CSCE 580 Artificial Intelligence Ch.12 [P]: Individuals and Relations Datalog

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CSCE 580Artificial IntelligenceCh.12 P
Individuals and RelationsDatalog

• Fall 2009
• Marco Valtorta
• mgv_at_cse.sc.edu

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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

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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.

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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

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Selection

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

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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.

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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!

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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.

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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
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

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Selection

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

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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.

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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!

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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.

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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.

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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.

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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

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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.

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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.

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Case Study University Rules

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

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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)

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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

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A Rule

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

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More Rules

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

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More Rules

• passed(St, C, MinPass) lt-
• grade(St, C, Gr)
• Gr gt MinPass.
• A recursive rule that traverses a list.
• passed_each(, S, M).
• passed_each(CR, St, MinPass) lt-
• passed(St, C, MinPass)
• passed_each(R, St, MinPass).

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More Rules

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

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A KB about rooms (Fig.12.2P)