55 Search Results for "Ouaknine, Joël"

Document
DOI: 10.4230/LIPIcs.CONCUR.2024.5
MITL Model Checking via Generalized Timed Automata and a New Liveness Algorithm

Authors: S. Akshay, Paul Gastin, R. Govind, and B. Srivathsan

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
The translation of Metric Interval Temporal Logic (MITL) to timed automata is a topic that has been extensively studied. A key challenge here is the conversion of future modalities into equivalent automata. Typical conversions equip the automata with a guess-and-check mechanism to ascertain the truth of future modalities. Guess-and-check can be naturally implemented via alternation. However, since timed automata tools do not handle alternation, existing methods perform an additional step of converting the alternating timed automata into timed automata. This "de-alternation" step proceeds by an intricate finite abstraction of the space of configurations of the alternating automaton. Recently, a model of generalized timed automata (GTA) has been proposed. The model comes with several powerful additional features, and yet, the best known zone-based reachability algorithms for timed automata have been extended to the GTA model, with the same complexity for all the zone operations. An attractive feature of GTAs is the presence of future clocks which act like timers that guess a time to an event and stay alive until a timeout. Future clocks seem to provide another natural way to implement the guess-and-check: start the future clock with a guessed time to an event and check its occurrence using a timeout. Indeed, using this feature, we provide a new concise translation from MITL to GTA. In particular, for the timed until modality, our translation offers an exponential improvement w.r.t. the state-of-the-art. Thanks to this conversion, MITL model checking reduces to checking liveness for GTAs. However, no liveness algorithm is known for GTAs. Due to the presence of future clocks, there is no finite time-abstract bisimulation (region equivalence) for GTAs, whereas liveness algorithms for timed automata crucially rely on the presence of the finite region equivalence. As our second contribution, we provide a new zone-based algorithm for checking Büchi non-emptiness in GTAs, which circumvents this fundamental challenge.
Cite as

S. Akshay, Paul Gastin, R. Govind, and B. Srivathsan. MITL Model Checking via Generalized Timed Automata and a New Liveness Algorithm. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{akshay_et_al:LIPIcs.CONCUR.2024.5,
  author =	{Akshay, S. and Gastin, Paul and Govind, R. and Srivathsan, B.},
  title =	{{MITL Model Checking via Generalized Timed Automata and a New Liveness Algorithm}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak 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.CONCUR.2024.5},
  URN =		{urn:nbn:de:0030-drops-207774},
  doi =		{10.4230/LIPIcs.CONCUR.2024.5},
  annote =	{Keywords: MITL model checking, timed automata, zones, liveness}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2024.17
Invariants for One-Counter Automata with Disequality Tests

Authors: Dmitry Chistikov, Jérôme Leroux, Henry Sinclair-Banks, and Nicolas Waldburger

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
We study the reachability problem for one-counter automata in which transitions can carry disequality tests. A disequality test is a guard that prohibits a specified counter value. This reachability problem has been known to be NP-hard and in PSPACE, and characterising its computational complexity has been left as a challenging open question by Almagor, Cohen, Pérez, Shirmohammadi, and Worrell (2020). We reduce the complexity gap, placing the problem into the second level of the polynomial hierarchy, namely into the class coNP^NP. In the presence of both equality and disequality tests, our upper bound is at the third level, P^NP^NP. To prove this result, we show that non-reachability can be witnessed by a pair of invariants (forward and backward). These invariants are almost inductive. They aim to over-approximate only a "core" of the reachability set instead of the entire set. The invariants are also leaky: it is possible to escape the set. We complement this with separate checks as the leaks can only occur in a controlled way.
Cite as

Dmitry Chistikov, Jérôme Leroux, Henry Sinclair-Banks, and Nicolas Waldburger. Invariants for One-Counter Automata with Disequality Tests. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chistikov_et_al:LIPIcs.CONCUR.2024.17,
  author =	{Chistikov, Dmitry and Leroux, J\'{e}r\^{o}me and Sinclair-Banks, Henry and Waldburger, Nicolas},
  title =	{{Invariants for One-Counter Automata with Disequality Tests}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{17:1--17:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak 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.CONCUR.2024.17},
  URN =		{urn:nbn:de:0030-drops-207898},
  doi =		{10.4230/LIPIcs.CONCUR.2024.17},
  annote =	{Keywords: Inductive invariant, Vector addition system, One-counter automaton}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2024.18
Weighted Basic Parallel Processes and Combinatorial Enumeration

