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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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Cited by 228 (34 self)
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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...
An ImplementationOriented Semantics for Module Composition
, 1997
"... This paper describes an approach to module composition by executing "module expressions" to build systems out of component modules; the paper also gives a novel semantics intended to aid implementers. The semantics is based on set theoretic notions of tuple set, partial signature, and institution, t ..."
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Cited by 32 (14 self)
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This paper describes an approach to module composition by executing "module expressions" to build systems out of component modules; the paper also gives a novel semantics intended to aid implementers. The semantics is based on set theoretic notions of tuple set, partial signature, and institution, thus avoiding more difficult mathematics theory. Language features include information hiding, both vertical and horizontal composition, and views for binding modules to interfaces. Vertical composition refers to the hierarchical structuring of a system into layers, while horizontal composition refers to the structure of a given layer. Modules may involve information hiding, and views may involve behavioral satisfaction of a theory by a module. Several "Laws of Software Composition" are given, which show how the various module composition operations relate. Taken together, this gives foundations for an algebraic approach to software engineering. 1.1 Introduction The approach to module compos...
Proving Correctness Of Refinement And Implementation
, 1996
"... The notions of state and observable behaviour are fundamental to many areas of computer science. Hidden sorted algebra, an extension of many sorted algebra, captures these notions through hidden sorts and the behavioural satisfaction of equations. This makes it a powerful formalisation of abstract m ..."
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Cited by 29 (4 self)
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The notions of state and observable behaviour are fundamental to many areas of computer science. Hidden sorted algebra, an extension of many sorted algebra, captures these notions through hidden sorts and the behavioural satisfaction of equations. This makes it a powerful formalisation of abstract machines, and many results suggest that it is also suitable for the semantics of the object paradigm. Another extension of many sorted algebra, namely order sorted algebra, has proved useful in system specification and prototyping because of the way it handles subtypes and errors. The combination of these two algebraic approaches, hidden order sorted algebra, has also been proposed as a foundation for object paradigm, and has much promise as a foundation for Software Engineering. This paper extends recent work on hidden order sorted algebra by investigating the refinement and implementation of hidden order sorted specifications. We present definitions of refinement and implementation for suc...
Hidden Congruent Deduction
 Automated Deduction in Classical and NonClassical Logics
, 1998
"... This paper presents some techniques of this kind in the area called hidden algebra, clustered around the central notion of coinduction. We believe hidden algebra is the natural next step in the evolution of algebraic semantics and its first order proof technology. Hidden algebra originated in [7], a ..."
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Cited by 27 (18 self)
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This paper presents some techniques of this kind in the area called hidden algebra, clustered around the central notion of coinduction. We believe hidden algebra is the natural next step in the evolution of algebraic semantics and its first order proof technology. Hidden algebra originated in [7], and was developed further in [8, 10, 3, 12, 5] among other places; the most comprehensive survey currently available is [12]
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 24 (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
Parameterized Programming and Software Architecture
 IN PROCEEDINGS, REUSE'96
, 1996
"... This paper discusses an approach to software architecture based on concepts from parameterized programming, particularly its language of "module expressions." A module expression describes the architecture of a system as an interconnection of component modules, and executing the expression actually ..."
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Cited by 19 (3 self)
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This paper discusses an approach to software architecture based on concepts from parameterized programming, particularly its language of "module expressions." A module expression describes the architecture of a system as an interconnection of component modules, and executing the expression actually builds the system. Language features include: modules parameterized by theories, which declare interfaces; a number of module composition operations; views for binding modules to interfaces; and both vertical and horizontal composition. Modules may involve information hiding, theories may declare semantic restrictions with axioms, and views assert behavioral satisfaction of axioms by a module. Some "laws of software composition" are given, showing how various module composition operations are related. We also show how a variety of architectural styles can be supported, and how this approach can be extended to support evolution and traceability. All this is intended to ease the development of large systems, and in particular, to make reuse more effective in practice.
Invariants, Bisimulations and the Correctness of Coalgebraic Refinements
 Techn. Rep. CSIR9704, Comput. Sci. Inst., Univ. of Nijmegen
, 1997
"... . Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the behaviour of ..."
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Cited by 12 (4 self)
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. Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the behaviour of the other, "abstract" class specification. Two (complete) prooftechniques are given to establish such refinements: one involving an invariant (a predicate that is closed under transitions) on the concrete class, and one involving a bisimulation (a relation that is closed under transitions) between the concrete and the abstract class. The latter can only be used if the abstract class is what we call totally specified. Parts of the underlying theory of invariants and bisimulations in a coalgebraic setting are included, involving least and greatest invariants and connections between invariants and bisimulations. Also, the proofprinciples are illustrated in examples (which are fully formalise...
A Hidden Herbrand Theorem: Combining the Object and Logic Paradigms
 Principles of Declarative Programming
, 1998
"... : The benefits of the object, logic (or relational), functional, and constraint paradigms ..."
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Cited by 11 (3 self)
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: The benefits of the object, logic (or relational), functional, and constraint paradigms
Hidden Algebra for Software Engineering
 Proceedings Combinatorics, Computation and Logic
, 1999
"... : This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, vectors, ma ..."
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Cited by 10 (0 self)
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: This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, vectors, matrices, and lists. Software engineering also needs changeable "abstract machines," recently called "objects," that can communicate concurrently with other objects through visible "attributes" and statechanging "methods." Hidden algebra is a new development in algebraic semantics designed to handle such systems. Equational theories are used in both cases, but the notion of satisfaction for hidden algebra is behavioral, in the sense that equations need only appear to be true under all possible experiments; this extra flexibility is needed to accommodate the clever implementations that software engineers often use to conserve space and/or time. The most important results in hidden algebra are ...
Interconnection of Object Specifications
 Formal Methods and Object Technology
, 1996
"... ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and ap ..."
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Cited by 8 (2 self)
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ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...