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38
Proof Methods for Corecursive Programs
 Fundamenta Informaticae Special Issue on Program Transformation
, 1999
"... This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion. ..."
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Cited by 21 (6 self)
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This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion.
Final Semantics for a Higher Order Concurrent Language
 CAAP'96 Conference Proceedings, H.Kirchner ed., Springer LNCS
, 1995
"... We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspond ..."
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Cited by 15 (11 self)
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We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspondingly, we derive various coinduction and mixed inductioncoinduction proof principles for establishing these equivalences.
A Coalgebraic Foundation for Linear Time Semantics
 In Category Theory and Computer Science
, 1999
"... We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for nondeterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised ..."
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Cited by 14 (1 self)
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We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for nondeterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category of the nonempty powerset monad. The corresponding final coalgebra semantics is fully abstract with respect to trace equivalence.
Proof Methods for Structured Corecursive Programs
, 1999
"... Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. Ho ..."
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Cited by 12 (4 self)
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Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather lowlevel, in the sense that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using an operator called unfold that encapsulates a common pattern of corecursive de nition, we can then use highlevel algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of either induction or coinduction.
Towards a Coalgebraic Semantics of UML: Class Diagrams and Use Cases
, 2003
"... Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.2 Multiple Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Semantics of Class ..."
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Cited by 9 (3 self)
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Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.2 Multiple Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Semantics of Class Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.6 Examples in Checking Consistency of Class Diagrams . . . . . . . . . . . . . . . . . . 35 5 Use Cases 37 5.1 Discussions on Advanced Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Related Work 43 7 Conclusion and Discussion 44 Acknowledgments 45 List of Figures iii List of Figures 1 Different representations of a class in UML . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Representation of an association class . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 8 Qualification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9 Aggregation and Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 10 An nary Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 11 The ternary association Record being decomposed . . . . . . . . . . . . . . . . . . . . 26 12 The decomposi...
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 9 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
Categorical Modelling of Structural Operational Rules  Case Studies
, 1997
"... . This paper aims at substantiating a recently introduced categorical theory of `wellbehaved' operational semantics. A variety of concrete examples of structural operational rules is modelled categorically illustrating the versatility and modularity of the theory. Further, a novel functorial notion ..."
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Cited by 8 (4 self)
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. This paper aims at substantiating a recently introduced categorical theory of `wellbehaved' operational semantics. A variety of concrete examples of structural operational rules is modelled categorically illustrating the versatility and modularity of the theory. Further, a novel functorial notion of guardedness is introduced which allows for a general and formal treatment of guarded recursive programs. Introduction The predominant approach to operational semantics is Plotkin's SOS [13], which is based on structural rules. One finds in the literature various formats of structural rules which guarantee a good behaviour such as having adequate denotational models and behavioural equivalence (eg bisimulation) being a congruence. In [17], it is shown that the rules in the best known of these formats, namely GSOS [5], are in 11 correspondence with natural transformations of a suitable type, depending on specific functorial notions of syntax and behaviour. This led to studying abstract ...
A Structural CoInduction Theorem
 PROC. MFPS '93, SPRINGER LNCS 802
, 1993
"... The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final c ..."
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Cited by 7 (1 self)
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The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order stronglyextensional (sometimes called internal full abstractness): the order is the union of all (ordered) Fbisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar coinduction theorem is given for a category of complete metric spaces and locally contracting functors.
Processes and Hyperuniverses
 Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science 1994, volume 841 of LNCS
, 1994
"... . We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses ar ..."
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Cited by 7 (0 self)
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. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard ZermeloFraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten. Introduction In the Semantics of ...
Final Dialgebras: From Categories to Allegories
 Workshop on Fixed Points in Computer Science
, 1999
"... The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is ..."
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Cited by 6 (3 self)
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The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets. The question thus arises of how to extend the standard categorical results on the existence of final objects in categories (for example, coalgebras and products) to their existence in allegories. The motivation is to streamline current work on generic programming, in which the use of a relational theory rather than a functional theory has proved to be desirable. In this paper, we define the notion of a relational final dialgebra and prove, for an important class of dialgebras, that a relational final dialgebra exists in an allegory if and only if a final dialgebra exists in the underlying category of map...