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31
Concurrent Transition Systems
 Theoretical Computer Science
, 1989
"... : Concurrent transition systems (CTS's), are ordinary nondeterministic transition systems that have been equipped with additional concurrency information, specified in terms of a binary residual operation on transitions. Each CTS C freely generates a complete CTS or computation category C , whose ..."
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Cited by 40 (5 self)
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: Concurrent transition systems (CTS's), are ordinary nondeterministic transition systems that have been equipped with additional concurrency information, specified in terms of a binary residual operation on transitions. Each CTS C freely generates a complete CTS or computation category C , whose arrows are equivalence classes of finite computation sequences, modulo a congruence induced by the concurrency information. The categorical composition on C induces a "prefix" partial order on its arrows, and the computations of C are conveniently defined to be the ideals of this partial order. The definition of computations as ideals has some pleasant properties, one of which is that the notion of a maximal ideal in certain circumstances can serve as a replacement for the more troublesome notion of a fair computation sequence. To illustrate the utility of CTS's, we use them to define and investigate a dataflowlike model of concurrent computation. The model consists of machines, which ...
SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 37 (3 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semiabelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar nonabelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...
An Algebraic Construction of Predicate Transformers
 Science of Computer Programming
, 1994
"... . In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isom ..."
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Cited by 20 (1 self)
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. In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isomorphic to the category of sets and relations; a second application yields a category isomorphic to the category of monotonic predicate transformers. This hierarchy cannot be extended further: the category of total functions is not itself an instance of the categorical construction, and can only be extended by it twice. 1 Introduction Predicate transformers were introduced originally by Dijkstra [8] in order to provide an elegant semantics for his programming language. Their strength lies in the fact that they can be used to model nondeterministic and nonterminating behaviour in terms of total functions, rather than relations. Not all monotonic predicate transformers represent programs in ...
Compositional Relational Semantics for Indeterminate Dataflow Networks
, 1989
"... Given suitable categories T; C and functor F : T ! C, if X; Y are objects of T, then we define an (X; Y )relation in C to be a triple (R; r; ¯ r), where R is an object of C and r : R ! FX and ¯ r : R ! FY are morphisms of C. We define an algebra of relations in C, including operations of "relabeli ..."
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Cited by 17 (6 self)
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Given suitable categories T; C and functor F : T ! C, if X; Y are objects of T, then we define an (X; Y )relation in C to be a triple (R; r; ¯ r), where R is an object of C and r : R ! FX and ¯ r : R ! FY are morphisms of C. We define an algebra of relations in C, including operations of "relabeling," "sequential composition," "parallel composition," and "feedback," which correspond intuitively to ways in which processes can be composed into networks. Each of these operations is defined in terms of composition and limits in C, and we observe that any operations defined in this way are preserved under the mapping from relations in C to relations in C 0 induced by a continuous functor G : C ! C 0 . To apply the theory, we define a category Auto of concurrent automata, and we give an operational semantics of dataflowlike networks of processes with indeterminate behaviors, in which a network is modeled as a relation in Auto. We then define a category EvDom of "event domains," a (non...
A Type System for Computer Algebra
 Journal of Symbolic Computation
, 1994
"... ing RationalFun from Rational yields a higher order type operator that, given a specification, forms the type of objects that satisfy it. Philip Santas DeclareDomain := (Fun: Type?Category) +? (((Rep: Type) +? with(Rep,Fun(Rep))) SomeRep) The type of Rational objects can now be expressed by applyi ..."
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Cited by 12 (0 self)
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ing RationalFun from Rational yields a higher order type operator that, given a specification, forms the type of objects that satisfy it. Philip Santas DeclareDomain := (Fun: Type?Category) +? (((Rep: Type) +? with(Rep,Fun(Rep))) SomeRep) The type of Rational objects can now be expressed by applying the DeclareDomain constructor to the specification RationalFun: Rational := DeclareDomain(RationalFun) or the shortcut: Rational : RationalFun In order to give proper treatment to the interaction between representations and subtyping, it is necessary to separate Rational into the specifications of its functions and the operators which capture the common structure of all object types. This separation is also important for the semantical construction of categories and the definition of the internal structures of the types. 2.1. Multiple Representations Rationals are created using the function box, which captures the semantics of dynamic objects in object oriented programming.A rational...
