Document

Early Ideas

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

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.

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

**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance of such a general result. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstraß theorem; this allows us to cover, for instance, behavioural ultrametrics.

Jonas Forster, Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, and Paul Wild. Quantitative Hennessy-Milner Theorems via Notions of Density. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 22:1-22:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{forster_et_al:LIPIcs.CSL.2023.22, author = {Forster, Jonas and Goncharov, Sergey and Hofmann, Dirk and Nora, Pedro and Schr\"{o}der, Lutz and Wild, Paul}, title = {{Quantitative Hennessy-Milner Theorems via Notions of Density}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {22:1--22:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.22}, URN = {urn:nbn:de:0030-drops-174836}, doi = {10.4230/LIPIcs.CSL.2023.22}, annote = {Keywords: Behavioural distances, coalgebra, characteristic modal logics, density, Hennessy-Milner theorems, quantale-enriched categories, Stone-Weierstra{\ss} theorems} }

Document

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

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.

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

**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

We introduce a framework for universal algebra in categories of relational structures given by finitary relational signatures and finitary or infinitary Horn theories, with the arity λ of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders (λ = ω) or metric spaces (λ = ω₁). We establish a bijective correspondence between λ-accessible enriched monads on the given category of relational structures and a notion of λ-ary algebraic theories (i.e. with operations of arity < λ), with the syntax of algebraic theories induced by the relational signature (e.g. inequations or equations-up-to-ε). We provide a generic sound and complete derivation system for such relational algebraic theories, thus in particular recovering (extensions of) recent systems of this type for monads on partial orders and metric spaces by instantiation. In particular, we present an ω₁-ary algebraic theory of metric completion. The theory-to-monad direction of our correspondence remains true for the case of κ-ary algebraic theories and κ-accessible monads for κ < λ, e.g. for finitary theories over metric spaces.

Chase Ford, Stefan Milius, and Lutz Schröder. Monads on Categories of Relational Structures. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 14:1-14:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ford_et_al:LIPIcs.CALCO.2021.14, author = {Ford, Chase and Milius, Stefan and Schr\"{o}der, Lutz}, title = {{Monads on Categories of Relational Structures}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio 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.CALCO.2021.14}, URN = {urn:nbn:de:0030-drops-153697}, doi = {10.4230/LIPIcs.CALCO.2021.14}, annote = {Keywords: monads, relational structures, Horn theories, relational logic} }

Document

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

Logics and automata models for languages over infinite alphabets, such as Freeze LTL and register automata, serve the verification of processes or documents with data. They relate tightly to formalisms over nominal sets, such as nondetermininistic orbit-finite automata (NOFAs), where names play the role of data. Reasoning problems in such formalisms tend to be computationally hard. Name-binding nominal automata models such as {regular nondeterministic nominal automata (RNNAs)} have been shown to be computationally more tractable. In the present paper, we introduce a linear-time fixpoint logic Bar-μTL} for finite words over an infinite alphabet, which features full negation and freeze quantification via name binding. We show by a nontrivial reduction to extended regular nondeterministic nominal automata that even though Bar-μTL} allows unrestricted nondeterminism and unboundedly many registers, model checking Bar-μTL} over RNNAs and satisfiability checking both have elementary complexity. For example, model checking is in 2ExpSpace, more precisely in parametrized ExpSpace, effectively with the number of registers as the parameter.

Daniel Hausmann, Stefan Milius, and Lutz Schröder. A Linear-Time Nominal μ-Calculus with Name Allocation. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 58:1-58:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hausmann_et_al:LIPIcs.MFCS.2021.58, author = {Hausmann, Daniel and Milius, Stefan and Schr\"{o}der, Lutz}, title = {{A Linear-Time Nominal \mu-Calculus with Name Allocation}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {58:1--58:18}, 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.2021.58}, URN = {urn:nbn:de:0030-drops-144987}, doi = {10.4230/LIPIcs.MFCS.2021.58}, annote = {Keywords: Model checking, linear-time logic, nominal sets} }

Document

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

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.

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

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

We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time 𝒪((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was 𝒪(m n).