Authors: Lorenzo Clemente

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
We study weighted basic parallel processes (WBPP), a nonlinear recursive generalisation of weighted finite automata inspired from process algebra and Petri net theory. Our main result is an algorithm of 2-EXPSPACE complexity for the WBPP equivalence problem. While (unweighted) BPP language equivalence is undecidable, we can use this algorithm to decide multiplicity equivalence of BPP and language equivalence of unambiguous BPP, with the same complexity. These are long-standing open problems for the related model of weighted context-free grammars. Our second contribution is a connection between WBPP, power series solutions of systems of polynomial differential equations, and combinatorial enumeration. To this end we consider constructible differentially finite power series (CDF), a class of multivariate differentially algebraic series introduced by Bergeron and Reutenauer in order to provide a combinatorial interpretation to differential equations. CDF series generalise rational, algebraic, and a large class of D-finite (holonomic) series, for which no complexity upper bound for equivalence was known. We show that CDF series correspond to commutative WBPP series. As a consequence of our result on WBPP and commutativity, we show that equivalence of CDF power series can be decided with 2-EXPTIME complexity. In order to showcase the CDF equivalence algorithm, we show that CDF power series naturally arise from combinatorial enumeration, namely as the exponential generating series of constructible species of structures. Examples of such species include sequences, binary trees, ordered trees, Cayley trees, set partitions, series-parallel graphs, and many others. As a consequence of this connection, we obtain an algorithm to decide multiplicity equivalence of constructible species, decidability of which was not known before. The complexity analysis is based on effective bounds from algebraic geometry, namely on the length of chains of polynomial ideals constructed by repeated application of finitely many, not necessarily commuting derivations of a multivariate polynomial ring. This is obtained by generalising a result of Novikov and Yakovenko in the case of a single derivation, which is noteworthy since generic bounds on ideal chains are non-primitive recursive in general. On the way, we develop the theory of WBPP series and CDF power series, exposing several of their appealing properties.
Cite as

Lorenzo Clemente. Weighted Basic Parallel Processes and Combinatorial Enumeration. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{clemente:LIPIcs.CONCUR.2024.18,
  author =	{Clemente, Lorenzo},
  title =	{{Weighted Basic Parallel Processes and Combinatorial Enumeration}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{18:1--18:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak 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.CONCUR.2024.18},
  URN =		{urn:nbn:de:0030-drops-207903},
  doi =		{10.4230/LIPIcs.CONCUR.2024.18},
  annote =	{Keywords: weighted automata, combinatorial enumeration, shuffle, algebraic differential equations, process algebra, basic parallel processes, species of structures}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2024.27
Inaproximability in Weighted Timed Games

Authors: Quentin Guilmant and Joël Ouaknine

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
We consider two-player, turn-based weighted timed games played on timed automata equipped with (positive and negative) integer weights, in which one player seeks to reach a goal location whilst minimising the cumulative weight of the underlying path. Although the value problem for such games (is the value of the game below a given threshold?) is known to be undecidable, the question of whether one can approximate this value has remained a longstanding open problem. In this paper, we resolve this question by showing that approximating arbitrarily closely the value of a given weighted timed game is computationally unsolvable.
Cite as

Quentin Guilmant and Joël Ouaknine. Inaproximability in Weighted Timed Games. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{guilmant_et_al:LIPIcs.CONCUR.2024.27,
  author =	{Guilmant, Quentin and Ouaknine, Jo\"{e}l},
  title =	{{Inaproximability in Weighted Timed Games}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak 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.CONCUR.2024.27},
  URN =		{urn:nbn:de:0030-drops-207998},
  doi =		{10.4230/LIPIcs.CONCUR.2024.27},
  annote =	{Keywords: Weighted timed games, approximation, undecidability}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2024.31
Bi-Reachability in Petri Nets with Data

Authors: Łukasz Kamiński and Sławomir Lasota

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
We investigate Petri nets with data, an extension of plain Petri nets where tokens carry values from an infinite data domain, and executability of transitions is conditioned by equalities between data values. We provide a decision procedure for the bi-reachability problem: given a Petri net and its two configurations, we ask if each of the configurations is reachable from the other. This pushes forward the decidability borderline, as the bi-reachability problem subsumes the coverability problem (which is known to be decidable) and is subsumed by the reachability problem (whose decidability status is unknown).
Cite as

