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A Complete Logic for Behavioural Equivalence in Coalgebras of Finitary Set Functors
[chapter]
2016
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-319-40370-0_10
fatcat:ure6ah7mvjhene4ejd7pc7dcma
Systematic Construction of Temporal Logics for Dynamical Systems via Coalgebra
[article]
2012
arXiv
pre-print
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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. ...
arXiv:1212.1734v1
fatcat:yvtvvaxkrfbzjosqzjopixpg2q
Ultrafilter Extensions for Coalgebras
[chapter]
2005
Lecture Notes in Computer Science
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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. ...
doi:10.1007/11548133_17
fatcat:bxsfjcbgwfbcnchuzmq3sxzqmi
On a Categorical Framework for Coalgebraic Modal Logic
2014
Electronical Notes in Theoretical Computer Science
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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. ...
doi:10.1016/j.entcs.2014.10.007
fatcat:pgjloiin3vc47fek5ho55jjmdu
Simulations and Bisimulations for Coalgebraic Modal Logics
[chapter]
2013
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-642-40206-7_19
fatcat:gprfvxutw5g4nkh24wxbjoy2nu
Definability, Canonical Models, Compactness for Finitary Coalgebraic Modal Logic
2002
Electronical Notes in Theoretical Computer Science
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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. ...
doi:10.1016/s1571-0661(04)80363-3
fatcat:zk4twquh3nbuniq2z7w4d2yasu
Coalgebraic semantics of modal logics: An overview
2011
Theoretical Computer Science
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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. ...
doi:10.1016/j.tcs.2011.04.023
fatcat:ffi53okdkzeqtbyqt5phjdrfaa
Strongly Complete Logics for Coalgebras
2012
Logical Methods in Computer Science
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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 : Set → Set. ...
doi:10.2168/lmcs-8(3:14)2012
fatcat:hic7twwz4rb4jbibiz2r2ebxyq
Equational Coalgebraic Logic
2009
Electronical Notes in Theoretical Computer Science
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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. ...
doi:10.1016/j.entcs.2009.07.097
fatcat:qwz6upbojjck3lsqgdy3vjrady
Coalgebraic Modal Logic Beyond Sets
2007
Electronical Notes in Theoretical Computer Science
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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 ...
doi:10.1016/j.entcs.2007.02.034
fatcat:ysh64itqcbcibl47fgwz3mosgm
Coequational Logic for Finitary Functors
2008
Electronical Notes in Theoretical Computer Science
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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. ...
doi:10.1016/j.entcs.2008.05.028
fatcat:4d3wbjngtvbv5ke7gvjhrpd2eu
Logic Column 15: Coalgebras and Their Logics
[article]
2006
arXiv
pre-print
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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. ...
arXiv:cs/0605128v1
fatcat:tkr62ke6rnaohapvpl4tz6elle
Bisimulations for coalgebras on Stone spaces
2018
Journal of Logic and Computation
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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. ...
doi:10.1093/logcom/exy001
fatcat:ks24sudl7zebpi3njgbzikle4q
Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions
[article]
2022
arXiv
pre-print
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; 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. ...
arXiv:2007.01033v5
fatcat:2p2vdpzofbgqphqrzg4srf6s7q
(Ω, Ξ)-Logic: On the Algebraic Extension of Coalgebraic Specifications
1999
Electronical Notes in Theoretical Computer Science
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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 ...
doi:10.1016/s1571-0661(05)80275-0
fatcat:hdapvjcvnnadzhb2weh7wy5ism
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