Relational Databases: A Tutorial for Statisticians Abstract ... - CiteSeerX

Relational Databases: A Tutorial for Statisticians Joe R. Hill

EDS Research, 5951 Jeerson St. NE, Albuquerque, NM 87109

Abstract

This tutorial will not cover  Anything about particular relational database management systems.  Network, hierarchical, or object-oriented database models.  Distributed databases. Basic references for relational databases include Codd (1970), Date (1986), Maier (1983), and Ullman (1982). More advanced references include Fagin (1977), Fagin, Mendelzon & Ullman (1982), Beeri, Fagin, Maier & Yannakakis (1983), and Beeri & Kifer (1986a, b, 1987). Connections to probability theory are mentioned in Pearl (1988), Geiger & Pearl (1988, 1990), Geiger, Paz & Pearl (1991), Lauritzen & Spiegelhalter (1988), and Thoma (1989).

This tutorial links relational database concepts to probability concepts. For example, the fundamental relational database concepts of an attribute (column heading), a relation scheme (unpopulated table), and a relation (populated table) correspond respectively to the probability concepts of a random variable, a random vector, and a multivariate probability distribution. The relational select and project operators correspond respectively to nding a conditional and marginal distribution. Functional dependencies, multivalued dependencies, and join dependencies correspond respectively to variable transformations, conditional independencies, and more general factorizations of distributions. These connections indicate that statisticians may know more about relational databases than they realize. Beyond these pedagogical bene ts, these connections between relational databases and statistics provide a bridge, both directions of which have proven to be useful for developing new theory.

2 Database Concepts and Probability Parallels

1 Introduction

This section de nes the basic database concepts and the parallel probability concepts. The de nitions are given in parallel because familiarity with the probability concepts might help the reader understand the essential ideas underlying the database concepts. Also, as sections 4, 5, and 6 show, there are parallel problems and results in the two elds. A relation scheme (table skeleton) R is a set of attributes (column headings). A relation (table) over relation scheme R is an indicator function for a set of tuples (rows), written r[R]: r[R](t) = 1 if the tuple t is in the relation; r[R](t) = 0 if t is not in the relation. When storing or writing out a relation, it is common to list only those tuples that are in the relation (i.e. that have r[R](t) = 1). The parallel concepts in probability theory are a random vector and a probability distribution. A random vector V is a set of random variables. A distribution for the random vector V is a probability function, written p[V ]. The distribution of V evaluated at v is written p[V ](v).

This tutorial will cover:  Relational database concepts and probability parallels (Section 2).  An introduction to database normalization theory (Section 3).  Parallel theorems for consistent databases and consistent sets of marginal distributions (Section 4).  Finding closures of sets of multivalued dependencies and sets of conditional independencies (Section 5).  Eliminating intersection anomalies in sets of conditional independencies and sets of multivalued dependencies (Section 6).  Concluding remarks (Section 7). 1

Airline example (Maier, 1983). Relation schedule contains scheduling information for an airline. Relation schedule is de ned over the relation scheme with attributes FLT, FROM, TO, DEP, and ARR. The rst tuple in schedule, t1 , maps FLT into 84, FROM into O'Hare, and DEP into 3:00pm. The projection of t1 onto fFROM; TOg is t1 [FROM; TO] = (O'Hare, JFK).

FLT 84 109 117 213 214

schedule FROM TO O'Hare JFK JFK Los Angeles Atlanta Boston JFK Boston Boston JFK

DEP ARR 3:00pm 5:55pm 9:40pm 2:42am 10:05pm 12:43am 11:43am 12:45pm 2:20pm 3:12pm

The basic operators on relations are a projection of a relation onto a subset of its attributes, a selection from a relation of the tuples having a speci c value for a subset of its attributes, and a join of two relations. These operators correspond to a marginal distribution, a conditional distribution, and a product of two functions. The projection of the relation r[R] onto X  R, written r[X ] or  (r[R]), is the indicator function: r[X ](x) = 1 if there is a tuple t such that r[R](t) = 1 and t[X ] = x; r[X ](x) = 0 otherwise. The marginal distribution of X  V based on p[V ], written p[X ], is found by summing p[V ] over the variables not in X ; that is, letting Y = V ? X , X

p[X ](x) =

X p[XY ](x; y): y

Airline example. The following tables show the projections of schedule onto fDEP, ARRg and onto FROM.

