603 Database Systems

Senior Lecturer: Laurie Webster II, M.S.S.E.,M.S.E.E., M.S.BME, Ph.D., P.E.

Lecture 25Lecture 25

A First Course in Database Systems

Deductive Databases

RULES:

Although formally all variables in a clause are universally quantified, we will sometimes refer to variables that occur in the body of the clause, but not in its head, as if they are existentially quantified inside the body.

Deductive DatabasesExample:

grandfather rule can be read:

“For all X and Y, X is the grandfather of Y if there exists a Z such that X is the father of Z and Z is the father of Y.”

(X,Y){(grandfather(X, Y)) ( (Z) | father (X, Z),

father (Z, Y))}.

Deductive Databases

Incorporate rules into our framework of logical deduction:

To incorporate rules into our framework of logical deduction, we need the law of modus ponens.

Modus ponens states that from B and A B we can deduce A.

Deductive DatabasesThe Law of Universal Modus Ponens:

RULE: R = (A B1 , B2 , ……….Bn )

FACTS: B’1 .

B’ 2 .

.

.

B’ n .

A’ can be deduced, if

A’ B’1 , B’2 , ……….B’n is an instance of R.

Universal Modus Ponens include identity and instantiation as a special cases.

Deductive DatabasesComplete definition of the concept of a logic program and its associated concept of logical consequence:

Definition: A logic program is a finite set of rules.

Definition: An existentially quantified goal G is a logical consequence of a program P if there is a clause in P with a ground instance A B1 , B2 , ……….Bn , n 0,

such that are logical consequences of P and A is an instance of G.

Deductive DatabasesUniversal Modus Ponens:

Consider the query son(S, haran)?

Rule: son(X,Y) father (Y,X), male(X).

The substitution {X=lot, Y=haran} applied to the rule gives the instance

son(lot, haran) father (haran,lot), male(lot).

Both goals in the body of this rule are facts in the database.

Deductive DatabasesUniversal Modus Ponens:

In the previous query the modus ponens implies the query with answer S=lot.

The rule for son is correct, but is an incomplete specification of the relationship.

A new rule expressing this relationship can be added, namely

son(X,Y) mother(Y,X), male(X).

Deductive Databases

Grandparent rules:

grandparent(X,Y) father(X,Y), father(Y,Z).

grandparent(X,Y) father(X,Y), mother(Y,Z).

grandparent(X,Y) mother(X,Y), father(Y,Z).

grandparent(X,Y) mother(X,Y), mother(Y,Z).

More precise way of expressing these rules.

The Deductive Databases

Rules for son and grandparent:

son(X,Y) parent(Y,X), male(X).

grandparent(X,Y) parent(X,Z), parent(X,Y).

Deductive Databases

parent(X,Y) father(X,Y).

parent(X,Y) mother(X,Y).

A collection of rules with the same predicate in the head, such as the pair of parent rules , is called a procedure.

Deductive DatabasesAn abstract interpreter for logic programs:Input: A ground query Q and a program P

Output: yes if a proof of Q form P was found,

no otherwise

Algorithm:

Initialize the resolvent to Q

while the resolvent A1 , A2 , ……….An is not empty

begin

choose a goal Ai , 1 i n, and a ground instance of a clause A B1 , B2 , ……….Bn , n 0, in P, such that

A = Ai (If no such clause exists, exit the while loop);

determine the new resolvent

A1 ,... Ai-1 , ………. , B1 , B2 , ……….Bk , Ai+1 ,... An

end

If the resolvent is empty, output yes, otherwise output no.

Deductive DatabasesSimple Problem :

If Fido goes wherever John goes and if John is at

School, where is Fido?

Rule: (X ) [At (John, X) At(Fido, X)]

Fact: At(John, School)

Query: ( X ) At (Fido, X)?

Where is Fido?

Rule

Facts

QueryResultant fact(answers)

Deductive Database

Deductive DatabasesResolution Process(rule of inference):

Applied to a pair of parent clauses

Produce a derived clause (inferred clause)

Deductive DatabasesConversion of Predicate Calculus Formula into clausal form includes the following steps:

Step 1. Remove Implications

X [X, man(X) human(X)]

becomes X [X, ~ man(X) # human(X)]

Step 2. Moving negations inward

~ (human (caesar) & living (caesar))

becomes ~ human(caesar) # ~ living (caesar)

Deductive Databases

Conversion of Predicate Calculus Formula into clausal form includes the following steps(continued):

Step 3. Skolemising - removing existential

quantifiers

X (X, female(X) & mother of (X, eve))

etc.

Deductive DatabasesOnce we have the PC formula in clausal form we can then apply the resolution process to prove the goal!!

In resolution refutation, we first negate the goal and then add the negation to the set.

The set is then converted to a set of clauses and resolution is used to derive a contradiction, represented by an empty clause, NIL

Deductive Databases

When we get NIL,

The answer is provided.

Deductive Databases

Fact 1: (X ) [At (John, X) At(Fido, X)] =>Axiom 1

Fact 2: At(John, School)=> Axiom 2

Query:( X ) At (Fido, X)? => conjecture

Where is Fido?

Deductive Databases

Resolution Refutation:

Conjecture: ( X ) At (Fido, X)?

Negation of conjecture: ( X ) ~At (Fido, X)?

( X ) ~At (Fido, X) in clausal form is ~At (Fido, X)

Deductive DatabasesRefutation TREE(negate the goal):~At (Fido, X) ~At (John, X) # At(Fido, X)

(negation of conjecture) (Axiom 1)

~At (John, X) At(John, School)

(Axiom 2)

NIL (X ) [At (John, X) At(Fido, X)]

in clausal form becomes ~At (John, X) # At(Fido, X) and ~At (Fido, X) + ~At (John, X) # At(Fido, X) becomes ~At (John, X)

~At (John, X) + At(John, School)

becomes NIL

Deductive DatabasesExtracting the Answer:

~At (Fido, X) # At(Fido, X) ~At (John, X) # At(Fido, X)

~At (John, X) # At(Fido, X)

At (John, School)

At (Fido, School)

X get bound to School

Deductive Databases

Company Database:

Facts: supervise(franklin, john).

supervise(franklin, ramesh).

supervise(franklin, joyce).

supervise(jennifer, alicia).

supervise(jennifer, ahmad).

supervise(james, franklin).

supervise(james, jennifer).

...

Deductive Databases

Rules:

superior(X,Y):- supervise(X,Y).

superior(X,Y):- supervise(X,Z), superior(Z,Y).

subordinate(X,Y):-superior(Y,X).

Queries:

superior(james, Y)?

superior(james,joyce)?

Deductive Databases

The supervisory TREE:

james

franklin jennifer

john ramesh joyce alicia ahmad

Deductive DatabasesFact: supervise predicate

Rules: superior predicate

superior predicate

subordinate predicate

head :- body

conclusion premise

Deductive DatabaseQueries:

superior (james, Y)?

superior (james, joyce)?

NOTE: predicate with constant as

argument => ground or instantiated predicate!!

Deductive Databases

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