CTFM2019 t ————————————– Introduction f Continuities of posets . . . Models of Computations Continuous . . . —Information Systems and Domains Generalized algebraic . . . a Weak algebraic . . . Categorical aspects Related topics r Luoshan Xu homepage D Outline School of Mathematics Science ◭◭ ◮◮ ◭ ◮ Yangzhou University Page 1 Total 54 Wuhan, 2019. 3. 24 Return Full screen Close Out
t Introduction f Continuities of posets . . . ⑧ Introduction Continuous . . . Generalized algebraic . . . a ⑧ Continuities of Posets and Scott topology Weak algebraic . . . Categorical aspects ⑧ Continuous information systems Related topics r ⑧ Generalized algebraic information systems homepage D Outline ⑧ Weak algebraic information systems ◭◭ ◮◮ ⑧ Categorical aspects ◭ ◮ Page 2 Total 54 ⑧ Related topics Return Full screen Close Out
1 Introduction t Introduction f Computations Continuities of posets . . . Continuous . . . • can be viewed as both functions and process. Generalized algebraic . . . a Weak algebraic . . . Categorical aspects Related topics • can be carried out by programs. r homepage D Outline • are changes of states (of Turing machines). ◭◭ ◮◮ ◭ ◮ • can be taken as maps from Input information to Output information. Page 3 Total 54 Return • can also be taken as modal logic of inferences (formula), special binary Full screen relations, partial orders. Close Out
t Introduction f • So, to study computations is to study posets, and study states of informa- Continuities of posets . . . Continuous . . . tion systems, what we should study for posets? Generalized algebraic . . . a Weak algebraic . . . • In order to assign meanings to programs written in high-level program- Categorical aspects Related topics r ming languages, Dana Scott invented continuous lattices [14] which is now homepage grown up as Domain Theory [1, 4]. D Outline ◭◭ ◮◮ ◭ ◮ • From states of computations, with continuity, domains can be taken as Page 4 Total 54 models of denotational semantics of computations. Return Full screen Close Out
t How to model computations? —By domains: Introduction f Continuities of posets . . . • Structures arising in theoretical computer science admit natural partial or- Continuous . . . Generalized algebraic . . . a ders of appropriate information content. Weak algebraic . . . Categorical aspects • The more information some state contains, the larger it is in the informa- Related topics r tion order. homepage D Outline • It is a common sense that the increasing sequence of information should ◭◭ ◮◮ give more (converges to) accurate states (of computation). ◭ ◮ • D. Scott lead to the discovery (1972): continuous lattices [14], now more Page 5 Total 54 Return generalized as domains = continuous dcpos. Full screen Close Out
t • Domain theory is one of the important research fields of theoretical com- Introduction f Continuities of posets . . . puter science. Mutual transformations and infiltration of order, topology Continuous . . . Generalized algebraic . . . a and logic are the basic features of this theory. Weak algebraic . . . Categorical aspects • Ways to characterize domains: not only by continuity, but also by Related topics r Stone duality [3], homepage D abstract bases [24], Outline formal topologies [25], ◭◭ ◮◮ ◭ ◮ information systems [2, 16], Page 6 Total 54 rough approximable concepts [6] and Return F-augmented closure spaces [5]. Full screen Close Out
t Introduction f How to model computations? —By information systems: Continuities of posets . . . Continuous . . . Generalized algebraic . . . a • From actions of a computation, Dana Scott in his seminal paper [15], in- Weak algebraic . . . Categorical aspects troduced information systems as a logic-oriented approach to denotational Related topics r semantics of programming languages, or, models of denotational seman- homepage tics of computations. D Outline ◭◭ ◮◮ • A large volume of work followed with information systems has been done ◭ ◮ [8, 9, 18, 19, 20, 21, 26, 27, 28]. Page 7 Total 54 Return Full screen Close Out
t Introduction f Continuities of posets . . . • In 1993, Hoofman [8] introduced continuous information systems (shortly, Continuous . . . Generalized algebraic . . . cis) in his sense that represent bc-domains (the continuous counterpart of a Weak algebraic . . . Categorical aspects Scott domains). Related topics r • In 2001, Bedregal [2] modified Hoofman’s definition of cis. homepage D Outline • In 2008, Spreen, Xu and Mao [16] first introduced a new concept of con- ◭◭ ◮◮ tinuous information systems (in short, C-inf). C-infs generate/represent ◭ ◮ Page 8 Total 54 exactly all the continuous (not necessarily pointed) domains. Return Full screen Close Out