DEP ARR (schedule) DEP ARR 3:00pm 5:55pm 9:40pm 2:42am 10:05pm 12:43am 11:43am 12:45pm 2:20pm 3:12pm ;

FROM (schedule) FROM O'Hare JFK Atlanta Boston

The selection from the relation r[R] of the tuples with X = x, X  R, written r[R j X = x] or  = (r[R]), is the indicator function: r[R j X = x](t) = 1 if t[X ] = x; r[R j X = x](t) = 0 otherwise. The Y -projection of the X = x selection from r[R] is written r[Y j X = x]. X

x

The conditional distribution of V given X = x, X  V , based on p[V ], written p[V j X = x], is the probability function: p[V j X = x](v) = p[V ](v)=p[X ](x) if p[X ](x) > 0 and v[X ] = x; p[V j X = x](v) = 0 otherwise. The Y -margin of the X = x conditional is written p[Y j X = x]. Airline example. The following table shows the data for ights from JFK.

FROM=JFK (schedule) FLT FROM TO DEP ARR 109 JFK Los Angeles 9:40pm 2:52am 213 JFK Boston 11:43am 12:45pm Let r1 [R1] and r2 [R2] be relations over relation schemes R1 and R2. Let X = R1 ? R2, Y = R1 \ R2, Z = R2 ?R1 . The join of r1 [R1] and r2 [R2] is the relation over R1 [ R2 = XY Z (XY Z is shorthand for X [ Y [ Z ) de ned by (r1 1 r2)[XY Z ](x; y; z ) = r1[XY ](x; y) r2[Y Z ](y; z ): Let h1 [V1] and h2 [V2] be functions over variable sets V1 and V2 . Let X = V1 ? V2 , Y = V1 \ V2 , Z = V2 ? V1. The product of h1[V1 ] and h2[V2] is the function over V1 [ V2 = XY Z de ned by (h1 h2)[XY Z ](x; y; z ) = h1 [XY ](x; y) h2[Y Z ](y; z ): Airline example. Relation usable contains the equipment requirements for each ight. Relation certified contains the equipment quali cations for each pilot. Suppose we want to know the pilots that can y each of the ights. To nd the answer to this query, we rst form options = usable 1 certified. Then we project options onto FLT and PILOT, providing the answer to the original query.

usable FLT EQPMT 83 727 83 747 84 727 84 747 109 707

certified PILOT EQPMT Simmons 707 Simmons 727 Barth 747 Hill 727 Hill 747

options = usable 1 certified FLT EQPMT PILOT 83 727 Simmons 83 727 Hill 83 747 Barth 83 747 Hill 84 727 Simmons 84 727 Hill 84 747 Barth 84 747 Hill 109 707 Simmons FLT PILOT (options) FLT PILOT 83 Simmons 83 Hill 83 Barth 84 Simmons 84 Hill 84 Barth 109 Simmons ;

Table 1: Basic database and probability parallels. DATABASE PROBABILITY CONCEPT CONCEPT Relation scheme (table skeleton) R, a set of attributes (column names)

Random vector V , a set of random variables

Relation (table) over R, r[R], an indicator function for a set of tuples (rows)

Distribution for V , p[V ], a probability function

Projection of r[R] onto X  R,  (r[R]), or r[X ]

Marginal distribution of X  V , p[X ]

Selection  = (r[R]), or r[R j X = x], XR

Conditional distribution p[V j X = x], XV

Join of 2 relations r1[R1] 1 r2[R2]

Product of 2 functions h1 [V1] h2[V2 ]

