Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, we further continue a project carried out to investigate connections between domain theory and quasi-metric spaces [1]. Here, we provide some domain theoretic (computational) models for the hyperspace of nonempty compact subsets of quasi-metric spaces. On one hand, the recent applications of quasi-metric spaces in different subjects of computer science, e.g. denotational semantics of programming languages, complexity and dual-complexity spaces and complexity distances between algorithms ([rom-val, rod-sch-val, gar-rom-sch, rom-san-val, rom-sch, rodri-rom-val]) and, on the other hand, a new insight of the domain theoretic point of view into the theory of hyperspaces and its new applications within mathematics, e.g. discrete dynamical systems, measure and integration theory ([edalat1, edalat2, edalat3]), motivate establishing computational models of these structures.

Finding a domain theoretic (computational) model for a topological space $(X,\tau)$ amounts to providing a suitable partially ordered set $(P,\sqsubseteq)$ together with a topological embedding $\phi$ from $(X,\tau)$ to $(P,\sqsubseteq)$ endowed with the Scott topology, denoted by $\sigma$. This is a variant of a fundamental problem in domain theory, called the maximal point space problem, which demands a homeomorphism between $(X,\tau)$ and the space of maximal point of $(P,\sqsubseteq)$. The study of computational models for various type of topological spaces goes back to the works of Edalat and respectively Blanck [edalat1, Blanck, Blanck1]. Later, the maximal point space problem was explicitly formulated and became a subject of intensive investigations by many authors [Law, Lutzer, Martin, Rutten]. Some special cases of this problem have satisfactory solutions [2, kop-kunz].

The domain theoretic construction $\mathbf{B}X$ of the space of formal balls, introduced by Edalat and Heckmann, provides a concrete (computational) model for a metric space $(X,d)$ [edalat]. The importance of this construction is that, first of all, it connects some metric properties of $(X,d)$ to the order theoretic properties of $\mathbf{B}X$. Secondly, it ties the above notion of computational model to the notion of computability for metric space $(X,d)$ [edalat4, Laws]. The notion of formal balls is also defined in the same way for a quasi-metric space $(X,d)$ and the order theoretic properties of ${\bf B}X$ are tightly connected to the topological properties of $(X,d)$ [1, Rom-Val, Rom-Val1]. In particular, for a $T_{1}$ quasi-metric space $(X,d)$ is sequentially Yoneda-complete if and only if ${\bf B}X$ is a directed complete partially ordered set.

Edalat and Heckmann also constructed the Plotkin powerdomain ${\mathcal{P}}{\bf B}X$ of the space of formal balls of a metric space $(X,d)$ and showed that there is a one-to-one correspondence between the nonempty compact subsets of $(X,d)$ and the maximal elements of the Plotkin powerdomain of ${\mathbf{B}}X$. As an application, a domain theoretic proof was given for a classical result of Hutchinson ([hutchinson]) which states that if $(X,d)$ is complete, then any hyperbolic iterated function system has a unique non-empty compact attractor. It can be shown that this construction is a computational model for the hyperspace of nonempty compact subsets of $X$, denoted by ${\mathcal{K}}_{0}(X)$, with the Vietoris or equivalently the Hausdorff topology. This fact was also proved in a different way by Martin in [martin1]. His interesting idea is based on the existence of a certain measurement, called Lebesgue measurement, on any domain $D$ which models the metric space $(X,d)$. Subsequently, Liang and Kou in [L-K] generalized these results to continuous dcpo’s which have the Lawson condition, i.e. the Lawson and Scott topologies coincide on the space of maximal points. Indeed, under the Lawson condition, it is proved that there is a homeomorphism between the space of nonempty compact subsets of maximal points of a continuous dcpo $D$ endowed with the Vietoris topology and the space of maximal points of Plotkin powerdomain $D$ equipped with the induced Scott topology. More recently, in another line of research, Berger et al. ([berger]) showed that for any $T_{1}$ topological space which is represented by an $\omega$-domain $D$, the hyperspace of its nonempty compact subsets can be represented by the Plotkin powerdomain ${\mathcal{P}}D$ of domain $D$. This result was made possible by a theorem of Smyth ([smyth]) which states that for any $\omega$-continuous dcpo $D$ the space $({\mathcal{P}}D,\sigma)$ is homeomorphic to the space of $D$-lenses, i.e. nonempty compact subsets of domain $D$ which are intersection of a closed set and a saturated set, endowed with the Vietoris topology.