Thorsten Wißmann, Stefan Milius, and Lutz Schröder. Explaining Behavioural Inequivalence Generically in Quasilinear Time. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 32:1-32:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{wimann_et_al:LIPIcs.CONCUR.2021.32, author = {Wi{\ss}mann, Thorsten and Milius, Stefan and Schr\"{o}der, Lutz}, title = {{Explaining Behavioural Inequivalence Generically in Quasilinear Time}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {32:1--32:18}, 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.2021.32}, URN = {urn:nbn:de:0030-drops-144094}, doi = {10.4230/LIPIcs.CONCUR.2021.32}, annote = {Keywords: bisimulation, partition refinement, modal logic, distinguishing formulae, coalgebra} }

Document

**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

Alternating-time temporal logic (ATL) and its extensions, including the alternating-time µ-calculus (AMC), serve the specification of the strategic abilities of coalitions of agents in concurrent game structures. The key ingredient of the logic are path quantifiers specifying that some coalition of agents has a joint strategy to enforce a given goal. This basic setup has been extended to let some of the agents (revocably) commit to using certain named strategies, as in ATL with explicit strategies (ATLES). In the present work, we extend ATLES with fixpoint operators and strategy disjunction, arriving at the alternating-time µ-calculus with disjunctive explicit strategies (AMCDES), which allows for a more flexible formulation of temporal properties (e.g. fairness) and, through strategy disjunction, a form of controlled non-determinism in commitments. Our main result is an ExpTime upper bound for satisfiability checking (which is thus ExpTime-complete). We also prove upper bounds QP (quasipolynomial time) and NP∩coNP for model checking under fixed interpretations of explicit strategies, and NP under open interpretation. Our key technical tool is a treatment of the AMCDES within the generic framework of coalgebraic logic, which in particular reduces the analysis of most reasoning tasks to the treatment of a very simple one-step logic featuring only propositional operators and next-step operators without nesting; we give a new model construction principle for this one-step logic that relies on a set-valued variant of first-order resolution.

Merlin Göttlinger, Lutz Schröder, and Dirk Pattinson. The Alternating-Time μ-Calculus with Disjunctive Explicit Strategies. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 26:1-26:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gottlinger_et_al:LIPIcs.CSL.2021.26, author = {G\"{o}ttlinger, Merlin and Schr\"{o}der, Lutz and Pattinson, Dirk}, title = {{The Alternating-Time \mu-Calculus with Disjunctive Explicit Strategies}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {26:1--26:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.26}, URN = {urn:nbn:de:0030-drops-134605}, doi = {10.4230/LIPIcs.CSL.2021.26}, annote = {Keywords: Alternating-time logic, multi-agent systems, coalitional strength} }

Document

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

Behavioural distances provide a fine-grained measure of equivalence in systems involving quantitative data, such as probabilistic, fuzzy, or metric systems. Like in the classical setting of crisp bisimulation-type equivalences, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy bisimulations that need not themselves be (pseudo-)metrics, in analogy to classical bisimulations (which need not be equivalence relations). The known instances of generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For non-expansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic.

Paul Wild and Lutz Schröder. Characteristic Logics for Behavioural Metrics via Fuzzy Lax Extensions. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 27:1-27:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{wild_et_al:LIPIcs.CONCUR.2020.27, author = {Wild, Paul and Schr\"{o}der, Lutz}, title = {{Characteristic Logics for Behavioural Metrics via Fuzzy Lax Extensions}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {27:1--27:23}, 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.2020.27}, URN = {urn:nbn:de:0030-drops-128394}, doi = {10.4230/LIPIcs.CONCUR.2020.27}, annote = {Keywords: Modal logic, behavioural distance, coalgebra, bisimulation, lax extension} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

The coalgebraic mu-calculus is a generic framework for fixpoint logics with varying branching types that subsumes, besides the standard relational mu-calculus, such diverse logics as the graded mu-calculus, the monotone mu-calculus, the probabilistic mu-calculus, and the alternating-time mu-calculus. In the present work, we give a local model checking algorithm for the coalgebraic mu-calculus using a coalgebraic variant of parity games that runs, under mild assumptions on the complexity of the so-called one-step satisfaction problem, in time p^k where p is a polynomial in the formula and model size and where k is the alternation depth of the formula. We show moreover that under the same assumptions, the model checking problem is in both NP and coNP, improving the complexity in all mentioned non-relational cases. If one-step satisfaction can be solved by means of small finite games, we moreover obtain standard parity games, ensuring quasi-polynomial run time. This applies in particular to the monotone mu-calculus, the alternating-time mu-calculus, and the graded mu-calculus with grades coded in unary.

