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Relative Information Capacity of Simple Relational Database Schemata Paper by: Richard Hull Presented by: Jose Picado

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Outline Problem: Data relativism and information capacity – Definition – Examples – Importance Hierarchy of dominance measures Basic results Discussion

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Data relativism Represent the same data in different ways

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Data relativism Represent the same data in different ways Represent the same data under different schemas

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Data relativism Represent the same data in different ways Represent the same data under different schemas Person namesexspouse Schema 1 Example taken from: Kosky, Anhony. Transforming Databases with Recursive Data Structures, 1996.

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Data relativism Represent the same data in different ways Represent the same data under different schemas Person namesexspouse Female name Male name Marriage husbandwife Schema 1 Schema 2 Example taken from: Kosky, Anhony. Transforming Databases with Recursive Data Sturctures, 1996.

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Relative information capacity Expressiveness of a schema Different schemas representing same data may have different information capacity

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Relative information capacity Expressiveness of a schema Different schemas representing same data may have different information capacity Person namesexspouse Female name Male name Marriage husbandwife Schema 1 Schema 2 Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

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Relative information capacity Expressiveness of a schema Different schemas representing same data may have different information capacity Person namesexspouse Female name Male name Marriage husbandwife Schema 1: Does not require that the spouse attribute of a man goes to a woman. Does not require that for each spouse attribute in one direction there is a corresponding spouse attribute in another direction. Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

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Relative information capacity Expressiveness of a schema Different schemas representing same data may have different information capacity Person namesexspouse Female name Male name Marriage husbandwife Schema 2: Allows unmarried people to be represented in the database. Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

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Relative information capacity Possible solution: – Transform existing schema to new schema by structural manipulations Person namesexspouse Female name Male name Marriage husbandwife transformation

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Relative information capacity Possible solution: – Transform existing schema to new schema by structural manipulations – Information capacity preserving? Person namesexspouse Female name Male name Marriage husbandwife transformation

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Importance Schema evolution – None of the information stored in the initial database is lost Person namesexspouse Female name Male name Marriage husbandwife

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Importance Data integration – All information in one of the component databases is reflected in the integrated database City namestate State namecapital City nameisCapitalcountry Country namelanguagecurrency City nameplace Country namelanguagecurrencycapital State namecapital Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

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Importance Database normalization theory User view construction Schema simplification Translation between data models

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Hulls paper Introduces theoretical tools for studying measures of relative information capacity – Theoretical frameworks at the time were complex – There was no clear definition about the concept – Hull introduced nice ways of comparing schemata and their information capacity Defines a hierarchy of measures to compare information capacity of schemata

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Hulls paper Gives some basic results concerning the previous measures Considers only non-keyed relations Person idnam e Person idnam e 123John 123Mary 123John 123Mary Non-keyed Keyed Instances: Relations:

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Definitions Schema P is a set of relations Relations composed of attributes, which may be of different basic types Basic types are domain designators (have a fixed domain of possible values) I(P) is the instances of P, usually infinite Person idnam e 111John 222Mary 123Anne 234Joe aaaJack bbbTed Schema P Instances I(P) …

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Transformation P and Q are relational schemata A transformation from P to Q is a map

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Transformation P and Q are relational schemata A transformation from P to Q is a map P Person idname Birth iddate

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Transformation P and Q are relational schemata A transformation from P to Q is a map P Q PersonInfo idnamebdate Person idname Birth iddate

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Transformation P and Q are relational schemata A transformation from P to Q is a map P Q PersonInfo idnamebdate Person idname Birth iddate PersonInfo(x,y,z) :- Person(x,y), Birth(x,z).

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Dominance P and Q are relational schemata Q dominates P via if the composition of followed by is the identity on P

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Dominance Person namesexspouse Female name Male name Marriage husbandwife P Q

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Dominance 1.Take instances of P: I(P) Person JohnmaleMary femaleJohn AnnefemaleJoe maleAnne

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Dominance 2.Apply to I(P) Male(x) :- Person(x,y,z), y=male. Female(x) :- Person(x,y,z), y=female. Marriage(x,y) :- Person(x,u,y), Person(y,v,x), u=male, v=female Male John Joe Female Mary Anne Marriage JohnMary JoeAnne Person JohnmaleMary femaleJohn AnnefemaleJoe maleAnne

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Dominance 3.Apply to (I(P)) Person(x,male,z) :- Male(x), Marriage(x,z). Person(x,female,z) :- Female(x), Marriage(x,z). Male John Joe Female Mary Anne Marriage JohnMary JoeAnne Person JohnmaleMary femaleJohn AnnefemaleJoe maleAnne

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( (I(P))) Dominance 4.Compare I(P) and ( (I(P))) Person JohnmaleMary femaleJohn AnnefemaleJoe maleAnne Person JohnmaleMary femaleJohn AnnefemaleJoe maleAnne I(P)

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Dominance P and Q are relational schemata Q dominates P via if the composition of followed by is the identity on P Q has at least as much capacity for storing information as P Information structured according to P can be restructured to fit into Q, and restructured again to fit into P

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Equivalence P and Q are equivalent (xxx) if they have equivalent information capacity P and Q are equivalent if – Q dominates P (xxx) and – P dominates Q (xxx)

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Information dominance measures 1.Calculous dominance 2.Generic dominance 3.Internal dominance 4.Absolute dominance More restrictive Less restrictive

