1. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering CSCE 580 Artificial Intelligence Ch.12 [P]: Individuals and Relations Datalog Fall 2009 Marco Valtorta mgv@cse.sc.edu

2. UNIVERSITY OF SOUTH CAROLINA 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

3. UNIVERSITY OF SOUTH CAROLINA 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.

4. UNIVERSITY OF SOUTH CAROLINA 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

5. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Selection Selection: –cs_course(X) <- department(X, comp_science). –math_course(X) <- department(X, math).

6. UNIVERSITY OF SOUTH CAROLINA 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.

7. UNIVERSITY OF SOUTH CAROLINA 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!

8. UNIVERSITY OF SOUTH CAROLINA 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.

9. UNIVERSITY OF SOUTH CAROLINA 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.

10. UNIVERSITY OF SOUTH CAROLINA 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

11. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Selection Selection: –cs_course(X) <- department(X, comp_science). –math_course(X) <- department(X, math).

12. UNIVERSITY OF SOUTH CAROLINA 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.

13. UNIVERSITY OF SOUTH CAROLINA 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!

14. UNIVERSITY OF SOUTH CAROLINA 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.

15. UNIVERSITY OF SOUTH CAROLINA 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.”

16. UNIVERSITY OF SOUTH CAROLINA 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.

17. UNIVERSITY OF SOUTH CAROLINA 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

18. UNIVERSITY OF SOUTH CAROLINA 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.

19. UNIVERSITY OF SOUTH CAROLINA 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.

20. UNIVERSITY OF SOUTH CAROLINA 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.

21. UNIVERSITY OF SOUTH CAROLINA 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)

22. UNIVERSITY OF SOUTH CAROLINA 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

23. UNIVERSITY OF SOUTH CAROLINA 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).

24. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering More Rules Covers_core_courses(St, Dept) <- core_courses(dept, CC, MinPass) & passed_each(CC, St, MinPass)

25. UNIVERSITY OF SOUTH CAROLINA 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).

26. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering More Rules passed_one(CL, St, MinPass) <- member(C, CL) & passed(St, C, MinPass).

27. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering A KB about rooms (Fig.12.2[P])