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Leibniz Transactions on Embedded Systems, Volume 8, Issue 2, 2022

LITES, Volume 8, Issue 2

Editors: Alessandro Abate, Uli Fahrenberg, and Martin Fränzle

Special Issue on Distributed Hybrid Systems

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2024.149

A Complete Quantitative Axiomatisation of Behavioural Distance of Regular Expressions

Authors: Wojciech Różowski

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

Abstract

Deterministic automata have been traditionally studied through the point of view of language equivalence, but another perspective is given by the canonical notion of shortest-distinguishing-word distance quantifying the of states. Intuitively, the longer the word needed to observe a difference between two states, then the closer their behaviour is. In this paper, we give a sound and complete axiomatisation of shortest-distinguishing-word distance between regular languages. Our axiomatisation relies on a recently developed quantitative analogue of equational logic, allowing to manipulate rational-indexed judgements of the form e ≡_ε f meaning term e is approximately equivalent to term f within the error margin of ε. The technical core of the paper is dedicated to the completeness argument that draws techniques from order theory and Banach spaces to simplify the calculation of the behavioural distance to the point it can be then mimicked by axiomatic reasoning.

Cite as

Wojciech Różowski. A Complete Quantitative Axiomatisation of Behavioural Distance of Regular Expressions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 149:1-149:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{rozowski:LIPIcs.ICALP.2024.149,
  author =	{R\'{o}\.{z}owski, Wojciech},
  title =	{{A Complete Quantitative Axiomatisation of Behavioural Distance of Regular Expressions}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{149:1--149:20},
  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.2024.149},
  URN =		{urn:nbn:de:0030-drops-202920},
  doi =		{10.4230/LIPIcs.ICALP.2024.149},
  annote =	{Keywords: Regular Expressions, Behavioural Distances, Quantitative Equational Theories}
}

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2024.157

Domain Reasoning in TopKAT

Authors: Cheng Zhang, Arthur Azevedo de Amorim, and Marco Gaboardi

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

Abstract

TopKAT is the algebraic theory of Kleene algebra with tests (KAT) extended with a top element. Compared to KAT, one pleasant feature of TopKAT is that, in relational models, the top element allows us to express the domain and codomain of a relation. This enables several applications in program logics, such as proving under-approximate specifications or reachability properties of imperative programs. However, while TopKAT inherits many pleasant features of KATs, such as having a decidable equational theory, it is incomplete with respect to relational models. In other words, there are properties that hold true of all relational TopKATs but cannot be proved with the axioms of TopKAT. This issue is potentially worrisome for program-logic applications, in which relational models play a key role. In this paper, we further investigate the completeness properties of TopKAT with respect to relational models. We show that TopKAT is complete with respect to (co)domain comparison of KAT terms, but incomplete when comparing the (co)domain of arbitrary TopKAT terms. Since the encoding of under-approximate specifications in TopKAT hinges on this type of formula, the aforementioned incompleteness results have a limited impact when using TopKAT to reason about such specifications.

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Cheng Zhang, Arthur Azevedo de Amorim, and Marco Gaboardi. Domain Reasoning in TopKAT. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 157:1-157:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{zhang_et_al:LIPIcs.ICALP.2024.157,
  author =	{Zhang, Cheng and de Amorim, Arthur Azevedo and Gaboardi, Marco},
  title =	{{Domain Reasoning in TopKAT}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{157:1--157:18},
  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.2024.157},
  URN =		{urn:nbn:de:0030-drops-203003},
  doi =		{10.4230/LIPIcs.ICALP.2024.157},
  annote =	{Keywords: Kleene algebra, Kleene Algebra With Tests, Kleene Algebra With Domain, Kleene Algebra With Top and Tests, Completeness, Decidability}
}

Document

Introduction

DOI: 10.4230/LITES.8.2.0

Introduction to the Special Issue on Distributed Hybrid Systems

Authors: Alessandro Abate, Uli Fahrenberg, and Martin Fränzle

Published in: LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2

Abstract

This special issue contains seven papers within the broad subject of Distributed Hybrid Systems, that is, systems combining hybrid discrete-continuous state spaces with elements of concurrency and logical or spatial distribution. It follows up on several workshops on the same theme which were held between 2017 and 2019 and organized by the editors of this volume. The first of these workshops was held in Aalborg, Denmark, in August 2017 and associated with the MFCS conference. It featured invited talks by Alessandro Abate, Martin Fränzle, Kim G. Larsen, Martin Raussen, and Rafael Wisniewski. The second workshop was held in Palaiseau, France, in July 2018, with invited talks by Luc Jaulin, Thao Dang, Lisbeth Fajstrup, Emmanuel Ledinot, and André Platzer. The third workshop was held in Amsterdam, The Netherlands, in August 2019, associated with the CONCUR conference. It featured a special theme on distributed robotics and had invited talks by Majid Zamani, Hervé de Forges, and Xavier Urbain. The vision and purpose of the DHS workshops was to connect researchers working in real-time systems, hybrid systems, control theory, formal verification, distributed computing, and concurrency theory, in order to advance the subject of distributed hybrid systems. Such systems are abundant and often safety-critical, but ensuring their correct functioning can in general be challenging. The investigation of their dynamics by analysis tools from the aforementioned domains remains fragmentary, providing the rationale behind the workshops: it was conceived that convergence and interaction of theories, methods, and tools from these different areas was needed in order to advance the subject.

