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  1. CM036: Advanced Database Lecture 3 [Self Study] Relational Calculus

  2. Relational Calculus • Declarative language based on predicate logic - checks what is true in the database rather then looking to get it • The main difference in comparison with the relational algebra is the introduction of variables which range over attributes or tuples • Two different variations of the relational calculus: • domain calculus - specification of formal properties for different data types, used for describing data • tuple calculus - querying and checking formal properties of stored relational data CM036: Advanced Databases Relational Calculus

  3. Predicate Calculus • Vocabulary of constant names, functional attributes and predicate properties, used for describing the information • Phrases for specification of constant, functionally dependant or predicated properties build using proper vocabulary terms • Statements about the world, formulated as meaningful phrases, connected through logical connectives in logical sentences • conjunction () • disjunction () • negation () • implication () • equivalence () • Possible quantification of the sentences using existential () or universal() quantifiers, ranging over variables Example:All units registered in the database have unit leaders among the lecturers (x).(Unit (x)  (y).(Lecturer(y)  Leader(y,x))) CM036: Advanced Databases Relational Calculus

  4. Relational DB as a Predicate Calculus model • Semantic interpretation of the calculus is given in a set, so either all type domains of the relational schema (domain calculus), or the set of all relations in the database (tuple calculus) can serve a model for it • All meaningful sentences in the model are true or false, so the relation tuples can be interpreted as truthful facts describing the world • The constraints describe logical regularities among the attribute values, so they can be expressed as logical sentences Example:All records of unit leaders have primary keys • Relation names are predicates with a place for each attribute Leader(Lno:INTEGER,Lname:CHAR,Uname:CHAR) - Lno, Lname and Uname are attributes of Leader • Data values in the relation tuples are constants from the domains Leader(2,‘Johnson’,’CS234’) - 2,Johnsonand CS234are constants forming one Leader tuple • Constraints can be stated using quantified variables over domains (x)(y,z) .(Leader(x,y,z)) - x stands for the primary key of Leader CM036: Advanced Databases Relational Calculus

  5. Notes: 1. In predicate calculus sentences contain quantified variables only 2. The sub-expressions following the quantified variables form the scope of quantification; it is usually closed in round parentheses Example:The table for unit leaders has foreign keys to both the lecturers and units tables • Relation names are predicates with place for each attribute Unit(Uno:INTEGER, Uname:CHAR) - Uno and Uname are attributes of relation Unit Lecturer(Lno:INTEGER, Lname:CHAR) - Lno and Lname are attributes of relation LecturerLeader(LUno:INTEGER,Lno:INTEGER,Uno:INTEGER) - LUno, Lno and Uno are attributes of relation Leader • All foreign key constraints can be stated logically using quantified sentences, in which variables range over the respective values ( LUno,Uno,Lno).(Leader(LUno,Uno,Lno)  ( Lname).(Lecturer(Lno,Lname))  ( Uname).(Unit(Uno,Uname)) ) CM036: Advanced Databases Relational Calculus

  6. Relational Calculus as database query language • Queries are logical expressions, which contain non-quantified (free) variables for tuples • The free variables are placeholders for the information, we are looking for from the database relations • The logical expressions, used in the query, are its conditions which need to be met during answering the query Example:Who are the employees with salary greater than 40 000? {e.FNAME,e.LNAME |Employee(e)  e.SALARY > 40 000} • An answer is a replacement of the free variables in the query with tuple attributes from the relations used to formulate the query conditions; they should make the statement of the query true in database (pattern matching) Answer:James Borg and Jennifer Wallace Free variablesMatchingvalues e.FNAMEJames Jennifer e.LNAMEBorg Wallace CM036: Advanced Databases Relational Calculus

  7. Notes: 1. The question variables are always listed in front of the expression, separated by | from the question condition; 2. Question condition can contain free variables only from the list in the beginning; all other variables should be quantified • Querying related tables requires check of the corresponding keys for equality Example:Retrieve the name and address of all employees who work for the ‘Research’ department {e.FNAME,e.LNAME,e.ADDRESS | Employee(e)  ( d).(Department(d)  d.DNAME = ‘Research’  d.DNUMBER = e.DNO ) } CM036: Advanced Databases Relational Calculus

  8. Equvalence between relational algebra and relational calculus • Relational algebra and relational calculus are equivalent with respect to their ability to formulate queries against relational databases (Codd 1972) • Relational algebra concentrates on the procedural (“how-to”)aspects and because of this it is used as an intermediate language for optimization of database queries • Relational calculus is more appropriate to specify declaratively the model properties (“what is true”), without worrying about the way it is achieved and as such it can be used as a specification or querying language Note:The relational language Query-By-Example (QBE), which is developed by IBM during 70ties and is implemented in Paradox and Access desktop database systems is based on the relational calculus CM036: Advanced Databases Relational Calculus