Relational Algebra and Calculus

Transcription

1 and Calculus Marek Rychly Strathmore & Brno University of Technology, Faculty of Information Technology Advanced Databases and Enterprise Systems 7 September 2015 Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

2 Outline 1 Relational Algebra 2 Tuple Domain Safety of Expressions 3 Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

3 1 st Assessment Relational Algebra (2 points) describe at least 4 relation schemas for different data sets utilized in your company (or at your university in the case you are full-time student), including description of domains used by attributes in the schemas (2 points) describe sample relations for each relation schema defined above with cardinality 2 and above (i.e., each relation should have at least 2 tuples) (3 points) for each of the relations describe at least three super-keys, two candidate keys (if possible), and one primary key with reasoning why the described primary key is the best choice (2 points) describe at lest two referential constraints between the relations (by describing their foreign keys and home relations) (1 point) describe at least one view on the base relations you defined The result should be submitted by to Marek Rychly <mrychly@strathmore.edu> by 14 September 2015 at the latest. Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

4 Relational Algebra and Calculus Relational algebra and calculus (RA&RC) as a basis for query languages. (serve as the formal basis for other, more user-friendly, relational query languages) s able to express RA&RC queries are relationally complete. (Structured Query (SQL), Query-By-Example (QBE), Microsoft Access) RA and RC define basic operations for all relational complete languages. (defined by Codd in 1971; did not consider Null values, multi-set semantics, etc.) Relational algebra is a procedural query language. (it describes how a query should be performed) Relational calculus is a declarative (non-procedural) query language. (it describes what will be the result of a query) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

5 Relational Algebra as a Relational RA expressions describes processes of transformation of some input relations into an output relation. The transformation is done by application of relational operations. (unary and binary operations that operate on one or two relations, respectively) The RA operations are set-theoretic operations: union, difference, intersection, Cartesian product relation-theoretic operations: projections, selection, renaming, join, division (including natural join, semi-join, (left) outer join, etc.) aggregation and grouping operations Five basic RA operations can express the other RA operations. (the basic ops are: projection, union and difference, selection, and Cartesian product) The operations can be composed into more complex operations. (in the similar way as a function composition in mathematics, i.e. op 2 (op 1 (R))) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

6 RelaX Relational Algebra Calculator online relational algebra calculator available on it calculates any relational algebra statement on a set of relations (the most common relational algebra operators are supported) suitable for practising the relational algebra language examples shown during this lecture will be demonstrated on RelaX the examples will use the following relational schemas (the examples are adopted from Database Systems: The Complete Book by H. Garcia-Molina, J. Ullman, and J. Widom, 2 nd ed., exercise on pp ) RS Product = fmaker : string; model : number ; type : fpc; laptop; printergg RS PC = fmodel : number ; speed : number ; ram : number ; hd : number ; price : numberg RS Laptop = fmodel : number ; speed : number ; ram : number ; hd : number ; screen : number ; price : numberg RS Printer = fmodel : number ; color : boolean; type : string; price : numberg RS Owner = fname : string; model : numberg Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

7 Relation-theoretic : Projection Definition (Projection Operation, Pi letter) Let R be a relation. Then, a1 ;:::;a n (R) is a relation that contains a vertical subset of R, extracting the values of specified attributes a 1 ; : : : ; a n and eliminating duplicates. the schema is based on the input but limited to the listed attributes (its degree is the degree of the input relation minus number of the listed attributes) the cardinality may be lower than the cardinality of the input (extracting values of the attributes may cause duplicate tuples that are eliminated) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

8 Examples of Projection Operation List all (distinct) makers of the products in our database. maker (Product) For all PCs and laptops, list their (distinct pairs of) model and speed numbers. model;speed (PC) [ model;speed (Laptop) List all (distinct) screen sizes of available laptops. screen (Laptop) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

