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Understanding Object Identities in Database Formalism

A presentation on object identity formalisms in databases based on Abiteboul and Kanellakis' work, covering classes, relations, typing, schemas, instances, examples, queries, IQL, and type inheritance.

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Understanding Object Identities in Database Formalism

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  1. Object Identities A presentation by Corey Anderson Inspired by Abiteboul and Kanellakis

  2. The big idea • Object-orientation in a database is good • But no one uniform formalism exists yet • So let’s use A&K’s formalism • Object identities as First Class Citizens

  3. Object Identity • A unique identity for each object oid o-value adam [ name: “Adam”, spouse: eve, children: {cain, abel, seth, …} ]

  4. Formalisms – basics • oid • An identifier • o-value • An oid, a constant, or a tuple of o-values • Schema v. Instance

  5. Formalisms – classes • Classes P = { P1, P2, … } • P = { 1st-gen, 2nd-gen } • (P) : P {oids} • (1st-gen) = { adam, eve }

  6. Formalisms – relations • Relations R = { R1, R2, … } • R = { ancestor-of-celebrity, founded-lineage } • (R) : R  {o-values} • (ancestor-of-celebrity) = {[seth, noah], …}

  7. Formalisms – typing • Everything has a type  •  = {} | constant | class | tuple | {} |  |  • Given , each type has an interpretation [](i.e., the set of things that are of that type) • [P] = (P) • [[1, 2]] = {[v1,v2] | v1  [1], v2  [2]}

  8. Formalisms – the last one! • A schema S is a triple (R,P,T) • Relations R, classes P, type mapping T : RP types(P) • An instance I is a triple (, , ) • o-value assignment , oid-assignment , oid-to-o-value partial function 

  9. An Example Schema • P = { 1st-gen, 2nd-gen } • R = { ancestor-of-celebrity, founded-lineage } • T(1st-gen) = [name: string, spouse: 1st-gen, children: {2nd-gen}] • T(2nd-gen) = [name: string, occupations: { string }] • T(ancestor-of-celebrity) = 2nd-gen • T(founded-lineage) = [anc: 2nd-gen, desc: (string or [spouse: string])]

  10. An Example Instance • (1st-gen) = { adam, eve } • (2nd-gen) = { seth, cain, abel, other } • (founded-lineage) = {cain, seth, other } • (ancestor-of-celebrity) = {[anc: seth, desc: noah], [anc: cain, desc: [spouse: “Ada”]] } • (adam) = [name: “Adam”, spouse: eve, children: {cain, abel, seth, other}] • (abel) = [name: “Abel”, occupations: {“shepherd”}] • (other) is undefined • …

  11. Posing Queries – IQL • The Identity Query Language • Rule-based, like Datalog • Formally, (S, Sin, Sout) • Sin, Sout are projections of S • Convert an instance of input schema Sin to an instance of output schema Sout • Typed

  12. Simple IQL example • Farmers who founded lineage • farmer-founders(x) 2nd-gen(x), founded-lineage(x), x = [n, o], o = “farmer”

  13. Less simple IQL example • Make a class of all the occupations S = { R, P {occupation}, T {T(occupation) = [name: string]} SinSSin from earlier SoutSSout = {{}, {occupation}, …} • What’s (S, Sin, Sout)?

  14. The query T(R1) = string T(R2) = [string, occupation] R1(o)  2nd-gen(x), x = [n,o] R2(o,z) R1(o) ;; Creates the oid z and ;; automagically sets ;; (occupation) = z R2(o,z) ;; Define z’s o-value

  15. Another example – nesting R = {R2, R3} T(R2) = [D, D] T(R3) = [D, {D}] R4(x) R2(x,y) R5(x,z) R4(x) ;; A new oid! (y) R5(x,z), R2(x,y) ;; z is set-valued R3(x, ) R5(x,z) G1 G2

  16. Why the authors like IQL • IQL can express (almost) all possible database transformations • oid invention was necessary here • IQL contains other cool query languages as sublanguages • Datalog, Datalog with negation, COL, …

  17. Type inheritance • Augment definition of schema to include type hierarchy • Schema S is a quadruple (R, P, T, ) where  is a partial order on P • Augment definition of instance to account for oids being a subclass object • (P) =  { (P’) | P’ P, P’ P }

  18. Take home messages • oids are unique identifiers for objects • oids and their use are formally defined • IQL is a language with which to query OODBs

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