Daniel Hausmann and Lutz Schröder. Game-Based Local Model Checking for the Coalgebraic mu-Calculus. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 35:1-35:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hausmann_et_al:LIPIcs.CONCUR.2019.35, author = {Hausmann, Daniel and Schr\"{o}der, Lutz}, title = {{Game-Based Local Model Checking for the Coalgebraic mu-Calculus}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.35}, URN = {urn:nbn:de:0030-drops-109373}, doi = {10.4230/LIPIcs.CONCUR.2019.35}, annote = {Keywords: Model checking, mu-calculus, coalgebraic logic, graded mu-calculus, probabilistic mu-calculus, parity games} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time - branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time - branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplary proofs of their expressiveness based on our generic criterion.

Ulrich Dorsch, Stefan Milius, and Lutz Schröder. Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 36:1-36:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorsch_et_al:LIPIcs.CONCUR.2019.36, author = {Dorsch, Ulrich and Milius, Stefan and Schr\"{o}der, Lutz}, title = {{Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {36:1--36:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.36}, URN = {urn:nbn:de:0030-drops-109384}, doi = {10.4230/LIPIcs.CONCUR.2019.36}, annote = {Keywords: Linear Time, Branching Time, Monads, System Equivalences, Modal Logics, Expressiveness} }

Document

**Published in:** LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula - the interpolant - to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. Examples of such logics include the modal logics K and KD, neighbourhood logic and its monotone variant, finite-monoid-weighted logics, and coalition logic. We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it has one-step interpolation. Moreover, we identify preservation of finite surjective weak pullbacks as a sufficient, and in the monotone case necessary, condition for one-step interpolation. We thus prove or reprove uniform interpolation for most of the examples listed above.

Fatemeh Seifan, Lutz Schröder, and Dirk Pattinson. Uniform Interpolation in Coalgebraic Modal Logic. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 21:1-21:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{seifan_et_al:LIPIcs.CALCO.2017.21, author = {Seifan, Fatemeh and Schr\"{o}der, Lutz and Pattinson, Dirk}, title = {{Uniform Interpolation in Coalgebraic Modal Logic}}, booktitle = {7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-033-0}, ISSN = {1868-8969}, year = {2017}, volume = {72}, editor = {Bonchi, Filippo 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.CALCO.2017.21}, URN = {urn:nbn:de:0030-drops-80415}, doi = {10.4230/LIPIcs.CALCO.2017.21}, annote = {Keywords: modal logic, coalgebraic logic, uniform interpolation, preservation of weak pullbacks} }

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**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in reactive verification; coalgebraic generality implies in particular that we cover not only classical relational systems but also various forms of weighted systems. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time O(m log n) where n and m are the numbers of nodes and edges, respectively. Instances of our generic algorithm thus match the runtime of the best known algorithms for unlabelled transition systems, Markov chains, and deterministic automata (with fixed alphabets), and improve the best known algorithms for Segala systems.

Ulrich Dorsch, Stefan Milius, Lutz Schröder, and Thorsten Wißmann. Efficient Coalgebraic Partition Refinement. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 32:1-32:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dorsch_et_al:LIPIcs.CONCUR.2017.32, author = {Dorsch, Ulrich and Milius, Stefan and Schr\"{o}der, Lutz and Wi{\ss}mann, Thorsten}, title = {{Efficient Coalgebraic Partition Refinement}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.32}, URN = {urn:nbn:de:0030-drops-77939}, doi = {10.4230/LIPIcs.CONCUR.2017.32}, annote = {Keywords: markov chains, deterministic finite automata, partition refinement, generic algorithm, paige-tarjan algorithm, transition systems} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We present a sound, complete, and optimal single-pass tableau algorithm for the alternation-free mu-calculus. The algorithm supports global caching with intermediate propagation and runs in time 2^O(n). In game-theoretic terms, our algorithm integrates the steps for constructing and solving the Büchi game arising from the input tableau into a single procedure; this is done on-the-fly, i.e. may terminate before the game has been fully constructed. This suggests a slogan to the effect that global caching = game solving on-the-fly. A prototypical implementation shows promising initial results.

Daniel Hausmann, Lutz Schröder, and Christoph Egger. Global Caching for the Alternation-free µ-Calculus. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 34:1-34:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hausmann_et_al:LIPIcs.CONCUR.2016.34, author = {Hausmann, Daniel and Schr\"{o}der, Lutz and Egger, Christoph}, title = {{Global Caching for the Alternation-free µ-Calculus}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.34}, URN = {urn:nbn:de:0030-drops-61724}, doi = {10.4230/LIPIcs.CONCUR.2016.34}, annote = {Keywords: modal logic, fixpoint logic, satisfiability, global caching, coalgebraic logic} }

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

Models of concurrent systems employ a wide variety of semantics inducing various notions of process equivalence, ranging from linear-time semantics such as trace equivalence to branching-time semantics such as strong bisimilarity. Many of these generalize to system types beyond standard transition systems, featuring, for example, weighted, probabilistic, or game-based transitions; this motivates the search for suitable coalgebraic abstractions of process equivalence that cover these orthogonal dimensions of generality, i.e. are generic both in the system type and in the notion of system equivalence. In recent joint work with Kurz, we have proposed a parametrization of system equivalence over an embedding of the coalgebraic type functor into a monad. In the present paper, we refine this abstraction to use graded monads, which come with a notion of depth that corresponds, e.g., to trace length or bisimulation depth. We introduce a notion of graded algebras and show how they play the role of formulas in trace logics.

