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Document

DOI: 10.4230/LIPIcs.STACS.2026.65

Demystifying Codensity Monads via Duality

Authors: Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat

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

Abstract

Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.

Cite as

Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat. Demystifying Codensity Monads via Duality. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lenke_et_al:LIPIcs.STACS.2026.65,
  author =	{Lenke, Fabian and Wittrock, Nico and Milius, Stefan and Urbat, Henning},
  title =	{{Demystifying Codensity Monads via Duality}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{65:1--65:20},
  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.2026.65},
  URN =		{urn:nbn:de:0030-drops-255549},
  doi =		{10.4230/LIPIcs.STACS.2026.65},
  annote =	{Keywords: Codensity, Monad, Duality}
}

Document

DOI: 10.4230/LIPIcs.CSL.2026.24

Well-Founded Coalgebras Meet Kőnig’s Lemma

Authors: Henning Urbat and Thorsten Wißmann

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

Kőnig’s lemma is a fundamental result about trees with countless applications in mathematics and computer science. In contrapositive form, it states that if a tree is finitely branching and well-founded (i.e. has no infinite paths), then it is finite. We present a coalgebraic version of Kőnig’s lemma featuring two dimensions of generalization: from finitely branching trees to coalgebras for a finitary endofunctor H, and from the base category of sets to a locally finitely presentable category ℂ, such as the category of posets, nominal sets, or convex sets. Our coalgebraic Kőnig’s lemma states that, under mild assumptions on ℂ and H, every well-founded coalgebra for H is the directed join of its well-founded subcoalgebras with finitely generated state space - in particular, the category of well-founded coalgebras is locally presentable. As applications, we derive versions of Kőnig’s lemma for graphs in a topos as well as for nominal and convex transition systems. Additionally, we show that the key construction underlying the proof gives rise to two simple constructions of the initial algebra (equivalently, the final recursive coalgebra) for the functor H: The initial algebra is both the colimit of all well-founded and of all recursive coalgebras with finitely presentable state space. Remarkably, this result holds even in settings where well-founded coalgebras form a proper subclass of recursive ones. The first construction of the initial algebra is entirely new, while for the second one our approach yields a short and transparent new correctness proof.

Cite as

Henning Urbat and Thorsten Wißmann. Well-Founded Coalgebras Meet Kőnig’s Lemma. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{urbat_et_al:LIPIcs.CSL.2026.24,
  author =	{Urbat, Henning and Wi{\ss}mann, Thorsten},
  title =	{{Well-Founded Coalgebras Meet K\H{o}nig’s Lemma}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{24:1--24:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.24},
  URN =		{urn:nbn:de:0030-drops-254485},
  doi =		{10.4230/LIPIcs.CSL.2026.24},
  annote =	{Keywords: K\H{o}nig’s Lemma, Well-Foundedness, Coalgebra}
}

Document

(Co)algebraic pearl

DOI: 10.4230/LIPIcs.CALCO.2025.16

Active Learning of Upward-Closed Sets of Words ((Co)algebraic pearl)

Authors: Quentin Aristote

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract

We give a new proof of a result from well quasi-order theory on the computability of bases for upwards-closed sets of words. This new proof is based on Angluin’s L* algorithm, that learns an automaton from a minimally adequate teacher. This relates in particular two results from the 1980s: Angluin’s L* algorithm, and a result from Valk and Jantzen on the computability of bases for upwards-closed sets of tuples of integers. Along the way, we describe an algorithm for learning quasi-ordered automata from a minimally adequate teacher, and extend a generalization of Valk and Jantzen’s result, encompassing both words and integers, to finitely generated monoids.

Cite as

Quentin Aristote. Active Learning of Upward-Closed Sets of Words ((Co)algebraic pearl). In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aristote:LIPIcs.CALCO.2025.16,
  author =	{Aristote, Quentin},
  title =	{{Active Learning of Upward-Closed Sets of Words}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{16:1--16:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.16},
  URN =		{urn:nbn:de:0030-drops-235751},
  doi =		{10.4230/LIPIcs.CALCO.2025.16},
  annote =	{Keywords: active learning, well quasi-orders, Valk-Jantzen lemma, piecewise-testable languages, monoids}
}

Document

DOI: 10.4230/LIPIcs.CALCO.2025.3

Terminal Coalgebras for Finitary Functors

Authors: Jiří Adámek, Stefan Milius, and Lawrence S. Moss

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract

We present a result that implies that an endofunctor on a category has a terminal coalgebra obtainable as a countable limit of its terminal-coalgebra sequence. It holds for finitary endofunctors preserving nonempty binary intersections on locally finitely presentable categories, assuming that the posets of strong quotients and subobjects of every finitely presentable object satisfy the descending chain condition. This allows one to adapt finiteness arguments that were originally advanced by Worrell concerning terminal coalgebras for finitary set functors. Examples include the categories of sets, posets, vector spaces, graphs, and nominal sets. A similar argument is presented for the category of metric spaces (although it is not locally finitely presentable).

