UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

CSCE 580Artificial Intelligence

Ch.12 [P]: Individuals and Relations

DatalogFall 2009

Marco Valtortamgv@cse.sc.edu

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Acknowledgment• The slides are based on [AIMA] and other

sources, including other fine textbooks• David Poole, Alan Mackworth, and Randy Goebel.

Computational Intelligence: A Logical Approach. Oxford, 1998– A second edition (by Poole and Mackworth) is

under development. Dr. Poole allowed us to use a draft of it in this course

• Ivan Bratko. Prolog Programming for Artificial Intelligence, Third Edition. Addison-Wesley, 2001– The fourth edition is under development

• George F. Luger. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Sixth Edition. Addison-Welsey, 2009

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Databases and Recursion

• Datalog is a subset of Prolog that can be used to define relational algebra

• Datalog is more powerful than relational algebra:– It has variables– It allows recursive definitions of relations– Therefore, e.g., datalog allows the definition

of the transitive closure of a relation.• Datalog has no function symbols: it is a

subset of definite clause logic.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Relational DB Operations

• A relational db is a kb of ground facts• datalog rules can define relational algebra

database operations• The examples refer to the database in

course.pl

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Selection

• Selection:– cs_course(X) <- department(X,

comp_science).– math_course(X) <- department(X, math).

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Union

• Union: multiple rules with the same head– cs_or_math_course(X) <- cs_course(X).– cs_or_math_course(X) <- math_course(X).

• In the example, the cs_or_math_course relation is the union of the two relations defined by the rules above.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Join

• Join: the join is on the shared variables, e.g.:– ?enrolled(S,C) & department(C,D). – One must find instances of the relations

such that the values assigned to the same variables unify• in a DB, unification simply means that the same

variables have the same value!

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Projection

• When there are variables in the body of a clause that don’t appear in the head, you say that the relation is projected onto the variables in the head, e.g.:– in_dept(S,D) <- enrolled(S,C) &

department(C,D).• In the example, the relation in_dept is the

projection of the join of the enrolled and department relations.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Databases and Recursion

• Datalog can be used to define relational algebra• Datalog is more powerful than relational algebra:

– It has variables– It allows recursive definitions of relations– Therefore, e.g., datalog allows the definition of

the transitive closure of a relation.• Datalog has no function symbols: it is a subset of

definite clause logic.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Relational DB Operations

• A relational db is a kb of ground facts• datalog rules can define relational algebra

database operations• The examples refer to the database in

course.pl

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Selection

• Selection:– cs_course(X) <- department(X,

comp_science).– math_course(X) <- department(X, math).

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Union

• Union: multiple rules with the same head– cs_or_math_course(X) <- cs_course(X).– cs_or_math_course(X) <- math_course(X).

• In the example, the cs_or_math_course relation is the union of the two relations defined by the rules above.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Join

• Join: the join is on the shared variables, e.g.:– ?enrolled(S,C) & department(C,D). – One must find instances of the relations

such that the values assigned to the same variables unify• in a DB, unification simply means that the same

variables have the same value!

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Projection

• When there are variables in the body of a clause that don’t appear in the head, you say that the relation is projected onto the variables in the head, e.g.:– in_dept(S,D) <- enrolled(S,C) &

department(C,D).• In the example, the relation in_dept is the

projection of the join of the enrolled and department relations.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Recursion

• Define a predicate in terms of simpler instances of itself– Simpler means: easier to prove

• Examples:– west in west.pl– live in elect.pl

• “Recursion is a way to view mathematical induction top-down.”

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Well-founded Ordering

• Each relation is defined in terms of instances that are lower in a well-founded ordering, a one-to-one correspondence between the relation instances and the non-negative integers.

• Examples:– west: induction on the number of doors to the

west---imm_west is the base case, with n=1.– live: number of steps away from the outside---

live(outside) is the base case.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Verification of Logic Programs

• Verifiability of logic programs is the prime motivation behind using semantics!

• If g is false in the intended interpretation and g is proved from the KB,– Find the clause used to prove g– If some atom in the body of the clause is

false in the intended interpretation, then debug it

– Else return the clause as the buggy clause instance

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Verifiability II

• Also need to show that all cases are covered: if an instance of a predicate is true in the intended interpretation, then one of the clauses is applicable to prove the predicate.

• Also need to show termination---this is in general impossible, due to semidicidability results, but it is possible in many practical situations.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Limitations

• No notion of complete knowledge!– Cannot conclude that something is

false.– Cannot conclude something from lack

of knowledge. Example:• The relation empty_course(X) with the

obvious intended interpretation cannot be defined from enrolled(S,C) relation.

• The Closed World Assumption (CWA) allows reasoning from lack of knowledge.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Case Study: University Rules

• univ.pl• DB of student records• DB of relations about the university• Rules about satisfying degree

requirements.

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

Case Study: University Rules

• This is an example of representing regulatory knowledge.– (Another great example: Sergot, M.J. et

al., “The British Nationality Act as a Logic Program.” CACM, 29, 5 (may 1986), 370-386.)

• DB of relations about the university (univ.pl)• Rules about satisfying degree requirements

(univ.pl)• Lists are used (lists.pl)

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

Some facts

• grade(St, Course, Mark)• dept(Dept,Fac)

– We would say College, not Faculty• course(Course, Dept, Level)• core_courses(Dept,CC, MinPass)

– CC is a list

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

A Rule

• satisfied_degree_requirements(St,Dept) <- covers_core_courses(St,Dept) & dept(Dept, Fac) & satisfies_faculty_req(St, Fac) & fulfilled_electives(St, Dept) & enough_units(St, Dept).

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

More Rules

• Covers_core_courses(St, Dept) <- core_courses(dept, CC, MinPass) & passed_each(CC, St, MinPass)

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

Engineering

Department of Computer Science and Engineering

More Rules

• passed(St, C, MinPass) <- grade(St, C, Gr) & Gr >= MinPass.

• A recursive rule that traverses a list.• passed_each([], S, M).• passed_each([C|R], St, MinPass) <-

passed(St, C, MinPass) & passed_each(R, St, MinPass).

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

More Rules

• passed_one(CL, St, MinPass) <-member(C, CL) &passed(St, C, MinPass).

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINADepartment of Computer Science and

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Department of Computer Science and Engineering

A KB about rooms (Fig.12.2[P])