The Hoare powerdomain plays an important role in modeling the programming semantics of nondeterminism, which is analogous to the powerset construction (see for example HECKMANN199177 ; HECKMANN1992 ; HECKMANN2013215 ; Plotkin1976 ; SMYTH197823 ; SMYTH1983 ). The Hoare power domain over a dcpo (directed complete poset) is the free inflationary semilattice where the inflationary operator is exactly the binary join operator. The standard construction of a Hoare powerdomain consists all nonempty Scott closed subsets of the dcpo , order given by the inclusion relation. In Yuan2014Consistent , Yuan and Kou introduced a new type of powerdomain, called the consistent Hoare powerdomain. The new powerdomain over a dcpo is a free algebra where the inflationary operator delivers joins only for consistent pairs. They provided a concrete way of constructing the consistent Hoare powerdomain over a domain (continuous dcpo): the family of all nonempty relatively consistent Scott closed subsets of the domain is such a powerdomain. Follows from the work, there is a natural problem: whether every dcpo has a consistent Hoare powerdomain. Geng and Kou GENG2017169 gave an affirmative answer to the question by showing that the Scott closure of in is a consistent Hoare powerdomain over the dcpo , where is the collection of all nonempty consistent Scott closed subsets of . One question arises naturally: can we give a direct characterization of the consistent Hoare powerdomain over every dcpo ? That is to say, what type of Scott closed subsets of is exactly the consistent Hoare powerdomain? This paper is mainly set to answer this question.
The consistent Hoare powerdomain can be viewed as a type of completion that embeds every dcpo into a dcpo which is also complete with respect to consistent pair joins. In zd , Zhao and Fan introduced a type of dcpo-completion embedding each poset into a dcpo, called -completion. They proved that the Scott closed set lattice of a poset is order isomorphic to that of the -completion. It is also the case for the sobrification, i.e., the topology lattice of a -space is isomorphic to the topology lattice of the sobrification of (see gg1 ). In this paper, we introduce a type of closed sets on every dcpo--semilattice, called -Scott closed sets. We prove that the Scott closed set lattice of a dcpo is isomorphic to the family of all -Scott closed subsets of its consistent Hoare powerdomain. It is known that two sober dcpos are isomorphic if and only if their Scott set lattices are isomorphic (see HZP ; Zhao2018U ). Consequently, the consistent Hoare powerdomains for sober dcpos are uniquely determined up to isomorphism.
This section is an introduction to the concepts and results about the consistent Hoare powerdomains over domains and dcpos. For more details, refer to GENG2017169 ; gg1 ; Yuan2014Consistent . In this paper, the order relation on each family of sets is the set-theoretic inclusion relation.
A nonempty subset of a poset is called directed if every pair has an upper bound in . If every directed has a supremum , then is called a dcpo. For , let for some , where . If , then is called a lower set. For , we say is way below , written , if implies for any directed . Let , and . A dcpo is called a domain if each element is the directed join of . A subset of a dcpo is Scott closed if it is a lower set and closed under directed sups. Let denote the Scott closure of .
Yuan2014Consistent A consistent inflationary semilattice is a dcpo with a Scott continuous binary partial operator defined only for consistent pairs of points that satisfies three equations for commutativity , associativity and idempotency together with the inequality for . The free consistent inflationary semilattice over a dcpo is called the consistent Hoare powerdomain over and denoted by .
For a consistent inflationary semilattice, the operator coincides with , the join operator defined only for consistent pairs. That is to say, a consistent inflationary semilattice is exactly a dcpo that is a -semilattice, also called a dcpo--semilattice. A Scott closed subset of a domain is called relatively consistent if is the Scott closure of the directed union , where
is a nonempty finite consistent subset of and .
The set of all nonempty relatively consistent Scott closed subsets of is denoted by .
Yuan2014Consistent Let be a domain. The consistent Hoare powerdomain over can be realized as where . The embedding of into is given by for .