Cite as

Jiří Adámek, Stefan Milius, and Lawrence S. Moss. Terminal Coalgebras for Finitary Functors. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{adamek_et_al:LIPIcs.CALCO.2025.3,
  author =	{Ad\'{a}mek, Ji\v{r}{\'\i} and Milius, Stefan and Moss, Lawrence S.},
  title =	{{Terminal Coalgebras for Finitary Functors}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.3},
  URN =		{urn:nbn:de:0030-drops-235623},
  doi =		{10.4230/LIPIcs.CALCO.2025.3},
  annote =	{Keywords: terminal coalgebra, countable iteration, descending chain condition}
}

Document

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

DOI: 10.4230/LIPIcs.ICALP.2025.165

Algebraic Language Theory with Effects

Authors: Fabian Lenke, Stefan Milius, Henning Urbat, and Thorsten Wißmann

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

Abstract

Regular languages - the languages accepted by deterministic finite automata - are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we generalize the correspondence between automata and monoids to automata with generic computational effects given by a monad, providing the foundations of an effectful algebraic language theory. We show that, under suitable conditions on the monad, a language is computable by an effectful automaton precisely when it is recognizable by (1) an effectful monoid morphism into an effect-free finite monoid, and (2) a monoid morphism into a monad-monoid bialgebra whose carrier is a finitely generated algebra for the monad, the former mode of recognition being conceptually completely new. Our prime application is a novel algebraic approach to languages computed by probabilistic finite automata. Additionally, we derive new algebraic characterizations for nondeterministic probabilistic finite automata and for weighted finite automata over unrestricted semirings, generalizing previous results on weighted algebraic recognition over commutative rings.

Cite as

Fabian Lenke, Stefan Milius, Henning Urbat, and Thorsten Wißmann. Algebraic Language Theory with Effects. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 165:1-165:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lenke_et_al:LIPIcs.ICALP.2025.165,
  author =	{Lenke, Fabian and Milius, Stefan and Urbat, Henning and Wi{\ss}mann, Thorsten},
  title =	{{Algebraic Language Theory with Effects}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{165:1--165:20},
  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.2025.165},
  URN =		{urn:nbn:de:0030-drops-235423},
  doi =		{10.4230/LIPIcs.ICALP.2025.165},
  annote =	{Keywords: Automaton, Monoid, Monad, Effect, Algebraic language theory}
}

Document

DOI: 10.4230/LIPIcs.CSL.2025.33

Quantitative Graded Semantics and Spectra of Behavioural Metrics

Authors: Jonas Forster, Lutz Schröder, Paul Wild, Harsh Beohar, Sebastian Gurke, Barbara König, and Karla Messing

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the linear-time/ branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics vary in granularity, and are often characterized by fragments of suitable modal logics. In the latter respect, the quantitative case is, however, more involved than the two-valued one; in fact, we show that probabilistic metric trace distance cannot be characterized by any compositionally defined modal logic with unary modalities. We go on to provide a unifying treatment of spectra of behavioural metrics in the emerging framework of graded monads, working in coalgebraic generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we introduce algebraic presentations of graded monads on the category of metric spaces. Moreover, we provide a general criterion for a given real-valued modal logic to characterize a given behavioural distance. As a case study, we apply this criterion to obtain a new characteristic modal logic for trace distance in fuzzy metric transition systems.

