Partial Continuous Functions Research Articles

Normal semicontinuity and the Dedekind completion of pointfree function rings

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

Decomposing Borel functions and structure at finite levels of the Baire hierarchy

We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of continuous functions with domains of type Πn0, for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2.

Čech-completeness and related properties of the generalized compact-open topology

The generalized compact-open topology τ C on partial continuous functions with closed domains in X and values in Y is studied. If Y is a non-countably compact Čech-complete space with a G δ -diagonal, then τ C is Čech-complete, sieve complete and satisfies the p -space property of Arhangel'skiǐ, respectively, if and only if X is Lindelöf and locally compact. Lindelöfness, paracompactness and normality of τ C is also investigated. New results are obtained on Čech-completeness, sieve completeness and the p -space property for the compact-open topology on the space of continuous functions with a general range Y .

Effectivity on Continuous Functions in Topological Spaces

In this paper we investigate aspects of effectivity and computability on partial continuous functions in topological spaces. We use the framework of TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. We generalize the representations introduced in [Weihrauch, K., “Computable Analysis,” Springer, Berlin, 2000] for the Euclidean case to computable T0-spaces and computably locally compact Hausdorff spaces respectively. We show their equivalence and in particular, prove an effective version of the Stone-Weierstrass approximation theorem.

Partial Continuous Functions and Admissible Domain Representations

It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison d'etre for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by a partial continuous function is Cartesian closed. Finally, we consider the question of effectivity.

Finitely Continuous Hamel Functions

A function $h : \mathbb{R}^n \to \mathbb{R}^k$ is called a Hamel function if it is a Hamel basis for $\mathbb{R}^{n+k}$. We prove that there exists a Hamel function which is finitely continuous (its graph can be covered by finitely many partial continuous functions). This answers the question posted in [3].

On simultaneous linear extensions of partial (pseudo)metrics

We consider the question of simultaneous extension of (pseudo) metrics defined on nonempty closed subsets of a compact metrizable space. The main result is a counterpart of the result due to Künzi and Shapiro for the case of extension operators of partial continuous functions and includes, as a special case, Banakh’s theorem on linear regular operators extending (pseudo)metrics.

A Relational Account of Call-by-Value Sequentiality

We construct a model for FPC, a purely functional, sequential, call-by-value language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.

The Computational Power of ℳω

We prove that the Kleene schemes for primitive recursion relative to the μ-operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene-Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳ︁ω we prove Nigg's conjecture that ℳ︁ω is strictly weaker than Plotkin's PCF + PA.

Computability over the partial continuous functionals

AbstractWe show that to every recursive total continuous functional Φ there is a PCF-definable representative Ψ of Φ in the hierarchy of partial continuous functionals, where PCF is Plotkin's programming language for computable functionals. PCF-definable is equivalent to Kleene's S1–S9-computable over the partial continuous functionals.

Mω considered as a programming language

The paper studies a simply typed term system M ω providing a primitive recursive concept of parallelism in the sense of Plotkin. The system aims at defining and computing partial continuous functionals. Some connections between denotational and operational semantics → for M ω are investigated. It is shown that → is correct with respect to the denotational semantics. Conversely, → is complete in the sense that if a program denotes some number k, then it is reducible to the numeral n k . Restricting to the primitive recursive kernel ℘R ω of M ω , it is shown that → is strongly normalising with uniquely determined normal forms. The twist is the design of fixed point style conversion rules for constants μ l accounting for parallelly bounded parallel search such that correctness and strong normalisation hold. Thereupon, minor alternations to → bring about that every reduction sequence for a program of ℘R ω terminates either in a numeral n k if the program denotes k, or in the term (⊂-1)0 if the program denotes the “undefined” object. Thus, ℘R ω can be considered a primitive recursive version of Plotkin's L PA+∃·.

A Relational Account of Call-by-Value Sequentiality

We construct a model for FPC, a purely functional, sequential, call-by-value&lt;br /&gt;language. The model is built from partial continuous functions,&lt;br /&gt;in the style of Plotkin, further constrained to be uniform with respect to&lt;br /&gt;a class of logical relations. We prove that the model is fully abstract.

Non-definability of the Ackermann function with type 1 partial primitive recursion

The paper builds on a simply typed term system ${\cal PR}^\omega $ providing a notion of partial primitive recursive functional on arbitrary Scott domains $D_\sigma$ that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension ${\cal PR}^{\omega e}$ is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist are certain type 1 Godel recursors ${\cal R}_k$ for simultaneous partial primitive recursion. Formally, the value ${\cal R}_k\vec{g}\vec{H} x i$ is a function $f_i$ of type $\iota \to \iota$ , however, ${\cal R}_k$ is modelled such that $f_i$ is a finite element of $D_{\iota\to\iota}$ or in other words, a partial sequence. It is shown that the Ackermann function is not definable in ${\cal PR}^{\omega e}$ .

Subrecursive hierarchies on Scott domains

We study a notion ofpartial primitive recursion (p.p.r.) including the concept ofparallelism in the context of partial continuous functions of type level one in the sense of [Krei], [Sco82], [Ers]. A variety of subrecursive hierarchies with respect top.p.r. is introduced and it turns out that they all coincide.