Let denote the family of all nonempty, consistent and Scott closed subsets of a dcpo . And let be the Scott closure of in . The family was introduced by Geng and Kou GENG2017169 , in order to declare that the consistent Hoare powerdomain over every dcpo exists.
GENG2017169 Let be a dcpo and is a dcpo--semilattice. If is a Scott continuous function, then for any , exists in .
GENG2017169 Let be a dcpo. Then the consistent Hoare powerdomain over can be realized as the , where whenever are consistent in . The embedding of into is given by for any .
3 A direct characterization of the consistent Hoare powerdomain
A subset of a dcpo is called existing if for any continuous function mapping into a dcpo--semilattice , always exists in .
For any directed subset of , is directed in since continuous functions are monotone, and then exists in the dcpo--semilattice . Hence every directed subset is existing. Similarly, every consistent finite subset is also -existing. Notice that an empty set is not existing because a dcpo--semilattice may not have a least element.
Let be a continuous function from dcpo to dcpo--semilattice , and . Then
(1) exists exists .
(2) is -existing if and only if is -existing.
(1) Assume that is an upper bound of . Then and . Hence , and then , i.e., is an upper bound of . The converse is clearly true. Thus exists iff exists, and both imply that .
(2) It is straightforward from (1) and Definition 3.1. ∎
For any dcpo , we write is -existing and Scott closed. By Lemma 2.3, we immediately have that . We shall show that the converse inclusion is also true, and then a characterization of the consistent Hoare powerdomain is obtained.
A subset of a dcpo--semilattice is called -Scott closed if it is Scott closed and for any consistent nonempty finite set , . We write , called the -Scott closure system on , for the set of all -Scott closed subsets of , and let denote the corresponding closure operator. A function between dcpo--semilattices is called -Scott continuous if is continuous with respect to the -Scott closure systems.
(1) A function between dcpo--semilattices is a dcpo--semilattice homomorphism iff is -Scott continuous.
(2) If a nonempty subset of a dcpo--semilattice is consistent, then .
(1) Suppose that is a dcpo--semilattice homomorphism. Let be -Scott closed. We shall show that is also -Scott closed. For any consistent nonempty finite , we have is also nonempty finite and consistent since is monotone. Then and then is in . Similarly, is also closed with respect to directed joins. Thus is -Scott continuous.
Conversely, let be -Scott continuous. Clearly, is monotone. Let be a consistent nonempty finite subset of . Then . The set is -Scott closed in . Then is also -Scott closed and . Hence and then . Thus . Analogously, for all directed . Therefore, is a dcpo--semilattice homomorphism.
(2) Let be any -Scott closed set with . Notice that each subset of is also consistent. Then is nonempty and finite is a directed subset of , and hence . Moreover, every -Scott closed set is a lower set. Then , and thus . ∎
Let be a dcpo--semilattice. A subset is called --existing if for any dcpo--semilattice homomorphism mapping into a dcpo--semilattice , exists in .
Let be a dcpo--semilattice. A subset is --existing iff is --existing.
The process is similar to that of Proposition 3.2. Notice that every principal ideal is -Scott closed. ∎
If a subset of a dcpo--semilattice is -Scott closed and --existing, then for some .
The identity function is a dcpo--semilattice homomorphism. We have that exists in . Then every nonempty finite subset is consistent, and hence exists. Since is -Scott closed, we obtain that is nonempty and finite is a directed subset of , and then . Thus , which completes the proof. ∎
A subset of a dcpo is -existing iff is an --existing subset of . (Notice that is the function in Theorem 2.4.)
Suppose that is -existing. Let be any dcpo--semilattice homomorphism mapping into a dcpo--semilattice . Then is a Scott continuous function from to , and hence exists in . Thus is an --existing subset of .
Now assume that is an --existing subset of . Let be a Scott continuous mapping into a dcpo--semilattice . Then by Theorem 2.4, there exists a unique dcpo--semilattice homomorphism such that . Then exists. Thus is a -existing subset of . ∎
We now come to the characterization of the consistent Hoare powerdomains by -existing Scott closed sets:
Let be a dcpo and . Then iff is Scott closed and -existing, i.e., .