Łukasz Kamiński and Sławomir Lasota. Bi-Reachability in Petri Nets with Data. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kaminski_et_al:LIPIcs.CONCUR.2024.31,
  author =	{Kami\'{n}ski, {\L}ukasz and Lasota, S{\l}awomir},
  title =	{{Bi-Reachability in Petri Nets with Data}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{31:1--31:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak 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.CONCUR.2024.31},
  URN =		{urn:nbn:de:0030-drops-208038},
  doi =		{10.4230/LIPIcs.CONCUR.2024.31},
  annote =	{Keywords: Petri nets, Petri nets with data, reachability, bi-reachability, reversible reachability, mutual reachability, orbit-finite sets}
}
Document
Invited Paper
DOI: 10.4230/LIPIcs.MFCS.2024.2
Challenges of the Reachability Problem in Infinite-State Systems (Invited Paper)

Authors: Wojciech Czerwiński

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The reachability problem is a central problem for various infinite state systems like automata with pushdown, with different kinds of counters or combinations thereof. Despite its centrality and decades of research the community still lacks a lot of answers for fundamental and basic questions of that type. I briefly describe my personal viewpoint on the current state of art and emphasise interesting directions, which are worth investigating in my opinion. I also formulate several easy to formulate and understand challenges, which might be pretty hard to solve but at the same time illustrate fundamental lack of our understanding in the area.
Cite as

Wojciech Czerwiński. Challenges of the Reachability Problem in Infinite-State Systems (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{czerwinski:LIPIcs.MFCS.2024.2,
  author =	{Czerwi\'{n}ski, Wojciech},
  title =	{{Challenges of the Reachability Problem in Infinite-State Systems}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{2:1--2:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.2},
  URN =		{urn:nbn:de:0030-drops-205582},
  doi =		{10.4230/LIPIcs.MFCS.2024.2},
  annote =	{Keywords: reachability problem, infinite-state systems, vector addition systems, pushdown}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.23
Minimizing Cost Register Automata over a Field

Authors: Yahia Idriss Benalioua, Nathan Lhote, and Pierre-Alain Reynier

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Weighted automata (WA) are an extension of finite automata that define functions from words to values in a given semiring. An alternative deterministic model, called Cost Register Automata (CRA), was introduced by Alur et al. It enriches deterministic finite automata with a finite number of registers, which store values, updated at each transition using the operations of the semiring. It is known that CRA with register updates defined by linear maps have the same expressiveness as WA. Previous works have studied the register minimization problem: given a function computable by a WA and an integer k, is it possible to realize it using a CRA with at most k registers? In this paper, we solve this problem for CRA over a field with linear register updates, using the notion of linear hull, an algebraic invariant of WA introduced recently by Bell and Smertnig. We then generalise the approach to solve a more challenging problem, that consists in minimizing simultaneously the number of states and that of registers. In addition, we also lift our results to the setting of CRA with affine updates. Last, while the linear hull was recently shown to be computable by Bell and Smertnig, no complexity bounds were given. To fill this gap, we provide two new algorithms to compute invariants of WA. This allows us to show that the register (resp. state-register) minimization problem can be solved in 2-ExpTime (resp. in NExpTime).
Cite as

Yahia Idriss Benalioua, Nathan Lhote, and Pierre-Alain Reynier. Minimizing Cost Register Automata over a Field. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{benalioua_et_al:LIPIcs.MFCS.2024.23,
  author =	{Benalioua, Yahia Idriss and Lhote, Nathan and Reynier, Pierre-Alain},
  title =	{{Minimizing Cost Register Automata over a Field}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.23},
  URN =		{urn:nbn:de:0030-drops-205798},
  doi =		{10.4230/LIPIcs.MFCS.2024.23},
  annote =	{Keywords: Weighted automata, Cost Register automata, Zariski topology}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2024.18
Representation of Peano Arithmetic in Separation Logic

Authors: Sohei Ito and Makoto Tatsuta

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract
Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the validity such as decidability and complexity for separation logic with numbers have not been fully studied yet. This paper presents the translation of Pi-0-1 formulas in Peano arithmetic to formulas in a small fragment of separation logic with numbers, which consists only of the intuitionistic points-to predicate, 0 and the successor function. Then this paper proves that a formula in Peano arithmetic is valid in the standard model if and only if its translation in this fragment is valid in the standard interpretation. As a corollary, this paper also gives a perspective proof for the undecidability of the validity in this fragment. Since Pi-0-1 formulas can describe consistency of logical systems and non-termination of computations, this result also shows that these properties discussed in Peano arithmetic can also be discussed in such a small fragment of separation logic with numbers.
Cite as

