On the structure of causal spaces

Kronheimer, E. H.; Penrose, Roger

doi:10.1017/s030500410004144x

Cited by 233 publications

(219 citation statements)

References 13 publications

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“…Similar questions arise in other applications of ordered sets, e.g., in space-time geometry where the causality ordering relation a ≤ b means that an event a can influence the event b; see, e.g., [1], [3], [4], [5], [6], [8], [11], [12].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

Zapata

Kosheleva

Villaverde

2011

2011 Annual Meeting of the North American Fuzzy Information Processing Society

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Abstract-In this paper, we describe how checking whether a given property F is true for a product A1 × A2 of partially ordered spaces can be reduced to checking several related properties of the original spaces Ai. I. FORMULATION OF THE PROBLEMDegrees of certainty: from {0, 1} to [0, 1] to general partially ordered sets. In the traditional 2-valued logic, every statement is either true or false. Thus, the set of possible truth values consists of two elements: true (1) and false (0).Fuzzy logic takes into account that people have different degrees of certainty in their statements; see, e.g., [2], [10].Traditionally, fuzzy logic uses values from the interval [0, 1] to describe uncertainty. In this interval, the order is total (linear) in the sense that for every two elements a, a More complex sets of possible degrees are also sometimes useful. Not to miss any new options, in this paper, we consider general partially ordered spaces.Need for product operations. Often, two (or more) experts evaluate a statement S. Then, our certainty in S is described by a pair (a 1 , a 2 ), where a i ∈ A i is the i-th expert's degree of certainty. To compare such pairs, we must therefore define a partial order on the set A 1 × A 2 of all such pairs.Two examples of product operations. One example of a partial order on A 1 × A 2 is a Cartesian product:This product corresponds to a cautious approach, when our confidence in S ′ is higher than in S if and only if it is higher for both experts.Another example is a lexicographic product:This product corresponds to the case when we have the absolute confidence in the first expert; then, we only use the opinion of the second expert when, to the first expert, the degrees of certainty are indistinguishable. We can have other product operations in which the relation between the pairs (a 1 , a 2 ) and (a A natural question. Once a product is defined, it is reasonable to ask when the resulting partially ordered set A 1 × A 2 it satisfies a certain property: is it a total order? is it a lattice order? etc. It is desirable to have some criteria that would transform the question about the product space into questions about related properties of component spaces.Some such criteria are known (see, e.g., [13] and references therein). For example:• A Cartesian product is a total order if and only if one of the components is a total order, and the other consists of a single element.• A lexicographic product is a total order if and only if both components are totally ordered.What we do in this paper. In this paper, we provide a general algorithm that reduced the question whether a certain property is satisfied for a product to several properties of component spaces. Applications beyond orders. Our algorithm does not use the fact that the original relations are orders (i.e., transitive antisymmetric relations). Thus, our algorithm is applicable to a general case when we have an arbitrary binary relationequivalence, similarity, etc. Moreover, this algorithm can be

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Section: Formulation Of the Problemmentioning

confidence: 99%

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

Zapata

Kosheleva

Villaverde

2011

2011 Annual Meeting of the North American Fuzzy Information Processing Society

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“…In a related vein there is Kronheimer and Penrose's axiomatic approach to space and time based entirely on causal relations [6]. This work is motivated more by general relativity than special, and attempts to encompass some of the unusual causal properties which can arise in that theory.…”

Section: Casual Spacesmentioning

confidence: 99%

“…Urquhart, attempted to give an axiomatic foundation of the kinematic part of Einstein's special theory of relativity [9]. This work was subsequently enlarged on by John Schutz [8].In a related vein there is Kronheimer and Penrose's axiomatic approach to space and time based entirely on causal relations [6]. This work is motivated more by general relativity than special, and attempts to encompass some of the unusual causal properties which can arise in that theory.…”

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confidence: 99%

Signal spaces—an axiomatic approach to space-time

Szekeres

1991

Bull. Austral. Math. Soc.

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Dedicated to my father, George Szekeres, on his eightieth birthday An axiomatic approach to relativity theory is proposed which is based entirely on a single reflexive relation called signalling. This generalises and simplifies earlier schemes of Kronheimer and Penrose and of Carter which are based on causality relations. Many of the features of space-times, such as photon paths and topology, have their counterparts in general signal spaces. CASUAL SPACESThere is a school of thought [7] which regards Euclidean geometry as an excellent piece of mathematical physics, that is, an axiomatic system which applies to the real world. It was in something of this spirit in 1968 that George Szekeres, following a suggestion of M.L. Urquhart, attempted to give an axiomatic foundation of the kinematic part of Einstein's special theory of relativity [9]. This work was subsequently enlarged on by John Schutz [8].In a related vein there is Kronheimer and Penrose's axiomatic approach to space and time based entirely on causal relations [6]. This work is motivated more by general relativity than special, and attempts to encompass some of the unusual causal properties which can arise in that theory. However it does allow for structures more general than manifolds, and may eventually pave the way to a theory of discrete space-times which could be reconciled with the paradoxes of quantum theory.One thing which is not permitted in the Kronheimer-Penrose theory (hereinafter referred to as KP) is closed timelike lines such as occur in the Godel universe [3]. Carter [1] has extended the idea of a causal space to include the possibility of such causal loops. The resulting structure is one he calls an etiological space. In this paper I shall adopt a less arcane terminology.By a pre-causal space we shall mean a triple (X, -<,

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“…After Urysohn's early death, this activity was continued in Russia by his student Vadim Efremovich, by Efremovich's student Revolt Pimenov, and by Pimenov's students -and also by H. Busemann in the US and by E. Kronheimer and R. Penrose in the UK [10,13,22] (see also [16]). …”

Section: Fig 1 Causality Relation Of the Special Relativity Theorymentioning

confidence: 99%

Metrization Theorem for Space-Times: From Urysohn’s Problem towards Physically Useful Constructive Mathematics

Kreinovich

2010

Fields of Logic and Computation

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Abstract. In the early 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as "first non-trivial result of point set topology"). Among other applications, this lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized. A few years before that, in 1919, a complex mathematical theory was experimentally proven to be extremely useful in the description of real world phenomena: namely, during a solar eclipse, General Relativity theory -that uses pseudo-Riemann spaces to describe space-timewas (spectacularly) experimentally confirmed. Motivated by this success, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. After Urysohn's early death in 1924, this activity was continued in Russia by his student Vadim Efremovich, Efremovich's student Revolt Pimenov, and by Pimenov's students (and also by H. Busemann in the US and by E. Kronheimer and R. Penrose in the UK). By the 1970s, reasonably general space-time versions of Urysohn's lemma and metrization theorem have been proven. However, these 1970s results are not constructive. Since one of the main objectives of this activity is to come up with useful applications to physics, we definitely need constructive versions of these theorems -versions in which we not only claim the theoretical existence of a pseudometric, but we also provide an algorithm enabling the physicist to generate such a metric based on empirical data about the causality relation. An additional difficulty here is that for this algorithm to be useful, we need a physically relevant constructive description of a causality-type ordering relation. In this paper, we propose such a description and show that, for this description, a combination of the existing constructive ideas with the known (non-constructive) proof leads to successful constructive spacetime versions of the Urysohn's lemma and of the metrization theorem.

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On the structure of causal spaces

Cited by 233 publications

References 13 publications

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

Signal spaces—an axiomatic approach to space-time

Metrization Theorem for Space-Times: From Urysohn’s Problem towards Physically Useful Constructive Mathematics

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