Formal Definitions for an Extended Relational Algebra

1.    Introduction

These are the formal definitions of an Extended Relational Algebra in a style similar to Algebra A in Appendix A of DTATRM by D&D. The operators of this ERA are used to construct shorthands, intended to form the basis of a relational programming language. It should be noted that Appendix A takes the reduction path from language shorthand to algebra operation. Here the path is taken in the opposite direction, defining algebraic operations from which the shorthands of a language may be constructed.

These operators do not provide or require any particular type system. Any use of the algebra is dependent on an external provider of a type system and functions over those types. Shorthands such as GROUP and UNGROUP, WRAP and UNWRAP require that relations and tuples be provided by that type system.

The naming conventions are adapted from Appendix A. In what follows:

A is an attribute name

T is a type

v is a value

<A,T> is an attribute (member of a heading)

<A,T,v> is an attribute value (member of a tuple)

r is a relation

Hr is the heading of r

Br is the body of r

tr is a tuple of r

2.    Basics from Appendix A

These are the basic 5 definitions adapted from Appendix A.

a)      NOT

Given a relation r then s = NOT(r) is defined as follows.

Hs = Hr

Bs = { ts : ∃ tr ∉ Br, ts = tr

The NOT operator yields the complement s of a given relation r. The heading of s is the heading of r. The body of s contains every tuple with that heading that is not in the body of r.

The NOT operator is used in shorthands that involve negation, such as ANTIJOIN, MINUS.

b)      REMOVE

Given a relation r and a type T such that <A,T> ∈ Hr then s = REMOVE(A) is defined as follows.

Hs = Hr minus { <A,T> }

Bs = { ts : ∃ tr ∉ Br, ∃ v ∈ T, <A,T,v> ∈ tr, ts = tr minus { <A,T,v> }

The REMOVE operator yields a relation s formed by removing a given attribute A from a given relation r. The operation is equivalent to taking the projection of r over all of its attributes except A. The heading of s is the heading of r minus the ordered pair <A,T>. The body of s contains every tuple that conforms to the heading of s and is a subset of some tuple of r.

The REMOVE operator is used in shorthands that remove one or more attributes, such as PROJECT.

c)      RENAME

Given a relation r, a type T such that <A,T> ∈ Hr and no type T such that <B,T> ∈ Hr then s = RENAME(A,B) is defined as follows.

Hs = ( Hr minus { <A,T> } ) union { <B,T> }

Bs = { ts : ∃ tr ∈ Br, ∃ v ∈ T, <A,T,v> ∈ tr,

   ts = ( tr minus { <A,T,v> } ) union { <B,T,v> } }

The RENAME operator yields a relation s that differs from a given relation r only in the name of one of its attributes, which is changed from A to B. The heading of s is the heading of r except that the ordered pair <A,T> is replaced by the ordered pair <B,T>. The body of s consists of every tuple of the body of r, except that in each such tuple the triple <A,T,v> is replaced by the triple <B,T,v>.

The RENAME operator is used in shorthands that rename one or more attributes, such as RENAME.

d)      AND

Given relations r1 and r2 such that <A,T1> ∈ Hr1 and <A,T2> ∈ Hr2 implies T1 = T2 then s = AND(r1,r2) is defined as follows.

Hs = Hr1 union Hr2

Bs = { ts : ∃ tr1 ∈ Br1, ∃ tr2 ∈ Br2, ts = tr1 union tr2 }

The AND operator is relational conjunction, which is usually known as a natural join. The heading of s is the union of the headings of r1 and r2. The body of s contains every tuple that conforms to the heading of s and is a superset of both some tuple in the body of r1 and some tuple in the body of r2.

The AND operator is used in join shorthands such as JOIN, SEMIJOIN, ANTIJOIN, COMPOSE, TIMES.

e)      OR

Given relations r1 and r2 such that <A,T1> ∈ Hr1 and <A,T2> ∈ Hr2 implies T1 = T2 then s = OR(r1,r2) is defined as follows.

Hs = Hr1 union Hr2

Bs = { ts : ∃ tr1, ∃ tr2, ( tr1 ∈ Br1 or tr2 ∈ Br2), ts = tr1 union tr2 }

The OR operator is relational disjunction, being a generalization of what is usually known as union. In the special case where the given relations r1 and r2 have the same heading, the result s is in fact the union of those two relations in the traditional sense. The heading of s is the union of the headings of r1 and r2. The body of s contains every tuple that conforms to the heading of s and is a superset of either some tuple in the body of r1 or some tuple in the body of r2.

The OR operator is used in set shorthands such as UNION, MINUS, SYMDIFF, INSERT.

3.    Relational Constant

This is the formalisation of a relational constant or ‘relcon’, and relies on some previously defined function f. Separate formalisations are provided for monadic and dyadic functions, in both predicate and value-returning forms. Extending them to n-adic functions is left as an exercise.

a)      Monadic predicate

Given attribute names X, types Tx and function f with the type signature Tx->Boolean then s = RELCON(X,f) is defined as follows.

Hs = {<X,Tx>}

Bs = { ts : ∃ vx ∈ Tx, f(vx) = true, ts = {<X,Tx,vx>} }

b)      Monadic value

Given attribute names X,Y, types Tx,Ty and function f with the type signature Tx->Ty then s = RELCON(X,Y,f) is defined as follows.

