192 research outputs found
Inductive Definition and Domain Theoretic Properties of Fully Abstract
A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF +
"parallel conditional function"), respectively, is presented. It is based on
general notions of sequential computational strategies and wittingly consistent
non-deterministic strategies introduced by the author in the seventies.
Although these notions of strategies are old, the definition of the fully
abstract models is new, in that it is given level-by-level in the finite type
hierarchy. To prove full abstraction and non-dcpo domain theoretic properties
of these models, a theory of computational strategies is developed. This is
also an alternative and, in a sense, an analogue to the later game strategy
semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong;
and Nickau. In both cases of PCF and PCF^+ there are definable universal
(surjective) functionals from numerical functions to any given type,
respectively, which also makes each of these models unique up to isomorphism.
Although such models are non-omega-complete and therefore not continuous in the
traditional terminology, they are also proved to be sequentially complete (a
weakened form of omega-completeness), "naturally" continuous (with respect to
existing directed "pointwise", or "natural" lubs) and also "naturally"
omega-algebraic and "naturally" bounded complete -- appropriate generalisation
of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
A Few Notes on Formal Balls
Using the notion of formal ball, we present a few new results in the theory
of quasi-metric spaces. With no specific order: every continuous
Yoneda-complete quasi-metric space is sober and convergence Choquet-complete
hence Baire in its -Scott topology; for standard quasi-metric spaces,
algebraicity is equivalent to having enough center points; on a standard
quasi-metric space, every lower semicontinuous -valued
function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the
continuous Yoneda-complete quasi-metric spaces are exactly the retracts of
algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete
quasi-metric space has a so-called quasi-ideal model, generalizing a
construction due to K. Martin. The point is that all those results reduce to
domain-theoretic constructions on posets of formal balls
Uniqueness of directed complete posets based on Scott closed set lattices
In analogy to a result due to Drake and Thron about topological spaces, this
paper studies the dcpos (directed complete posets) which are fully determined,
among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will
be called -unique).
We introduce the notions of down-linear element and quasicontinuous element
in dcpos, and use them to prove that dcpos of certain classes, including all
quasicontinuous dcpos as well as Johnstone's and Kou's examples, are
-unique. As a consequence, -unique dcpos with their
Scott topologies need not be bounded sober.Comment: 12 pages. arXiv admin note: substantial text overlap with
arXiv:1607.0357
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 and mirror properties in inverse semigroups
Inverse semigroups are a class of semigroups whose structure induces a
compatible partial order. This partial order is examined so as to establish
mirror properties between an inverse semigroup and the semilattice of its
idempotent elements, such as continuity in the sense of domain theory.Comment: 15 pages. The final publication is available at www.springerlink.com.
See http://link.springer.com/article/10.1007%2Fs00233-012-9392-4?LI=tru
- …