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Documents authored by Vahanwala, Mihir

Document
DOI: 10.4230/LIPIcs.LICS.2026.7
On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic

Authors: Piotr Bacik, Joris Nieuwveld, Joël Ouaknine, Mihir Vahanwala, Madhavan Venkatesh, and Emil Rugaard Wieser

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

Abstract
We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form 2n³-5n+3, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability of the corresponding theory follows from the solvability of hyperelliptic Diophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.
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Piotr Bacik, Joris Nieuwveld, Joël Ouaknine, Mihir Vahanwala, Madhavan Venkatesh, and Emil Rugaard Wieser. On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bacik_et_al:LIPIcs.LICS.2026.7,
  author =	{Bacik, Piotr and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Vahanwala, Mihir and Venkatesh, Madhavan and Wieser, Emil Rugaard},
  title =	{{On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.7},
  URN =		{urn:nbn:de:0030-drops-267947},
  doi =		{10.4230/LIPIcs.LICS.2026.7},
  annote =	{Keywords: Presburger arithmetic, Diophantine equations, decidability, B\"{u}chi’s conjecture}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2026.165
Automata on S-Adic Words

Authors: Valérie Berthé, Toghrul Karimov, and Mihir Vahanwala

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
A fundamental question in logic and verification is the following: for which unary predicates P_1, …, P_k is the monadic second-order theory of ⟨ℕ;<,P_1,…,P_k⟩ decidable? Equivalently, for which infinite words α can we decide whether a given Büchi automaton 𝒜 accepts α? Carton and Thomas showed decidability in the case that α is a fixed point of a letter-to-word substitution σ, i.e., σ(α) = α. However, abundantly more words, e.g., Sturmian words, are characterised by a broader notion of self-similarity that involves a set S of substitutions. A word α is said to be directed by a sequence s = (σ_n)_{n ∈ ℕ} over S if there is a sequence of words (α_n)_{n ∈ ℕ} such that α₀ = α and α_n = σ_n(α_{n+1}) for all n; such α are called S-adic. We study the automaton acceptance problem for such words and prove, among others, the following: given finite S and an automaton 𝒜, we can compute an automaton ℬ that accepts s ∈ S^ω if and only if s directs a word α accepted by 𝒜. Thus we can algorithmically answer questions of the form "Which S-adic words are accepted by a given automaton 𝒜?"
Cite as

Valérie Berthé, Toghrul Karimov, and Mihir Vahanwala. Automata on S-Adic Words. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 165:1-165:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{berthe_et_al:LIPIcs.ICALP.2026.165,
  author =	{Berth\'{e}, Val\'{e}rie and Karimov, Toghrul and Vahanwala, Mihir},
  title =	{{Automata on S-Adic Words}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{165:1--165:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.165},
  URN =		{urn:nbn:de:0030-drops-265534},
  doi =		{10.4230/LIPIcs.ICALP.2026.165},
  annote =	{Keywords: Sturmian words, S-adic words, automata theory, word combinatorics}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.17
Robust Positivity Problems for Linear Recurrence Sequences: The Frontiers of Decidability for Explicitly Given Neighbourhoods

Authors: Mihir Vahanwala

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
Linear Recurrence Sequences (LRS) are a fundamental mathematical primitive for a plethora of applications such as the verification of probabilistic systems, model checking, computational biology, and economics. Positivity (are all terms of the given LRS non-negative?) and Ultimate Positivity (are all but finitely many terms of the given LRS non-negative?) are important open number-theoretic decision problems. Recently, the robust versions of these problems, that ask whether the LRS is (Ultimately) Positive despite small perturbations to its initialisation, have gained attention as a means to model the imprecision that arises in practical settings. However, the state of the art is ill-equipped to reason about imprecision when its extent is explicitly specified. In this paper, we consider Robust Positivity and Ultimate Positivity problems where the neighbourhood of the initialisation, expressed in a natural and general format, is also part of the input. We contribute by proving sharp decidability results: decision procedures at orders our techniques are unable to handle for general LRS would entail significant number-theoretic breakthroughs.
Cite as

Mihir Vahanwala. Robust Positivity Problems for Linear Recurrence Sequences: The Frontiers of Decidability for Explicitly Given Neighbourhoods. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{vahanwala:LIPIcs.FSTTCS.2023.17,
  author =	{Vahanwala, Mihir},
  title =	{{Robust Positivity Problems for Linear Recurrence Sequences: The Frontiers of Decidability for Explicitly Given Neighbourhoods}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{17:1--17:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.17},
  URN =		{urn:nbn:de:0030-drops-193909},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.17},
  annote =	{Keywords: Dynamical Systems, Verification, Robustness, Linear Recurrence Sequences, Positivity, Ultimate Positivity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.5
On Robustness for the Skolem and Positivity Problems

Authors: S. Akshay, Hugo Bazille, Blaise Genest, and Mihir Vahanwala

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: The best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variant. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighborhood is given as part of the input, then robust Skolem and robust positivity are Diophantine-hard, i.e., solving either would entail major breakthrough in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine-hardness result on a variant of the Skolem problem, to the best of our knowledge. On the other hand, if one asks whether such a neighborhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations.
Cite as

S. Akshay, Hugo Bazille, Blaise Genest, and Mihir Vahanwala. On Robustness for the Skolem and Positivity Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akshay_et_al:LIPIcs.STACS.2022.5,
  author =	{Akshay, S. and Bazille, Hugo and Genest, Blaise and Vahanwala, Mihir},
  title =	{{On Robustness for the Skolem and Positivity Problems}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.5},
  URN =		{urn:nbn:de:0030-drops-158156},
  doi =		{10.4230/LIPIcs.STACS.2022.5},
  annote =	{Keywords: Skolem problem, verification, dynamical systems, robustness}
}
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