Cite as

Jonas Forster, Lutz Schröder, Paul Wild, Harsh Beohar, Sebastian Gurke, Barbara König, and Karla Messing. Quantitative Graded Semantics and Spectra of Behavioural Metrics. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{forster_et_al:LIPIcs.CSL.2025.33,
  author =	{Forster, Jonas and Schr\"{o}der, Lutz and Wild, Paul and Beohar, Harsh and Gurke, Sebastian and K\"{o}nig, Barbara and Messing, Karla},
  title =	{{Quantitative Graded Semantics and Spectra of Behavioural Metrics}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{33:1--33:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.33},
  URN =		{urn:nbn:de:0030-drops-227907},
  doi =		{10.4230/LIPIcs.CSL.2025.33},
  annote =	{Keywords: transition systems, modal logics, coalgebras, behavioural metrics}
}

Document

Early Ideas

DOI: 10.4230/LIPIcs.CALCO.2023.24

Higher-Order Mathematical Operational Semantics (Early Ideas)

Authors: Sergey Goncharov, Stefan Milius, Lutz Schröder, Stelios Tsampas, and Henning Urbat

Published in: LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

Abstract

We present a higher-order extension of Turi and Plotkin’s abstract GSOS framework that retains the key feature of the latter: for every language whose operational rules are represented by a higher-order GSOS law, strong bisimilarity on the canonical operational model is a congruence with respect to the operations of the language. We further extend this result to weak (bi-)similarity, for which a categorical account of Howe’s method is developed. It encompasses, for instance, Abramsky’s classical compositionality theorem for applicative similarity in the untyped λ-calculus. In addition, we give first steps of a theory of logical relations at the level of higher-order abstract GSOS.

Cite as

Sergey Goncharov, Stefan Milius, Lutz Schröder, Stelios Tsampas, and Henning Urbat. Higher-Order Mathematical Operational Semantics (Early Ideas). In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 24:1-24:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goncharov_et_al:LIPIcs.CALCO.2023.24,
  author =	{Goncharov, Sergey and Milius, Stefan and Schr\"{o}der, Lutz and Tsampas, Stelios and Urbat, Henning},
  title =	{{Higher-Order Mathematical Operational Semantics}},
  booktitle =	{10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)},
  pages =	{24:1--24:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-287-7},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{270},
  editor =	{Baldan, Paolo and de Paiva, Valeria},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.24},
  URN =		{urn:nbn:de:0030-drops-188213},
  doi =		{10.4230/LIPIcs.CALCO.2023.24},
  annote =	{Keywords: Abstract GSOS, lambda-calculus, applicative bisimilarity, bialgebra}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2023.48

Positive Data Languages

Authors: Florian Frank, Stefan Milius, and Henning Urbat

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

Abstract

Positive data languages are languages over an infinite alphabet closed under possibly non-injective renamings of data values. Informally, they model properties of data words expressible by assertions about equality, but not inequality, of data values occurring in the word. We investigate the class of positive data languages recognizable by nondeterministic orbit-finite nominal automata, an abstract form of register automata introduced by Bojańczyk, Klin, and Lasota. As our main contribution we provide a number of equivalent characterizations of that class in terms of positive register automata, monadic second-order logic with positive equality tests, and finitely presentable nondeterministic automata in the categories of nominal renaming sets and of presheaves over finite sets.

Cite as

Florian Frank, Stefan Milius, and Henning Urbat. Positive Data Languages. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{frank_et_al:LIPIcs.MFCS.2023.48,
  author =	{Frank, Florian and Milius, Stefan and Urbat, Henning},
  title =	{{Positive Data Languages}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{48:1--48:15},
  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.2023.48},
  URN =		{urn:nbn:de:0030-drops-185828},
  doi =		{10.4230/LIPIcs.MFCS.2023.48},
  annote =	{Keywords: Data Languages, Register Automata, MSO, Nominal Sets, Presheaves}
}

Document

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

DOI: 10.4230/LIPIcs.ICALP.2023.114

Nominal Topology for Data Languages

Authors: Fabian Birkmann, Stefan Milius, and Henning Urbat

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

Abstract

We propose a novel topological perspective on data languages recognizable by orbit-finite nominal monoids. For this purpose, we introduce pro-orbit-finite nominal topological spaces. Assuming globally bounded support sizes, they coincide with nominal Stone spaces and are shown to be dually equivalent to a subcategory of nominal boolean algebras. Recognizable data languages are characterized as topologically clopen sets of pro-orbit-finite words. In addition, we explore the expressive power of pro-orbit-finite equations by establishing a nominal version of Reiterman’s pseudovariety theorem.

