6 research outputs found
Domain Theory in Constructive and Predicative Univalent Foundations
We develop domain theory in constructive univalent foundations without
Voevodsky's resizing axioms. In previous work in this direction, we constructed
the Scott model of PCF and proved its computational adequacy, based on directed
complete posets (dcpos). Here we further consider algebraic and continuous
dcpos, and construct Scott's model of the untyped
-calculus. A common approach to deal with size issues in a predicative
foundation is to work with information systems or abstract bases or formal
topologies rather than dcpos, and approximable relations rather than Scott
continuous functions. Here we instead accept that dcpos may be large and work
with type universes to account for this. For instance, in the Scott model of
PCF, the dcpos have carriers in the second universe and suprema
of directed families with indexing type in the first universe .
Seeing a poset as a category in the usual way, we can say that these dcpos are
large, but locally small, and have small filtered colimits. In the case of
algebraic dcpos, in order to deal with size issues, we proceed mimicking the
definition of accessible category. With such a definition, our construction of
Scott's again gives a large, locally small, algebraic dcpo with
small directed suprema.Comment: A shorter version of this paper will appear in the proceedings of CSL
2021, volume 183 of LIPIc
Apartness, sharp elements, and the Scott topology of domains
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets
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.
Exponentiation of Scott formal topologies
We prove that Scott formal topologies are exponentiable in the category of inductively generated formal topologies. From an impredicative point of view, this means that Scott domains are exponentiable in the category of open locales