Generalising Relational Algebra Set Operators

Transcription

1 Generalising Relational lgebra Set Operators Introduction The relational algebra join operators, Natural Join and Generalised (or Theta) Join, can both be generalised to so that they incorporate semi joins left and right and outer joins using default tuples 1 left, right, and full. Relational divide (based on set comparisons) can also be generalised so that it incorporates semi divides left and right and outer divides using default tuples left, right, and full and is thereby seen to be analogous to joins. If all dyadic operators whose function is to merge their operands by attribute can be generalised to include semi and outer versions, the question arises as to whether all the dyadic operators whose function is to merge their operands by tuple i.e. the set operators, Union, Intersect, and Difference - can also be generalised to include semi and outer versions using default tuples; and if not, why not. The question is important for RQUEL because in expressing the relational model, RQUEL 2 aims to utilise as few concepts as possible and make them as general as possible. Therefore it is preferable to incorporate the same generality as regards semi and outer versions into merge-by-tuple operators as merge-byattribute operators already possess, so that the two kinds of operators are consistent with each other and one kind is not a special case compared to the other. The purpose of this document is to review the question in order to consider whether merge-by-tuple operators can be generalised in a way comparable to that of the merge-by-attribute operators. Fundamental Principles to be Used 1. Just as the join operators and divide are generalised from their original inner versions to include semi and outer versions, so should the inner versions of the set operators Union, Intersect, and Difference be generalised to include semi and outer versions. 2. The equivalent of the merge-by-attribute approach applicable to the joins and divide should be used in the merge-by-tuple approach applicable to the set operators. Therefore : s the results of semi merge-by-attribute operators contain only the attributes of one operand, so the results of semi merge-by-tuple operators should contain only the tuples of one operand. 1 Outer joins are sometimes considered to be dependent on the use of NULLs, possibly because this is the case in SQL. This is incorrect. Outer joins are dependent on the use of Default Tuples (in RQUEL terminology). default tuple is a tuple of attribute values, identical in type to the tuples appearing in a relational value. In a NULL-permitting system, such as SQL, NULLs may appear in the attributes of a default tuple instead of values if required. 2 RQUEL does not permit the use of NULLs. Page 1 of 10

2 s the results of outer merge-by-attribute operators include the tuples of one or both operands that would be missing from an inner result, so the results of outer merge-by-tuple operators should include the attributes of one or both operands that would be missing from an inner result. Example Relations for Illustrative Purposes R1 R R3 R4 C 6 x 8 y 15 m C x y z m D aa bb cc dd bove is shown the relational values (= relvalues) of 4 relational variables (= relvars). ttributes of the same name have the same type, and attributes of different names have different types. Thus only R1 and R2 have the same relational type (= reltype). Generalisation of Operators Traditionally the operands of the set operators are required to have the same reltype. So of the 4 example relvars, only R1 and R2 could be the operands of set operators. It is proposed to generalise this so that the set operators can be applied to any operands that have attributes in common 3, and that the result contains only the 3 n attribute common to both operands has the same name and type in both operands. Page 2 of 10

3 common attributes. For each operand, a projection of the common attributes would be taken and the set operator applied to the 2 projections to yield a result. The generalisation leaves unchanged the application of the operators to operands of the same reltype. If the operands have no attributes in common, two possibilities arise. One is that this is treated as an error. The other is that this is treated as part of the generalisation; in which case it would be the application of the set operator to 2 relvalues of no attributes, which had either no tuples or one tuple containing the empty set (depending on the results of the projections on the operands). With the example relvars described above : set operator applied to R1 and R3, or to R2 and R3, would yield a result containing only attribute. set operator applied to R3 and R4 would yield a result containing only attribute C. set operator applied to R1 and R4, or to R2 and R4, would yield either an error or a result containing no attributes. The relvalues returned by the set operators when applied to R2 and R3 are :- R2 Union R3 R2 Intersect R3 R2 Diff R Page 3 of 10

4 Semi Set Operators semi set operator returns the set of tuples from the relevant operand that participate in the result of the inner operation. This is illustrated using relvars R2 and R3. Union Consider a left semi union 4 :- R2 Union[ R3 Tuples in the result of the inner union that do not originate from the left hand operand, R2, are removed from the result, which is then expanded by adding the remaining attributes of the left hand operand, i.e. attribute, with their values in the left hand operand. Hence the result is the same value as that of the left hand operand, R2. The right semi union R2 Union] R3 correspondingly returns the value of R3, the right hand operand. Intersect Consider a left semi intersect : R2 Intersect[ R3 4 For asymmetric joins and divide, RQUEL syntax uses a single bracket to indicate the asymmetry, [ for left semi and outer versions, and ] for right semi and outer versions. These appear to the left and right respectively of the [.. ] brackets which surround the parameter to the operator. Set operators require no parameter, but for consistency of syntax [ and ] are still used to indicate left and right asymmetry. Page 4 of 10

5 To the result of the inner intersect on attribute, attribute of the left-hand operand has been added. The right semi intersect R2 Intersect] R3 correspondingly yields the result C 6 x 8 y Difference Consider a left semi difference :- R2 Diff[ R3 To the result of the inner difference on attribute, attribute of the left-hand operand has been added. The right semi difference R2 Diff] R does not make sense, since by definition the purpose of a difference operation is to return tuples from the left hand operand; therefore there is nothing from the right hand operand to put in the result. It seems preferable not to have a right semi difference operator, rather than have one which returns an error or which returns a relvalue of no tuples with the attributes of the right hand operand. Page 5 of 10

