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23
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 269 (36 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
Generic trace semantics via coinduction
 Logical Methods in Comp. Sci
, 2007
"... Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace ..."
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Cited by 35 (10 self)
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Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace
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.
Towards a Duality Result in the Modal Logic of Coalgebras
 In Coalgebraic Methods in Computer Science, volume 33 of ENTCS
, 2000
"... This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations ta ..."
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Cited by 22 (1 self)
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This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations take the form of an adjunction. The BAO associated with a coalgebra can be used for specification, e.g. of classes in objectoriented languages.
Proof Principles for Datatypes with Iterated Recursion
, 1997
"... . Data types like trees which are finitely branching and of (possibly) infinite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of [14, 15]. The technica ..."
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Cited by 16 (3 self)
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. Data types like trees which are finitely branching and of (possibly) infinite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of [14, 15]. The technical contribution of this paper involves a description of initial algebras and terminal coalgebras in total categories of fibrations for lifted "datafunctors". These lifted functors are used to formulate our proof principles. We test these principles by proving some elementary results for four kinds of trees (with finite or infinite breadth or depth) using the proof tool pvs. 1 Introduction Algebras and coalgebras are of wellestablished importance in computer science, notably in the theory of datatypes, where especially initial algebras and terminal coalgebras play a distinguished role. Over the past decade there is more and more interest in the logic associated with initial algebras and ter...
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 14 (12 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Generic trace theory
 International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci
, 2006
"... Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms ..."
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Cited by 14 (5 self)
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Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of “trace semantics” are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in
Scalars, monads and categories
 Quantum Physics and Linguistics. A Compositional, Diagrammatic Discourse
, 2013
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Freyd is Kleisli, for arrows
 In C. McBride, T. Uustalu, Proc. of Wksh. on Mathematically Structured Programming, MSFP 2006, Electron. Wkshs. in Computing. BCS
, 2006
"... Arrows have been introduced in functional programming as generalisations of monads. They also generalise comonads. Fundamental structures associated with (co)monads are Kleisli categories and categories of (EilenbergMoore) algebras. Hence it makes sense to ask if there are analogous structures for ..."
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Cited by 6 (2 self)
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Arrows have been introduced in functional programming as generalisations of monads. They also generalise comonads. Fundamental structures associated with (co)monads are Kleisli categories and categories of (EilenbergMoore) algebras. Hence it makes sense to ask if there are analogous structures for Arrows. In this short note we shall take first steps in this direction, and identify for instance the Freyd
Probabilities, Distribution Monads, and Convex Categories
"... Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again g ..."
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Cited by 4 (2 self)
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Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]probabilities. It will be shown that there are translations backandforth, in the form of an adjunction, between effect monoids and “convex ” monads. This convexity property is formalised, both for monads and for categories. In the end this leads to “triangles of adjunctions ” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1