In the present work, we study computational models of the hyperspace $\mathcal{K}_{0}(X)$ of a $T_{1}$ quasi-metric space $(X,d)$ equipped with the Vietoris and respectively the Hausdorff topology, proving that whenever $(X,d)$ satisfies certain completeness properties, e.g. Yoneda and respectively Smyth completeness, this space has a computational model. It is worth mentioning that the space of formal balls of a quasi-metric space does not generally satisfy the Lawson nor countable based conditions. Therefore, the results of Liang and Kou [L-K] and Berger et al. [berger] do not apply to the present context. Also, unlike the metric case, there is no natural candidate for a measurement on the space of formal balls of a quasi-metric space and hence the method used by Martin in [martin1] cannot be applied here either.

Edalat and Heckmann used the Plotkin powerdomain ${\mathcal{P}}{\bf B}X$ given as the ideal completion of the abstract basis of finite subset of ${\mathbf{B}}X$, ${\mathcal{P}}_{fin}{\bf B}X$, with respect to the Egli-milner relation, $\prec_{EM}$, to present a computational model of $\mathcal{K}_{0}(X)$, for every metric space $(X,d)$. To this end, they employed the symmetry axiom of metric $d$, to get a key fact that any maximal ideal has a cofinal $\omega$-chain. In the case of quasi-metric spaces, the Plotkin powerdomain ${\mathcal{P}}{\bf B}X$ can also be defined, though, the lack of symmetry for the quasi-metric $(X,d)$ prevents us from finding cofinal $\omega$-chains in maximal ideals. That is why we prefer to work directly with the $\omega$-chains and this leads us to the chain-completion construction instead.

So, for a $T_{1}$ quasi-metric space $(X,d)$, we consider the space ${\mathbf{B}}X$ of formal balls and let ${\mathbf{C}\mathbf{B}}X$ be the chain completion of $({\mathcal{P}}_{fin}{\bf B}X,\prec_{EM})$. This construction is called the $\omega$-Plotkin domain. By the general construction of chain completion, ${\mathbf{C}\mathbf{B}}X$ is a continuous $\omega$-dcpo, i.e. a continuous poset in which every $\omega$-chain has a least upper bound. Now, to achieve our purpose in finding a computational model, we define a one-to-one map $\phi:\mathcal{K}_{0}(X)\rightarrow{\mathbf{C}\mathbf{B}}X$, which is an embedding if we consider the Vietoris topology on $\mathcal{K}_{0}(X)$ and assume that $(X,d)$ is a sequentially Yoneda-complete $T_{1}$ quasi-metric space. Moreover, $\phi$ is an embedding with respect to the topology of the Hausdorff quasi-metric $H_{d}$ on $\mathcal{K}_{0}(X)$ if any compact subset of $X$ is $d^{-1}$-precompact. Therefore, $({\mathbf{C}\mathbf{B}}X,\phi)$ serves as an $\omega$-computational model for $\mathcal{K}_{0}(X)$ endowed with the mentioned topologies. Although it is not known whether ${\mathbf{C}\mathbf{B}}X$ is a dcpo and therefore a computational model of $\mathcal{K}_{0}(X)$, nevertheless, thanks to Fact 1 and Theorem LABEL:dcpo, the ideal completion of ${\mathbf{C}\mathbf{B}}X$ gives a computational model for $\mathcal{K}_{0}(X)$.