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LITES, Volume 8, Issue 2: Special Issue on Distributed Hybrid Systems, pp. 0:i-0:iii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{abate_et_al:LITES.8.2.0,
  author =	{Abate, Alessandro and Fahrenberg, Uli and Fr\"{a}nzle, Martin},
  title =	{{Introduction to the Special Issue on Distributed Hybrid Systems}},
  journal =	{Leibniz Transactions on Embedded Systems},
  pages =	{00:1--00:3},
  ISSN =	{2199-2002},
  year =	{2022},
  volume =	{8},
  number =	{2},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES.8.2.0},
  doi =		{10.4230/LITES.8.2.0},
  annote =	{Keywords: Distributed hybrid systems}
}

Document

DOI: 10.4230/LITES.8.2.3

Higher-Dimensional Timed and Hybrid Automata

Authors: Uli Fahrenberg

Published in: LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2

Abstract

We introduce a new formalism of higher-dimensional timed automata, based on Pratt and van Glabbeek’s higher-dimensional automata and Alur and Dill’s timed automata. We prove that their reachability is PSPACE-complete and can be decided using zone-based algorithms. We also extend the setting to higher-dimensional hybrid automata.The interest of our formalism is in modeling systems which exhibit both real-time behavior and concurrency. Other existing formalisms for real-time modeling identify concurrency and interleaving, which, as we shall argue, is problematic.

Cite as

Uli Fahrenberg. Higher-Dimensional Timed and Hybrid Automata. In LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2, pp. 03:1-03:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{fahrenberg:LITES.8.2.3,
  author =	{Fahrenberg, Uli},
  title =	{{Higher-Dimensional Timed and Hybrid Automata}},
  journal =	{Leibniz Transactions on Embedded Systems},
  pages =	{03:1--03:16},
  ISSN =	{2199-2002},
  year =	{2022},
  volume =	{8},
  number =	{2},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES.8.2.3},
  doi =		{10.4230/LITES.8.2.3},
  annote =	{Keywords: timed automaton, higher-dimensional automaton, precubical set, real time, non-interleaving concurrency, hybrid automaton}
}

Document

DOI: 10.4230/LIPIcs.CONCUR.2022.29

A Kleene Theorem for Higher-Dimensional Automata

Authors: Uli Fahrenberg, Christian Johansen, Georg Struth, and Krzysztof Ziemiański

Published in: LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Abstract

We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders are used as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.

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Uli Fahrenberg, Christian Johansen, Georg Struth, and Krzysztof Ziemiański. A Kleene Theorem for Higher-Dimensional Automata. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fahrenberg_et_al:LIPIcs.CONCUR.2022.29,
  author =	{Fahrenberg, Uli and Johansen, Christian and Struth, Georg and Ziemia\'{n}ski, Krzysztof},
  title =	{{A Kleene Theorem for Higher-Dimensional Automata}},
  booktitle =	{33rd International Conference on Concurrency Theory (CONCUR 2022)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-246-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{243},
  editor =	{Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.29},
  URN =		{urn:nbn:de:0030-drops-170925},
  doi =		{10.4230/LIPIcs.CONCUR.2022.29},
  annote =	{Keywords: higher-dimensional automata, interval posets, Kleene theorem, concurrency theory, labelled precube categories}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2015.394

An omega-Algebra for Real-Time Energy Problems

Authors: David Cachera, Uli Fahrenberg, and Axel Legay

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Abstract

We develop a *-continuous Kleene omega-algebra of real-time energy functions. Together with corresponding automata, these can be used to model systems which can consume and regain energy (or other types of resources) depending on available time. Using recent results on *-continuous Kleene omega-algebras and computability of certain manipulations on real-time energy functions, it follows that reachability and Büchi acceptance in real-time energy automata can be decided in a static way which only involves manipulations of real-time energy functions.

