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15
Convenient Categories of Smooth Spaces
, 2008
"... A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual ..."
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Cited by 38 (0 self)
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A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual
Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Cited by 12 (1 self)
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps
The Representation of Discrete Multiresolution Spatial Knowledge
 PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING: PROCEEDINGS OF KR2000
, 2000
"... The representation of spatial knowledge has been widely studied leading to formal systems such as the RegionConnection Calculus, and various formal accounts of mereotopology. The majority of this work has been in the context of spaces which are continuous and innitely divisible. However, dis ..."
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Cited by 11 (6 self)
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The representation of spatial knowledge has been widely studied leading to formal systems such as the RegionConnection Calculus, and various formal accounts of mereotopology. The majority of this work has been in the context of spaces which are continuous and innitely divisible. However, discrete spaces are of practical importance, and arise in several application areas. The present paper contributes to the representation of discrete spatial knowledge on two fronts. Firstly it provides an algebraic calculus for discrete regions, and demonstrates its expressiveness by showing how the concepts of interior, closure, boundary, exterior, and connection can all be formulated in the calculus. The second advance presented is an account of discrete spaces at multiple levels of detail. This is achieved by developing a general theory of coarsening for sets which is then extended to the case of discrete regions.
Introduction to categories and categorical logic
 In: New Structures for Physics, B. Coecke (ed), Springer Lecture Notes in Physics
, 2009
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A Topos Perspective on StateVector Reduction
"... A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all Msets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘ ..."
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A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all Msets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘neorealist’ interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of statevector reduction. 1
Emergence and Evolution of Meaning: The General Definition of Information (GDI) Revisiting Program—Part I: The Progressive Perspective: TopDown
, 2012
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Topological Completeness of FirstOrder Modal Logic
"... As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topo ..."
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As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary theories of full firstorder logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full firstorder S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems. Keywords: Firstorder modal logic, topological semantics, completeness.
Holography, Quantum . . .
, 2000
"... We interpret the Holographic Conjecture in terms of quantum bits (qubits). Nqubit states are ..."
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We interpret the Holographic Conjecture in terms of quantum bits (qubits). Nqubit states are
A Critical Approach on Reusing Ontologies
, 1995
"... operations Concrete operations Exists a / I Exists c / D Theory Discourse universum Discourse universum Figure 2: Our interpretation of Aristote's causal reasoning model ffl Induction is an operation for hypothesis building. It consists in finding a method for achieving a goal. For instance, ..."
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operations Concrete operations Exists a / I Exists c / D Theory Discourse universum Discourse universum Figure 2: Our interpretation of Aristote's causal reasoning model ffl Induction is an operation for hypothesis building. It consists in finding a method for achieving a goal. For instance, given a set of objects, inductive learning systems show the underlying structure allowing their classification. if A and B then A ! B In an ontological point of view, this would be expressing an organisation from a set of classes. ffl Abduction choses hypothesis. Given a method and a goal, abduction searches an initial state on wich the method can provide the goal. if B and A ! B then A Abduction on an ontology consists in finding objects on the real world domain that, represented as classes, would provide a domain model useful for solving a problem. ffl Deduction identifies premices. Deductive reasonning consists in applying a given method on a given system's state, and providing a solution. ...