t • Later, Xu and Mao [26] introduced the concept of algebraic information Introduction system (in short, A-inf). f Continuities of posets . . . Continuous . . . • In 2012, Spreen in [17] introduced L -information systems which represent Generalized algebraic . . . a Weak algebraic . . . all pointed L -domains. Categorical aspects Related topics • In 2013, Wu and Li [20] proposed new algebraic information systems (e- r homepage quivalent to A-infs) with briefer conditions to represent algebraic domains. D Outline ◭◭ ◮◮ • In 2016, by adding new conditions to C-infs, Wu, Guo and Li [21] provid- ◭ ◮ Page 9 Total 54 ed a kind of information systems which serve as representations of general Return L -domains. Full screen Close Out
t Introduction f Continuities of posets . . . Continuous . . . Generalized algebraic . . . a Weak algebraic . . . • Since domains and information systems are all models of computations, Categorical aspects Related topics they are closely linked. r • We will see that homepage D Outline —- all the states of a C-inf forms a domain, and ◭◭ ◮◮ —- every domain can induce an information system in a standard manner. ◭ ◮ Page 10 Total 54 Return Full screen Close Out
t Introduction f We for details, in this talk, Continuities of posets . . . Continuous . . . • introduce basic concepts for domains and information systems. Generalized algebraic . . . a Weak algebraic . . . Categorical aspects • introduce results for domains and information systems. Related topics r homepage • give relationships of the two kinds of models. D Outline • and propose some further topics. ◭◭ ◮◮ ◭ ◮ Page 11 Total 54 Some of them are newly obtained by our group. Return Full screen Close Out
2 Continuities of posets and Scott topology t One of the important things for posets is the way-below relation, or approxi- Introduction f Continuities of posets . . . mation order. Continuous . . . Generalized algebraic . . . a Weak algebraic . . . Categorical aspects Definition 2.1. (Way-below relation) Let P be a poset, x, y ∈ P . We say that Related topics r x approximates y , written x ≪ y , if whenever D is directed with sup D � y , homepage then x � d for some d ∈ D . We use ↓ ↓ x to denote the set { a ∈ P : a ≪ x } . D Outline ◭◭ ◮◮ If for every element x ∈ P , the set ↓ ↓ x := { a ∈ P : a ≪ x } is directed and ◭ ◮ Page 12 Total 54 sup ↓ ↓ x = x , then P is called a continuous poset . A continuous poset which Return is also a dcpo (resp., bounded complete dcpo, complete lattice) is called a Full screen continuous domain or briefly a domain (resp., bc-domain, continuous lattice). Close Out
t Introduction f Continuities of posets . . . “ ≪ ” also known as way-below relation. Continuous . . . Generalized algebraic . . . a Weak algebraic . . . Example 2.2. Some examples and counterexamples: Categorical aspects Related topics • Continuous posets: discrete sets, (0, 1), R , N , . . . . r homepage • Domains: half open unit interval (0,1], finite posets, . . . . D Outline • Continuous lattices: CD-lattices, topologies of compact Hausdroff spaces. ◭◭ ◮◮ ◭ ◮ • NOT continuous: complete lattice shaped “ ♦ ” . Page 13 Total 54 Return Full screen Close Out
Definition 2.3. Let P be a poset, B ⊆ P . The set B is called a basis for P if ∀ a ∈ P , there is a directed set D a ⊆ B such that ∀ d ∈ D a , d ≪ P a and t sup P D a = a . Introduction f Continuities of posets . . . Theorem 2.4. A poset P is continuous iff it has a basis. Continuous . . . Generalized algebraic . . . a To clarify relationships of continuous posets and domains, the concept of Weak algebraic . . . Categorical aspects embedded basis for posets is useful. Related topics r Definition 2.5. (Xu, 2006, [23]) Let B and P be posets. If there is a map homepage D j : B → P satisfying Outline (1) j preserves existing directed sups, ◭◭ ◮◮ ◭ ◮ (2) j : B → j ( B ) is an order isomorphism, Page 14 Total 54 (3) j ( B ) is a basis for P , Return then ( B, j ) is called an embedded basis for P . If B ⊆ P and ( B, i ) is an Full screen embedded basis for P , where i is the inclusion map, then we say also that B Close is an embedded basis for P . Out
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