X

X

Table 1 summarizes the basic database/probability parallels covered to this point. Database design concepts involve putting constraints on the data that can populate a table. There are three basic kinds of constraints: a functional dependency, a multivalued dependency, and a join dependency. These correspond to three constraints on probablity distributions: transformation constraints, conditional independencies, and general factorization constraints. A relation r[R] satis es the functional dependency FD: X ! Y if for each X -value x with r[X ](x) = 1, there is a unique Y -value y such that r[Y j X = x](y) = 1 if y = y and r[Y j X = x](y) = 0 otherwise. A distribution p[V ] satis es the transformation constraint TC: X ! Y if for each X -value x with p[X ](x) > 0, there is a unique Y -value y such that p[Y j X = x](y) = 1 if y = y and p[Y j X = x](y) = 0 otherwise. Airline example. The relation schedule satis es the FD FLT ! fFROM; TO; DEP; ARRg. The FLTvalue of a tuple uniquely determines the rest of the tuple. The relation schedule does not satisfy the FD FROM ! TO because t2[FROM] = t4 [FROM] = JFK, but t2 [TO] = Los Angeles 6= Boston = t4 [TO]. A random vector V satis es a constraint if all distributions for V must satisfy the constraint. Likewise, a relation scheme R satis es a data dependency if all relax

x

x

x

x

tions over R must satisfy the dependency. Airline example. The functional dependency FLT ! fFROM, TO, DEP, ARRg remains true over time. As a result, FLT is a candidate key for the relation schedule. A relation r[R] satis es the multivalued dependency MVD: Z !!X j Y if r[XY Z ](x1 ; y1; z ) r[XY Z ](x2; y2 ; z ) = r[XY Z ](x1; y2; z ) r[XY Z ](x2 ; y1; z ): Similarly, a distribution p[V ] satis es the conditional independency CI: X ??Y j Z if p[XY Z ](x1 ; y1; z ) p[XY Z ](x2; y2 ; z ) = p[XY Z ](x1; y2; z ) p[XY Z ](x2 ; y1; z ): Multivalued dependencies are equivalent to binary join dependencies. That is, a relation satis es an MVD i

it can be recovered as the join of two relations de ned on \smaller" relation schemes. In symbols, a relation r[R] satis es MVD: Z !!X j Y i there exist relations r1[XZ ] and r2[Y Z ] such that r = r1 1 r2: that is,

r[XY Z ](x; y; z ) = r1[XZ ](x; z ) r2[Y Z ](y; z ): If such r 's exist, then r[R] is said to satisfy the binary join dependency BJD: 1 fXZ; Y Z g. Also, if such r 's exist, then they can be taken to be r1[XZ ] = r[XZ ] and r2[Y Z ] = r[Y Z ]. Similarly, conditional independencies are equivalent to binary factorization constraints. That is, a probability distribution satis es a CI i it can be recovered as the product of two functions de ned on \smaller" random vectors. In symbols, a distribution p[V ] satis es CI: X ??Y j Z i there exist nonnegative functions h1[XZ ] and h2[Y Z ] such that p = h1 h2: that is, j

j

p[XY Z ](x; y; z ) = h1 [XZ ](x; z ) h2 [Y Z ](y; z ): If such h 's exist, then p[V ] is said to satisfy the binary factorization constraint BFC: fXZ; Y Z g. Airline example. Relation service satis es the MVD: FLT!!DAY OF WEEK j PLANE TYPE because service = servday 1 servtype where servday = service[FLT; DAY OF WEEK] and servtype = service[FLT; PLANE TYPE]. Relation service2, which has the same two projections as service, does not satisfy this MVD because it lacks the tuple (106, Thursday, 1011). j

servday FLT DAY OF WEEK (service) FLT DAY OF WEEK 106 Monday 106 Thursday 204 Wednesday ;

servtype = FLT PLANE TYPE (service) FLT PLANE TYPE 106 747 106 1011 204 707 204 727 ;

