Interval Domains and Computable Sequences: A Case Study of Domain Reductions

J. Blanck
2012 Computer journal  
The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions. The effectivity theory induced by these variations is not stable, and this paper investigates some of the rich structure found. We follow Mostowski (On computable sequences. Fund. Math., 44, 37-51) and use computable sequences to exhibit this structure. Handling editor: Leonid Libkin 2. BACKGROUND
more » ... omains We will briefly give some background to domain theory. For a complete background on domains, we refer the readers to [4, 5] . Let D = (D, ) be a partially ordered set. A subset A ⊆ D is an upper set (dual lower set) if x ∈ A and x y implies y ∈ A. Let ↑A = {y ∈ D : ∃x ∈ A(x y)}. We will abbreviate ↑{x} by ↑x. A subset A ⊆ D is directed if A = ∅ and whenever x, y ∈ A then there is z ∈ A such that x z and y z. The supremum, or least upper bound, of A (if it exists) is denoted by A. A (directed) complete partial order, abbreviated CPO, is a partial order, D = (D; , ⊥), such that ⊥ is the least element in D and any directed set A ⊆ D has a supremum, Downloaded from Definition 2.5. A retract representation of X is a quadruple (D, D R , ρ, η) where (D, D R , ρ) is a representation, and η : X → D R is a continuous function such that ρη = id X .
doi:10.1093/comjnl/bxs121 fatcat:hx2g6bxfjrevvgpvrufrszikcq