Functorial Data Migration. Categorical Informatics, Inc. December 1, C i

Transcription

1 Functorial Data Migration Categorical Informatics, Inc. December 1, 2015 C i

2 Outline The Functorial Data Model (FDM) The FDM is based on category theory, which was designed to migrate theorems from one area of mathematics to another. Functorial Data Migration Now, researchers at MIT use category theory to migrate data from one computer system to another. The Functorial Query Language (FQL) tool The FDM research has culminated in a prototype ETL software tool, FQL, available at catinf.com. Categorical Informatics (C i ) Because the FQL tool is based on a principled theoretical foundation, it gives the best possible answer to many ETL problems, and is being commercialized by C i. 2 / 1

3 Category theory Knowledge of category theory is not required to understand this talk, functorial data migration, or use the FQL tool. But for completeness we include a brief description of category theory. A category is an algebraic structure similar to a group, ring, or field. A category is a multi-sorted monoid and an algebra of functions. A functor is a homomorphism of categories. In the FDM, schemas, instances, schema mappings, and even data migration operations are expressed as categories and functors. Category theory has informed the design of: Functional programming languages (Haskell, ML, Scala) Query languages (nested relational calculus, LINQ) Proof assistants (Coq, Agda, HoTT) Mathematics itself (algebraic topology, set theory, model theory) 3 / 1

4 Category theory A category C consists of a set of objects for all objects X, Y a set CpX, Y q of arrows for all objects X an arrow id P CpX, Xq for all objects X, Y, Z a function : CpY, Zq ˆ CpX, Y q Ñ CpX, Zq such that f id id and id f f and pf gq h f pg hq A functor F : C Ñ D is a function taking objects in C to objects in D and arrows f : X Ñ Y in C to arrows F pfq: F pxq Ñ F py q in D such that F pidq id and F pf gq F pfq F pgq. A category presentation C consists of a set of nodes for all nodes X, Y a set CpX, Y q of edges a set of path equations A functor presentation F : C Ñ D is a function taking nodes in C to nodes in D and edges f : X Ñ Y in C to paths F pfq: F pxq Ñ F py q in D such that C $ p q implies D $ F ppq F pqq. 4 / 1

5 The Functorial Data Model manager Emp works secretary Dept last first Dom name Emp ID mgr works first last q10 Al Akin x02 Bob Bo q10 Carl Cork Emp.manager.works Emp.works Dept.secretary.works Dept Dept ID sec name q CS x Math Dom ID Al Akin Bob Bo Carl Cork CS Math 5 / 1

6 Attributes Omit Dom table, and draw edges Ñ f Dom as f : manager Emp works secretary Dept last first Dom name manager Emp works secretary Dept first last name 6 / 1

7 The Functorial Data Model manager Emp works secretary Dept first last name Emp.manager.works Emp.works Dept.secretary.works Dept Emp ID mgr works first last q10 Al Akin x02 Bob Bo q10 Carl Cork Dept ID sec name q CS x Math 7 / 1

8 Functorial Data Migration A functor F : S Ñ T is a constraint-respecting mapping: nodespsq Ñ nodespt q edgespsq Ñ pathspt q and it induces three adjoint data migration functors: F : T -inst Ñ S-inst (like project) S F T I F piq : IF Set ΠF : S-inst Ñ T -inst (like join) F % Π F Σ F : S-inst Ñ T -inst (like outer disjoint union then quotient) Σ F % F 8 / 1

9 (Project) Name Name Salary N1 N2 F ÝÝÝÑ N Salary Age Age N1 ID Name Salary 1 Alice $100 2 Bob $250 3 Sue $300 N2 ID Age F ÐÝÝ N ID Name Salary Age a Alice $ b Bob $ c Sue $ / 1

10 Π (Join) Name Name Salary N1 N2 F ÝÝÝÑ N Salary Age Age N1 ID Name Salary 1 Alice $100 2 Bob $250 3 Sue $300 N2 ID Age Π F ÝÝÑ N ID Name Salary Age a Alice $ b Alice $ c Alice $ d Bob $ e Bob $ f Bob $ g Sue $ h Sue $ i Sue $ / 1

11 Σ (Union) Name Name Salary N1 N2 F ÝÝÝÑ N Salary Age Age N1 ID Name Salary 1 Alice $100 2 Bob $250 3 Sue $300 N2 ID Age Σ F ÝÝÑ N ID Name Salary Age a Alice $100 null 1 b Bob $250 null 2 c Sue $300 null 3 d null 4 null 5 20 e null 6 null 7 20 f null 8 null / 1

12 Foreign keys Name Name Salary N1 f N2 F ÝÝÝÑ N Salary Age Age N1 ID Name Salary f 1 Alice $ Bob $ Sue $300 6 N2 ID Age F ÐÝÝ Π F,Σ F ÝÝÝÝÑ N ID Name Salary Age a Alice $ b Bob $ c Sue $ / 1

13 Expressivity of Functorial Data Migration - Example mgr Emp F ÝÝÝÑ SelfMgr F will copy SelfMgr into Emp, and put the identity into mgr. Π F will migrate into SelfMgr those Emps who are their own mgr. Σ F will migrate into SelfMgr representatives of the management groups of Emp, i.e. equivalence classes of Emps modulo the equivalence relation generated by mgr. 13 / 1

14 Pivot (Instance ô Schema) CS q name first works last Al Akin Math name x02 works mgr works first mgr first last Bob last Carl Bo Cork mgr Emp ID mgr works first last q10 Al Akin x02 Bob Bo q10 Carl Cork Dept ID name q10 CS x02 Math 14 / 1

15 Positives of the functorial data model The category of categories is bi-cartesian closed (model of STLC). Schemas support 0, 1, `, ˆ,p. For each category C, the category C-inst is a topos (model of HOL). Instances support 0, 1, `, ˆ,p D, ^, _,, Ñ, J, K. Data integrity constraints (path equations) are built-in to schemas. In progress: more expressive constraints ( EDs ) in schemas. Data migrations transform entire instances. Easy to pivot. Σ has better semantics than TGD-only systems (e.g., Clio). 15 / 1

16 FQL - A Functorial Query Language FQL is an open-source, graphical IDE available at catinf.com. It translates data migrations of the form Σ F Π G H into SQL and executes via JDBC whenever possible. Otherwise, FQL executes the migration directly. FQL also includes a (partial) translator SQL Ñ FQL, as well as RDF/JSON input/output. Some FQL queries can be written using built-in SELECT-FROM-WHERE syntax. Demo 16 / 1

17 Conclusion There are deep connections between the FDM and other data models, including relational, RDF, and XML. MIT and C i have had initial success using FQL on a data integration scenario identified by the National Institute of Standards and Technology (NIST) and are looking for academic collaborators, customers, test cases, and employees. Visit catinf.com for more information. See categoricaldata.net and appliedcategorytheory.org for other interesting categorical projects. 17 / 1