1. Logics for Data and Knowledge Representation
Query languages
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. Outline Relational and Algebraic Structures Answer set
Relational schemas and Databases
Query languages
Domain Relational Calculus
Tuple Relational Calculus
SQL
Changed 2

3. <N, ≤> is a relational structure
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A relational structure is a mathematical structure of the form
S = <D, R1, …, Rn> where:
- D is a non-empty domain (a set of objects)
- Ri with 1≤ i ≤ n is a relation over D of any arity
<N, ≤> is a relational structure

4. <N, +> is an algebraic structure
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A mathematical structure S is an algebraic structure if
S = <D, f1, …, fn> where each fi with 1≤ i ≤ n is a function.
The term structure is therefore used to denote either a relational structure or an algebraic structure. If D is finite, we say that the structure S is finite
<N, +> is an algebraic structure 4

5. Answer Set FOL can be used as query language on structures
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
FOL can be used as query language on structures
Let be γ = γ(x1, …, xn) a FOL-formula, M a structure, a an assignment
then γ is a query over M.
Qγ = {(a1, …, an)  Dn | M ⊨ γ [a1, …, an]}
Qγ is called the answer set (or retrieval set) of γ.
Since model checking takes polynomial time, also the problem of finding the answer set takes polynomial time.
NOTE: A database is a relational structure. [Codd, 1970] Tables in databases are finite relations.

6. Example (I)
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
ATTEND
STUDENT
COURSE
PROFESSOR
Enzo
LDKR
Fausto
Max
Mary ML
Alex
DB = <D, Attend> D = {Enzo, Max, Mary, LDKR, ML, Fausto, Alex}
Attend = {(Enzo, LDKR, Fausto), (Max, LDKR, Fausto), (Mary, ML, Alex), (Mary, LDKR, Fausto)}
γ = Attend(Enzo, LDKR, x) Qγ = {(Fausto)}
γ = Attend(Mary, x, Alex) Qγ = {(ML)}
γ = Attend(x, LDKR, Fausto) Qγ = {(Enzo), (Max), (Mary)}

7. Example (II)
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
ATTEND
STUDENT
COURSE
PROFESSOR
Enzo
LDKR
Fausto
Max
Mary ML
Alex
DB = <D, Attend> D = {Enzo, Max, Mary, LDKR, ML, Fausto, Alex}
Attend = {(Enzo, LDKR, Fausto), (Max, LDKR, Fausto), (Mary, ML, Alex), (Mary, LDKR, Fausto)}
γ = ∃z Attend(Mary, x, z) Qγ = {(ML), (LDKR)}
Qγ is such that DB ⊨ ∃z Attend(Mary, x, z) [a(x) = ML] or
DB ⊨ ∃z Attend(Mary, x, z) [a(x) = LDKR]

8. Relational schema and database schema
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A relational schema consists of a name and an arity
A relational schema is a FOL-formula.
A database schema is a collection of relational schemas plus a domain
Attend(x, y, z) has name “Attend” and arity 3.
D + Attend(x, y, z) + Student(x, y) + Course(x, y)

9. Relational instance and database instance
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A n-ary relational instance is an n-ary relation R  Dn
A database instance is a collection of relational instances over a domain
NOTE: a database instance is a relational structure
Attend(Enzo, LDKR, Fausto)
DB + the collection of rows in the tables

10. Relational database
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A relational database is a finite relational structure DB = <D, R1, …, Rn>
In particular a DB is said to be in first normal form if:
- the objects in D are atomic (i.e. are not sets)
- the relations R1, …, Rn are defined over D
- the order of the tuples in each Ri does not matter (is a set)

11. Domain Relational Calculus (DRC)
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
EMPLOYEE
NAME
POSITION
MANAGER
Enzo
PhD
Fausto
Professor
Bassi
Feroz
“List name, position and manager of the employees”
γ = Employee(x, y, z)
Qγ = All the tuples in the Relation
“List managers of the employees named Enzo”
γ = ∃y Employee(Enzo, y, z)
Qγ = {(Fausto)}

12. SELECT
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
EMPLOYEE
NAME
POSITION
MANAGER
Enzo
PhD
Fausto
Professor
Bassi
Feroz
“List name and position of the employees having a manager”
γ = ∃z Employee(x, y, z)
Qγ = {(Enzo, PhD), (Fausto, Professor), (Feroz, PhD)}
NOTE: The ∀ quantifier is used to state a condition that must be true for all the tuples in the database, i.e. no missing values.

13. FILTER
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
EMPLOYEE
NAME
POSITION
MANAGER
AGE
Enzo
PhD
Fausto 35
Professor
Bassi 40
Feroz 20
“List name and position of the employees with Fausto as manager”
γ = ∃z Employee(x, y, Fausto, z)
Qγ = {(Enzo, PhD)}
“List name and position of the employees with age > 28”
γ = ∃w ∃z (Employee(x, y, z, w) ∧ (w > 28))
Qγ = {(Enzo, PhD), (Fausto, Professor)}

14. JOIN FACULTY DEPT “List name of faculty members and their department”
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
FACULTY
DEPT
NAME
POSITION
MANAGER
Enzo
PhD
Fausto
Professor
Bassi
Feroz
NAME
DEPT
Enzo
DISI
Fausto
Feroz
MATH
“List name of faculty members and their department”
γ = ∃y ∃z ( Faculty(x, y, z) ∧ Dept(x, w) )
Qγ = {(Enzo, DISI), (Fausto, DISI), (Feroz, MATH)}
“List name of faculty members who belong to all departments”
γ = ∀w (∃x Dept(x, w) → ∃y ∃z (Faculty(x, y, z) ∧ Dept(x, w)))

15. Problems with DRC
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
Atomic objects in the domain cannot be distinguished by column
Faculty(Phd, 28, Enzo, Fausto) is a valid (but unwanted) query
The order of the columns is important. The DB must be queried by remembering the order of the columns
To overcome these problems
we use the Tuple Relational Calculus (see next slide)

16. Tuple Relational Calculus (TRC)
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
We define a set of types {t1, …, tn}. The set of objects D is then partitioned as follows:
D = Dt1 ∪ … ∪ Dtn where Dti is the set of objects of type ti
We introduce a set of attributes, i.e. pairs of the form (name, type). Each pair is a typed attribute and the name is the attribute (that correspond to the name of the columns).
A relational schema is a relation name with a tuple of typed attributes.
Employee(Name String, Position String, Manager String, Age Int)

17. Queries in Tuple Relational Calculus
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
“List all employees who have an age of exactly 35”
γ = Employee(t) ∧ (t.age = 35)
“List name and position of the employees who have age 35”
γ = ∃ t : {name, position} (Employee(t) ∧ t.age = 35)
We write t.age to extract from the tuple t the content of the attribute called “age”
t is called a tuple variable
In TRC variables range over tuples and not over the domain D

18. Structured Query Language (SQL)
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
A friendly face over the Tuple Relational Calculus
SELECT {T1.attrib, …, T2.attrib} FROM {relation} T1, {relation} T2, … WHERE {predicates} 18

19. Queries in Structured Query Language
STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES
EMPLOYEE
EMPLOYEE
POSITION
MANAGER
Enzo
PhD
Fausto
Professor
Bassi
Feroz
“List name and position of the employees with Fausto as manager”
SELECT name, position
FROM employee
WHERE manager = ‘Fausto’