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19
Semantical Principles in the Modal Logic of Coalgebraic
"... Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natur ..."
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Cited by 34 (9 self)
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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.
Automata and fixed point logics: a coalgebraic perspective
 Electronic Notes in Theoretical Computer Science
, 2004
"... This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton a ..."
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Cited by 26 (11 self)
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This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite twoplayer graph game. We also introduce a language of coalgebraic fixed point logic for Fcoalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an Fautomaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed Fcoalgebras in which p holds.
Logical Relations for Monadic Types
, 2002
"... Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambdacal ..."
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Cited by 24 (7 self)
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Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambdacalculus. The treatment is categorical, and is based on notions of subsconing and distributivity laws for monads. Our approach has a number of interesting applications, including cases for lambdacalculi with nondeterminism (where being in logical relation means being bisimilar), dynamic name creation, and probabilistic systems.
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of thi ..."
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Cited by 24 (2 self)
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A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
Foundational, Compositional (Co)datatypes for HigherOrder Logic  Category Theory Applied to Theorem Proving
"... Higherorder logic (HOL) forms the basis of several popular interactive theorem provers. These follow the definitional approach, reducing highlevel specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in H ..."
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Cited by 16 (10 self)
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Higherorder logic (HOL) forms the basis of several popular interactive theorem provers. These follow the definitional approach, reducing highlevel specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes. We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a rich type constructor—a functor satisfying specific properties preserved by interesting categorical operations. Our ideas are formalized in Isabelle and implemented as a new definitional package, answering a longstanding user request.
Calculating invariants as coreflexive bisimulations
, 2008
"... Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as ..."
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Cited by 6 (6 self)
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Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants ’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds ’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical ParkMilner definition popular in process algebra.
Bisimulation for Neighbourhood Structures
, 2007
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notio ..."
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Cited by 5 (0 self)
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notions of 2 2bisimulation, behavioural equivalence and neighbourhood bisimulation (a notion based on pushouts), with the aim of finding the logically correct notion of equivalence on neighbourhood structures. Our results include relational characterisations for 2 2bisimulation and neighbourhood bisimulation, and an analogue of Van Benthem’s characterisation theorem for all three equivalence notions. We also show that behavioural equivalence gives rise to a HennessyMilner theorem, and that this is not the case for the other two equivalence notions.
Witnessing (Co)datatypes
"... Abstract. Datatypes and codatatypes are very useful for specifying and reasoning about (possibly infinite) computational processes. The interactive theorem prover Isabelle/HOL has been extended with a definitional package that supports both. Here we describe a complete procedure for deriving nonempt ..."
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Cited by 4 (4 self)
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Abstract. Datatypes and codatatypes are very useful for specifying and reasoning about (possibly infinite) computational processes. The interactive theorem prover Isabelle/HOL has been extended with a definitional package that supports both. Here we describe a complete procedure for deriving nonemptiness witnesses in the general mutually recursive, nested case—nonemptiness being a proviso for introducing new types in higherorder logic. The nonemptiness problem also provides an illuminating case study that shows the package in action, tracing its journey from abstract category theory to handson functionality. 1