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46
Equational and implicational classes of coalgebras
, 2001
"... If F: Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of Fcoalgebras is a covariety, i.e., closed under H (homomorphic images), S (subcoalgebras) and � (sums), if and only if it can be de ned by a set of “coequations”. Similarly, quasicovarieties, i.e ..."
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Cited by 8 (3 self)
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If F: Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of Fcoalgebras is a covariety, i.e., closed under H (homomorphic images), S (subcoalgebras) and � (sums), if and only if it can be de ned by a set of “coequations”. Similarly, quasicovarieties, i.e., classes closed under H and � , can be characterized by implications of coequations. These results are analogous to the theorems of Birkhoff and of Mal’cev in classical
Quotients of fully nonlinear control systems
 SIAM Journal on Control and Optimization
, 2001
"... Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and charac ..."
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Cited by 7 (5 self)
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Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and characterize the structure of the quotient state/input space. This allows one to understand how states and inputs of the quotient system are related to states and inputs of the original system. We also introduce a notion of projectability which turns out to be equivalent to controlled invariance. This allows one to regard previous work on symmetries, partial symmetries, and controlled invariance as leading to special types of quotients. We also show the existence of quotients that are not induced by symmetries or controlled invariance. Such decompositions have a potential use in a theory of hierarchical control based on quotients.
Iterative circular coinduction for CoCasl in Isabelle/HOL
 FUNDAMENTAL APPROACHES TO SOFTWARE ENGINEERING, VOLUME 3442 OF LECT. NOTES COMPUT. SCI
, 2005
"... Coalgebra has in recent years been recognized as the framework of choice for the treatment of reactive systems at an appropriate level of generality. Proofs about the reactive behavior of a coalgebraic system typically rely on the method of coinduction. In comparison to ‘traditional ’ coinduction, ..."
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Cited by 6 (1 self)
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Coalgebra has in recent years been recognized as the framework of choice for the treatment of reactive systems at an appropriate level of generality. Proofs about the reactive behavior of a coalgebraic system typically rely on the method of coinduction. In comparison to ‘traditional ’ coinduction, which has the disadvantage of requiring the invention of a bisimulation relation, the method of circular coinduction allows a higher degree of automation. As part of an effort to provide proof support for the algebraiccoalgebraic specification language CoCasl, we develop a new coinductive proof strategy which iteratively constructs a bisimulation relation, thus arriving at a new variant of circular coinduction. Based on this result, we design and implement tactics for the theorem prover Isabelle which allow for both automatic and semiautomatic coinductive proofs. The flexibility of this approach is demonstrated by means of examples of (semi)automatic proofs of consequences of CoCasl specifications, automatically translated into Isabelle theories by means of the Bremen heterogeneous Casl tool set Hets.
A Coalgebraic Perspective on Linear Weighted Automata
, 2011
"... Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either ( ..."
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Cited by 4 (1 self)
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Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of statebased systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on
Dialgebraic Specification and Modeling
"... corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube ..."
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Cited by 4 (4 self)
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corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube
Modal Predicates and Coequations
, 2002
"... We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach. ..."
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Cited by 4 (2 self)
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We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach.
Completeness for flat modal fixpoint logics
 Annals of Pure and Applied Logic, 162(1):55 – 82
, 2010
"... This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for ..."
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Cited by 3 (1 self)
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This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for each γ ∈ Γ. The term ♯γ(ϕ1,..., ϕn) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, ϕ1,..., ϕn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L♯(Γ) on Kripke frames. We prove two results that solve this problem. First, let K♯(Γ) be the logic obtained from the basic polymodal K by adding a KozenPark style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ♯γ. Provided that each indexing formula γ satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K + ♯ (Γ) of K♯(Γ). This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ♯γ, of size bounded by the length of γ. Thus the axiom system K + (Γ) is finite whenever Γ is finite.
This research was supported by NASA grant NAG54301 and NSF grant AST9619552 to AW. MK was supported by KBN grant 2.P03D.006.1. Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under cont
, 2005
"... A partial component is a process which fails or dies at some stage, thus exhibiting a finite, more ephemeral behaviour than expected (e.g. operating system crash). Partiality — which is the rule rather than exception in formal modelling — can be treated mathematically via totalization techniques. In ..."
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Cited by 3 (2 self)
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A partial component is a process which fails or dies at some stage, thus exhibiting a finite, more ephemeral behaviour than expected (e.g. operating system crash). Partiality — which is the rule rather than exception in formal modelling — can be treated mathematically via totalization techniques. In the case of partial functions, totalization involves error values and exceptions. In the context of a coalgebraic approach to component semantics, this paper argues that the behavioural counterpart to such functional techniques should extend behaviour with tryagain cycles preventing from component collapse, thus extending totalization or transposition from the algebraic to the coalgebraic context. We show that a refinement relationship holds between original and totalized components which is reasoned about in a coalgebraic approach to component refinement expressed in the pointfree binary relation calculus. As part of the pragmatic aims of this research, we also address the factorization of every such totalized coalgebra into two coalgebraic components — the original one and an added frontend — which cooperate in a clientserver style. Key words: partial components, tryagain cycles, refinement, coalgebra 1
Pointwise Extensions of GSOSDefined Operations
"... Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for ..."
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Cited by 2 (2 self)
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Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for the endofunctor F = B × − on Set. If B comes with a binary operation +, one can define an addition operation ⊕ on streams coinductively: hd(σ ⊕ τ) = hd(σ) + hd(τ) tl(σ ⊕ τ) = tl(σ) ⊕ tl(τ). It is easy to see that these equations define a distributive law, i.e., a natural transformation λ: ΣF ⇒ FΣ, where ΣX = X 2 is the signature endofunctor corresponding to a single binary operation. The operation ⊕: B ω × B ω → B ω is now defined as the unique morphism to the final coalgebra as in: ΣB ω B ω