Hs = {<X,Tx>, <Y,Ty>}

Bs = { ts : ∃ vx ∈ Tx, ∃ vy ∈ Ty, f(vx) = vy,

   ts = {<X,Tx,vx>, <Y,Ty,vy>} }

c)      Dyadic predicate

Given attribute names X,Y, types Tx,Ty and function f with the type signature Tx->Ty->Boolean then s = RELCON(X,Y,f) is defined as follows.

Hs = {<X,Tx>, <Y,Ty>}

Bs = { ts : ∃ vx ∈ Tx, ∃ vy ∈ Ty, f(vx,vy) = true),

   ts = {<X,Tx,vx>, <Y,Ty,vy} }

d)      Dyadic value

Given attribute names X,Y,Z, types Tx,Ty,Tz and function f with the type signature Tx->Ty->Tz then s = RELCON(X,Y,Z,f) is defined as follows.

Hs = {<X,Tx>, <Y,Ty>, <Z,Tz>}

Bs = { ts : ∃ vx ∈ Tx, ∃ vy ∈ Ty, ∃ vz ∈ Tz, f(vx,vy) = vz),

   ts = {<X,Tx,vx>, <Y,Ty,vy, <Z,Tz,vz>} }

[Note: for the relcon PLUS in App-A, types Tx, Ty and Tz are all INTEGER, and the function f is the scalar operator “+”.]

The RELCON operator predicate forms are used together with AND in shorthands such as WHERE/RESTRICT.

The RELCON operator value-returning forms are used together with AND in shorthands such as EXTEND, SUMMARIZE, UPDATE and DELETE.

4.    Aggregation

This is the formalisation of simple aggregation, similar to SUM, MAX or MIN but allowing for a wider range of functions. It relies on some previously defined aggregating function f. An aggregating function has an argument which is a list of values (a ‘bag’ or ‘multiset’ rather than a ‘set’) and returns a single value. Extending this approach to more complex aggregations is left as an exercise.

Given a relation r, a type T such that <A,T> ∈ Hr and an aggregating function f with the type signature T[]->Ta then s = AGGREGATE(r,A,f) is defined as follows.

Hs = ( Hr minus {<A,T>} ) union {<A,Ta>}

Bs = { ts : ∃ v ∈ T, ∃ tr ∈ r, ∃ <A,T,v> ∈ tr, v ∈ v[],

   ts = tr minus {<A,T,v>} union {<A,Ta,f(v[])} }

Note: the term v[] should be read as: a list of the values of v for some value of ts.

The AGGREGATION operator is used in shorthands such as SUMMARIZE.

5.    Transitive Closure

As transitive closure cannot be defined directly in set-builder notation, this formalisation defines it indirectly by means of a recurrence relation. This consists of a starting value (given) and a sequence of successors (defined by set-builder). The transitive closure is the fix-point union of that sequence.

The starting value (‘seed’) represents known edges in a directed graph; the end value is all the possible paths through the graph.

a)      Simple Transitive Closure

Given a relation r with the heading {<A,T>,<B,T>} for some type T then the successor relation r’ is defined as follows.

Hr’ = Hr

Br’ = { tr’ : tr’ ∈ Br or

   ∃ v1 ∈ T, ∃ v2 ∈ T, ∃ v3 ∈ T, ∃ v4 ∈ T,

   ∃ tr1 ∈ Br, tr1 = {<A,T,v1>, <B,T,v2>},

   ∃ tr2 ∈ Br, tr2 = {<A,T,v2>, <B,T,v3>},

   tr’ = {<A,T,v1>, <B,T,v3>} }

Then the transitive closure s = TCLOSE(r) is defined as follows.

S = r1 U r2 U r3 … r¥

This is a linear recurrence, which can be shown to reach a fix-point termination.

The TCLOSE operator is used in a shorthand of the same name.

b)      Extended Transitive Closure

In this case each tuple is associated with a value, and this definition relies on some previously defined dyadic function f that takes values of that type as its argument and return value. The value computed represents in some sense the cost of that path.

Given a relation r with heading {<A,T>,<B,T>,<C,Tv>} for some types T and Tv, and a dyadic function f with the type signature Tv->Tv->Tv then the successor relation r’ is defined as follows.

Hr’ = Hr
Br’ = { tr’ : tr’ ∈ Br or
 ( ∃ v1 ∈ T, ∃ v2 ∈ T, ∃ v3 ∈ T, ∃ v4 ∈ T,
   ∃ w1 ∈ Tv, ∃ w2 ∈ Tv,
   ∃ tr1 ∈ Br, tr1 = {<A,T,v1>, <B,T,v2>, <C,Tv,w1>},
   ∃ tr2 ∈ Br, tr2 = {<A,T,v2>, <B,T,v3>, <C,Tv,w2>},
   tr’ = {<A,T,v1>, <B,T,v3>, <C,Tv,f(w1,w2)} ) }

Then the extended transitive closure s = ETCLOSE(r,f) is defined as follows.

s = r1 U r2 U r3 … r¥

This is a linear recurrence, which can be shown to reach a fix-point termination.

The ETCLOSE operator is used in combination with AGGREGATE in a shorthand to perform Generalised Transitive Closure as defined by D&D (RM VSS 5).

Leave a Comment

Filed under Language, Relational Algebra

Leave a Reply

Your email address will not be published.

This site uses Akismet to reduce spam. Learn how your comment data is processed.