9 Set-theoretic : Union Definition (Union Operation) Let R and S be relations with the same schema (they are union-compatible ). Then, R [ S is a relation (with the same schema as R and S) that contains all the tuples of R, or S, or both R and S (duplicate tuples are eliminated). the result has the same schema as input relations (same degree) the result s cardinality is the sum of cardinalities of input relations input relations have to be union-compatible (to get the union-compatible inputs, the projection operation can be used, see later) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

10 Set-theoretic : Difference Definition (Difference Operation) Let R and S be relations with the same schema (they are union-compatible ). Then, R S is a relation (with the same schema as R and S) consisting of the tuples that are in R, but not in S relation. the result has the same schema as input relations (same degree) the result s cardinality is cardinality of R minus cardinality of S input relations have to be union-compatible (to get the union-compatible inputs, the projection operation can be used, see later) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

11 Set-theoretic : Intersection Definition (Intersection Operation) Let R and S be relations with the same schema (they are union-compatible ). Then, R \ S is a relation (with the same schema as R and S) consisting of the tuples that are both in R and S relations. the result has the same schema as input relations (same degree) the result s cardinality is not greater than cardinalities of R and S input relations have to be union-compatible (to get the union-compatible inputs, the projection operation can be used, see later) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

12 Examples of Elementary Set List all RAM sizes that are either in available PCs or in available laptops. ram (PC) [ ram (Laptop) List all HD sizes that are in available laptops but not in available PCs. hd (Laptop) hd (PC) List all CPU speeds where there is both a PC and a laptop with such speed. speed (PC) \ speed (Laptop) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

13 Relation-theoretic : Selection (Restriction) Definition (Selection Operation, sigma letter) Let R be a relation and C be a selection criteria involving constants and attributes of R and logical operations and predicates (it is a formulae of attributive logic). Then, C (R) is a relation that contains only those tuples of R that satisfy the specified selection criteria C. the result has the same schema as the input (same degree) the cardinality is often lower than the cardinality of the input (lowered by tuples that do not match the specified selection criteria) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

14 Examples of Selection Operation List only those PCs that have RAM size at least 2GB and HD size at least 200GB. ram2048^hd200 (PC) List all models of PCs or laptops where their price is up to $700. or alternatively price700 ( model;price(pc) [ model;price(laptop)) price700 ( model;price(pc)) [ price700 ( model;price(laptop)) or alternatively model;price( price700 (PC)) [ model;price( price700 (Laptop)) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

15 Set-theoretic : Cartesian Product Definition (Cartesian Product Operation) Let R and S be relations. Then, R S is a relation that is the concatenation of every tuple of relation R with every tuple of relation S. the result s schema is union of schemas of input relations (its degree is the sum of degrees of the input relations) the result s cardinality is the product of cardinalities of R and S there is no union-compatible restriction Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

16 Examples of Cartesian Product Operation List all possible combinations of PC models and printer details. model (PC) Printer List product models, prices, and vendors of all laptops. Laptop:model;Laptop:price;Product:maker ( Product:model=Laptop:model (Product Laptop)) List all makers that make laptops with 17-inch screen Product:maker ( Product:model=Laptop:model^Laptop:screen=17(Product Laptop)) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

17 Relation-theoretic : Renaming Definition (Renaming Operation, rho letter) Let R be a relation and S; a 1 ; : : : ; a n be names. Then, S(a 1 ;:::;a n)(r) provides a new name S for relation R, and (optionally) also names the attributes of S as a 1 ; : : : ; a n. the schema is based on the input but with the renamed attributes the result consists of the same tuples as the input relation (it has the same degree and the same cardinality as the input) there is an alternative notation for the rename operation: S(a 1 ; : : : ; a n ) R Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

18 Examples of Renaming Operation List all makers that make laptops with 17-inch screen 1 SeventeenLaptops screen=17(laptop) SeventeenVendors(name) Product:maker ( Product:model=SeventeenLaptops:model (Product SeventeenLaptops)) 1 in RelaX, SeventeenVendors(name) : : : is SeventeenVendors ( name maker (: : :)) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

