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ManySorted Coalgebraic Modal Logic: a Modeltheoretic Study
 Theoretical Informatics and Applications
, 2001
"... This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants an ..."
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Cited by 53 (3 self)
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This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Manysorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of manysorted predicate logic.
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 25 (15 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.
What is the Coalgebraic Analogue of Birkhoff's Variety Theorem?
 THEORETICAL COMPUTER SCIENCE
, 2000
"... Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. T ..."
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Cited by 7 (4 self)
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Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A and A + validate the same observable formulas. It is shown that a class of coalgebras is de nable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra) lter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations. Examples are
Coalgebras and Monads in the Semantics of Java
 Theoretical Computer Science
, 2002
"... This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at th ..."
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Cited by 4 (0 self)
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This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at the University of Nijmegen. It is shown how the monadic perspective gives rise to the relevant computational structure in Java (composition, extension and repetition), and how the coalgebraic perspective o#ers an associated program logic (with invariants, bisimulations, and Hoare logics) for reasoning about the computational structure provided by the monad.
A modal proof theory for final polynomial coalgebras. Theoret
 Comput. Sci
"... An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maxim ..."
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Cited by 4 (1 self)
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An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal ” sets of formulas that have natural syntactic closure properties. The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination ” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if if the deducibility relation is generated by countably many inference rules. A counterexample to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 1
Coalgebras and modal logics for parameterised endofunctors
, 2000
"... for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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Cited by 3 (3 self)
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for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
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 ..."
Abstract

Cited by 1 (0 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.