670 research outputs found
A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes
The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational model
Topos theory and `neo-realist' quantum theory
Topos theory, a branch of category theory, has been proposed as mathematical
basis for the formulation of physical theories. In this article, we give a
brief introduction to this approach, emphasising the logical aspects. Each
topos serves as a `mathematical universe' with an internal logic, which is used
to assign truth-values to all propositions about a physical system. We show in
detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop
"Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200
A Coalgebraic View on Reachability
Coalgebras for an endofunctor provide a category-theoretic framework for
modeling a wide range of state-based systems of various types. We provide an
iterative construction of the reachable part of a given pointed coalgebra that
is inspired by and resembles the standard breadth-first search procedure to
compute the reachable part of a graph. We also study coalgebras in Kleisli
categories: for a functor extending a functor on the base category, we show
that the reachable part of a given pointed coalgebra can be computed in that
base category
The physical interpretation of daseinisation
We provide a conceptual discussion and physical interpretation of some of the
quite abstract constructions in the topos approach to physics. In particular,
the daseinisation process for projection operators and for self-adjoint
operators is motivated and explained from a physical point of view.
Daseinisation provides the bridge between the standard Hilbert space formalism
of quantum theory and the new topos-based approach to quantum theory. As an
illustration, we will show all constructions explicitly for a three-dimensional
Hilbert space and the spin-z operator of a spin-1 particle. This article is a
companion to the article by Isham in the same volume.Comment: 39 pages; to appear in "Deep Beauty", ed. Hans Halvorson, Cambridge
University Press (2010
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
Coalgebra Learning via Duality
Automata learning is a popular technique for inferring minimal automata
through membership and equivalence queries. In this paper, we generalise
learning to the theory of coalgebras. The approach relies on the use of logical
formulas as tests, based on a dual adjunction between states and logical
theories. This allows us to learn, e.g., labelled transition systems, using
Hennessy-Milner logic. Our main contribution is an abstract learning algorithm,
together with a proof of correctness and termination
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