service2 FLT DAY OF WEEK PLANE TYPE 106 Monday 747 106 Thursday 747 106 Monday 1011 204 Wednesday 707 204 Wednesday 727 A distribution p[V ] satis es the factorization constraint FC: V , V = fV1 ; :::; V g, V  V , if there exist nonnegative functions h1 [V1]; :::; h [V ] such that k

j

k

k

p[V ] = h1 [V1]


h [V ]: The set of margins V is a hypergraph over V . Factorization constraints generalize loglinear models which must be strictly positive. Example. The relation r[ABC ] satis es the JD: 1 fAB; BC; AC g but does not satisfy any nontrivial MVD. k

service FLT DAY OF WEEK PLANE TYPE 106 Monday 747 106 Thursday 747 106 Monday 1011 106 Thursday 1011 204 Wednesday 707 204 Wednesday 727 A relation r[R] satis es the join dependency JD: 1 R, R = fR1; :::; R g, R  R, if there exist relations r1 [R1], ..., r [R ] such that k

k

j

k

r[R] = r1[R1] 1


1 r [R ]: k

k

If such r 's exist, then they can be taken to be r [R ] = r[R ]; j = 1; :::; k. The set of relation schemes R is a set of subsets of R; in other words, R is a hypergraph over R. j

j

j

j

r[ABC ] A B C 1 1 1 1 2 2 2 3 3 3 3 4 4 4 5 5 5 5

r[AB ] A B 1 1 1 2 2 3 3 3 4 4 5 5

r[BC ] B C 1 1 2 2 3 3 3 4 4 5 5 5

k

r[AC ] A C 1 1 1 2 2 3 3 4 4 5 5 5

Finally, databases correspond to sets of marginal distributions. A database scheme over attribute set R is a set of relation schemes with attributes from R: R = fR1; :::; R g, R  R. The database scheme R is a hypergraph over R. A database over database scheme R is a set of relations over the relation schemes in R: r[R] = fr1[R1]; :::; r [R ]g. A set of margins of random vector V is a set of random vectors with variables from V : V = fV1 ; :::; V g, V  V . The set of margins V is a hypergraph over V . A set of marginals over set of margins V is a set of distributions for the margins in V : p[V ] = fp1[V1 ]; :::; p [V ]g. Table 2 summarizes this second collection of parallels. There are many more parallels between database theory and probability. Sections 4, 5, and 6 discuss, very brie y, three parallel problems and solutions. k

