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21
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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Cited by 408 (42 self)
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
An Application Of Coinductive Stream Calculus To Signal Flow Graphs
, 2003
"... This report contains a set of lecture notes that were used in the spring of 2003 for a mini course of six lectures on the subject of streams, coinduction and signal flow graphs. It presents an application of coinductive stream calculus (as introduced in Technical Report SEN-R0023, CWI, Amsterdam, ..."
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Cited by 17 (5 self)
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This report contains a set of lecture notes that were used in the spring of 2003 for a mini course of six lectures on the subject of streams, coinduction and signal flow graphs. It presents an application of coinductive stream calculus (as introduced in Technical Report SEN-R0023, CWI, Amsterdam, 2000) to signal flow graphs. In comparison to existing approaches, which are usually based on Laplace and Z-transforms, the model presented in these notes is very elementary. From a didactical point of view, the formal treatment of flow graphs is interesting because it deals with two fundamental phenomena in the theory of computation: memory (in the form of register or delay elements) and infinite behaviour (in the form of feedback).
Brzozowski’s algorithm (co)algebraically
"... Abstract. We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on the duality between reachability and observability. This leads to a simple proof of its correctn ..."
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Cited by 10 (2 self)
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Abstract. We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on the duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. 1
Algebra-Coalgebra Duality in Brzozowski’s Minimization Algorithm
"... We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simpl ..."
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Cited by 7 (2 self)
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We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal, language equivalent automata from Moore, non-deterministic and weighted automata.
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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Cited by 7 (1 self)
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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 (F-coalgebras) 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 non-deterministic 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
MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS
"... ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursively-defined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad ..."
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Cited by 5 (0 self)
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ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursively-defined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad. 1.
SHAPE: A Machine Learning System from Examples
, 1995
"... This paper presents a new machine learning system called SHAPE. The input data are vectors of properties (represented as attribute-value pairs) which are used to describe individual cases, examples or observations in a given world. Each case belongs to exactly one of a set of classes, and the aim ..."
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Cited by 4 (3 self)
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This paper presents a new machine learning system called SHAPE. The input data are vectors of properties (represented as attribute-value pairs) which are used to describe individual cases, examples or observations in a given world. Each case belongs to exactly one of a set of classes, and the aim is to produce a collection of decision rules concluding the class according to the properties observed.
INTERCONNECTION OF PROBABILISTIC SYSTEMS
, 2000
"... There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of ..."
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Cited by 3 (3 self)
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There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of
On partially additive Kleene algebras
- In Proc. 8th Int. Conf. Relational Methods in Computer Science (RelMiCS 8
, 2005
"... We define the notion of a partially additive Kleene algebra, which is a Kleene algebra where the + operation need only be partially defined. These structures formalize a number of examples that cannot be handled directly by Kleene algebras. We relate partially additive Kleene algebras to existing al ..."
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Cited by 2 (0 self)
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We define the notion of a partially additive Kleene algebra, which is a Kleene algebra where the + operation need only be partially defined. These structures formalize a number of examples that cannot be handled directly by Kleene algebras. We relate partially additive Kleene algebras to existing algebraic structures, by exhibiting categorical connections with Kleene algebras, partially additive categories, and closed semirings. 1
Algebraic representation of dynamics and behavior for continuous-time linear systems
- Math. Systems Theory
, 1991
"... continuous-time systems An algebraic approach to continuous-time linear systems is presented which closely parallels the discrete-time decomposable systems approach of Arbib and Manes, as well as the older k[z]module theory of linear systems of Kalman. The focal point of the presentation is a class ..."
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Cited by 1 (1 self)
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continuous-time systems An algebraic approach to continuous-time linear systems is presented which closely parallels the discrete-time decomposable systems approach of Arbib and Manes, as well as the older k[z]module theory of linear systems of Kalman. The focal point of the presentation is a class of topological rings, termed R+-rings, which play the same role for continuous time that k[z] does for discrete-time. Each such ring R defines a class of toplogical modules, termed the (R)-modules, which may be naturally identified with a class of locally equicontinuous semigroups, called the (R)semigroups. Thus, just as discrete-time linear dynamics are coextensive with k[z]-modules, so too are continuous-time linear dynamics coextensive with (R)-modules. This identification underlies the development of a purely algebraic theory of behavior and realization for continuous-time linear systems. The specific choice of R determines the type of dynamics allowed. For example, taking R to be the ring of all measures on the nonnegative reals yields dynamics described by the class of all semigroups, while choosing R to be the ring of all L 1 measures yields dynamics whose responses vanish at infinity.