Stefan Milius, Dirk Pattinson, and Lutz Schröder. Generic Trace Semantics and Graded Monads. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 253-269, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{milius_et_al:LIPIcs.CALCO.2015.253, author = {Milius, Stefan and Pattinson, Dirk and Schr\"{o}der, Lutz}, title = {{Generic Trace Semantics and Graded Monads}}, booktitle = {6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)}, pages = {253--269}, 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.2015.253}, URN = {urn:nbn:de:0030-drops-55389}, doi = {10.4230/LIPIcs.CALCO.2015.253}, annote = {Keywords: transition systems, monads, coalgebra, trace logics} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid
logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding.

Lutz Schröder and Dirk Pattinson. Named Models in Coalgebraic Hybrid Logic. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 645-656, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{schroder_et_al:LIPIcs.STACS.2010.2492, author = {Schr\"{o}der, Lutz and Pattinson, Dirk}, title = {{Named Models in Coalgebraic Hybrid Logic}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {645--656}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2492}, URN = {urn:nbn:de:0030-drops-24920}, doi = {10.4230/LIPIcs.STACS.2010.2492}, annote = {Keywords: Logic in computer science, semantics, deduction, modal logic, coalgebra} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.

Lutz Schröder and Dirk Pattinson. Strong Completeness of Coalgebraic Modal Logics. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 673-684, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{schroder_et_al:LIPIcs.STACS.2009.1855, author = {Schr\"{o}der, Lutz and Pattinson, Dirk}, title = {{Strong Completeness of Coalgebraic Modal Logics}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {673--684}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1855}, URN = {urn:nbn:de:0030-drops-18559}, doi = {10.4230/LIPIcs.STACS.2009.1855}, annote = {Keywords: Logic in computer science, Semantics, Deduction, Modal logic, Coalgebra} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5491, Spatial Cognition: Specialization and Integration (2007)

In the domain of qualitative constraint reasoning, a subfield of AI which has evolved in the past 25 years, a large number of calculi for efficient reasoning about spatial and temporal entities has been developed. Reasoning techniques developed for these constraint calculi typically rely on so-called composition tables of the calculus at hand, which allow for replacing semantic reasoning by symbolic operations. Often these composition tables are developed in a quite informal, pictorial manner and hence composition tables are prone to errors. In view of possible safety critical applications of
qualitative calculi, however, it is desirable to formally verify these composition tables. In general, the verification of composition tables is a tedious task, in particular in cases where the semantics of the calculus depends on higher-order constructs such as sets. In this paper we address this problem by presenting a heterogeneous proof method that allows for combining a higher-order proof assistance system (such as Isabelle) with an automatic (first order) reasoner (such as SPASS or VAMPIRE). The benefit of this method is that the number of proof obligations that is to be proven interactively with a semi-automatic reasoner can be minimized to an acceptable level.

Stefan Wölfl, Till Mossakowski, and Lutz Schröder. Qualitative Constraint Calculi: Heterogeneous Verification of Composition Tables. In Spatial Cognition: Specialization and Integration. Dagstuhl Seminar Proceedings, Volume 5491, pp. 1-12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2007)

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@InProceedings{wolfl_et_al:DagSemProc.05491.9, author = {W\"{o}lfl, Stefan and Mossakowski, Till and Schr\"{o}der, Lutz}, title = {{Qualitative Constraint Calculi: Heterogeneous Verification of Composition Tables}}, booktitle = {Spatial Cognition: Specialization and Integration}, pages = {1--12}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {5491}, editor = {Anthony G. Cohn and Christian Freksa and Bernhard Nebel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05491.9}, URN = {urn:nbn:de:0030-drops-9798}, doi = {10.4230/DagSemProc.05491.9}, annote = {Keywords: Knowledge representation and reasoning, geometric and spatial reasoning, qualitative reasoning, automated versus interaction proving, heterogeneous} }

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