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### Positive fragments of coalgebraic logics

Adriana Balan, Alexander Kurz, Jiří Velebil, Reiko Heckel
2015 Logical Methods in Computer Science
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic.  ...  In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for  ...  POSITIVE FRAGMENTS OF COALGEBRAIC LOGICS 9 Remark 2.7. Alg(L ′ ) is the category of positive modal algebras of Dunn  . We shall later see in Corollary 5.17 that it is isomorphic to Alg(L ′ ).  ...

### Positive Fragments of Coalgebraic Logics [chapter]

Adriana Balan, Alexander Kurz, Jiří Velebil
2013 Lecture Notes in Computer Science
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic.  ...  To see the relevance of a presentation result specific to monotone operations, observe that in the example of positive modal logic it is indeed the case that both and ♦ are monotone.  ...  POSITIVE FRAGMENTS OF COALGEBRAIC LOGICS 9 Remark 2.7. Alg(L ′ ) is the category of positive modal algebras of Dunn  . We shall later see in Corollary 5.17 that it is isomorphic to Alg(L ′ ).  ...

### Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective [chapter]

Yde Venema
2014 Outstanding Contributions to Logic
Keywords modal fixpoint logic, bisimulation invariance, coalgebra, coalgebraic modal logic, predicate liftings, monadic second-order logic.  ...  One of van Benthem's seminal results is the Bisimulation Theorem characterizing modal logic as the bisimulation-invariant fragment of first-order logic.  ...  Acknowledgements The research of this author has been made possible by VICI grant 639.073.501 of the Netherlands Organization for Scientific Research (NWO).  ...

### An institutional approach to positive coalgebraic logic

Adriana Balan, Alexander Kurz, Jiří Velebil
2015 Journal of Logic and Computation
T (L, δ) and (L , δ ) are the (strongly finitary) logics of T and T Then L is the positive fragment of L.  ...  fragments of coalgebraic logics, CALCO2013, arXiv:1402.5922 (2014) Cîrstea06 An institution of modal logics for coalgebras, J.  ...  Poset-functors and their coalgebras T : Poset → Poset locally monotone functor T -coalgebras Partially ordered set of states X = (X , ≤) Examples -ordered automata: X → X A × 2, with A the (discrete) input  ...

### Logics Admitting Final Semantics [chapter]

Alexander Kurz
2002 Lecture Notes in Computer Science
It is shown that a logic admits nal semantics i the formulas of the logic are preserved under coproducts (disjoint unions) and quotients (homomorphic images).  ...  A logic for coalgebras is said to admit nal semantics i | up to some technical requirements|all de nable classes contain a fully abstract nal coalgebra.  ...  still can have distinct bisimilar points, and to Falk Bartels for suggesting P ne when looking for a category of coalgebras with a nal but not all cofree coalgebras.  ...

### Subsumption Checking in Conjunctive Coalgebraic Fixpoint Logics [article]

Daniel Gorín, Lutz Schröder
2014 arXiv   pre-print
We work in the framework of coalgebraic logic as a unifying semantic setting.  ...  Most notably we find tractable fragments of game logic and the alternating-time mu-calculus.  ...  Catesbeiana for unsolicited remarks regarding the absence of unsatisfiable formulas in EL.  ...

### Model Checking Linear Coalgebraic Temporal Logics: An Automata-Theoretic Approach [chapter]

Corina Cîrstea
2011 Lecture Notes in Computer Science
We then consider linear coalgebraic temporal logics interpreted over non-deterministic coalgebras, and show how to reduce the model checking problem for such logics to the problem of deciding the winner  ...  We extend the theory of maximal traces of pointed nondeterministic coalgebras by providing an automata-based characterisation of the set of maximal traces for finite such coalgebras.  ...  Such logics arise as certain fragments of the path-based coalgebraic temporal logics defined in  , and are interpreted over coalgebras of type P + •F .  ...

### Flat Coalgebraic Fixed Point Logics [chapter]

Lutz Schröder, Yde Venema
2010 Lecture Notes in Computer Science
Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus  ...  We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.  ...  Flat Coalgebraic Fixed Point Logics We briefly recall the generic framework of coalgebraic modal logic [21, 28] , and define its extension with flat fixed point operators, a fragment of the coalgebraic  ...

### Modular Games for Coalgebraic Fixed Point Logics

2008 Electronical Notes in Theoretical Computer Science
The results concerning the implicit negation and the alternation-free fragments of our logics make use of the notion of an S-modality (of some finite arity), with S a syntax constructor with an associated  ...  Coalgebraic fixed point logics were first considered in the work of Venema  , where a finitary version of the coalgebraic logic of Moss  was used as the underlying modal language.  ...  Macros As in the propositional μ-calculus, one can distinguish fragments of our coalgebraic fixed point logics that have desirable properties, for example the independent fixed point fragment IμL S (V)  ...

### Coalgebraic Predicate Logic [chapter]

Tadeusz Litak, Dirk Pattinson, Katsuhiko Sano, Lutz Schröder
2012 Lecture Notes in Computer Science
We study the expressive power of our language, contrasting it with both coalgebraic modal logic and existing first-order proposals for special classes of Set-coalgebras (apart for relational structures  ...  We propose a generalization of first-order logic originating in a neglected work by C.C.  ...  Section 4 gives both syntactic and semantic characterizations of coalgebraic modal logic as a fragment of CPL.  ...

### Structure and Power: an emerging landscape [article]

Samson Abramsky
2022 arXiv   pre-print
induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter  ...  One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic; the equivalences on structures  ...  considered: full modal logic of depth ≤ k, the diamond-only positive fragment, and graded modal logic  for the counting case. • The coalgebra number in this case recovers the property of being a  ...

### Coalgebraic Correspondence Theory [chapter]

Lutz Schröder, Dirk Pattinson
2010 Lecture Notes in Computer Science
invariant fragment of first-order logic.  ...  We lay the foundations of a first-order correspondence theory for coalgebraic logics that makes the transition structure explicit in the first-order modelling.  ...  Catesbeiana for suggesting the use of ⊥.  ...

### Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras

Sebastian Enqvist, Fatemeh Seifan, Yde Venema
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
Similar to well-known results for monadic second-order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of T-coalgebras that admit a tree-like supporting  ...  We then consider invariance under behavioral equivalence of MSO T -formulas; more in particular, we investigate whether the coalgebraic µ-calculus is the bisimulation-invariant fragment of MSO T .  ...  Introduction Logic, automata and coalgebra The aim of this paper is to strengthen the link between the areas of logic, automata and coalgebra.  ...

### Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras [article]

Sebastian Enqvist, Fatemeh Seifan, Yde Venema
2015 arXiv   pre-print
Similar to well-known results for monadic second-order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of coalgebras that admit a tree-like supporting  ...  We then consider invariance under behavioral equivalence of formulas; more in particular, we investigate whether the coalgebraic mu-calculus is the bisimulation-invariant fragment of monadic second-order  ...  Introduction Logic, automata and coalgebra The aim of this paper is to strengthen the link between the areas of logic, automata and coalgebra.  ...

### Coalgebraic Expressions

Robert S. R. Myers
2009 Fixed Points in Computer Science
There is an associated complete equational logic, analogous to Kleene algebra, which may now be seen as an equational presentation of a fragment of the coalgebraic µ-calculus.  ...  These expressions include the regular expressions, the free Kleene algebra with tests and fragments of CCS and Linear Temporal Logic.  ...  The equational logic ≡ K may be understood as an interesting equational presentation of a fragment of the coalgebraic µ-calculus.  ...
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