Cite as

Fabian Birkmann, Stefan Milius, and Henning Urbat. Nominal Topology for Data Languages. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 114:1-114:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{birkmann_et_al:LIPIcs.ICALP.2023.114,
  author =	{Birkmann, Fabian and Milius, Stefan and Urbat, Henning},
  title =	{{Nominal Topology for Data Languages}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{114:1--114:21},
  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.2023.114},
  URN =		{urn:nbn:de:0030-drops-181662},
  doi =		{10.4230/LIPIcs.ICALP.2023.114},
  annote =	{Keywords: Nominal sets, Stone duality, Profinite space, Data languages}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2022.30

Stateful Structural Operational Semantics

Authors: Sergey Goncharov, Stefan Milius, Lutz Schröder, Stelios Tsampas, and Henning Urbat

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Abstract

Compositionality of denotational semantics is an important concern in programming semantics. Mathematical operational semantics in the sense of Turi and Plotkin guarantees compositionality, but seen from the point of view of stateful computation it applies only to very fine-grained equivalences that essentially assume unrestricted interference by the environment between any two statements. We introduce the more restrictive stateful SOS rule format for stateful languages. We show that compositionality of two more coarse-grained semantics, respectively given by assuming read-only interference or no interference between steps, remains an undecidable property even for stateful SOS. However, further restricting the rule format in a manner inspired by the cool GSOS formats of Bloom and van Glabbeek, we obtain the streamlined and cool stateful SOS formats, which respectively guarantee compositionality of the two more abstract equivalences.

Cite as

Sergey Goncharov, Stefan Milius, Lutz Schröder, Stelios Tsampas, and Henning Urbat. Stateful Structural Operational Semantics. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goncharov_et_al:LIPIcs.FSCD.2022.30,
  author =	{Goncharov, Sergey and Milius, Stefan and Schr\"{o}der, Lutz and Tsampas, Stelios and Urbat, Henning},
  title =	{{Stateful Structural Operational Semantics}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.30},
  URN =		{urn:nbn:de:0030-drops-163111},
  doi =		{10.4230/LIPIcs.FSCD.2022.30},
  annote =	{Keywords: Structural Operational Semantics, Rule Formats, Distributive Laws}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2021.78

Syntactic Minimization Of Nondeterministic Finite Automata

Authors: Robert S. R. Myers and Henning Urbat

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

Abstract

Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality. We prove that converting a monoid recognizing a regular language into a small subatomic acceptor corresponds to an NP-complete problem. The NP certificates are solutions of simple equations involving relations over the syntactic monoid. We also consider the subclass of atomic nondeterministic automata introduced by Brzozowski and Tamm. Given a deterministic automaton and another one for the reversed language, computing small atomic acceptors is shown to be NP-complete with analogous certificates. Our complexity results emerge from an algebraic characterization of (sub)atomic acceptors in terms of deterministic automata with semilattice structure, combined with an equivalence of categories leading to succinct representations.

Cite as

Robert S. R. Myers and Henning Urbat. Syntactic Minimization Of Nondeterministic Finite Automata. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{myers_et_al:LIPIcs.MFCS.2021.78,
  author =	{Myers, Robert S. R. and Urbat, Henning},
  title =	{{Syntactic Minimization Of Nondeterministic Finite Automata}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{78:1--78:16},
  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.2021.78},
  URN =		{urn:nbn:de:0030-drops-145186},
  doi =		{10.4230/LIPIcs.MFCS.2021.78},
  annote =	{Keywords: Algebraic language theory, Nondeterministic automata, NP-completeness}
}

Document

DOI: 10.4230/LIPIcs.CONCUR.2021.4

Nominal Büchi Automata with Name Allocation

Authors: Henning Urbat, Daniel Hausmann, Stefan Milius, and Lutz Schröder

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Abstract

Infinite words over infinite alphabets serve as models of the temporal development of the allocation and (re-)use of resources over linear time. We approach ω-languages over infinite alphabets in the setting of nominal sets, and study languages of infinite bar strings, i.e. infinite sequences of names that feature binding of fresh names; binding corresponds roughly to reading letters from input words in automata models with registers. We introduce regular nominal nondeterministic Büchi automata (Büchi RNNAs), an automata model for languages of infinite bar strings, repurposing the previously introduced RNNAs over finite bar strings. Our machines feature explicit binding (i.e. resource-allocating) transitions and process their input via a Büchi-type acceptance condition. They emerge from the abstract perspective on name binding given by the theory of nominal sets. As our main result we prove that, in contrast to most other nondeterministic automata models over infinite alphabets, language inclusion of Büchi RNNAs is decidable and in fact elementary. This makes Büchi RNNAs a suitable tool for applications in model checking.

