10 Search Results for "Kenison, George"

Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2026.2
Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk)

Authors: Laura Kovács

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
With substantial progress in automated reasoning, algebraic approaches emerged to automatically analyse program loops in an exact manner. In this invited talk, we discuss recent results in characterizing the functional behaviour of loops with polynomial arithmetic and probabilistic updates. This problem remains unsolved even when we restrict consideration to loops that are non-nested, without conditionals, and/or without exit conditions [Ehud Hrushovski et al., 2023; Julian Müllner and others, 2024]. We are motivated by applications of computer-aided verification, in particular to assess the safety, security, and sensitivity of computer systems [M. Z. Kwiatkowska et al., 2011; Gilles Barthe et al., 2012; Gilles Barthe and others, 2018; Marcel Moosbrugger et al., 2023; Alessandro Abate et al., 2023; Andrey Kofnov and others, 2024]. We are interested in modeling, deciding, and solving loop analysis. The key to our work are moment-computable loops [L. Kovács, 2008; Marcel Moosbrugger et al., 2022] which allow us to set limits on what is decidable and solvable in loop analysis. Our approach combines algebra, statistics, and automated reasoning to mechanize loop analysis. Various techniques, such as martingale theory and quantifier elimination, can be seen as examples of moment-computable loop analysis. This talk is structured within three inter-connected parts. We first bring moment-based loop analysis into the landscape of {loop invariant synthesis} and extend moment-computable loops with {termination guarantees}. We next automate the reasoning about (probabilistic) loops by summarizing loop semantics as (probabilistic) algebraic recurrences, whose closed-form solutions capture (higher-order) moments, and hence invariants, among loop variables. These recurrences together with loop tests yield moment-based (super)martingales necessary to prove loop termination and compute probability bounds on termination. We finally describe moment-computable loops whose invariant synthesis {decidable} or as {hard} as open problems, such as the Skolem problem [Graham Everest et al., 2003; Terrence Tao, 2008].
Cite as

Laura Kovács. Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk). In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kovacs:LIPIcs.STACS.2026.2,
  author =	{Kov\'{a}cs, Laura},
  title =	{{Moments in Time: Algebraic Analysis for Solvable Loops}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.2},
  URN =		{urn:nbn:de:0030-drops-254910},
  doi =		{10.4230/LIPIcs.STACS.2026.2},
  annote =	{Keywords: program analysis, algebraic reasoning, symbolic computation, loop invariants}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.79
Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space

Authors: Eike Neumann

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We study the problem of deciding whether a point escapes a closed subset of ℝ^d under the iteration of a continuous map f : ℝ^d → ℝ^d in the bit-model of real computation. We give a sound partial decision method for this problem which is complete in the sense that its halting set contains the halting set of all sound partial decision methods for the problem. Equivalently, our decision method terminates on all problem instances whose answer is robust under all sufficiently small perturbations of the function. We further show that the halting set of our algorithm is dense in the set of all problem instances. While our algorithm applies to general continuous functions, we demonstrate that it also yields complete decision methods for much more rigid function families: affine linear systems and quadratic complex polynomials. In the latter case, completeness is subject to the density of hyperbolicity conjecture in complex dynamics. This in particular yields an alternative proof of Hertling’s (2004) conditional answer to a question raised by Penrose (1989) regarding the computability of the Mandelbrot set.
Cite as

Eike Neumann. Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{neumann:LIPIcs.MFCS.2025.79,
  author =	{Neumann, Eike},
  title =	{{Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{79:1--79:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.79},
  URN =		{urn:nbn:de:0030-drops-241866},
  doi =		{10.4230/LIPIcs.MFCS.2025.79},
  annote =	{Keywords: Dynamical Systems, Computability in Analysis, Non-Linear Functions}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.159
The Ultimate Signs of Second-Order Holonomic Sequences

Authors: Fugen Hagihara and Akitoshi Kawamura

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
A real-valued sequence f = {f(n)}_{n ∈ ℕ} is said to be second-order holonomic if it satisfies a linear recurrence f (n + 2) = P (n) f (n + 1) + Q (n) f (n) for all sufficiently large n, where P, Q ∈ ℝ(x) are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of f (n) follow for sufficiently large n. For each P, Q we determine all ultimate signs that f can have, and show how they partition the space of initial values of f. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when P, Q have rational coefficients, f either has an ultimate sign of length 1, 2, 3, 4, 6, 8 or 12, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of f (or tells that there is none) in almost all cases.
Cite as

Fugen Hagihara and Akitoshi Kawamura. The Ultimate Signs of Second-Order Holonomic Sequences. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 159:1-159:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hagihara_et_al:LIPIcs.ICALP.2025.159,
  author =	{Hagihara, Fugen and Kawamura, Akitoshi},
  title =	{{The Ultimate Signs of Second-Order Holonomic Sequences}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{159:1--159:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.159},
  URN =		{urn:nbn:de:0030-drops-235363},
  doi =		{10.4230/LIPIcs.ICALP.2025.159},
  annote =	{Keywords: Holonomic sequences, ultimate signs, Skolem Problem, Positivity Problem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.145
The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