j

k

k

k

k

j

k

Table 2: Further database and probability parallels. DATABASE CONCEPT

PROBABILITY CONCEPT

Functional dependency X !Y X; Y  R

Variable transformation X!Y X; Y  V

Multivalued dependency Z !!X j Y X; Y; Z  R

Conditional independency X ??Y j Z X; Y; Z  V

Join dependency 1R R = fR1; :::; R g, R  R

Factorization constraint

V V = fV1 ; :::; V g, V  V

MVDs are binary JDs 1 fXZ; Y Z g

CIs are binary FCs

fXZ; Y Z g

Database scheme over R R = fR1; :::; R g, R  R

Set of margins of V V = fV1 ; :::; V g, V  V

k

k

j

j

Database over R

r[R] = fr1[R1]; :::; rk[Rk ]g

k

k

j

j

Set of marginals on V

p[V ] = fp1 [V1]; :::; pk[Vk ]g

3 A Brief Introduction to Normalization Theory

Here is a very tiny bit of normalization theory, an important standard topic in database theory with no useful parallels in probability theory. The basic reason for normalizing a database is to automatically eliminate possible inconsistencies that might otherwise arise. A set of attributes K is a candidate key of R if K ! R. One of the candidate keys of relation R is designated the primary key and the other attributes are called non-keys. A set of attributes Y is fully dependent on another set of attributes X if X ! Y and there is no Z  X such that Z ! Y . If there is such a Z then Y is partially dependent on X . A set of attributes Z is transitively dependent on X if there is a Y such that X ! Y and Y ! Z . The normal forms are:  First Normal Form (1NF): A relation is in 1NF if all the values in its tuples are atomic. There are no repeating groups.  Second Normal Form (2NF): A relation is in 2NF if it is in 1NF and every non-key is fully dependent on the primary key. A relation in 2NF has no partial dependencies.  Third Normal Form (3NF): A relation is in 3NF if it is in 2NF and no non-key is transitively dependent on the primary key. A relation in 3NF has no partial or transitive dependencies. All the non-keys in a 3NF relation are mutually independent (i.e. no nonkey is functionally dependent on another nonkey).  Boyce/Codd Normal Form (BCNF): A relation is in BCNF if every FD is a consequence of the candidate keys. Date: \Each eld must represent a fact about the key, the whole key, and nothing but the key."  Fourth Normal Form (4NF): A relation is in 4NF if every MVD is a consequence of the candidate keys. All dependencies (MVDs and FDs) of a 4NF relation are FDs from a candidate key to another attribute. A relation is in 4NF if it is in BCNF and all its MVDs are FDs.  Fifth Normal Form (5NF): A relation is in 5NF if every JD is a consequence of the candidate keys. 5NF is also called project/join normal form. There are rules for converting database schemes that do not satisfy normal forms into ones that do. The interested reader should consult Maier (1983) or Ullman (1982), for example.

4 Parallel Theorems for Consistent Databases and Consistent Sets of Marginal Distributions

It was noted earlier that database schemes and sets of margins are hypergraphs. There are strong connections between relational databases and graph theory and between probability theory and graph theory. Often, properties of databases and properties of probability distributions are determined by the underlying graphical structure. This section gives an example of the kind of parallel results that arise because of these connections to graph theory. A database r[R] is pairwise consistent if r [R \ R ] = r [R \R ]. A database r[R] is globally consistent if there exists a single relation r[R] such that r [R ] = r[R ]; if such an r[R] exists, then it can be taken to be r[R] = r1[R1] 1


1 r [R ]. A set of marginals p[V ] is pairwise consistent if p [V \ V ] = p [V \ V ]. A set of marginals p[V ] is globally consistent (or extendable) if there exists a single distribution p[V ] such that p [V ] = p[V ]. Consider the following two examples. Example 1 (Vorob'ev, 1962). Let V = fAB; BC; AC g be a set of margins of the random vector ABC . Let p = fp1; p2; p3g be the set of marginals over V de ned by p1 [AB ](0; 0) = p1 [AB ](1; 1) = 1=2; p2[BC ](1; 0) = p2 [BC ](0; 1) = 1=2; and p3 [AC ](0; 0) = p3 [AC ](1; 1) = 1=2: There is no distribution p[ABC ] such that p[AB ] = p1 [AB ], p[BC ] = p2[BC ], and p[AC ] = p3 [AC ]. Such a p[ABC ] would have p[ABC ](0; 0; 0) = 0 because p2 [BC ](0; 0) = 0, and p[ABC ](0; 0; 1) = 0 because p3 [AC ](0; 1) = 0, so p[AB ](0; 0) = 0, contradicting p1 [AB ] = 1=2. This same example can be given as a database example with p's replaced by r's and 1=2's replaced by 1's. Example 2. Let R = fABD; BCD; BCE g be a database scheme over ABCDE . For every pairwise consistent database r = fr1; r2; r3g over R, there is a single relation r[ABCDE ] such that r[ABD] = r1 [ABD], r[BCD] = r2[BCD], and r[BCE ] = r3[BCE ]. The parallel statement holds for probability distributions. The dierence between these examples is that the hypergraph in Example 2 is acyclic but the one in Example 1 is not acyclic. There are many ways to de ne an acyclic hypergraph. The following de nition, referred to as the running intersection property, does not require de nitions for any other concepts. A hypergraph H is acyclic i

j

i

i

j

j

j

k

j

j

k

i

j

j

i

j

j

j

j

i

if its elements can be ordered so that for each i = 2; :::; k, there is a j < i with

H \ (H1 [


[ H ?1)  H : i

i

j

The two results can now be stated. Vorob'ev (1962) proved that every pairwise consistent set of marginals over a set of margins V is extendable if and only if the hypergraph V is acyclic (see also Lauritzen, Speed & Vijayan, 1984). Beeri, Fagin, Maier & Yannakakis (1983) proved the parallel result for relational databases: that is, every pairwise consistent database over a database scheme R is globally consistent if and only if the hypergraph R is acyclic.