Cite as

Henning Urbat, Daniel Hausmann, Stefan Milius, and Lutz Schröder. Nominal Büchi Automata with Name Allocation. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{urbat_et_al:LIPIcs.CONCUR.2021.4,
  author =	{Urbat, Henning and Hausmann, Daniel and Milius, Stefan and Schr\"{o}der, Lutz},
  title =	{{Nominal B\"{u}chi Automata with Name Allocation}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.4},
  URN =		{urn:nbn:de:0030-drops-143813},
  doi =		{10.4230/LIPIcs.CONCUR.2021.4},
  annote =	{Keywords: Data languages, infinite words, nominal sets, inclusion checking}
}

Document

DOI: 10.4230/LIPIcs.CONCUR.2020.44

Residual Nominal Automata

Authors: Joshua Moerman and Matteo Sammartino

Published in: LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

Abstract

We are motivated by the following question: which nominal languages admit an active learning algorithm? This question was left open in previous work, and is particularly challenging for languages recognised by nondeterministic automata. To answer it, we develop the theory of residual nominal automata, a subclass of nondeterministic nominal automata. We prove that this class has canonical representatives, which can always be constructed via a finite number of observations. This property enables active learning algorithms, and makes up for the fact that residuality - a semantic property - is undecidable for nominal automata. Our construction for canonical residual automata is based on a machine-independent characterisation of residual languages, for which we develop new results in nominal lattice theory. Studying residuality in the context of nominal languages is a step towards a better understanding of learnability of automata with some sort of nondeterminism.

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Joshua Moerman and Matteo Sammartino. Residual Nominal Automata. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 44:1-44:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{moerman_et_al:LIPIcs.CONCUR.2020.44,
  author =	{Moerman, Joshua and Sammartino, Matteo},
  title =	{{Residual Nominal Automata}},
  booktitle =	{31st International Conference on Concurrency Theory (CONCUR 2020)},
  pages =	{44:1--44:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-160-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{171},
  editor =	{Konnov, Igor and Kov\'{a}cs, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.44},
  URN =		{urn:nbn:de:0030-drops-128563},
  doi =		{10.4230/LIPIcs.CONCUR.2020.44},
  annote =	{Keywords: nominal automata, residual automata, derivative language, decidability, closure, exact learning, lattice theory}
}

Document

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

DOI: 10.4230/LIPIcs.ICALP.2019.130

Varieties of Data Languages (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors: Henning Urbat and Stefan Milius

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract

We establish an Eilenberg-type correspondence for data languages, i.e. languages over an infinite alphabet. More precisely, we prove that there is a bijective correspondence between varieties of languages recognized by orbit-finite nominal monoids and pseudovarieties of such monoids. This is the first result of this kind for data languages. Our approach makes use of nominal Stone duality and a recent category theoretic generalization of Birkhoff-type theorems that we instantiate here for the category of nominal sets. In addition, we prove an axiomatic characterization of weak pseudovarieties as those classes of orbit-finite monoids that can be specified by sequences of nominal equations, which provides a nominal version of a classical theorem of Eilenberg and Schützenberger.

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Henning Urbat and Stefan Milius. Varieties of Data Languages (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 130:1-130:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{urbat_et_al:LIPIcs.ICALP.2019.130,
  author =	{Urbat, Henning and Milius, Stefan},
  title =	{{Varieties of Data Languages}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{130:1--130:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.130},
  URN =		{urn:nbn:de:0030-drops-107063},
  doi =		{10.4230/LIPIcs.ICALP.2019.130},
  annote =	{Keywords: Nominal sets, Stone duality, Algebraic language theory, Data languages}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2017.43

Eilenberg Theorems for Free

Authors: Henning Urbat, Jiri Adámek, Liang-Ting Chen, and Stefan Milius

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract

Eilenberg-type correspondences, relating varieties of languages (e.g., of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. We show that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on infinity-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two categorical approaches of Bojanczyk and of Adamek et al. In addition we derive new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.

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Henning Urbat, Jiri Adámek, Liang-Ting Chen, and Stefan Milius. Eilenberg Theorems for Free. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{urbat_et_al:LIPIcs.MFCS.2017.43,
  author =	{Urbat, Henning and Ad\'{a}mek, Jiri and Chen, Liang-Ting and Milius, Stefan},
  title =	{{Eilenberg Theorems for Free}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.43},
  URN =		{urn:nbn:de:0030-drops-81032},
  doi =		{10.4230/LIPIcs.MFCS.2017.43},
  annote =	{Keywords: Eilenberg's theorem, variety of languages, pseudovariety, monad, duality}
}

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