62 research outputs found

    Unsharp Values, Domains and Topoi

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    The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a classical system, and a quantity-value object, generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -- hence the `values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as `unsharp values'. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.Comment: 32 pages, no figures; to appear in Proceedings of Quantum Field Theory and Gravity, Regensburg (2010

    Domain Theory in Constructive and Predicative Univalent Foundations

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    We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. Domain theory studies so-called directed complete posets (dcpos) and Scott continuous maps between them and has applications in programming language semantics, higher-type computability and topology. A common approach to deal with size issues in a predicative foundation is to work with information systems, abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. In our type-theoretic approach, we instead accept that dcpos may be large and work with type universes to account for this. A priori one might expect that complex constructions of dcpos result in a need for ever-increasing universes and are predicatively impossible. We show that such constructions can be carried out in a predicative setting. We illustrate the development with applications in the semantics of programming languages: the soundness and computational adequacy of the Scott model of PCF and Scott's D∞D_\infty model of the untyped λ\lambda-calculus. We also give a predicative account of continuous and algebraic dcpos, and of the related notions of a small basis and its rounded ideal completion. The fact that nontrivial dcpos have large carriers is in fact unavoidable and characteristic of our predicative setting, as we explain in a complementary chapter on the constructive and predicative limitations of univalent foundations. Our account of domain theory in univalent foundations is fully formalised with only a few minor exceptions. The ability of the proof assistant Agda to infer universe levels has been invaluable for our purposes.Comment: PhD thesis, extended abstract in the pdf. v5: Fixed minor typos in 6.2.18, 6.2.19 and 6.4.

    A Recipe for State-and-Effect Triangles

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    In the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains

    A theory of quantale-enriched dcpos and their topologization

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    There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral and commutative, then Q-valued domains are Q-enriched, and every Q-enriched domain is sober in its Scott Q-valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of Q-enriched dcpos and Q-enriched domains in the general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right subdistributive quasi-magmas on Q in the sense of the category Cat(Q). It is important to point out that our quasi-magmas on Q are in tune with the «intersection axiom» of Q-enriched topologies. When Q is involutive, each Q-enriched domain becomes sober in its Q-enriched Scott topology. This paper also offers a perspective to apply Q-enriched dcpos to quantale computatio

    Continuous R-valuations

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    We introduce continuous RR-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags RR. Like the valuation monad V\mathbf{V} introduced by Jones and Plotkin, we show that the construction of continuous RR-valuations extends to a strong monad VR\mathbf{V}^R on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad VmR\mathbf{V}^R_m out of it, whose elements we call minimal RR-valuations. We also show that continuous RR-valuations have close connections to measures when RR is taken to be IR+⋆\mathbf{I}\mathbb{R}^\star_+, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded τ\tau-smooth measure μ\mu (defined on the Borel σ\sigma-algebra), canonically determines a continuous IR+⋆\mathbf{I}\mathbb{R}^\star_+-valuation; and (2) such a continuous IR+⋆\mathbf{I}\mathbb{R}^\star_+-valuation is the most precise (in a certain sense) continuous IR+⋆\mathbf{I}\mathbb{R}^\star_+-valuation that approximates μ\mu, when the support of μ\mu is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous IR+⋆\mathbf{I}\mathbb{R}^\star_+-valuation. Additionally, we show that the latter is minimal

    06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality

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    From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

    A duality of generalized metric spaces

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    AbstractWe develop a duality theory for Lawvereʼs generalized metric spaces that extends the Lawson duality for continuous dcpos and open filter reflecting maps: we prove that the category of relatively cocomplete and continuous [0,∞]-categories considered with open filter reflecting maps is self-dual
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