In section 5, we take another well-known notion of computational model, called the quantitative $\omega$-computational model. This is an $\omega$-computational model $(P,\sqsubseteq,\phi)$ carrying an additional quasi-metric $D$ such that $\phi$ is an isometry from $(X,d)$ into $(P,D)$ together with some extra conditions which capture the order structure of $(P,\sqsubseteq)$ (Definition LABEL:quantitative). A modified version of this notion can be found in [Rom-Val, Rutten, Schellekens, Was]. We prove that in fact ${\mathbf{C}\mathbf{B}}X$ is a quantitative $\omega$-computational model for $(\mathcal{K}_{0}(X),H_{d})$, by constructing a quasi-metric $D$ on ${\mathbf{C}\mathbf{B}}X$. To this end, we consider a quasi-metric $q$ defined by Romaguera and Valero ([Rom-Val]) on ${\bf B}X$. The primary reason to choose this quasi-metric on $\mathbf{B}X$ is that $(\mathbf{B}X,q)$ is a quantitative computational model for $(X,d)$. Therefore, its specialization order $\sqsubseteq_{q}$ is equivalent to the partial order of $\mathbf{B}X$. Consequently, one could naturally extend $q$ to the Hausdorff quasi-metric $H_{q}$ on ${\mathcal{P}}_{fin}{\bf B}X$ of the finite subsets of ${\bf B}X$, whose main property is that it induces the Egli-Milner relation on ${\mathcal{P}}_{fin}{\bf B}X$. Subsequently, the quasi-metric $H_{q}$ can be lifted up to a quasi-metric $D$ on ${\mathbf{C}\mathbf{B}}X$ in such a way that the ordered structures $({\mathbf{C}\mathbf{B}}X,\sqsubseteq_{D})$ and $({\mathbf{C}\mathbf{B}}X,\sqsubseteq)$ coincide. Once $D$ is established one can show that $({\mathbf{C}\mathbf{B}}X,D)$ is a Yoneda-complete space and in fact Yoneda-completion of $({\mathcal{P}}_{fin}{\bf B}X,H_{q})$. This makes $({\bf C}{\bf B}X,\sqsubseteq,\phi,D)$ a quantitative $\omega$-computational model for $({\mathcal{K}}_{0}(X),H_{d})$.

We, finally, conclude this paper by comparing the Plotkin powerdomain and the $\omega$-Plotkin domain constructions. We prove that if $(X,d)$ is either, Smyth-complete and all of its compact subsets are $d^{-1}$-precompact, or an $\omega$-algebraic Yoneda-complete space, then the Plotkin Powerdomain ${\mathcal{P}}\mathbf{B}X$ is order-isomorphic to $\mathbf{C}\mathbf{B}X$.

We assume the reader is familiar with the basic definitions and facts about domain theory which can be found in ([abramsky, compendium]), though, we briefly explain some of the definitions and facts which are more crucial in this note.

Let $(P,\sqsubseteq)$ be a partially ordered set (abbr. by poset). The binary relation $\prec$ is called an auxiliary relation on the poset $(P,\sqsubseteq)$ if (1) $p\prec p$ implies $p\sqsubseteq p$, (2) $p\sqsubseteq s\prec r\sqsubseteq q$ implies $p\prec q$ and (3) satisfies the interpolation property, i.e. for any finite subset $M$ of $P$ and $p\in P$, if for every $m\in M$, $m\prec p$ then there exists some $q\in P$ such that $m\prec q\prec p$, for every $m\in M$. The pair $(P,\prec)$ is called an abstract basis, if $\prec$ is a transitive relation which also satisfies the interpolation property. A nonempty directed lower subset $I$ of $P$ is called a round ideal if for any $x\in I$ there is $y\in I$ such that $x\prec y$. The set of all round ideals of $P$ partially ordered by $\subseteq$ is called the ideal completion of $P$, denoted by $Idl(P)$. Let $\hbox to0.0pt{$\uparrow$\hss}\raise 1.72218pt\hbox{$\uparrow$}p=\{q:\ p\prec q\}$ and $\hbox to0.0pt{$\downarrow$\hss}\raise 1.72218pt\hbox{$\downarrow$}p=\{q:\ q\prec p\}$. An auxiliary relation is called approximating if $\hbox to0.0pt{$\downarrow$\hss}\raise 1.72218pt\hbox{$\downarrow$}p\subseteq\hbox to0.0pt{$\downarrow$\hss}\raise 1.72218pt\hbox{$\downarrow$}q$ implies $p\sqsubseteq q$. One can see that the set $\{\hbox to0.0pt{$\uparrow$\hss}\raise 1.72218pt\hbox{$\uparrow$}p:\ p\in P\}$ forms a basis for a topology called the pseudoScott topology on $P$, denoted by $\mathbf{P}\sigma$. The following fact is needed for the proof of Theorem LABEL:domrep.