Sohei Ito and Makoto Tatsuta. Representation of Peano Arithmetic in Separation Logic. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ito_et_al:LIPIcs.FSCD.2024.18,
  author =	{Ito, Sohei and Tatsuta, Makoto},
  title =	{{Representation of Peano Arithmetic in Separation Logic}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.18},
  URN =		{urn:nbn:de:0030-drops-203476},
  doi =		{10.4230/LIPIcs.FSCD.2024.18},
  annote =	{Keywords: First order logic, Separation logic, Peano arithmetic, Presburger arithmetic}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.136
Improved Algorithm for Reachability in d-VASS

Authors: Yuxi Fu, Qizhe Yang, and Yangluo Zheng

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

Abstract
An 𝖥_{d} upper bound for the reachability problem in vector addition systems with states (VASS) in fixed dimension is given, where 𝖥_d is the d-th level of the Grzegorczyk hierarchy of complexity classes. The new algorithm combines the idea of the linear path scheme characterization of the reachability in the 2-dimension VASSes with the general decomposition algorithm by Mayr, Kosaraju and Lambert. The result improves the 𝖥_{d + 4} upper bound due to Leroux and Schmitz (LICS 2019).
Cite as

Yuxi Fu, Qizhe Yang, and Yangluo Zheng. Improved Algorithm for Reachability in d-VASS. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 136:1-136:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fu_et_al:LIPIcs.ICALP.2024.136,
  author =	{Fu, Yuxi and Yang, Qizhe and Zheng, Yangluo},
  title =	{{Improved Algorithm for Reachability in d-VASS}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{136:1--136:18},
  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.2024.136},
  URN =		{urn:nbn:de:0030-drops-202799},
  doi =		{10.4230/LIPIcs.ICALP.2024.136},
  annote =	{Keywords: Petri net, vector addition system, reachability}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.139
T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations

Authors: Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah

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

Abstract
We introduce a powerful termination algorithm for structurally recursive functions that improves on the core ideas behind lexicographic termination algorithms for functional programs. The algorithm generates linear-lexicographic combinations of primitive measure functions measuring the recursive structure of terms. We introduce a measure language that enables the simplification and comparison of measures and we prove meta-theoretic properties of our measure language. Moreover, we demonstrate our algorithm, on an untyped first-order functional language and prove its soundness and that it runs in polynomial time. We also provide a Haskell implementation. As part of this work, we also show how to solve the maximisation of negative vector-components as a linear program.
Cite as

Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah. T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{giles_et_al:LIPIcs.ICALP.2024.139,
  author =	{Giles, Raphael Douglas and Jackson, Vincent and Rizkallah, Christine},
  title =	{{T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{139:1--139:19},
  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.2024.139},
  URN =		{urn:nbn:de:0030-drops-202827},
  doi =		{10.4230/LIPIcs.ICALP.2024.139},
  annote =	{Keywords: Termination, Recursive functions}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.140
The 2-Dimensional Constraint Loop Problem Is Decidable

Authors: Quentin Guilmant, Engel Lefaucheux, Joël Ouaknine, and James Worrell

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

Abstract
A linear constraint loop is specified by a system of linear inequalities that define the relation between the values of the program variables before and after a single execution of the loop body. In this paper we consider the problem of determining whether such a loop terminates, i.e., whether all maximal executions are finite, regardless of how the loop is initialised and how the non-determinism in the loop body is resolved. We focus on the variant of the termination problem in which the loop variables range over ℝ. Our main result is that the termination problem is decidable over the reals in dimension 2. A more abstract formulation of our main result is that it is decidable whether a binary relation on ℝ² that is given as a conjunction of linear constraints is well-founded.
Cite as

Quentin Guilmant, Engel Lefaucheux, Joël Ouaknine, and James Worrell. The 2-Dimensional Constraint Loop Problem Is Decidable. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 140:1-140:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{guilmant_et_al:LIPIcs.ICALP.2024.140,
  author =	{Guilmant, Quentin and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Worrell, James},
  title =	{{The 2-Dimensional Constraint Loop Problem Is Decidable}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{140:1--140:21},
  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.2024.140},
  URN =		{urn:nbn:de:0030-drops-202831},
  doi =		{10.4230/LIPIcs.ICALP.2024.140},
  annote =	{Keywords: Linear Constraints Loops, Minkowski-Weyl, Convex Sets, Asymptotic Expansions}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.144
On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words