By Lemma 2.3, we have that . Now suppose that . Then, by Lemma 3.8, the set is an --existing subset of . And by Lemma 3.6, the -Scott closed set is also --existing. Then has a supremum which is in itself by Lemma 3.7. Since the order on is the inclusion relation, we have . For each , , and then . Thus because is a lower set of , which completes the proof. ∎
Let be a dcpo. The family of Scott closed sets is order isomorphic to the -Scott closure system on the consistent Hoare powerdomain , i.e., .
Define a function by
where is the -Scott closure of in . We shall show that is an order isomorphism. Obviously, is monotone.
Firstly, we prove that is injective. Suppose that with . Without loss of generality, we may assume that there is . Let . Clearly, and . We claim that is -Scott closed in . Then , and hence . But . Thus , i.e., is injective. Indeed, let be directed, then , the supremum of in , is the Scott closure of in , and hence since each element of is contained in , i.e., . And if , then , thus (notice that is closed under finite unions), which proves the claim.
We next prove that is surjective. Suppose that is -Scott closed in . We claim that is Scott closed in . Indeed, if is directed, then is directed, and hence , i.e., . We shall show that . Since , we have . Conversely, assume that . Then is consistent in . By Proposition 3.4(2), we have . Thus , i.e., is surjective, which completes the proof. ∎
A dcpo is called sober if every Scott closed irreducible set is of the form . It is known that for sober dcpos and , iff (see HZP ; Zhao2018U for more results about uniqueness of dcpos based on the Scott closed set lattices). In particularly, the Scott topology of every quasicontinuous domain is sober (each domain is a quasicontinuous domain, see gg1 for the detailed definition of a quasicontinuous domain). By applying the above theorem, we immediately have that for sober dcpos, the consistent Hoare powerdomains are uniquely determined up to isomorphism:
If dcpos and are sober, then iff .
By Theorem 3.10, implies iff iff implies . ∎
This work is supported by National Natural Science Foundation of China (No.11801491,11771134) and Shandong Provincial Natural Science Foundation, China (No. ZR2018BA004).
- (1) J. Geng and H. Kou. Consistent Hoare powerdomains over dcpos. Topology and its Applications, 232:169 – 175, 2017.
- (2) G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott. Continuous Lattices and Domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2003.
- (3) R. Heckmann. Power domain constructions. Science of Computer Programming, 598(1):77 – 117, 1991.
- (4) R. Heckmann. An upper power domain construction in terms of strongly compact sets. Lect. Notes Comput. Sci., 598(1):272 – 293, 1992.
- (5) R. Heckmann and K. Keimel. Quasicontinuous domains and the Smyth powerdomain. Electronic Notes in Theoretical Computer Science, 298:215 – 232, 2013.
- (6) W. K. Ho, J. Goubault-Larrecq, A. Jung, and X. Xi. The Ho-Zhao problem. Logical Methods in Computer Science, 14(1:7):1–19, 2018.
- (7) G. Plotkin. A powerdomain construction. SIAM Journal on Computing, 5(3):452–487, 1976.
- (8) M.B. Smyth. Power domains. Journal of Computer and System Sciences, 16(1):23 – 36, 1978.
- (9) M.B. Smyth. Power domains and predicate transformers: A topological view. Lecture Notes in Computer Science, 154:662–675, 1983.
- (10) Y. Yuan and H. Kou. Consistent Hoare powerdomains ☆. Topology and its Applications, 178:40–45, 2014.
- (11) D. Zhao and T. Fan. Dcpo-completion of posets. Theoretical Computer Science, 411(22):2167–2173, 2010.
- (12) D. Zhao and L. Xu. Uniqueness of directed complete posets based on Scott closed set lattices. Logical Methods in Computer Science, 14(2:10):1–12, 2018.