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19
Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation
, 2002
"... We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the bigstep semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and ..."
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Cited by 98 (17 self)
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We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the bigstep semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the EgliMilner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conservative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski
Abstract syntax and variable binding (extended abstract
 In Proc. 14 th LICS
, 1999
"... Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the ..."
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Cited by 21 (0 self)
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Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.
Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory
"... ' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebra ..."
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Cited by 15 (0 self)
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' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebraization of flowchart schemes, including [CaU82, BlEs85, Ste86/90, Bar87a, CaS88a, CaS90b]. With different sets of operators various algebras for flowgraphs appear in [Mil79, Parr87, CaS90b, CaS88b]. In the classical algebraic calculus for regular languages it is often the case that certain abstract semirings are used instead of the Boolean f0; 1g semiring, e.g. by using formal series with such coefficients. 5 This property is similar to the universal property of the polynomials over a ring. Chapter 6 Graph isomorphism with various constants In this chapter we extend the axiomatistion for flowgraphs modulo isomorphism to the case where more constants for generating relations are present i...
Coinductive Models of Finite Computing Agents
 Electronic Notes in Theoretical Computer Science
, 1999
"... This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points ..."
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Cited by 13 (6 self)
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This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points are shown to play a role in models of observation that parallels minimal xed points in inductive mathematics. The impact of interactive (coinductive) models on Church's thesis and the connection between incompleteness and greater expressiveness are examined. A nal section shows that actual software systems are interactive rather than algorithmic. Coinductive models could become as important as inductive models for software technology as computer applications become increasingly interactive.
Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
Rewriting On Cyclic Structures: Equivalence Between The Operational And The Categorical Description
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, fo ..."
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Cited by 12 (6 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...
Atomic Actions
, 1989
"... We give a formal specification of the semantics of atomic actions. We show that adding atomic action constructs to a lowlevel imperative language allows one to program higherlevel synchronization mechanisms. 1. Introduction. This note intends to provide an alternative view on the socalled "actio ..."
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Cited by 6 (0 self)
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We give a formal specification of the semantics of atomic actions. We show that adding atomic action constructs to a lowlevel imperative language allows one to program higherlevel synchronization mechanisms. 1. Introduction. This note intends to provide an alternative view on the socalled "action refinement" concept, which is by now widely studied (cf. for instance [4], and the references therein). Action refinement is formalized as the replacement of an action by a possibly complex process. Many recent papers on this topic advocate the idea that a sensible notion of process should be robust with respect to action refinement. For lack of an abstract mathematical notion, a process is understood as an equivalence class, and the problem is to find equivalences which are congruences with respect to the operation of substituting processes for actions. It is not my intention to question the theoretical work that has been done in this direction. However, one may wonder whether this substi...
Analogy categories, virtual machines and , ,structured programming
 GI2 Jahrestagung, Lecture Notes in Computer Science
, 1975
"... Abstract This paper arises from a number of studies of machine/problem relationships, software development techniques, language and machine design. It develops a categorytheoretic framework for the analysis of the relationships between programmer, virtual machine, and problem that are inherent in d ..."
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Cited by 5 (5 self)
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Abstract This paper arises from a number of studies of machine/problem relationships, software development techniques, language and machine design. It develops a categorytheoretic framework for the analysis of the relationships between programmer, virtual machine, and problem that are inherent in discussions of "ease of programming", "good programming techniques", "structured programming", and so ono The concept of "analogy " is introduced as an expllcatum of the comprehensibility of the relationship between two systems. Analogy is given a formal definition in terms of a partially ordered structure of analogy categories whose minimal element is a "truth " c ~ "proof " category ° The theory is constructive and analogy relationships are computable between defined systems, c ~ classes of system. Thus the structures developed may be used to study the relationships between programmer, problem, and virtual machine in practical situations. io
Interaction, Computability, and Church's Thesis
, 1999
"... : This article formalizes the claim that interactive finite computing agents are more expressive than Turing machines. The impact of models of interaction on Church's thesis and Godel's incompleteness result is explored. The evolution from algorithmic to interactive models of computation in computer ..."
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Cited by 2 (1 self)
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: This article formalizes the claim that interactive finite computing agents are more expressive than Turing machines. The impact of models of interaction on Church's thesis and Godel's incompleteness result is explored. The evolution from algorithmic to interactive models of computation in computer architecture, software engineering, and AI is considered in a final section. Contents 1. Interaction Machines 1.1. Sequential Interaction Machines (SIMs) 1.2. Interactive Behavior and Expressiveness 1.3. MultiStream Interaction Machines (MIMs) 2. Extensions of Expressiveness 2.1. Interactive Extensions of Machines, Sets, and Algebras 2.2. Interactive Extensions of the ChurchTuring Thesis 3. Mathematical Models of Interaction 3.1. NonWellFounded Set Theory 3.2. Coalgebras 3.3. Beyond NonWellFounded Sets 4. From Induction to Coinduction 4.1. The Inductive Modeling Paradigm 4.2. Coinduction and Greatest Fixed Points 5. Metamathematics of Coinduction 5.1. From Formal Mode...
First steps in the construction of the Geometric Machine, em “Seleta do XXIV
 Tendências em Matemática Aplicada e Computacional
, 2002
"... Abstract. This work introduces the Geometric Machine (GM) – a computational model for the construction and representation of concurrent and nondeterministic processes, preformed in a synchronized way, with infinite memory whose positions are labelled by the points of a geometric space. The ordered ..."
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Cited by 2 (2 self)
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Abstract. This work introduces the Geometric Machine (GM) – a computational model for the construction and representation of concurrent and nondeterministic processes, preformed in a synchronized way, with infinite memory whose positions are labelled by the points of a geometric space. The ordered structure of the GM model is based on Girard’s Coherence Spaces. Starting with a coherence space of elementary processes, the inductive domaintheoretic structure of this model is stepwise and systematically constructed and the procedure completion ensures the existence of temporally and spatially infinite computations. A particular aim of our work is to apply this coherencespacebased interpretation to the semantic modelling parallelism and distributed computation over array structures. 1.