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Generalizing the powerset construction, coalgebraically
, 2010
"... Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a ..."
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Cited by 23 (9 self)
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Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.
The microcosm principle and concurrency in coalgebras
 I. HASUO, B. JACOBS, AND A. SOKOLOVA
, 2008
"... Coalgebras are categorical presentations of statebased systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final ..."
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Cited by 16 (10 self)
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Coalgebras are categorical presentations of statebased systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final coalgebra as an object in it. This phenomenon is what Baez and Dolan have called the microcosm principle, a prototypical example of which is “a monoid in a monoidal category.” In this paper we obtain a formalization of the microcosm principle in which such a nested model is expressed categorically as a suitable lax natural transformation. An application of this account is a general compositionality result which supports modular verification of complex systems.
Semantics of higherorder quantum computation via geometry of interaction,” Extended version
, 2011
"... Abstract—While much of the current study on quantum computation employs lowlevel formalisms such as quantum circuits, several highlevel languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by ..."
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Abstract—While much of the current study on quantum computation employs lowlevel formalisms such as quantum circuits, several highlevel languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by providing interactionbased semantics of a functional quantum programming language; the latter is based on linear lambda calculus and is equipped with features like the! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we also prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard’s Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques—largely due to their categorical formulation—is exploited for our move from classical to quantum. Keywordsquantum computation; lambda calculus; categorical semantics; geometry of interaction; realizability I.
Tracebased coinductive operational semantics for While; Bigstep and smallstep, relational and functional styles
 In Theorem Proving in Higher Order Logics, 22nd International Conference, TPHOLs 2009, volume 5674 of LNCS
, 2009
"... Abstract. We present four coinductive operational semantics for the While language accounting for both terminating and nonterminating program runs: bigstep and smallstep relational semantics and bigstep and smallstep functional semantics. The semantics employ traces (possibly infinite sequences ..."
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Abstract. We present four coinductive operational semantics for the While language accounting for both terminating and nonterminating program runs: bigstep and smallstep relational semantics and bigstep and smallstep functional semantics. The semantics employ traces (possibly infinite sequences of states) to record the states that program runs go through. The relational semantics relate statementstate pairs to traces, whereas the functional semantics return traces for statementstate pairs. All four semantics are equivalent. We formalize the semantics and their equivalence proofs in the constructive setting of Coq. 1
GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS
"... The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an ..."
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The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioural equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construction, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready
Complete iterativity for algebras with effects
 In Algebra and Coalgebra in Computer Science (Proc. Third International Conference
"... Abstract. Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a ..."
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Abstract. Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a canonical distributive law over any commutative monad M, hence a lifting of that endofunctor to the Kleisli category ofM. Then, for an arbitrary distributive law λ of an endofunctor H on Set over a monad M we introduce λcias. The cias for the corresponding lifting ofH (called Kleislicias) form a full subcategory of the category of λcias. For various monads of interest we prove that free Kleislicias coincide with free λcias, and these free algebras are given by free algebras for H. Finally, for three concrete examples of monads we prove that Kleislicias and λcias coincide and give a characterisation of those algebras. Key words: iterative algebra, monad, distributive law, initial algebra, terminal coalgebra 1
Generic forward and backward simulations II: Probabilistic simulations
 International Conference on Concurrency Theory (CONCUR 2010), Lect. Notes Comp. Sci
, 2010
"... Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincid ..."
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Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincides with the second, and that the second is a special case of the last. We investigate implications of this characterization; notably the JonssonLarsen simulation is shown to be sound, i.e. its existence implies trace inclusion. 1
A Hoare Logic for the Coinductive TraceBased BigStep Semantics of While
"... Abstract. In search for a foundational framework for reasoning about observable behavior of programs that may not terminate, we have previously devised a tracebased bigstep semantics for While. In this semantics, both traces and evaluation (relating initial states of program runs to traces they pr ..."
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Cited by 2 (0 self)
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Abstract. In search for a foundational framework for reasoning about observable behavior of programs that may not terminate, we have previously devised a tracebased bigstep semantics for While. In this semantics, both traces and evaluation (relating initial states of program runs to traces they produce) are defined coinductively. On terminating runs, it agrees with the standard inductive statebased semantics. Here we present a Hoare logic counterpart of our coinductive tracebased semantics and prove it sound and complete. Our logic subsumes both the partial correctness Hoare logic and the total correctness Hoare logic: they are embeddable. Since we work with a constructive underlying logic, the range of expressible program properties has a rich structure; in particular, we can distinguish between termination and nondivergence, e.g., unbounded total search fails to be terminating but is nonetheless nondivergent. Our metatheory is entirely constructive as well, and we have formalized it in Coq. 1
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
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Cited by 2 (1 self)
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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.