1. Lecture 3 [Self Study] Relational Calculus
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,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

5. 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

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 ?
{e.FNAME,e.LNAME | Employee(e)  e.SALARY > }
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

7. 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

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