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David Cachera, Uli Fahrenberg, and Axel Legay. An omega-Algebra for Real-Time Energy Problems. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 394-407, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cachera_et_al:LIPIcs.FSTTCS.2015.394,
  author =	{Cachera, David and Fahrenberg, Uli and Legay, Axel},
  title =	{{An omega-Algebra for Real-Time Energy Problems}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{394--407},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.394},
  URN =		{urn:nbn:de:0030-drops-56511},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.394},
  annote =	{Keywords: Energy problem, Real time, Star-continuous Kleene algebra}
}

Document

DOI: 10.4230/LIPIcs.CALCO.2015.101

Partial Higher-dimensional Automata

Authors: Uli Fahrenberg and Axel Legay

Published in: LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

Abstract

We propose a generalization of higher-dimensional automata, partial HDA. Unlike HDA, and also extending event structures and Petri nets, partial HDA can model phenomena such as priorities or the disabling of an event by another event. Using open maps and unfoldings, we introduce a natural notion of (higher-dimensional) bisimilarity for partial HDA and relate it to history-preserving bisimilarity and split bisimilarity. Higher-dimensional bisimilarity has a game characterization and is decidable in polynomial time.

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Uli Fahrenberg and Axel Legay. Partial Higher-dimensional Automata. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 101-115, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fahrenberg_et_al:LIPIcs.CALCO.2015.101,
  author =	{Fahrenberg, Uli and Legay, Axel},
  title =	{{Partial Higher-dimensional Automata}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{101--115},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-84-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{35},
  editor =	{Moss, Lawrence S. and Sobocinski, Pawel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.101},
  URN =		{urn:nbn:de:0030-drops-55295},
  doi =		{10.4230/LIPIcs.CALCO.2015.101},
  annote =	{Keywords: higher-dimensional automata, bisimulation}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2011.103

The Quantitative Linear-Time--Branching-Time Spectrum

Authors: Uli Fahrenberg, Axel Legay, and Claus Thrane

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract

We present a distance-agnostic approach to quantitative verification. Taking as input an unspecified distance on system traces, or executions, we develop a game-based framework which allows us to define a spectrum of different interesting system distances corresponding to the given trace distance. Thus we extend the classic linear-time--branching-time spectrum to a quantitative setting, parametrized by trace distance. We also provide fixed-point characterizations of all system distances, and we prove a general transfer principle which allows us to transfer counterexamples from the qualitative to the quantitative setting,showing that all system distances are mutually topologically inequivalent.

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Uli Fahrenberg, Axel Legay, and Claus Thrane. The Quantitative Linear-Time--Branching-Time Spectrum. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 103-114, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fahrenberg_et_al:LIPIcs.FSTTCS.2011.103,
  author =	{Fahrenberg, Uli and Legay, Axel and Thrane, Claus},
  title =	{{The Quantitative Linear-Time--Branching-Time Spectrum}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{103--114},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.103},
  URN =		{urn:nbn:de:0030-drops-33324},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.103},
  annote =	{Keywords: Quantitative verification, System distance, Distance hierarchy, Linear time, Branching time}
}

Document

DOI: 10.4230/DROPS.MEMICS.2009.2345

A Quantitative Characterization of Weighted Kripke Structures in Temporal Logic

Authors: Kim G. Larsen, Uli Fahrenberg, and Claus Thrane

Published in: OASIcs, Volume 13, Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09) (2009)

Abstract

We extend the usual notion of Kripke Structures with a weighted transition relation, and generalize the usual Boolean satisfaction relation of CTL to a map which assigns to states and temporal formulae a real-valued distance describing the degree of satisfaction. We describe a general approach to obtaining quantitative interpretations for a generic extension of the CTL syntax, and show that, for one such interpretation, the logic is both adequate and expressive with respect to quantitative bisimulation.

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Kim G. Larsen, Uli Fahrenberg, and Claus Thrane. A Quantitative Characterization of Weighted Kripke Structures in Temporal Logic. In Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09). Open Access Series in Informatics (OASIcs), Volume 13, pp. 10-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{larsen_et_al:OASIcs:2009:DROPS.MEMICS.2009.2345,
  author =	{Larsen, Kim G. and Fahrenberg, Uli and Thrane, Claus},
  title =	{{A Quantitative Characterization of Weighted Kripke Structures in Temporal Logic}},
  booktitle =	{Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09)},
  pages =	{10--17},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-15-6},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{13},
  editor =	{Hlinen\'{y}, Petr and Maty\'{a}\v{s}, V\'{a}clav and Vojnar, Tom\'{a}\v{s}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DROPS.MEMICS.2009.2345},
  URN =		{urn:nbn:de:0030-drops-23454},
  doi =		{10.4230/DROPS.MEMICS.2009.2345},
  annote =	{Keywords: Quantitative analysis, Kripke structures, characteristic formulae, bisimulation distance, weighted CTL}
}

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