Using Definite Knowledge

1
Using Definite Knowledge

• Notes for Ch.3 of Poole et al.
• CSCE 580
• Marco Valtorta

2
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.

3
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

4
Selection

• Selection
• cs_course(X) lt- department(X, comp_science).
• math_course(X) lt- department(X, math).

5
Union

• Union multiple rules with the same head
• cs_or_math_course(X) lt- cs_course(X).
• cs_or_math_course(X) lt- math_course(X).
• In the example, the cs_or_math_course relation is
  the union of the two relations defined by the
  rules above.

6
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!

7
Projection

• When there are variables in the body of a clause
  that dont appear in the head, you say that the
  relation is projected onto the variables in the
  head, e.g.
• in_dept(S,D) lt- 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.

8
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.

9
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 n1.
• live number of steps away from the
  outside---live(outside) is the base case.

10
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

11
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.

12
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.

13
Case Study University Rules

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