Definitions%20:%20Consistent,%20Refutation

1
Definitions Consistent, Refutation

• A set of formulas T is inconsistent iff there
  exists a deduction of False from T (T - False).
• A refutation from a set of clauses S is a
  deduction of False from S (S - False) .

2
Proof as a search task

• State representation
• a set of wffs (considered to be true)
• Operators inference rules
• Initial state an initial set of wffs
• (what is initially considered to be true)
• Goal state the wff to prove is in our states
  set of known wffs

3
Implicative Normal Form (INF)

• More natural formeach clause is an implication
• Body ? Head
• where Body is a conjunction of atoms
• and Head is a disjunction of atoms
• Notice that ? (negation) is not used
• Examples
• 1) ? P
• 2) P ? R
• 3) P ? R ? T V S V Q
• 4) P ?

4
Converting arbitrary wffs to INF

• After converting to CNF add the following step to
  convert it to INF
• Convert disjunctions to implications Body ? Head
• Gather up the negative literals into one
  list,take them as a conjuntion (Body) gather up
  the positive literals into another, take them as
  a disjunction (Head).
• CNF INF
• P ? P
• ? P V R P ? R
• ? P V ? R V T V S V Q P ? R ? T V S V Q
• ? P V ? R P ? R ?

5
Resolution in INF

• P1 ? ... ? Pi ? ... ? Pn1 ? R1 v ... v Rn2
• S1 ? ... ? Sn3 ? Q1 v ... v Pi v... v Qn4
• --------------------------------------------------
  -------
• P1 ? ... ? Pi-1 ? Pi1... ? Pn1 ? S1 ? ... ? Sn3
  ? R1 v ... v Rn2 v Q1 v...v Qj-1 v Qj1... v Qn4
  
• Example
• P ? R ? T V S V Q
• U ? Z ? R
• --------------------------------------------------
  
• P ? U ? Z ? T V S V Q

6
Horn Clauses

• A Horn clause is a clause that has at most one
  positive literal
• Examples P ? P V ? R ? P V R ? P V ? R V T
  
• P V R is not a Horn clause
• Usually written in Implicative Normal Form (INF)
• at most one atom in the head
• Examples
• ? P Fact
• P ? R Rule
• P ? R ? T Rule
• P ? R ? Goal
• Definite Clauses Facts and Rules
• Objective (or goal) clause goal

7
SLD Resolution

• Language Horn clauses in INF
• S - Selection Function - selects the atom in the
  goal clause to resolve
• L- Linear Resolution
• D - Definite Clauses
• The set of clauses S
• a set of definite clauses representing KB,
  together with one goal clause representing ? Q

8
SLD Resolution

• Resolution rule in INF (order of atoms is
  important!)
• P1 ? ... ? Pi ? ... ? Pn1 ?
• S1 ? ... ? Sn3 ? Pi
• --------------------------------------------------
  -------
• P1 ? ... ? Pi-1 ? S1 ? ... ? Sn3 ? Pi1... ? Pn1
  ?

9
SLD Resolution

• Standard selection function (PROLOG) leftmost
  atom
• The above rule becomes
• P1 ? ... ? Pn1 ?
• S1 ? ... ? Sn3 ? P1
• --------------------------------------------------
  -------
• S1 ? ... ? Sn3 ? P2... ? Pn1 ?
• Remember that the order of atoms is important!
• resolvent - play DO
• body of definite clause then body of the
  objective clause

10
SLDNF Resolution

• Negation in PROLOG Negation as (finite) Failure
• Not P if an SLD tree starting from
• P ?
• finitely fails.