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Modal Logics for Coalgebras
, 2003
"... This presentation is to introduce basic notions and recent results on an approach to specify behaviors of coalgebras by means of modal logics, based on the lecture by Dirk Pattinson (Pattinson, 2003) in NASSLLI 2003. Contents 1 ..."
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This presentation is to introduce basic notions and recent results on an approach to specify behaviors of coalgebras by means of modal logics, based on the lecture by Dirk Pattinson (Pattinson, 2003) in NASSLLI 2003. Contents 1
Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
 Theoret. Comput. Sci
, 2002
"... This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the ca ..."
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Cited by 74 (29 self)
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This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations. 1
On Generalised Coinduction and Probabilistic Specification Formats: Distributive Laws in Coalgebraic Modelling
, 2004
"... ..."
Streams from a typetheoretic perspective
, 2010
"... A stream is a paradigm for an ‘infinite object ’ in a certain sense. There are two ‘poles ’ in the notion of infinity: µ transfinite entities, having a wellfounded structure, typified by the ordinal ω. We define functions on such: folds, primitive recursion,.... ν coinductive entities, explorable ‘i ..."
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. We define functions on such: folds, primitive recursion,.... ν coinductive entities, explorable ‘in perpetuity’, typified by streams B ω. We define functions to such: unfolds,.... The work presented here (joint with Ghani and Pattinson) concerns an interplay between initiality µ and finality ν
A note on expressive coalgebraic logics for finitary set functors
 J. Log. Comput
"... This paper has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L ∗ of the limit L of the first ω terms in the final sequence. L ∗ is the set of points in L which arise from all coalgebras using their canonical morphisms ..."
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Cited by 4 (0 self)
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morphisms into L, and it was used earlier for different purposes in Kurz and Pattinson [5]. Viglizzo in [11] showed that the same set L ∗ carried a final coalgebra structure for functors in a certain inductively defined family. Our first goal is to generalize this to all finitary endofunctors; the result
ON MINIMAL COALGEBRAS
"... Abstract. We define an outdegree for Fcoalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all Fcoalgebras, this class has a terminal object, which for many problems can stand in for the terminal Fcoalgebra, which need not exist in general. As exam ..."
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Cited by 4 (1 self)
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logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition. For every automaton A there exists a minimal automaton ∇(A), which displays
PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
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Cited by 37 (19 self)
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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
Expressive Logics for Coalgebras via Terminal Sequence Induction
 Notre Dame J. Formal Logic
, 2002
"... This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras. ..."
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Cited by 39 (12 self)
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This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras.
Completeness for the coalgebraic cover modality
 Logical Methods in Computer Science
"... We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics ..."
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Cited by 3 (1 self)
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and completeness proof is algebraic, and we employ Pattinson’s stratification method, showing that our derivation system can be stratified in ω many layers, corresponding to the modal depth of the formulas involved. In the proof of our main result we identify some new concepts and obtain some auxiliary results
A Finite Model Construction For Coalgebraic Modal Logic
"... In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness result ..."
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Cited by 36 (17 self)
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In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard HennessyMilner logic, graded modal logic and probabilistic modal logic.
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