Logic Programming

1
Logic Programming

• Robert Kowalskis equation
• Algorithm Logic Control
• Definite logic program
• A ? B1, B2, , Bm (program clause)
• head body
• ? A1, A2, , An (goal clause)

2
Logic Programming

• SLD-resolution
• Resolvent
• ? A1, , Ak-1, Ak, Ak1, , An A ? B1, B2, ,
  Bm
• ? ?(A1, , Ak-1, B1, B2, , Bm, Ak1, ,
  An) ?Ak ?A

SL-resolution for Definite clauses - SL Linear
resolution with Selection function
3
Logic Programming

• SLD-resolution
• SLD-derivation
• G0 G, G1, G2, , Gn
• C1, C2, , Cn
• ?1, ?2, , ?n (mgus)
• ?i1Ci1 ?i1Gi
• SLD-refutation Gn empty
• Computed answer ?n ?2?1

4
Logic Programming

• SLD-resolution
• Soundness
• if ?n ?2?1 is a computed answer, then P ?n
  ?2?1G
• Completeness
• if P ?G, then there exists a computed answer
  s such that ? s? for some ?
• Example p(X,Z) ? q(X,Y), p(Y,Z)
• p(X,X)
• q(a,b)

5
Logic Programming
SLD-tree
? p(X,b)
? q(X,Y), p(Y,b)
X/b success
? p(b,b)
? q(b,U), p(U,b) failure
X/b success
Each branch is an SLD-derivation
6
Logic Programming

• PROLOG (Alain Colmerauer 1972)
• Only Horn sentences are acceptable
• The occur-check is omitted from the unification ?
  unsound
• Example test ? p(X,X)
• p(X,f(X))
• Backward chaining with depth-first search ?
  incomplete
• Example p(X,Y) ? q(X,Y)
• p(X,X)
• q(X,Y) ? q(Y,X)

7
Logic Programming
Infinite SLD-tree
? p(X,b)
? q(X,b)
X/b success
? q(b,X)
? q(X,b)
8
Logic Programming

• PROLOG (Alain Colmerauer 1972)
• Unsafe cut ? incomplete
• Example A ? B, C ? A
• B ? D, !, E
• D ? ? B, C
• ? D, !, E, C
• ? !, E, C
• Negation as failure ? P if fails to prove P

9
Theorem Provers

• Accepts full first-order logic
• Keeps control distinct from logic
• A ? B ? C
• A ? C ? B

10
Production Systems

• Working memory contains a set of positive
  literals with no variables
• Rule memory contains a set of reference rules of
  the form p1 ? p2 ? act1 ? act2
• pi are literals
• acti are actions to delete or add elements to
  the working memory

11
Production Systems

• Forwarding chaining in each cycle
• Match phasecomputes the subset of rules whose
  left-hand side is satisfied by the current
  contents of the working memory
• Conflict resolution phase decides which of the
  rules to be executed
• Act phase executes the actions in the chosen
  rules

12
Production Systems

• Example
• Working memory A(1), A(2), B(2), B(3), B(4),
  C(5)
• Rule memory
• A(x) ? B(x) ? C(y) ? add D(x)
• A(x) ? B(y) ? D(x) ? add E(x)
• A(x) ? B(x) ? E(x) ? delete A(x)

13
Production Systems
RETE network
D
AD
add E
A
B
AB
C
add D
C(5)
D(2)
A(1), A(2)
B(2), B(3), B(4)
A(2) B(2)
E
delete A
14
Exercises

• In Russell Norvigs AIMA (Chapter 10) 10.2,
  10.4