Context Institutions
, 1996
"... . The paper introduces a notion of a context institution. The notion is explicitly illustrated by two standard examples. Morphism between context institutions are introduced, thus yielding a category of context institutions. Some expected constructions on context institutions are presented as functo ..."
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Cited by 7 (2 self)
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. The paper introduces a notion of a context institution. The notion is explicitly illustrated by two standard examples. Morphism between context institutions are introduced, thus yielding a category of context institutions. Some expected constructions on context institutions are presented as functors from this category. The potential usefulness of these notions is illustrated by one such a construction, yielding a Hoare logic for an arbitrary small context institution satisfying mild extra assumptions. 1 Introduction The theory of institutions ([4], [6]) has proved its usefulness in the area of foundations of software specification and development. The modeltheoretic view of logical systems advocated in the theory of institutions captures very well the idea that in computer science applications of logic what we are really interested in are models. We always try to specify (logical) properties of concrete objects standard examples can be programs, database management systems or ...
Combination Problems for Commutative/Monoidal Theories or How Algebra Can Help in Equational Unification
 J. Applicable Algebra in Engineering, Communication and Computing
, 1996
"... We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the au ..."
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Cited by 7 (7 self)
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We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the authors independently of each other as "commutative theories " (Baader) and "monoidal theories" (Nutt). We show that commutative theories and monoidal theories indeed define the same class (modulo a translation of the signature), and we prove that it is undecidable whether a given theory belongs to it. In the remainder of the paper we investigate combinations of commutative/monoidal theories with other theories. We show that finitary commutative/monoidal theories always satisfy the requirements for applying general methods developed for the combination of unification algorithms for disjoint equational theories. Then we study the adjunction of monoids of homomorphisms to commutative /monoidal t...
Semantics of Dynamic Variables in Algollike Languages
, 1997
"... A denotational semantic model of an Algollike programming language with local variables, providing fully functional dynamic variable manipulation is presented. Along with the other usual language features, the standard operations with pointers, that is reattachement and dereferencing, and dynamic v ..."
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Cited by 5 (1 self)
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A denotational semantic model of an Algollike programming language with local variables, providing fully functional dynamic variable manipulation is presented. Along with the other usual language features, the standard operations with pointers, that is reattachement and dereferencing, and dynamic variables, that is creation and assignment, are explicated using a possible worlds, functor category, location oriented model. It is shown that the model used to explicate local variables in Algollike languages can be extended to dynamic variables and pointers. Such a model allows for an analytic comparison of the properties of local and dynamic variables and, at the same time, validates several equivalences that, by common computational and operational intuition, are expected to hold. Two fundamental types of equivalences for linked data structures created using pointers are defined, observational equivalence and aeisomorphism, and it is contended that they are both the extensional respect...
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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Cited by 5 (4 self)
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
Unit and kernel systems in algebraic frames
 Alg. Univ
"... Abstract. One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of delements. However, unless the frame is already compact there is no largest such quotient. Wit ..."
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Cited by 5 (5 self)
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Abstract. One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of delements. However, unless the frame is already compact there is no largest such quotient. With the additional assumption of disjointification on the frame, one then studies the maximal ideal spaces of these quotients and the relationship to covers of compact spaces. Several applications are considered, with considerable attention to the frame quotients defined by extension of ideals of a commutative ring A to a ring extension; this type of frame quotient is considered both with and without an underlying lattice structure on the rings. Our interest in this subject springs from a fascination with the theory of covers in compact spaces, and, in particular, those covers which can be constructed as a structure space. Algebraically, the ideas which will be presented here borrow heavily from the context and language of rings of fractions of a commutative ring with identity. To be more precise, given a commutative ring A with identity, it is the relationship between a multiplicative set S of elements of A which are not zerodivisors and the ideals of A which are disjoint to S which we propose to view more generally. Some of the ideas of this paper have been germinating for a decade. Recently, however, it has become apparent that the appropriate context for this discussion is that of frame theory. In this we follow the modus operandi of the recent articles [MZ03] and [M05], by presenting a general theory in algebraic frames, followed by applications to a number of latticealgebraic situations. We devote some attention to the role of maximal ideals in the interplay of units and kernels. In Sections 5 and 6 the socalled structure space covers are considered. The discussion given here is not the full story of these structure space covers, however. Further atttention will be paid to the subject elsewhere.