K. Kuratowski: Sur une m'ethode de m'etrisation compl`ete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43, pp 114-138, 1956. Home/Search Document Not in Database Summary Related Articles Check |
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Topological Models for Higher Order Control Flow - de Bakker, van Breugel (Correct)
....on s. Definition 1. 8 The denotational semantics D : Res 1 P 1 is defined by 3 If (X; dX ) is a 1 bounded complete ultrametric space then the set of nonempty and compact subsets of X, Pnc (X) endowed with the Hausdorff metric based on dX is a 1 bounded complete ultrametric space (cf. [Kur56]) 4 Let (X; dX ) and (X 0 ; d X 0 ) be metric spaces. A function f : X X 0 is called nonexpansive if, for all x and x 0 , d X 0 (f (x) f (x 0 ) dX (x; x 0 ) 6 Introduction D (e) oe : f g D (s : r) D (s) D (r) The operational and denotational semantics are ....
K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43:114--138, 1956.
Comparative Semantics for a Real-Time Programming Language with .. - van Breugel (1991) (1 citation) (Correct)
....) ffl (X 1 [ X 2 ; d X 1 [ X 2 ) ffl (id 1 2 (X) d id 1 2 (X) and ffl (P nc (X) d Pnc (X) are also complete metric spaces. End 2.2 All proofs but the proof of the last case of the above theorem are straightforward. The proof of the last case, Kuratowski s theorem, can be found in [38]. The next theorem, Banach s fixed point theorem [17] states that a contraction on a complete metric space has a unique fixed point. We will use this theorem to define semantic operators and models and to compare the semantic models developed below. Theorem 2.3 If (X; d X ) is a complete metric ....
K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae 42 (1956), 114-138.
A Theory of Metric Labelled Transition Systems - van Breugel (1995) (Correct)
....the induced compact operational semantics. To prove the nonexpansiveness of O, a sequence (On ) n of nonexpansive functions converging to O is introduced. Because the set of nonexpansive functions C 1 P nc (A 1 ) is closed (a consequence of the completeness of A and Lemma 3 of Kuratowski s [Kur56]) we can conclude that O is nonexpansive. The function On : C P n (A 1 ) is defined by On (c) f a 1 a 2 Delta Delta Delta a k Gamma1 j c = c 1 a1 Gamma Gamma c 2 a2 Gamma Gamma Delta Delta Delta ak Gamma1 Gamma Gamma Gamma Gamma c k 6 k n 1 g[ f a 1 a 2 Delta Delta ....
K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43(1):114--138, 1956.
Domain Equations for Probabilistic Processes - Baier, Kwiatkowska (1997) (22 citations) (Correct)
.... sequence in D 1 (M) which does not have a limit in D 1 (M) If M is a complete ultrametric space then comp (M) equipped with the Hausdorff metric d(X; Y ) def = max ae sup x2X d(x; Y ) sup y2Y d(y; X) oe (where d(w; Z) def = inf z2Z d(w; z) is a complete ultrametric space (see [26]) If f : M M 0 is a non expansive function then comp (f) comp (M) comp (M 0 ) comp (f) X) def = f(X) is non expansive. Hence, we get a functor comp : CUM CUM which is locally non expansive (see [38] We define the functor Act 1 2 : CUM CUM as follows. If M is a complete ....
K. Kuratowski: Sur une m'ethode de m'etrisation compl`ete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43, pp 114-138, 1956.
Terminal Metric Spaces of Finitely Branching and Image Finite.. - van Breugel (1997) (2 citations) (Correct)
....words is endowed with the induced Hausdorff metric [Hau14] This space is not a metric space, but only a pseudometric space. The restriction to the subspaces P nk (A 1 ) of nonempty and compact sets of words and P nc (A 1 ) of nonempty and closed sets of words gives us a complete metric space [Kur56, Hah32]. Like in automata theory, one can associate to a labelled transition system hS; A; #i where S is the (possibly infinite) set of states, A is the (possibly infinite) set of actions, is the transition relation, and # tells us in which states a computation may (but not necessarily has to) ....
.... Phi sup w12W1 inf w22W2 dA 1 (w 1 ; w 2 ) sup w22W2 inf w12W1 dA 1 (w 2 ; w 1 ) Psi : Note that in the compact case, we can replace sup and inf by max and min, respectively. Proposition 4 (Kuratowski and Hahn) P nk (A 1 ) and P nc (A 1 ) are complete metric spaces. Proof See [Kur56, Lemma 3] and [Hah32, x 9.6 and x 18.10] 2 We conclude this section with some simple operations on complete metric spaces. We start with an elementary Example 5 The set 1 = f0g with the obvious metric d 1 is a complete metric space. The operation that leaves the set unchanged and multiplies the metric by ....
K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43(1):114--138, 1956.
Electronic Notes in Theoretical Computer Science 7 (1997) - Url Http Www (Correct)
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K. Kuratowski: Sur une m'ethode de m'etrisation compl`ete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43, pp 114-138, 1956.
Under consideration for publication in Math. Struct. in.. - Domain Equations For (Correct)
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K. Kuratowski: Sur une m'ethode de m'etrisation compl`ete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43, pp 114-138, 1956.
Generalizing Finiteness Conditions of Labelled Transition Systems - van Breugel (1993) (Correct)
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K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43:114--138, 1956.
From Branching to Linear Metric Domains (and back) - van Breugel (1995) (Correct)
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K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43(1):114--138, 1956.
Metric Semantics for Second Order Communication - de Bakker, al. (1995) (1 citation) (Correct)
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K. Kuratowski. Sur une M'ethode de M'etrisation Compl`ete des Certains Espaces d'Ensembles Compacts. Fundamenta Mathematicae, 43(1):114--138, 1956.
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