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From Branching to Linear Metric Domains (and back
 6th Nordic Workshop on Programming Theory, NWPT '6 Proceedings
, 1995
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
Terminal Metric Spaces of Finitely Branching and Image Finite Linear processes
, 1997
"... Wellknown metric spaces for modelling finitely branching and image finite systems are shown to be (the carrier of) terminal coalgebras. Introduction In the area of metric semantics, various metric structures have been proposed to model a wide spectrum of programming notions (see, e.g., [BV96]). In ..."
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Wellknown metric spaces for modelling finitely branching and image finite systems are shown to be (the carrier of) terminal coalgebras. Introduction In the area of metric semantics, various metric structures have been proposed to model a wide spectrum of programming notions (see, e.g., [BV96]). In this paper, we focus on metric structures for modelling nondeterministic systems which may give rise to both terminating and nonterminating computations. The systems we have in mind are labelled transition systems [Kel76]. A large variety of programming notions can be modelled by means of these systems (see, e.g., [Plo81]). The models we consider are linear (cf. [Pnu85]). In these models, the locations in a computation where a nondeterministic choice is made are not visible. These linear models are usually contrasted with branching models (cf. [Gla90]). In those models, the positions in the computation where a nondeterministic choice is made are administrated. Typical examples of linear me...
Looking through newly to the amazing irrationals
"... We assume basic familiarity with the theory of metric spaces, in particular, the space of real numbers with its usual metric. In this section, we review some notation and a few basic definitions and theorems concerning metric spaces (including the concept of dimension zero). In §3 we use the complet ..."
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We assume basic familiarity with the theory of metric spaces, in particular, the space of real numbers with its usual metric. In this section, we review some notation and a few basic definitions and theorems concerning metric spaces (including the concept of dimension zero). In §3 we use the completeness (or least upper bound) axiom of the real numbers in the form that says the space of real numbers is locally compact. 1.1. Basic definitions. We use standard notation: R denotes the set of real numbers, P the set of irrational numbers, Q the set of rational numbers, Z the set of all integers, N the set of all positive integers, and ω the set of nonnegative integers (the first infinite ordinal). Definition 1.1. A pair (X, d) is called a metric space provided X is a set and d is a metric, i.e., a function d: X × X → [0, ∞) which satisfies the following three properties for every x,y,z in X: (a) d(x, y) = 0 if and only if x = y (b) d(x, y) = d(y, x) (c) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) Let (X, d) be a metric space, x ∈ X, and r> 0. The set N(x, r) = {y ∈ X: d(x, y) < r} is called a basic neighborhood of x (of radius r). A set U ⊂ X is called an open set provided U is a union of basic neighborhoods, i.e., for every x ∈ U there exists r = r(x)> 0 such that N(x, r) ⊂ U. The set T = T(X, d) = {U ⊂ X: U is an open set} is called the topology of (X, d), or the topology on X induced by the metric d. A family B ⊂ T. is called a base for the topology T provided every element of T. is a union of elements from B. A set F ⊂ X is called closed (in X) provided X − F is open in X. If U is open and F is closed, then U \ F is open and F \ U is closed (exercise). For example, the function d(x, y) = x − y  (the absolute value) can easily be seen to be a metric on R, and is called the usual metric on R. If (X, d) is a metric space and Y ⊂ X, the induced metric on Y is d(Y × Y). Thus (Y, d(Y ×Y)) is a metric space and is called a subspace of (X, d). For example, the set of rational numbers Q is a subspace of the metric space R, where R has its usual metric, and likewise the set of irrational numbers P, is a subspace of R. In 1 c○My pleasure if you need it
A Theory of Metric Labelled Transition Systems
 Papers on General Topology and Applications: 11th Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences
, 1995
"... Labelled transition systems are useful for giving semantics to programming languages. Kok and Rutten have developed some theory to prove semantic models defined by means of labelled transition systems to be equal to other semantic models. Metric labelled transition systems are labelled transition sy ..."
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Labelled transition systems are useful for giving semantics to programming languages. Kok and Rutten have developed some theory to prove semantic models defined by means of labelled transition systems to be equal to other semantic models. Metric labelled transition systems are labelled transition systems with the configurations and actions endowed with metrics. The additional metric structure allows us to generalize the theory developed by Kok and Rutten. Introduction The classical result due to Banach [Ban22] that a contractive function from a nonempty complete metric space to itself has a unique fixed point plays an important role in the theory of metric semantics for programming languages. Metric spaces and Banach's theorem were first employed by Nivat [Niv79] to give semantics to recursive program schemes. Inspired by the work of Nivat, De Bakker and Zucker [BZ82] gave semantics to concurrent languages by means of metric spaces. The metric spaces they used were defined as solutio...
Comment.Math.Univ.Carolin. 46,1 (2005)43–54 43
"... A remark on a theorem of Solecki ..."
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Metric Semantics for Second Order Communication
, 1995
"... An operational and a denotational semantics are presented for a simple imperative language. The main feature of the language is second order communication: sending and receiving of statements rather than values. The operational semantics is based on a transition system. A complete 1bounded ultramet ..."
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An operational and a denotational semantics are presented for a simple imperative language. The main feature of the language is second order communication: sending and receiving of statements rather than values. The operational semantics is based on a transition system. A complete 1bounded ultrametric space is used in the denotational semantics. In establishing the connection between the two semantics fruitful use is made of Banach's fixed point theorem, Rutten's processes as terms technique, and Van Breugel's metric transition systems. Introduction In recent years the study of higher order programming notions has become a central topic in the field of semantics. Seminal in this development have been two schools of research, viz that of typed calculus in the area of functional programming (see, e.g., Barendregt's survey [Bar92]), and that of higher order processes in the theory of concurrency (see, e.g., the theses by Sangiorgi [San92] and Thomson [Tho90]). The aim of the present pa...
the hard way.
"... Departing from the usual author’s statementI would like to say that I am not responsible for any of the mistakes in this document. Any mistakes here are the responsibility of the reader. If anybody wants to point out a mistake to me, I promise to respond by saying “but you know what I meant to say, ..."
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Departing from the usual author’s statementI would like to say that I am not responsible for any of the mistakes in this document. Any mistakes here are the responsibility of the reader. If anybody wants to point out a mistake to me, I promise to respond by saying “but you know what I meant to say, don’t you?” These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Some knowledge of forcing is assumed as well as a modicum of elementary Mathematical Logic, for example, the LowenheimSkolem Theorem. Many of the classical theorems of descriptive set theory are presented “justintime ” for when they are needed. Questions like “Who proved what?” always interest me, so I have included my best guess here. Hopefully, I have managed to offend a large number of mathematicians. The results in section 14 and 15 are new and answer a questions from my thesis. I have also included (without permission) an unpublished result of