5 Closures of Sets of MVDs and Sets of CIs

Let M be a set of MVDs over R. The closure M of M is the set of MVDs implied by the MVDs in M, that is, if a relation satis es the MVDs in M, then it also satis es the MVDs in M . The closure M of M can be found as follows. Let M (X ) = fY  R ? X : X !!Y 2 M g. The dependency basis of X , DEPM (X ), is the partition of R ? X such that Y 2 M (X ) i Y is a union of sets in DEPM (X ). DEPM (X ) can be found using the following algorithm: (0) Start with partition P = fV ? X g. (1) If Y 2 P and there is an MVD: Z !!W in M such that Y \ Z = ;, then replace Y by the 2 sets Y \ W and Y ? W . (2) Repeat (1) until it no longer changes P . The nal partition is DEPM (X ). Example. Let M = fBC !!AD j E; BD!!A j CE g. To nd DEPM (BCD), (0) let P = fAE g, (1) AE 2 P , BC !!AD j E 2 M, and BC \ AE = ;, so replacing AE by AE \ AD = A and AE \ E = E gives DEPM (BCD) = fA; E g. Geiger & Pearl (1988, 1990) and Geiger, Paz & Pearl (1991) proved that the same algorithm can be used to nd the closure of a set of conditional independencies with respect to arbitrary (i.e. not necessarily strictly positive) distributions. They also derived a graph-based approach for nding the closure with respect to strictly positive distributions.

6 Eliminating Intersection Anomalies

The two CIs X ??Y j Z and X ??Z j Y imply the third X ??Y Z for strictly positive distributions. The same is not true for arbitrary distributions. For example, the distribution p[XY Z ](0; 0; 0) = p[XY Z ](1; 1; 1) = 1=2, p[XY Z ](x; y; z ) = 0 otherwise, satis es the rst two of these CIs, but does not satisfy the third. The set of CIs fX ??Y j Z; X ??Z j Y g is said to have an intersection anomaly. After reviewing several statistical arguments that were awed because they ignored intersection anomalies, Dawid (1979) showed that it is possible to x up this anomaly by adding a variable W such that W is functionally determined by each of Y and Z individually (i.e. Y ! W , Z ! W ) and X ??Y Z j W . The variable W represents the information that Y and Z have in common. Beeri and Kifer (1986a, b, 1987) and others have written extensively about the same issue for sets of MVDs. Their solution, which has implications for database design, is the same as Dawid's. They only apply the method to sets of MVDs that do not have split left hand sides, so after eliminating intersection anomalies they have a con ict-free set of MVDs which is equivalent to a single (acyclic) JD.

7 Concluding Remarks This tutorial reviewed basic parallels between database theory and probability theory. It discussed three parallel problems and corresponding solutions in the two areas. It mentioned some of the connections to graph theory which provide another bridge between results in database theory and those in probability theory. For example, acyclic databases and decomposable models (distributions that satisfy acyclic factorization constraints) have many desirable properties (Beeri, Fagin, Maier & Yannakakis, 1983; Darroch, Lauritzen & Speed, 1980). One particularly interesting connection concerns the positivity condition of the Gibbs-Markov equivalence theorem. It is possible to relax the positivity condition using concepts from relational database theory. Results on this topic and others will be given in future papers.

Acknowledgements

I thank Mathis Thoma and Dan Geiger for many insightful discussions on the connections between database theory, probability theory, and graph theory. I also thank

Jon Kettenring for encouraging me to give the tutorial.

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