Authors: George Kenison

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, ⟨u_n⟩_{n=0}^∞ is hypergeometric if it satisfies a first-order linear recurrence of the form p(n)u_{n+1} = q(n)u_n with polynomial coefficients p,q ∈ ℤ[x] and u₀ ∈ ℚ. In this paper, we consider the Threshold Problem for hypergeometric sequences: given a hypergeometric sequence ⟨u_n⟩_{n=0}^∞ and a threshold t ∈ ℚ, determine whether u_n ≥ t for each n ∈ ℕ₀. We establish decidability for the Threshold Problem under the assumption that the coefficients p and q are monic polynomials whose roots lie in an imaginary quadratic extension of ℚ. We also establish conditional decidability results; for example, under the assumption that the coefficients p and q are monic polynomials whose roots lie in any number of quadratic extensions of ℚ, the Threshold Problem is decidable subject to the truth of Schanuel’s conjecture. Finally, we show how our approach both recovers and extends some of the recent decidability results on the Membership Problem for hypergeometric sequences with quadratic parameters.
Cite as

George Kenison. The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 145:1-145:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kenison:LIPIcs.ICALP.2024.145,
  author =	{Kenison, George},
  title =	{{The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{145:1--145:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.145},
  URN =		{urn:nbn:de:0030-drops-202882},
  doi =		{10.4230/LIPIcs.ICALP.2024.145},
  annote =	{Keywords: Threshold Problem, Membership Problem, Hypergeometric Sequences}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.41
Linear Loop Synthesis for Quadratic Invariants

Authors: S. Hitarth, George Kenison, Laura Kovács, and Anton Varonka

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
Invariants are key to formal loop verification as they capture loop properties that are valid before and after each loop iteration. Yet, generating invariants is a notorious task already for syntactically restricted classes of loops. Rather than generating invariants for given loops, in this paper we synthesise loops that exhibit a predefined behaviour given by an invariant. From the perspective of formal loop verification, the synthesised loops are thus correct by design and no longer need to be verified. To overcome the hardness of reasoning with arbitrarily strong invariants, in this paper we construct simple (non-nested) while loops with linear updates that exhibit polynomial equality invariants. Rather than solving arbitrary polynomial equations, we consider loop properties defined by a single quadratic invariant in any number of variables. We present a procedure that, given a quadratic equation, decides whether a loop with affine updates satisfying this equation exists. Furthermore, if the answer is positive, the procedure synthesises a loop and ensures its variables achieve infinitely many different values.
Cite as

S. Hitarth, George Kenison, Laura Kovács, and Anton Varonka. Linear Loop Synthesis for Quadratic Invariants. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hitarth_et_al:LIPIcs.STACS.2024.41,
  author =	{Hitarth, S. and Kenison, George and Kov\'{a}cs, Laura and Varonka, Anton},
  title =	{{Linear Loop Synthesis for Quadratic Invariants}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{41:1--41:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.41},
  URN =		{urn:nbn:de:0030-drops-197512},
  doi =		{10.4230/LIPIcs.STACS.2024.41},
  annote =	{Keywords: program synthesis, loop invariants, verification, Diophantine equations}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2023.4
Algebraic Reasoning for (Un)Solvable Loops (Invited Talk)

Authors: Laura Kovács

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
Loop invariants describe valid program properties that hold before and after every loop iteration. As such, loop invariants are the workhorses in formalizing loop semantics and automating the formal analysis and verification of programs with loops. While automatically synthesizing loop invariants is, in general, an uncomputable problem, when considering only single-path loops with linear updates (linear loops), the strongest polynomial invariant is in fact computable [Michael Karr, 1976; Markus Müller-Olm and Helmut Seidl, 2004; Laura Kovács, 2008; Ehud Hrushovski et al., 2018]. Yet, already for loops with "only" polynomial updates, computing the strongest invariant has been an open challenge since 2004 [Markus Müller-Olm and Helmut Seidl, 2004]. In this invited talk, we first present computability results on polynomial invariant synthesis for restricted polynomial loops, called solvable loops [Rodríguez-Carbonell and Kapur, 2004]. Key to solvable loops is that one can automatically compute invariants from closed-form solutions of algebraic recurrence equations that model the loop behaviour [Laura Kovács, 2008; Andreas Humenberger et al., 2017]. We also establish a technique for invariant synthesis for classes of loops that are not solvable, termed unsolvable loops [Daneshvar Amrollahi et al., 2022]. We next study the limits of computability in deriving the (strongest) polynomial invariants for arbitrary polynomial loops. We prove that computing the strongest polynomial invariant of arbitrary, single-path polynomial loops is very hard [Julian Müllner, 2023] - namely, it is at least as hard as the Skolem problem [Graham Everest et al., 2003; Terrence Tao, 2008], a prominent algebraic problem in the theory of linear recurrences. Going beyond single-path loops, we show that the strongest polynomial invariant is uncomputable already for multi-path polynomial loops with arbitrary quadratic polynomial updates [Laura Kovács and Anton Varonka, 2023].
Cite as

