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OrderSorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations
 Theoretical Computer Science
, 1992
"... This paper generalizes manysorted algebra (hereafter, MSA) to ordersorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of objectoriented programming), several forms of pol ..."
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Cited by 226 (34 self)
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This paper generalizes manysorted algebra (hereafter, MSA) to ordersorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of objectoriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including Quotient, Homomorphism, and Initiality Theorems. The paper's major mathematical results include a notion of OSA deduction, a Completeness Theorem for it, and an OSA Birkhoff Variety Theorem. We also develop conditional OSA, including Initiality, Completeness, and McKinseyMalcev Quasivariety Theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive runtime error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like STACK and LIST, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
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Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach
, 1998
"... . The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a ..."
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Cited by 97 (15 self)
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. The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation. Keywords. Bisimulation, probabilistic transition system, coalgebra, ultrametric space, Borel measure, final coalgebra. 1 Introduction For discrete probabilistic transition systems the notion of probabilistic bisimilarity of Larsen and Skou [LS91] is regarded as the basic process equivalence. The definition was given for reactive systems. However, Van Glabbeek, Smolka and Steffen s...
Completeness of manysorted equational logic
 Houston Journal of Mathematics
, 1985
"... ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and chara ..."
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Cited by 72 (6 self)
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ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and characterizes when the onesorted rules can still be used safely; it also characterizes the related question of when manysorted algebras can be represented as onesorted algebras. The paper contains a detailed introduction to Hall's theory of clones (later developed into "algebraic theories " by Lawvere and Benabou); this allows a full algebraization of manysorted equational deduction that is not possible with the usual fully invariant congruences on the free algebra on countably many generators. 1. Introduction. The
Objects and Classes, Coalgebraically
 ObjectOrientation with Parallelism and Persistence
, 1995
"... The coalgebraic perspective on objects and classes in objectoriented programming is elaborated: objects consist of a (unique) identifier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) specifications of objects. The creation of a "n ..."
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Cited by 71 (18 self)
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The coalgebraic perspective on objects and classes in objectoriented programming is elaborated: objects consist of a (unique) identifier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) specifications of objects. The creation of a "new" object of a class is described in terms of the terminal coalgebra satisfying the specification. We present a notion of "totally specified" class, which leads to particularly simple terminal coalgebras. We further describe local and global operational semantics for objects. Associated with the local operational semantics is a notion of bisimulation (for objects belonging to the same class), expressing observational indistinguishability. AMS Subject Classification (1991): 18C10, 03G30 CR Subject Classification (1991): D.1.5, D.2.1, E.1, F.1.1, F.3.0 Keywords & Phrases: object, class, (terminal) coalgebra, coalgebraic specification, bisimulation 1. Introduction Within the objectoriente...
Initial Algebra and Final Coalgebra Semantics for Concurrency
, 1994
"... The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial ..."
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Cited by 56 (9 self)
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The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.
Type Theories and ObjectOriented Programming
 ACM Computing Surveys
, 1988
"... Objectoriented programming is becoming a popular approach to the construction of complex software systems. Benefits of object orientation include support for modular design, code sharing, and extensibility. In order to make the most of these advantages, a type theory for objects and their interacti ..."
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Cited by 52 (0 self)
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Objectoriented programming is becoming a popular approach to the construction of complex software systems. Benefits of object orientation include support for modular design, code sharing, and extensibility. In order to make the most of these advantages, a type theory for objects and their interactions should be developed to aid checking and
Abstract versus concrete computation on metric partial algebras
 ACM Transactions on Computational Logic
, 2004
"... Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We pr ..."
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Cited by 29 (17 self)
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Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations. With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ‘while ’ programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability. The theory of abstract models is stable: there are many models of computation, and
ProtocolIndependent Secrecy
 In 2000 IEEE Symposium on Security and Privacy. IEEE Computer Society
, 2000
"... Inductive proofs of secrecy invariants for cryptographic protocols can be facilitated by separating the protocol dependent part from the protocolindependent part. Our secrecy theorem encapsulates the use of induction so that the discharge of protocolspecific proof obligations is reduced to firsto ..."
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Cited by 28 (0 self)
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Inductive proofs of secrecy invariants for cryptographic protocols can be facilitated by separating the protocol dependent part from the protocolindependent part. Our secrecy theorem encapsulates the use of induction so that the discharge of protocolspecific proof obligations is reduced to firstorder reasoning. Also, the verification conditions are modularly associated with the protocol messages. Secrecy proofs for OtwayRees and the corrected NeedhamSchroeder protocol are given.
Hidden Coinduction: Behavioral Correctness Proofs for Objects
 Mathematical Structures in Computer Science
, 1999
"... This paper unveils and motivates an ambitious programme of hidden algebraic research in software engineering, beginning with our general goals, continuing with an overview of results, and including some future plans. The main contribution is powerful hidden coinduction techniques for proving behavio ..."
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Cited by 27 (8 self)
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This paper unveils and motivates an ambitious programme of hidden algebraic research in software engineering, beginning with our general goals, continuing with an overview of results, and including some future plans. The main contribution is powerful hidden coinduction techniques for proving behavioral correctness of concurrent systems; several mechanical proofs are given using OBJ3. We also show how modularization, bisimulation, transition systems, concurrency and combinations of the functional, constraint, logic and object paradigms fit into hidden algebra. 1. Introduction