19 Relation-theoretic Ops.: Theta-join and Equi-join Definition (Theta-join and Equi-join Operation) Let R and S be relations and C be a join criteria. Then, R on C S is a relation that is the concatenation of such pairs of tuples of relations R and S that satisfy the join criteria C. The join criteria C can be a b or a v where a; b are attributes of R and S, respectively, is one of binary relational operators <; ; =; ; >, and v is a value constant. In the case = as, we use term Equi-join for the operation. the resulting schema is the union of schemas of the input relations (its degree is the sum of degrees of the input relations) maximal cardinality is the product of cardinalities of R and S (lowered by tuples that do not match the specified join criteria) can be rewritten in terms of Cartesian product and selection ops.: R on C S = C (R S) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

20 Relation-theoretic : Natural Join Definition (Natural Join Operation) Let R and S be relations. Then, R on S is an Equi-join of the two relations R and S over all common attributes. One occurrence of each common attribute is eliminated from the result.. the resulting schema is the union of schemas of the input relations (its deg. is the sum of degs. of the relations minus the number of common attrs.) maximal cardinality is the product of cardinalities of R and S (lowered by tuples whose values do not match in the common attributes) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

21 Examples of Theta-, Equi-, and Natural Join Operation List all makers that make laptops with 17-inch screen Product:maker ( Laptop:screen=17(Product on Product:model=Laptop:model Laptop)) or alternatively (because attributes in join criteria of the equi-join have the same name) Product:maker ( Laptop:screen=17(Product on Laptop)) List all pairs of PCs and laptops where the PCs are faster than the laptops PC on PC:speed>Laptop:speed Laptop Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

22 Relation-theoretic : (Left) Outer Join Definition ((Left) Outer Join Operation) Let R and S be relations. Then, R on S is a variant of natural join in which tuples from R that do not have matching values in the common attributes of S are also included in the result relation (in this case, missing values in the second relation are set to null). the resulting schema is the union of schemas of the input relations (its deg. is the sum of degs. of the input relations minus the no. of common attrs.) maximal cardinality is the product of cardinalities of R and S (lowered by tuples not matching in the common attributes in the second relation) there exists a right outer join and a full outer join denoted by R on S and R on S, respectively R on S = S on R R on S = R on S [ R on S Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

23 Relation-theoretic : Semi(Theta)-join Definition (Semi(Theta)-join Operation) Let R and S be relations and C be a join criteria. Then, R / C S defines a relation that contains the tuples of R that participate in R on C S. the resulting schema is the schema of the first relation (its degree is the degree of the first relation) cardinality is not greater than the cardinality of the first relation can be rewritten in terms of Theta-join and Projection: R / C S = A (R on C S), where A are all attributes of R there are also semi- variants of equi-join and natural join Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

24 Examples of Outer Join and Semi-equi-join Operation For the following examples, we introduce relation 2 Owners = ffname : John; model : 2003g; fname : Jane; model : 2007gg For each product, list its owner if possible, or Null otherwise. Product on Owner List products that have owners. Product / Owner 2 in RelaX, use, without spaces around the colons, the following Owner (fname : string; model : number John; 2003 Jane; 2007g) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

25 Relation-theoretic : Division Definition (Division Operation) Let R and S be relations defined over set of attributes A and B, respectively, such that B A. Let C = A B, that is, C is the set of attributes of R that are not attributes of S. Then, R S defines a relation over the attributes C that consists of the set of tuples from R that match the combination of every tuple in S. the resulting schema is the first relation schema limited to attrs. C (its degree is the cardinality of set C) cardinality is not greater than the cardinality of the first relation can be rewritten by Cartesian product, difference, and projection: R S = C (R) C (( C (R) S) R) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

26 Examples of Division Operation For the following examples, we introduce relation 3 Type = fftype : pcg; ftype : laptopgg List all makers that provide both PCs and laptops. maker;type(product) Type 3 in RelaX, use, without spaces around the colons, the following Type(ftype : string pc laptopg) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