Laura Kovács. Algebraic Reasoning for (Un)Solvable Loops (Invited Talk). In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 4:1-4:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kovacs:LIPIcs.MFCS.2023.4,
  author =	{Kov\'{a}cs, Laura},
  title =	{{Algebraic Reasoning for (Un)Solvable Loops}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{4:1--4:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.4},
  URN =		{urn:nbn:de:0030-drops-185385},
  doi =		{10.4230/LIPIcs.MFCS.2023.4},
  annote =	{Keywords: Symbolic Computation, Formal Methods, Loop Analysis, Polynomial Invariants}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.130
Positivity Problems for Reversible Linear Recurrence Sequences

Authors: George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
It is a longstanding open problem whether there is an algorithm to decide the Positivity Problem for linear recurrence sequences (LRS) over the integers, namely whether given such a sequence, all its terms are non-negative. Decidability is known for LRS of order 5 or less, i.e., for those sequences in which every new term depends linearly on the previous five (or fewer) terms. For simple LRS (i.e., those sequences whose characteristic polynomials have no repeated roots), decidability of Positivity is known up to order 9. In this paper, we focus on the important subclass of reversible LRS, i.e., those integer LRS ⟨u_n⟩_{n=0}^∞ whose bi-infinite completion ⟨u_n⟩_{n=-∞}^∞ also takes exclusively integer values; a typical example is the classical Fibonacci (bi-)sequence ⟨ … , 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, … ⟩. Our main results are that Positivity is decidable for reversible LRS of order 11 or less, and for simple reversible LRS of order 17 or less.
Cite as

George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell. Positivity Problems for Reversible Linear Recurrence Sequences. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 130:1-130:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kenison_et_al:LIPIcs.ICALP.2023.130,
  author =	{Kenison, George and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Worrell, James},
  title =	{{Positivity Problems for Reversible Linear Recurrence Sequences}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{130:1--130:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.130},
  URN =		{urn:nbn:de:0030-drops-181821},
  doi =		{10.4230/LIPIcs.ICALP.2023.130},
  annote =	{Keywords: The Positivity Problem, Linear Recurrence Sequences, Verification}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.61
On the Skolem Problem for Reversible Sequences

Authors: George Kenison

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
Given an integer linear recurrence sequence ⟨X_n⟩, the Skolem Problem asks to determine whether there is a natural number n such that X_n = 0. Recent work by Lipton, Luca, Nieuwveld, Ouaknine, Purser, and Worrell proved that the Skolem Problem is decidable for a class of reversible sequences of order at most seven. Here we give an alternative proof of their result. Our novel approach employs a powerful result for Galois conjugates that lie on two concentric circles due to Dubickas and Smyth.
Cite as

George Kenison. On the Skolem Problem for Reversible Sequences. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kenison:LIPIcs.MFCS.2022.61,
  author =	{Kenison, George},
  title =	{{On the Skolem Problem for Reversible Sequences}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{61:1--61:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.61},
  URN =		{urn:nbn:de:0030-drops-168590},
  doi =		{10.4230/LIPIcs.MFCS.2022.61},
  annote =	{Keywords: The Skolem Problem, Linear Recurrences, Verification}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2021.3
Holonomic Techniques, Periods, and Decision Problems (Invited Talk)

Authors: Joël Ouaknine

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades (see, e.g., [Doron Zeilberger, 1990; Petkovšek et al., 1997]). In this talk, I give an overview of the area, and in particular present a select survey of known and original results on decision problems for holonomic sequences and functions. I also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier. Parts of this exposition draws upon [George Kenison et al., 2021].
Cite as

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ouaknine:LIPIcs.MFCS.2021.3,
  author =	{Ouaknine, Jo\"{e}l},
  title =	{{Holonomic Techniques, Periods, and Decision Problems}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.3},
  URN =		{urn:nbn:de:0030-drops-144431},
  doi =		{10.4230/LIPIcs.MFCS.2021.3},
  annote =	{Keywords: Holonomic and hypergeometric sequences, Inequality problems, Continued fractions, Periods}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.67
On Positivity and Minimality for Second-Order Holonomic Sequences

Authors: George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞. In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.
Cite as

George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell. On Positivity and Minimality for Second-Order Holonomic Sequences. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kenison_et_al:LIPIcs.MFCS.2021.67,
  author =	{Kenison, George and Klurman, Oleksiy and Lefaucheux, Engel and Luca, Florian and Moree, Pieter and Ouaknine, Jo\"{e}l and Whiteland, Markus A. and Worrell, James},
  title =	{{On Positivity and Minimality for Second-Order Holonomic Sequences}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{67:1--67:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.67},
  URN =		{urn:nbn:de:0030-drops-145071},
  doi =		{10.4230/LIPIcs.MFCS.2021.67},
  annote =	{Keywords: Holonomic sequences, Minimal solutions, Positivity Problem}
}
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