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15
The Temporal Logic of Coalgebras via Galois Algebras
, 1999
"... This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computatio ..."
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Cited by 33 (7 self)
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This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...
Elements Of The General Theory Of Coalgebras
, 1999
"... . Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intend ..."
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Cited by 30 (7 self)
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. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and coinduction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...
Covarieties and Complete Covarieties
, 1999
"... We present two ways to de ne covarieties and complete covarieties, i.e. covarieties that are closed under total bisimulation: by closure operators and by subcoalgebras of coalgebras. ..."
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Cited by 13 (3 self)
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We present two ways to de ne covarieties and complete covarieties, i.e. covarieties that are closed under total bisimulation: by closure operators and by subcoalgebras of coalgebras.
Coalgebraic Structure From Weak Limit Preserving Functors
, 1999
"... Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and ..."
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Cited by 13 (7 self)
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Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and subcoalgebras. We give a characterization of monos in Set F in terms of congruences and bisimulations, which explains, under which conditions monos must be injective maps.
A Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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Cited by 13 (5 self)
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkho# [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
MonoidLabeled Transition Systems
"... Given a # complete (semi)lattice L, we consider Llabeled transition systems as coalgebras of a functor L () , associating with a set X the set L X of all Lfuzzy subsets. We describe simulations and bisimulations of Lcoalgebras to show that L () weakly preserves nonempty kern ..."
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Cited by 11 (2 self)
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Given a # complete (semi)lattice L, we consider Llabeled transition systems as coalgebras of a functor L () , associating with a set X the set L X of all Lfuzzy subsets. We describe simulations and bisimulations of Lcoalgebras to show that L () weakly preserves nonempty kernel pairs i# it weakly preserves nonempty pullbacks i# L is join infinitely distributive (JID). Exchanging L for a commutative monoid M, we consider the functor M () # which associates with a set X all finite multisets containing elements of X with multiplicities m # M . The corresponding functor weakly preserves nonempty pullbacks along injectives i# 0 is the only invertible element of M, and it preserves nonempty kernel pairs i# M is refinable, in the sense that two sum representations of the same value, r 1 + . . . + r m = c 1 + . . . + c n , have a common refinement matrix (m i,j ) whose kth row sums to r k and whose lth column sums to c l for any 1 # k # m and 1 # l # n.
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 9 (4 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 Birkho and of Mal’cev
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.
From Tcoalgebras to filter structures and transition systems
 Algebra and Coalgebra in Computer Science
, 2005
"... Abstract. For any setendofunctor T: Set → Set there exists a largest subcartesian transformation µ to the filter functor F: Set → Set. Thus we can associate with every Tcoalgebra A a certain filtercoalgebra AF. Precisely, when T weakly preserves preimages, µ is natural, and when T weakly preserve ..."
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Cited by 4 (2 self)
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Abstract. For any setendofunctor T: Set → Set there exists a largest subcartesian transformation µ to the filter functor F: Set → Set. Thus we can associate with every Tcoalgebra A a certain filtercoalgebra AF. Precisely, when T weakly preserves preimages, µ is natural, and when T weakly preserves intersections, µ factors through the covariant powerset functor P, thus providing for every Tcoalgebra A a Kripke structure AP. The paper characterizes weak preservation of preimages, of intersections, and preservation of both preimages and intersections by a functor T via the existence of transformations from T to either F or P. Moreover, we define for arbitrary Tcoalgebras A a nexttime operator ○A with associated modal operators ✷ and ✸ and relate their properties to weak limit preservation properties of T. In particular, for any Tcoalgebra A there is a transition system K with ○A = ○K if and only if T weakly preserves intersections. 1.
State Based Systems Are Coalgebras
 Cubo  Matematica Educacional 5
, 2003
"... Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by corecursion. In this (entirely selfcontained) pap ..."
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Cited by 2 (0 self)
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Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by corecursion. In this (entirely selfcontained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment  from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the inputoutput behavior of the system. A simple example of a state based system is a digital watch with several buttons and a display. Clearly, the buttons that are pressed do not by themselves determine the output  it also depends on the internal state, which might include the current time, the mode (time/alarm/stopwatch), and perhaps the information which buttons have been pressed previously. The user of a system is normally not interested in knowing precisely, what the internal states of the system are, nor how they are represented. Of course, he might try to infer all possible states by testing various inputoutput combinations and attribute di#erent behaviors to di#erent states. Some states might not be distinguishable by their outside behavior. It is therefore natural to define an appropriate indistinguishability relation "#...