62 research outputs found
Unsharp Values, Domains and Topoi
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
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 model of the untyped
-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
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
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
We introduce continuous -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 .
Like the valuation monad introduced by Jones and Plotkin, we
show that the construction of continuous -valuations extends to a strong
monad 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 out of it, whose elements
we call minimal -valuations.
We also show that continuous -valuations have close connections to
measures when is taken to be , the interval
domain of the extended nonnegative reals: (1) On every coherent topological
space, every non-zero, bounded -smooth measure (defined on the
Borel -algebra), canonically determines a continuous
-valuation; and (2) such a continuous
-valuation is the most precise (in a certain
sense) continuous -valuation that approximates
, when the support of 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 -valuation.
Additionally, we show that the latter is minimal
06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality
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
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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