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Types of equivalency 1.P and Q are equivalent (calc) 2.P and Q are equivalent (gen) 3.P and Q are equivalent (int) 4.P and Q are equivalent (abs) More restrictive Less restrictive

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Level 1: Calculous dominance Only allow transformations to be relational calculus expressions Relational calculus: – First order logic or predicate calculus – Predicates: atom, – Each query Q(x1, …, xn) is a predicate P

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Level 1: Calculous dominance Only allow transformations to be relational calculus expressions are relational calculus expressions Q dominates P calculously

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Level 2: Generic dominance Only allow transformations that treat domain elements as essentially uninterpreted objects Treat all elements as equals except some set of constants Property of all query languages, such as SQL and Datalog

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Level 2: Generic dominance Only allow transformations that treat domain elements as essentially uninterpreted objects treat all elements as equals Q dominates P generically

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Level 3: Internal dominance Only allow transformations that do not invent any data Invent data: numerical computations or string manipulations playergoalsgamesplayerperformance performance = goals/games

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Level 3: Internal dominance Only allow transformations that do not invent any data do not invent data Q dominates P internally

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Level 4: Absolute dominance Some set of values : instances of P that contain only values in Y, where : cardinality of instances of P containing only values in Y If then Q dominates P absolutely Easy to compute: based on counting of instances, instead of transformations

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Basic results Q dominates P calculously Q dominates P generically Q dominates P internally Q dominates P absolutely

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Basic results Sometimes absolute and internal dominance hold, but generic and calculous dominance dont AA BB AB Q P Q dominates P (abs, int) and transformation (int) does not invent data Q does not dominate P (gen, calc) There is no transformation (gen, calc) that takes instances of P to Q and then back to P

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Basic results Absolute dominance useful for verifying calculous (not) dominance AB AC ABC Q P Q dominates P calculously Q dominates P absolutely P does not dominate Q absolutely P does not dominates Q calculously *under certain constraints

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Basic results Dominance is preserved by re-namings of basic types (homomorphism) – h(P): homomorphism of P – If Q dominates P then h(Q) dominates h(P) for any measure of dominance (calc, gen, int, abs)

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Basic results Calculous dominance does not accurately measure the presence of semantic correspondence

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Basic results Calculous dominance does not accurately measure the presence of semantic correspondence namepositiongoals namegoalsminutes S1 R1 NAMENUMBER NAME NUMBER titlepublisherpages titlepagesedition S2 R2 P

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Basic results Calculous dominance does not accurately measure the presence of semantic correspondence NAME NUMBER T P Q namepositiongoals namegoalsminutes S1 R1 NAMENUMBER NAME NUMBER titlepublisherpages titlepagesedition S2 R2

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Basic results Calculous dominance does not accurately measure the presence of semantic correspondence NAME NUMBER T P Q Q dominates P (calc), but there is not semantic mapping from P to Q namepositiongoals namegoalsminutes S1 R1 NAMENUMBER NAME NUMBER titlepublisherpages titlepagesedition S2 R2

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Basic results If only non-keyed relational schemata with only one basic type, then all types of dominance are equivalent Theorem: Let P and Q be non-keyed relational schemata over a single basic type B. Then the following are equivalent: a.Q dominates P (calc) b.Q dominates P (gen) c.Q dominates P (int) d.Q dominates P (abs)

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Basic results With any reasonable measure of relative information capacity, two non-keyed relational schemata are equivalent iff they are identical In the relational model (non-keyed), there is essentially at most one way to represent a given data set

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Discussion Strong points: – ???

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Discussion Strong points: 1.Provides a theory to study relative information capacity

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Discussion Strong points: 1.Provides a theory to study relative information capacity 2.Data relativism is important as it arises in many areas

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Discussion Strong points: 1.Provides a theory to study relative information capacity 2.Data relativism is important as it arises in many areas 3.Defines a hierarchy of dominance measures

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Discussion Strong points: 1.Provides a theory to study relative information capacity 2.Data relativism is important as it arises in many areas 3.Defines a hierarchy of dominance measures 4.Gives important results about the relational model

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Discussion Weak points: – ???

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Discussion Weak points: 1.Does not support dependencies/constraints Hierarchy of dominance measures Basic results

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Discussion Functional dependency (FD): Given attributes in relation R, the functional dependency means that all tuples in R that agree on attributes must also agree on. idnameaddress 123John21 Kings St. 234Mary31 Kings St.

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Discussion Multivalued dependency (MVD): For MVD, if two tuples of R agree on all the attributes of X, then their components in Y may be swapped, and the result will be two tuples that are also in the relation. coursebooklecturer Machine Learning Pattern Recognition John Artificial Intelligence AIMAMary

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Discussion Inclusion dependency (IND): For, for any tuple t1 in R1, there must exist a tuple t2 in R2, such that idtitle 111Pattern Recognition 222AIMA bookidcustomer 111John 222Mary Book Order

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Discussion Weak points: 1.Does not support dependencies/constraints Hierarchy of dominance measures Basic results Dependencies change the final result of the paper

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Discussion Weak points: 1.Does not support dependencies/constraints Hierarchy of dominance measures Basic results 2.Open questions: Absolute dominance implies internal dominance? Generic dominance implies calculous dominance? Is there a measure for semantic correspondence?

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Thank you

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Quiz What are the four formal measures of relative information capacity defined by Hull? Write them in order from most restrictive to less restrictive.

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