pages 409–420. Springer, 2013.
- On the existential theories of Büchi arithmetic and linear p𝑝pitalic_p-adic fields. In Proc. Symposium on Logic in Computer Science, LICS, pages 1–10, 2019. doi:10.1109/LICS.2019.8785681.
- Three lectures on automatic structures. In Proc. of Logic Colloquium, volume 35, pages 132–176, 2007.
- Automatic presentations of structures. In Logical and Computational Complexity, LCC, volume 960 of Lect. Notes Comp. Sci., pages 367–392. Springer, 1995. doi:10.1007/3-540-60178-3_93.
- Françoise Point. On the expansion of (ℕ,+,2x)ℕsuperscript2𝑥(\mathbb{N},+,2^{x})( blackboard_N , + , 2 start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) of Presburger arithmetic. Unpublished manuscript, 2010.
- Julien Reichert. Reachability games with counters: decidability and algorithms. PhD thesis, École normale supérieure de Cachan-ENS Cachan, 2015.
- A. L. Semënov. On certain extensions of the arithmetic of addition of natural numbers. Mathematics of the USSR-Izvestiya, 15(2):401, 1980. doi:10.1070/IM1980v015n02ABEH001252.
- On the construction of automata from linear arithmetic constraints. In Proc. Tools and Algorithms for the Construction and Analysis of Systems, TACAS, pages 1–19, 2000. doi:10.1007/3-540-46419-0_1.