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

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

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

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,Johnson and CS234 are 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

1. In predicate calculus sentences contain quantified variables only 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 Lecturer Leader(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

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 variables Matching values e.FNAME James Jennifer e.LNAME Borg Wallace CM036: Advanced Databases Relational Calculus

1. The question variables are always listed in front of the 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

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