27 Aggregate Operation Relational Algebra Definition (Aggregate Operation, gamma letter) Let R be a relation and L be an aggregate function list of pairs (f ; a) where f is an aggregate function and a is an attribute of relation R. Then, L (R)S applies the aggregate function list, L, to the relation R to define a relation over the aggregate list. the resulting schema consists of attributes of the aggr. function list (its degree is the cardinality of the list) cardinality is always 1, i.e., the result is always one tuple there exists several aggregate functions applicable on an associated attribute in the aggregate function list, for example: COUNT returns the number of values SUM returns the sum of the values AVG returns the average of the values MIN returns the smallest value MAX returns the largest value Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

28 Grouping Operation Relational Algebra Definition (Grouping Operation, gamma letter) Let R be a relation, L be an aggregate function list of pairs (f ; a) where f is an aggregate function and a is an attribute of relation R, and G be a grouping attributes list of attributes of relation R that are not in list L. Then, G L (R)S groups the tuples of relation R by the grouping attributes, G, and then applies the aggregate function list L to define a new relation. The resulting relation contains the grouping attributes, G, along with the results of each of the aggregate functions. the resulting schema consists of attributes of the both lists (its degree is the sum of cardinalities of the lists) cardinality is a number of distinct values of tuples of grouping attrs. (each such the tuple is, in the resulting relation, concatenated with a tuple of results of the aggregate functions applied on the attributes in the aggregate function list) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

29 Examples of Aggregate and Grouping Count number of available PCs. f(count;model)g(pc) Count number of individual types of products 4 R(mycount)( ftypeg f(count;model)g(product)) Get maximal RAM size for available laptops. f(max;ram)g(laptop) 4 in RelaX, use type; count(model)! mycount (Product) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

30 Relational Algebra Tuple Domain Safety of Expressions relational calculus (RC) is a declarative (non-procedural) language (it does not describe how to get resulting relation, contrary to the relational algebra) it is based on the first order predicate calculus (a language of formulas of predicate and function symbols and logical symbols) it describes relations as sets of tuples with particular properties (tuples for which is a given predicate symbol true when applied on the tuples) fx j P(x)g, i.e., a set of all x such that P(x) is true we distinguish two types of relational calculus tuple relational calculus (x above is a tuple) (based on tuple variables ranging over relations, i.e., variables whose only permitted values are tuples of the relations) domain relation calculus (x above is a list of attributes) (based on domain variables with values from domains of the attributes of the relations) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

31 Tuple Domain Safety of Expressions Expressions of Tuple an expression of tuple relational calculus is fs 1 :a 1 ; S 2 :a 2 ; : : : ; S n :a n j F (S 1 ; S 2 ; : : : ; S m )g, where n 6= m, F is a formula, S 1 ; : : : ; S m are tuple variables, and a 1 ; : : : ; a m, respectively, are their attributes a (well-formed) formula F is made of other formulas in logical connectives f_; ^; :g, quantifier symbols f8; 9g, and of atoms that are R(S i ), where S i is a tuple variable and R is a relation S i :a 1 S j :a 2, where S i and S j are tuple variables; is one of the comparison operators f<; ; >; ; =; 6=g; and a 1 and a 2 are attributes of the relations over which S i and S j ranges, respectively, whose domains are comparable by operator S i :a 1 c, where S i is a tuple variable; c is a constant; is a comparison operator; and a 1 is an attribute of the relation over which S i ranges whose domain is comparable by operator with c Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

32 Tuple Domain Safety of Expressions Examples of Expressions in Tuple RC List only those PCs that have RAM size at least 3GB and HD size at least 200GB. fr j PC(R) ^ R:ram 2048 ^ R:hd 200g List all models of PCs or laptops where their price is up to $700. fr:model; R:price j Laptop(R) ^ R:price 700g [fs:model; S:price j PC(S) ^ S:price 700g List product models, prices, and vendors of all laptops. fr:model; R:price; S:maker j Laptop(R) ^ Product(S) ^ R:model = S:modelg List all makers that make laptops with 17-inch screen. fr:maker j Product(R) ^ (9S)(Laptop(S) ^ R:model = S:model ^ S:screen = 17)g Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

