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13
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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. 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...
A Hidden Agenda
 Theoretical Computer Science
, 2000
"... This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behaviora ..."
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Cited by 120 (23 self)
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This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behavioral properties of concurrent systems, especially renements; some proofs are given using OBJ3. We also discuss where modularization, bisimulation, transition systems and combinations of the object, logic, constraint and functional paradigms t into our hidden agenda. 1 Introduction Algebra can be useful in many dierent ways in software engineering, including specication, validation, language design, and underlying theory. Specication and validation can help in the practical production of reliable programs, advances in language design can help improve the state of the art, and theory can help with building new tools to increase automation, as well as with showing correctness of the whole e...
Abstract CMCS’03 Preliminary Version Inductive Behavioral Proofs by Unhiding
"... We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, ope ..."
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We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, operations and equations), use it, and then hide it. We develop a procedure, called unhiding, which takes a finite B and produces a finite ˜ B. The practical aspect of this procedure is that one can use any standard equational inductive theorem prover to derive behavioral theorems, even if neither equational reasoning nor induction is sound for behavioral satisfaction. 1
Behavioral Abstraction is Hiding Information Abstract
"... We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, ope ..."
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We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, operations and equations), use it, and then hide it. We develop a procedure, called unhiding, which takes a finite B and produces a finite ˜ B. The practical aspect of this procedure is that one can use any standard equational inductive theorem prover to derive behavioral theorems, even if neither equational reasoning nor induction is sound for behavioral satisfaction.
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"... of Grigore Ro,su is approved, and it is acceptable in quality and form for publication on microfilm: ..."
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of Grigore Ro,su is approved, and it is acceptable in quality and form for publication on microfilm:
CMCS'03 Preliminary Version Inductive Behavioral Proofs by Unhiding
"... Abstract We show that for any behavioral \Sigmaspecification B there is an ordinary algebraic specification ~B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the \Sigmatheorems of ~B. The idea is to add machinery for contexts and experi ..."
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Abstract We show that for any behavioral \Sigmaspecification B there is an ordinary algebraic specification ~B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the \Sigmatheorems of ~B. The idea is to add machinery for contexts and experiments (sorts, operations and equations), use it, and then hide it. We develop a procedure, called unhiding, which takes a finite B and produces a finite ~B. The practical aspect of this procedure is that one can use any standard equational inductive theorem prover to derive behavioral theorems, even if neither equational reasoning nor induction is sound for behavioral satisfaction.
Behavioral Abstraction is Hiding Information Abstract
"... We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, ope ..."
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We show that for any behavioral Σspecification B there is an ordinary algebraic specification ˜ B over a larger signature, such that a model behaviorally satisfies B iff it satisfies, in the ordinary sense, the Σtheorems of ˜ B. The idea is to add machinery for contexts and experiments (sorts, operations and equations), use it, and then hide it. We develop a procedure, called unhiding, which takes a finite B and produces a finite ˜ B. The practical aspect of this procedure is that one can use any standard equational inductive theorem prover to derive behavioral theorems, even if neither equational reasoning nor induction is sound for behavioral satisfaction.
Dialgebraic Logics (Extended Abstract)
, 1998
"... Horst Reichel Institute: Theoretical Computer Science Dresden University of Technology D01062 Dresden, Germany Abstract We extend the notion of a firstorder signature in such a way that the type constructors used to define domain and codomain of the fundamental operations are taken to be a c ..."
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Horst Reichel Institute: Theoretical Computer Science Dresden University of Technology D01062 Dresden, Germany Abstract We extend the notion of a firstorder signature in such a way that the type constructors used to define domain and codomain of the fundamental operations are taken to be a constituent part of the signature. Using the generative power of the type constructors and the fundamental types and operations we obtain a general construction of a category of typed terms which will be called syntactic category. Functors into the category of set from a syntactic category preserving the used type constructors represent models and terms with the constant type Bool as codomain represent properties.
www.elsevier.com/locate/tcs A hidden agenda(
"... This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behaviora ..."
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This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behavioral properties of concurrent systems, especially renements; some proofs are given using OBJ3. We also discuss where modularization, bisimulation, transition systems and combinations of the object, logic, constraint and functional paradigms t into our hidden agenda.