LDK R Logics for Data and Knowledge Representation Query languages Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,

LLogics for DData and KKnowledgeRRepresentation

Query languages

Originally by Alessandro Agostini and Fausto GiunchigliaModified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

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Outline Relational and Algebraic Structures Answer set Relational schemas and Databases

Query languages Domain Relational Calculus Tuple Relational Calculus SQL

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Relational Structure 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

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<N, ≤> is a relational structure

STRUCTURES :: ANSWER SET :: RELATIONAL SCHEMAS AND DATABASES :: QUERY LANGUAGES

Algebraic Structure 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

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<N, +> is an algebraic structure


Answer Set 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.

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Example (I)

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)}

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ATTENDSTUDENT COURSE PROFESSOR

Enzo LDKR Fausto

Max LDKR Fausto

Mary ML Alex

Mary LDKR Fausto


Example (II)

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]

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STUDENT COURSE PROFESSOR

Enzo LDKR Fausto

Max LDKR Fausto

Mary ML Alex

Mary LDKR Fausto

ATTEND


Relational schema and database schema 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

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Attend(x, y, z) has name “Attend” and arity 3.

D + Attend(x, y, z) + Student(x, y) + Course(x, y)


Relational instance and database instance 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

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Attend(Enzo, LDKR, Fausto)

DB + the collection of rows in the tables


Relational database 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)

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Domain Relational Calculus (DRC)

“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)}

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NAME POSITION MANAGER

Enzo PhD Fausto

Fausto Professor Bassi

Feroz PhD Enzo

EMPLOYEE


SELECT

“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.

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Enzo PhD Fausto


Feroz PhD Enzo

EMPLOYEE


FILTER

“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)}

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NAME POSITION MANAGER AGE

Enzo PhD Fausto 35

Fausto Professor Bassi 40

Feroz PhD Enzo 20

EMPLOYEE


JOIN

“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)))

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Enzo PhD Fausto


Feroz PhD Enzo

NAME DEPT

Enzo DISI

Fausto DISI

Feroz MATH

FACULTY DEPT


Problems with DRC 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)

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Tuple Relational Calculus (TRC) 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.

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Employee(Name String, Position String, Manager String, Age Int)


Queries in Tuple Relational Calculus “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

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Structured Query Language (SQL) A friendly face over the Tuple Relational Calculus

SELECT {T1.attrib, …, T2.attrib}FROM {relation} T1, {relation} T2, …WHERE {predicates}

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Queries in Structured Query Language

“List name and position of the employees with Fausto as manager”

SELECT name, position

FROM employee

WHERE manager = ‘Fausto’

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EMPLOYEE POSITION MANAGER

Enzo PhD Fausto


Feroz PhD Enzo

EMPLOYEE


Documents

LDK R Logics for Data and Knowledge Representation Query languages Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,