33 Tuple Domain Safety of Expressions Expressions of Domain an expression of domain relational calculus is fd 1 ; d 2 ; : : : ; d n j F (d 1 ; d 2 ; : : : ; d m )g, where n m, F is a formula, d 1 ; : : : ; d m are domain variables a (well-formed) formula F is made of other formulas in logical connectives f_; ^; :g, quantifier symbols f8; 9g, and of atoms that are R(d 1 ; : : : ; d n), where R is a relation of degree n and each d i is a domain variable d i d j, where is one of the comparison operators f<; ; >; ; =; 6=g; and d i and d j are domain variables whose members are comparable by operator d i c, where is a comparison operator; c is a constant; and d i is a domain variable whose members are comparable by operator with c Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

34 Tuple Domain Safety of Expressions Examples of Expressions in Domain RC List only those PCs that have RAM size at least 3GB and HD size at least 200GB. fmo; sp; ra; hd; pr j PC(mo; sp; ra; hd; pr )^ra 2048^hd 200g List all models of PCs or laptops where their price is up to $700. fmo; pr j Laptop(mo; sp; ra; hd; sc; pr ) ^ pr 700g [fmp; pr j PC(mo; sp; ra; hd; pr ) ^ pr 700g List product models, prices, and vendors of all laptops. flmo; pr ; ma j Laptop(lmo; sp; ra; hd; pr ) ^ Product(ma; pmo; ty ) ^ lmo = pmog List all makers that make laptops with 17-inch screen. fma j Product(ma; pmo; ty ) ^ (9lmo)(Laptop(lmo; sp; ra; hd; pr ) ^ pmo = lmo ^ sc = 17)g Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

35 Tuple Domain Safety of Expressions Safety of Expressions it is possible for a calculus expression to generate an infinite set (for example, fs j :MyRelation(S)g is an infinite set for finite MyRelation) expression generating infinite sets are unsafe (such data can not be stored in a computer due to its finite memory) safe expressions have restricted resulting values to their domains Definition (Domain of a Expression) For a relational calculus expression E, the domain of E, denoted dom(e), is the set of all values that appear explicitly in E or that appear in one or more relations whose names appear in E. Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

36 Tuple Domain Safety of Expressions Equivalence of Relational Algebra and Calculus Codd s theorem Three languages are precisely equivalent in expressive power: the relational algebra without aggregation and grouping operations the domain relational calculus restricted to safe expressions the tuple relational calculus restricted to safe expressions That is, a database query can be formulated in one language if and only if it can be expressed in the other. (e.g., division in tuple RC is R S = ft j (8u)(S(u) _ R(t :u))g, wh.. is concatenation) Query languages that are equivalent in expressive power to relational algebra are called relationally complete. (e.g., SQL has greater expressive power, it allows aggregation and grouping operations, Null values, duplicate rows, etc., which are not defined in (basic) relational algebra) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

37 Relational Algebra continue with the results of the previous assessment (you can modify the relations from the previous assessment if necessary) (2 points) specify at least three queries by a textual description and in tuple or domain relational calculus (the queries should be non-trivial; free choice between tuple and domain RC) (3 points) for each query above, provide a relational algebra expression that produces the results of the query (i.e., the same output relation as defined by the relational calculus expression) The result should be submitted by to Marek Rychly <mrychly@strathmore.edu> by 21 September 2015 at the latest. Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

38 Summary Summary Query languages to define desired results of database queries. Relational algebra (RA) is a procedural language. Relational calculus (RC) is a declarative language. RA and safe RC are equivalent in their expressive power. RA and RC serve as formal bases for (practical) query languages. In the next lecture: Structured Query (SQL) (db. schema definition, insert/update/delete, select queries, aggregate functions, joins, sub-queries, etc.) Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42

39 Thanks Thank you for your attention! Marek Rychly Note: All definitions provided in this presentation have been adopted from Connolly & Begg: Database Systems, 2005, p ISBN Marek Rychly Relational Algebra and Calculus ADES, 7 September / 42