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144
Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
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Cited by 162 (4 self)
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Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Semantic Domains
, 1990
"... this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype fu ..."
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Cited by 148 (3 self)
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this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as welland that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the socalled untyped calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary wayavoiding heavy prerequisites (such as category theory) but still maintaining some level of abstractionwith the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...
Towards a Mathematical Operational Semantics
 In Proc. 12 th LICS Conf
, 1997
"... We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation ..."
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Cited by 134 (9 self)
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We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets both an operational model and a canonical, internally fully abstract denotational model for free; moreover, both models satisfy the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of wellbehaved rules for structural operational semantics, such as GSOS.
Hierarchical Finite State Machines with Multiple Concurrency Models
 IEEE Transactions on Computeraided Design of Integrated Circuits and Systems
, 1999
"... This paper studies the semantics of hierarchical finite state machines (FMS's) that are composed using various concurrency models, particularly dataflow, discreteevents, and synchronous/reactive modeling. It is argued that all three combinations are useful, and that the concurrency model can be sel ..."
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Cited by 115 (36 self)
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This paper studies the semantics of hierarchical finite state machines (FMS's) that are composed using various concurrency models, particularly dataflow, discreteevents, and synchronous/reactive modeling. It is argued that all three combinations are useful, and that the concurrency model can be selected independently of the decision to use hierarchical FSM's. In contrast, most formalisms that combine FSM's with concurrency models, such as Statecharts (and its variants) and hybrid systems, tightly integrate the FSM semantics with the concurrency semantics. An implementation that supports three combinations is described.
The Discoveries of Continuations
, 1993
"... We give a brief account of the discoveries of continuations and related concepts by, A. Van Wijngaarden , A. W. Mazurkiewicz , F. L. Morris , C. P. Wadsworth , J. H. Morris , M. J. Fischer , and S. K. Abdali. ..."
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Cited by 110 (2 self)
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We give a brief account of the discoveries of continuations and related concepts by, A. Van Wijngaarden , A. W. Mazurkiewicz , F. L. Morris , C. P. Wadsworth , J. H. Morris , M. J. Fischer , and S. K. Abdali.
Continuations: A Mathematical Semantics for Handling Full Jumps
, 1974
"... Abstract. This paper describes a method of giving the mathematical semantics of programming languages which include the most general form of jumps. 1. ..."
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Cited by 110 (0 self)
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Abstract. This paper describes a method of giving the mathematical semantics of programming languages which include the most general form of jumps. 1.
Projections for Strictness Analysis
, 1987
"... Contexts have been proposed as a means of performing strictness analysis on nonflat domains. Roughly speaking, a context describes how much a subexpression will be evaluated by the surrounding program. This paper shows how contexts can be represented using the notion of projection from domain theo ..."
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Cited by 98 (4 self)
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Contexts have been proposed as a means of performing strictness analysis on nonflat domains. Roughly speaking, a context describes how much a subexpression will be evaluated by the surrounding program. This paper shows how contexts can be represented using the notion of projection from domain theory. This is clearer than the previous explanation of contexts in terms of continuations. In addition, this paper describes finite domains of contexts over the nonflat list domain. This means that recursive context equations can be solved using standard fixpoint techniques, instead of the algebraic manipulation previously used. Praises of lazy functional languages have been widely sung, and so have some curses. One reason for praise is that laziness supports programming styles that are inconvenient or impossible otherwise [Joh87, Hug84, Wad85a]. One reason for cursing is that laziness hinders efficient implementation. Still, acceptable efficiency for lazy languages is at last being achieved...
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
Dynamical systems, Measures and Fractals via Domain Theory
 Information and Computation
, 1995
"... We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L ar ..."
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Cited by 69 (19 self)
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We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L are respectively the Vietoris hyperspace, the upper hyperspace and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, U f). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be omegacontinuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of nondeterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.
The origins of structural operational semantics
 Journal of Logic and Algebraic Programming
, 2004
"... We review the origins of structural operational semantics. The main publication ‘A Structural Approach to Operational Semantics, ’ also known as the ‘Aarhus Notes, ’ appeared in 1981 [G.D. Plotkin, A structural approach to operational semantics, DAIMI FN19, Computer Science Department, Aarhus Unive ..."
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Cited by 64 (0 self)
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We review the origins of structural operational semantics. The main publication ‘A Structural Approach to Operational Semantics, ’ also known as the ‘Aarhus Notes, ’ appeared in 1981 [G.D. Plotkin, A structural approach to operational semantics, DAIMI FN19, Computer Science Department, Aarhus University, 1981]. The development of the ideas dates back to the early 1970s, involving many people and building on previous work on programming languages and logic. The former included abstract syntax, the SECD machine, and the abstract interpreting machines of the Vienna school; the latter included the λcalculus and formal systems. The initial development of structural operational semantics was for simple functional languages, more or less variations of the λcalculus; after that the ideas were gradually extended to include languages with parallel features, such as Milner’s CCS. This experience set the ground for a more systematic exposition, the subject of an invited course of lectures at Aarhus University; some of these appeared in print as the 1981 Notes. We discuss the content of these lectures and some related considerations such as ‘small state’ versus ‘grand state, ’ structural versus compositional semantics, the influence of the Scott–Strachey approach to denotational semantics, the treatment of recursion and jumps, and static semantics. We next discuss relations with other work and some immediate further development. We conclude with an account of an old, previously unpublished, idea: an alternative, perhaps more readable, graphical presentation of systems of rules for operational semantics.