6 Outer Set Operators n outer set operator returns an inner result expanded to include all the attributes of the relevant operand(s). Where no attribute values are derivable from the operand(s) for these attributes, a default tuple(s) must be provided containing the attribute values to be used. This is illustrated using relvars R2 and R3. Union Consider a left outer union, using a default tuple 5 :- R2 Union{ d } [ R3 The result has all the tuples of the inner result, expanded with the attributes of the left hand operand, i.e. attribute. Since no attribute values are derivable for for those tuples originating in the right hand operand, a default tuple is provided that contains the value to be used for attribute in such cases. Therefore the default tuple relates to the left hand operand. This differs from the merge-by-attribute operators for which the default tuple must be relevant, not to the operand whose tuples are to be retained in the result, but the other operand. The right outer union d 15 R2 Union] { e } R3 correspondingly yields the result 5 To conform to RQUEL syntax, a default tuple is here designated by { d }, where d represents the set of attribute values in the tuple. The type of { d } must conform to that of tuples from the operand that it will be unioned with in the result. Page 6 of 10

7 gain the default tuple relates to the operand whose tuples are to be retained, in this case the right hand operand. The full outer union C 12 e 14 e 6 x 8 y 0 e 15 m R2 Union{ d } [ ] { e } R3 correspondingly yields the result Default tuples are required for both operands. Intersect Consider a left outer intersect :- C e e x y e d 15 m R2 Intersect[ R3 Page 7 of 10

8 No default tuple is needed, because the result of an intersection contains tuples common to both operands. Therefore when expanding the result to include attribute, values of attribute already exist in the left operand to be used in the result The result is the same as that for the left semi intersect. The same syntax is used because there is no default tuple, and so nothing to distinguish the semi and outer intersects. The right outer intersect R2 Intersect] R behaves in the corresponding way, yielding the result The full outer intersect R2 Intersect[ ] R3 also behaves in the corresponding way, yielding the result C 6 x 8 y C x y Difference Consider a left outer difference :- R2 Diff[ R gain a default tuple is not needed. The reason is analogous to that for intersection, namely that the result of a difference contains tuples already in the left hand operand. Therefore when expanding the result to include attribute, values of attribute already exist in the left operand to be used in the result. Page 8 of 10

9 The result is the same as that for the left semi difference. The same syntax is used because there is no default tuple, and so nothing to distinguish the semi and outer differences. The right outer difference R2 Diff] R3 does not make sense, for the same reason as with the right semi difference. Consequently there can be no full outer difference, since a full outer operator combines the functionality of both left outer and right outer operators. Summary of Semi and Outer Operators Union The semi unions return the value of the relevant operand, so their worth might be questioned. However if operands with no common attributes are permitted, then results with no attributes can arise. ll the outer unions return worthwhile results. The default tuple must correspond to the same operand as the left or right asymmetry, and not the opposite operand as is the case with merge-by-attribute operators. (The full outer union requires a default tuple for both operands). Intersect oth the semi intersects return worthwhile results. ll three outer intersects return worthwhile results. However default tuples are not needed. The semantics of intersection mean that only tuples common to both operands appear in the result; therefore required attribute values are already in the operands and do not need to be supplied via default tuples. nother consequence of this is that the left and right outer intersects return the same values as the left and right semi intersects. Difference Only the left semi difference is meaningful. The right semi difference is irrational and cannot exist, because the semantics of difference mean that the result can only contain tuples pertaining to the left operand. The left outer difference is meaningful but not the right outer difference, for the same reason as applies with semi differences. Consequently there cannot be a full outer difference as a full outer operator combines the functionality of left and right outer operators. The left outer difference does not need a default tuple. s with intersection, the semantics of difference mean that the left outer difference returns the same value as the left semi difference. Page 9 of 10

10 Conclusions 1. Generalising the inner set operators so that they work on operands with common attributes is worthwhile, because it adds functionality without adding any conceptual complexity. The concepts remain as they were but are now more generally applicable. The traditional cases still apply, just as before, but they are now special cases of more general functionality. 2. It is important to organise the semi and outer set versions of operators into a coherent set so that they are seen to be expressions of the minimum number of simple underlying general principles. For intersect and difference, there is no point in having both semi and outer operators because they duplicate each other. The absence of a need for default tuples suggests that semi operators be retained and outer operators abandoned. On the other hand, a full outer intersect yields a worthwhile result. Therefore let it be re-classified as a full semi intersect. t first sight this is a contradiction in terms, but it is entirely logical in the context and maintains the concept of semi operators having no default tuples. It enables the retention of a worthwhile operator. For union, semi operators should be retained for consistency with intersect and difference; they are also potentially useful. ll the outer operators should be retained because they provide worthwhile results. Having outer versions only for union stems from the fact that only the union operator can incorporate tuples in its result whose set of attribute values are not all derivable from tuples in its operands. This does not arise with intersect and difference because the former merges tuples from its operands and the latter takes tuples from only one operand. The above generalisations lead to the following versions of the set operators :- Generalised Inner Semi Outer Union Left, Right Left, Right, Full Intersect Difference Left, Right, Full Left The table shows that the incidence of the semi and outer versions of the mergeby-tuple operators is not uniform between them all. This is a consequence of the differing semantic principles used by the set operators to merge their operands. y comparison the merge-by-attribute operators all apply the same semantic principle to merge their operands, namely a cartesian product of comparisons of the tuples of their operands. Thus the incidence of the inner, semi and outer versions is the same for all the operators. (The difference between the operators is the nature of the comparisons they make and the logical consequences of the comparisons as regards the attributes to be put in the results). Page 10 of 10