scholarly journals Processes as terms: non-well-founded models for bisimulation

1992 ◽  
Vol 2 (3) ◽  
pp. 257-275 ◽  
Author(s):  
J. J. M. M. Rutten
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Operational Semantics ◽  
Transition System ◽  
Recursive Equation ◽  
Compositional Semantics ◽  
Complete Metric Spaces ◽  
Bisimulation Equivalence ◽  
Nonstandard Set Theory ◽  
Standard Set

A compositional semantics characterizing bisimulation equivalence is derived from transition system specifications in the SOS style, satisfying certain syntactic syntactic conditions. We use Aczel's nonstandard set theory for solving a recursive equation for a domain fo processes. It contains non-well-founded elements modelling possibly infinite behaviour. Semantic interpretations of syntactic operators are obtained by defining the operational semantics for terms consisting of both syntactic and semantic (processes)entities. Finally, we return to standard set theory by observing that a similar, though less general, result can be obtained with the use of complete metric spaces.

Domain equations for probabilistic processes

2000 ◽  
Vol 10 (6) ◽  
pp. 665-717 ◽  
Author(s):  
CHRISTEL BAIER ◽  
MARTA KWIATKOWSKA
Keyword(s):  
Metric Spaces ◽  
Operational Semantics ◽  
Denotational Semantics ◽  
Semantic Model ◽  
Transition Systems ◽  
Probabilistic Choice ◽  
Bisimulation Equivalence ◽  
Metric Semantics ◽  
Continuous Domains ◽  
Probabilistic Processes

In this paper we consider Milner's calculus CCS enriched by a probabilistic choice operator. The calculus is given operational semantics based on probabilistic transition systems. We define operational notions of preorder and equivalence as probabilistic extensions of the simulation preorder and the bisimulation equivalence respectively. We extend existing category-theoretic techniques for solving domain equations to the probabilistic case and give two denotational semantics for the calculus. The first, ‘smooth’, semantic model arises as a solution of a domain equation involving the probabilistic powerdomain and solved in the category CONT⊥ of continuous domains. The second model also involves an appropriately restricted probabilistic powerdomain, but is constructed in the category CUM of complete ultra-metric spaces, and hence is necessarily ‘discrete’. We show that the domain-theoretic semantics is fully abstract with respect to the simulation preorder, and that the metric semantics is fully abstract with respect to bisimulation.

Standard foundations for nonstandard analysis

1992 ◽  
Vol 57 (2) ◽  
pp. 741-748 ◽  
Author(s):  
David Ballard ◽  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Natural Place ◽  
Nonstandard Set Theory ◽  
Standard Set ◽  
The Subject ◽  
Internal Sets ◽  
Well Foundedness

In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).

scholarly journals A comparison of concepts from computable analysis and effective descriptive set theory

2016 ◽  
Vol 27 (8) ◽  
pp. 1414-1436 ◽  
Author(s):  
VASSILIOS GREGORIADES ◽  
TAMÁS KISPÉTER ◽  
ARNO PAULY
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Computable Analysis ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Complete Metric Spaces ◽  
Cauchy Completion ◽  
Precise Relationship ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

Some Common Fixed Point Theorem in Complete Metric Space of Integral Type

2018 ◽  
Vol 6 (8) ◽  
pp. 297
Author(s):  
Jagdish C. Chaudhary ◽  
Shailesh T. Patel
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Contractive Condition ◽  
Integral Type ◽  
Complete Metric Spaces ◽  
Common Fixed Point Theorems ◽  
Common Fixed Point Theorem

In this paper, we prove some common fixed point theorems in complete metric spaces for self mapping satisfying a contractive condition of Integral  type.

A generalization of Reich’s fixed point theorem for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3295-3305 ◽  
Author(s):  
Antonella Nastasi ◽  
Pasquale Vetro
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Complete Metric Spaces ◽  
New Class ◽  
Banach Contraction ◽  
The Banach Contraction Principle

Motivated by a problem concerning multi-valued mappings posed by Reich [S. Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 57 (1974) 194-198] and a paper of Jleli and Samet [M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014:38 (2014) 1-8], we consider a new class of multi-valued mappings that satisfy a ?-contractive condition in complete metric spaces and prove some fixed point theorems. These results generalize Reich?s and Mizoguchi-Takahashi?s fixed point theorems. Some examples are given to show the usability of the obtained results.

scholarly journals On Some New Contractive Conditions in Complete Metric Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 118
Author(s):  
Jelena Vujaković ◽  
Eugen Ljajko ◽  
Mirjana Pavlović ◽  
Stojan Radenović
Keyword(s):  
Current Literature ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Spaces ◽  
Nonlinear Anal ◽  
Increasing Mapping ◽  
Strictly Increasing Mapping

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.

scholarly journals Discussions on the almost 𝒵-contraction

2020 ◽  
Vol 18 (1) ◽  
pp. 448-457
Author(s):  
Erdal Karapınar ◽  
V. M. L. Hima Bindu
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Complete Metric Spaces ◽  
The Given

Abstract In this paper, we introduce a new contraction, namely, almost {\mathcal{Z}} contraction with respect to \zeta \in {\mathcal{Z}} , in the setting of complete metric spaces. We proved that such contraction possesses a fixed point and the given theorem covers several existing results in the literature. We consider an example to illustrate our result.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

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