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### A Complete Logic for Behavioural Equivalence in Coalgebras of Finitary Set Functors [chapter]

David Sprunger
2016 Lecture Notes in Computer Science
Acknowledgements I owe thanks to Alexandra Silva for helpful conversations at WoLLIC at the start of this project and to Larry Moss for his expertise and encouragement throughout.  ...  Thanks also to the anonymous referees for their careful reading and thoughtful comments.  ...  Introduction In this paper, we propose a logic for detecting bisimilar states in coalgebras of finitary Set-endofunctors.  ...

### Systematic Construction of Temporal Logics for Dynamical Systems via Coalgebra [article]

Baltasar Trancón y Widemann
2012 arXiv   pre-print
Here, a systematic approach to the construction of temporal logics for dynamical systems is proposed: Firstly, it is noted that dynamical systems can be seen as coalgebras in various ways.  ...  Secondly, a straightforward standard construction of modal logics out of coalgebras, namely Moss's coalgebraic logic, is applied.  ...  The Kripke modalities , ♦ and the Moss modality ∇ for finitary P-coalgebraic logics are equivalent.  ...

### Ultrafilter Extensions for Coalgebras [chapter]

C. Kupke, A. Kurz, D. Pattinson
2005 Lecture Notes in Computer Science
It is wellknown that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence.  ...  the bisimulation-somewhereelse theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of  ...  This provides a model-theoretic characterisation of logical equivalence for finitary logics. Related Work.  ...

### On a Categorical Framework for Coalgebraic Modal Logic

Liang-Ting Chen, Achim Jung
2014 Electronical Notes in Theoretical Computer Science
Under common assumptions, the logic of all predicate liftings together with a complete axiomatisation exists for any type of coalgebras, and it is one-step expressive for finitary functors.  ...  With regards to the example above, expressiveness of multi-modal logic for P A ω -coalgebras is inherited from modal logic for P ω -coalgebras.  ...  Acknowledgement We would like to thank Paul Levy, Clemens Kupke and Alexander Kurz for insightful discussions and feedback on the current work.  ...

### Simulations and Bisimulations for Coalgebraic Modal Logics [chapter]

Daniel Gorín, Lutz Schröder
2013 Lecture Notes in Computer Science
We define a notion of Λ-simulation for coalgebraic modal logics, parametric in the choice of a set Λ of monotone predicate liftings for a functor T .  ...  That is, for finitary set-based coalgebras, Λ-bisimulation works under strictly more general assumptions than T -bisimulation in the sense of Aczel and Mendler.  ...  Acknowledgements We wish to thank Helle Hvid Hansen for useful pointers to the literature. Erwin R. Catesbeiana has voiced vague support for empty bisimulations.  ...

### Definability, Canonical Models, Compactness for Finitary Coalgebraic Modal Logic

Alexander Kurz, Dirk Pattinson
2002 Electronical Notes in Theoretical Computer Science
We then investigate definability and compactness for finitary coalgebraic modal logic, show that the final object in Beh ω (T ) generalises the notion of a canonical model in modal logic, and study the  ...  Based on the terminal sequence, notions of finitary behaviours and finitary predicates are introduced. A category Beh ω (T ) of coalgebras with morphisms preserving finitary behaviours is defined.  ...  Also thanks to Clemens Kupke for the careful reading of a previous draft.  ...

### Coalgebraic semantics of modal logics: An overview

Clemens Kupke, Dirk Pattinson
2011 Theoretical Computer Science
Coalgebras can be seen as a natural abstraction of Kripke frames. In the same sense, coalgebraic logics are generalised modal logics.  ...  , expressiveness and complexity in these settings, both for basic languages as well as a number of extensions, such as hybrid languages and fixpoints.  ...  Acknowledgements We thank our colleagues in the field for many lively discussions and the editors of this volume for their continued patience.  ...

### Strongly Complete Logics for Coalgebras

Alexander Kurz, Jiri Rosicky, Yde Venema
2012 Logical Methods in Computer Science
Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras.  ...  In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.  ...  The Finitary Modal Logic of Set-Coalgebras The aim of this section is to associate a modal logic to an arbitrary functor T : SetSet.  ...

### Equational Coalgebraic Logic

Alexander Kurz, Raul Leal
2009 Electronical Notes in Theoretical Computer Science
A fundamental question in this area is how to obtain, for an arbitrary functor T , a logic for T -coalgebras.  ...  The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics fall apart in general.  ...  Namely, if we are working over BA, all expressive coalgebraic logics for a finitary functor that preserves finite sets are equivalent, i.e. mutually translatable.  ...

### Coalgebraic Modal Logic Beyond Sets

Bartek Klin
2007 Electronical Notes in Theoretical Computer Science
The first abstract approach to logics for coalgebras was coalgebraic logic of Moss [25] , providing expressive logics for essentially all functors on Set.  ...  Polyadic coalgebraic modal logic is studied in the setting of locally presentable categories.  ...  for comments on a  ...

### Coequational Logic for Finitary Functors

Daniel Schwencke
2008 Electronical Notes in Theoretical Computer Science
Furthermore we prove a statement for semantical consequences of sets of coequations in the case of accessible functors.  ...  Coequations, which are subsets of a cofree coalgebra, can be viewed as properties of systems. In case of a polynomial functor, a logic of coequations was formulated by J. Adámek.  ...  Since the elements of a cofree coalgebra for a fixed functor can be viewed as "behaviour patterns" of coalgebras for this functor, coequations describe certain properties of such coalgebras.  ...

### Logic Column 15: Coalgebras and Their Logics [article]

Alexander Kurz
2006 arXiv   pre-print
By giving a suitably abstract description of transition systems as coalgebras, it is possible to derive logics for capturing properties of these transition systems in an elegant way.  ...  This article describes recent work on the topic of specifying properties of transition systems.  ...  ' X ∼ = T X has a canonical solution (in sets or classes), namely the final coalgebra, which is fully abstract wrt behavioural equivalence.  ...

### Bisimulations for coalgebras on Stone spaces

Sebastian Enqvist, Sumit Sourabh
2018 Journal of Logic and Computation
The main result of our paper is that bisimulation for a Stone coalgebra is the topological closure of bisimulation for the underlying Set coalgebra.  ...  We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our notion of bisimulation is sound and complete for behavioural equivalence, and generalizes Vietoris bisimulations [4].  ...  If L-bisimilarity coincides with behavioural equivalence in every pair of coalgebras, then we say that L-bisimilarity is sound and complete for behavioural equivalence.  ...

### Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions [article]

Paul Wild, Lutz Schröder
2022 arXiv   pre-print
; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic.  ...  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  ...  To prove Theorem 4.12 in the case of non-finitary T , we make use of unravellings of coalgebras: Every coalgebra is behaviourally equivalent to its unravelling: Lemma 4.14.  ...

### (Ω, Ξ)-Logic: On the Algebraic Extension of Coalgebraic Specifications

Rolf Hennicker, Alexander Kurz
1999 Electronical Notes in Theoretical Computer Science
We present an extension of standard coalgebraic specification techniques for statebased systems which allows us to integrate constants and n-ary operations in a smooth way and, moreover, leads to a simplification  ...  The framework of (Ω, Ξ)-logic can be considered as the result of a translation of concepts of observational logic (cf. [9]) into the coalgebraic world.  ...  Acknowledgements We want to thank Pietro Cenciarelli for the many discussions initiating our work on the connection between algebraic and coalgebraic specifications and also Alexander Knapp for carefully  ...
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