Authors: Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell

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

Abstract
We consider numbers of the form S_β(u): = ∑_{n=0}^∞ (u_n)/(βⁿ), where u = ⟨u_n⟩_{n=0}^∞ is an infinite word over a finite alphabet and β ∈ ℂ satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_β(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise ̅{ℚ}-linear independence of sets of the form {1, S_β(u₁),…,S_β(u_k)}, where u₁,…,u_k are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.
Cite as

Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell. On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 144:1-144:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kebis_et_al:LIPIcs.ICALP.2024.144,
  author =	{Kebis, Pavol and Luca, Florian and Ouaknine, Jo\"{e}l and Scoones, Andrew and Worrell, James},
  title =	{{On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{144:1--144:15},
  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.2024.144},
  URN =		{urn:nbn:de:0030-drops-202873},
  doi =		{10.4230/LIPIcs.ICALP.2024.144},
  annote =	{Keywords: Transcendence, Subspace Theorem, Fibonacci Word, Tribonacci Word}
}
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
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.156
Verification of Population Protocols with Unordered Data

Authors: Steffen van Bergerem, Roland Guttenberg, Sandra Kiefer, Corto Mascle, Nicolas Waldburger, and Chana Weil-Kennedy

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

Abstract
Population protocols are a well-studied model of distributed computation in which a group of anonymous finite-state agents communicates via pairwise interactions. Together they decide whether their initial configuration, i. e., the initial distribution of agents in the states, satisfies a property. As an extension in order to express properties of multisets over an infinite data domain, Blondin and Ladouceur (ICALP'23) introduced population protocols with unordered data (PPUD). In PPUD, each agent carries a fixed data value, and the interactions between agents depend on whether their data are equal or not. Blondin and Ladouceur also identified the interesting subclass of immediate observation PPUD (IOPPUD), where in every transition one of the two agents remains passive and does not move, and they characterised its expressive power. We study the decidability and complexity of formally verifying these protocols. The main verification problem for population protocols is well-specification, that is, checking whether the given PPUD computes some function. We show that well-specification is undecidable in general. By contrast, for IOPPUD, we exhibit a large yet natural class of problems, which includes well-specification among other classic problems, and establish that these problems are in ExpSpace. We also provide a lower complexity bound, namely coNExpTime-hardness.
Cite as

Steffen van Bergerem, Roland Guttenberg, Sandra Kiefer, Corto Mascle, Nicolas Waldburger, and Chana Weil-Kennedy. Verification of Population Protocols with Unordered Data. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 156:1-156:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanbergerem_et_al:LIPIcs.ICALP.2024.156,
  author =	{van Bergerem, Steffen and Guttenberg, Roland and Kiefer, Sandra and Mascle, Corto and Waldburger, Nicolas and Weil-Kennedy, Chana},
  title =	{{Verification of Population Protocols with Unordered Data}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{156:1--156:20},
  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.2024.156},
  URN =		{urn:nbn:de:0030-drops-202993},
  doi =		{10.4230/LIPIcs.ICALP.2024.156},
  annote =	{Keywords: Population protocols, Parameterized verification, Distributed computing, Well-specification}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.27
Nonnegativity Problems for Matrix Semigroups

Authors: Julian D'Costa, Joël Ouaknine, and James Worrell

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

Abstract
The matrix semigroup membership problem asks, given square matrices M,M₁,…,M_k of the same dimension, whether M lies in the semigroup generated by M₁,…,M_k. It is classical that this problem is undecidable in general, but decidable in case M₁,…,M_k commute. In this paper we consider the problem of whether, given M₁,…,M_k, the semigroup generated by M₁,…,M_k contains a non-negative matrix. We show that in case M₁,…,M_k commute, this problem is decidable subject to Schanuel’s Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mⁿ)_{n = 0}^∞ is ultimately nonnegative. This answers a problem posed by S. Akshay [S. Akshay et al., 2022]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i,j), whether the sequence (Mⁿ)_{i,j} is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem.
Cite as

Julian D'Costa, Joël Ouaknine, and James Worrell. Nonnegativity Problems for Matrix Semigroups. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dcosta_et_al:LIPIcs.STACS.2024.27,
  author =	{D'Costa, Julian and Ouaknine, Jo\"{e}l and Worrell, James},
  title =	{{Nonnegativity Problems for Matrix Semigroups}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{27:1--27:16},
  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.2024.27},
  URN =		{urn:nbn:de:0030-drops-197371},
  doi =		{10.4230/LIPIcs.STACS.2024.27},
  annote =	{Keywords: Decidability, Linear Recurrence Sequences, Schanuel’s Conjecture}
}
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