Converting arbitrary wffs to CNF

1
Converting arbitrary wffs to CNF

• A wff is in prenex form iff it consists of a
  string of quantifiers (called a prefix) followed
  by a quantifier free formula called a matrix.
• 5. Convert to Prenex Form
• At this stage there is no remaining existential
  quantifiers and each universal quantifier has its
  own variable symbol.
• We may now move all of the universal quantifiers
  to the front of the wff and let the scope of each
  quantifier include the entirety of the wff
  following it. The resulting wff is in
• prenex form.
• From (1) ?X(?P(X) v ?Y((? P(Y) v P(f(X, Y))) ?
  Q(X, h(X)) ? ?P(h(X))))
• ?X ?Y(?P(X) v ((? P(Y) v P(f(X, Y))) ? Q(X, h(X))
  ? ?P(h(X))))

2
Converting arbitrary wffs to CNF

• 6. Eliminate universal quantifiers
• Since all the variables in the wff we use must be
  within the scope of a quantifier, we are assured
  that all the variables remaining at this step are
  universally quantified.
• Furthermore, the order of universal
  quantification is unimportant, so we may
  eliminate the explicit occurrence of universal
  quantifiers and assume, by convention, that
• all variables in the matrix are universally
  quantified.

3
Converting arbitrary wffs to CNF

• 7. Put the Matrix in CNF
• Distribute ? over v
• (A ? B) v C becomes (A v C) ? (B v C)
• Flatten nested conjunctions and disjunctions
• (A v B) v C becomes (A v B v C)
• (A ? B) ? C becomes (A ? B ? C)
• At this point we have a conjunction of clauses
• We must have a set of clauses!
• separate the conjuncts

4
Substitution

• A substitution ? is a set of ordered pairs
• ? X1 / t1 ,...., Xn / tn
• where Xi is a variable and ti is a term such that
  all Xi are distinct.
• A simple expression is either an atom or a term.
• Example E1 P(X, X, Y), ? X / bill, Y / Z
• E1 ? P(bill, bill , Z)
• Means that for each pair xi / ti , the term ti
  is substituted simultaneoulsy for every
  occurrence of the variable xi throughout the
  scope of the substitution (E1)

5
Composition

• Let ? X1 / s1 ,...., Xm / sm and a Y1 /
  t1 ,...., Yn / tn be substitutions.
• The composition of ? and a, denoted by ?a, is the
  substitution consisting of the bindings in the
  resulting sequence
• a) consider the sequence of bindings (apply a to
  the terms of ? and concatenate the result with a)
  
• X1 / (s1 a),...., Xm / (sm a) , Y1 / t1 ,...., Yn
  / tn
• b)delete from this sequence
• any binding Xi / (si a) for which Xi (si a) and
  
• any binding Yj / tj for which Yj ? X1 ,...., Xm
  .

6
Composition

• Examples
• 1) Let ? X / f(Y), Z / U and a Y / b, U
  / Z,
• then ?a X / f(b) , Y / b , U / Z
• 2) Let ? X / Y and a X / a, Y / a,
• then ?a X / a, Y /a

7
Unification

• Let S be a finite set of simple expressions.
• A unifier for the set S E1 , E2 is a
  substitution ? such that S? is a singleton
• E1 ? E2 ?
• S is said to be unifiable iff there exists such
  ?.
• If ? is a unifier for S, and if for any unifier a
  for S there exists a substitution f such that a
  ? f,
• then ? is called a most general unifier (mgu).
• A mgu is unique except for variables renaming
  (alphabetic variants).

8
Unification

• Example Let S R(X, f(Y), B), R(Z, f(B), B).
• Then ? X /Z , Y / B is a mgu for S.
• Although X / A , Z / A, Y / B is a unifier for
  S it is not a mgu (the simplest) since there does
  not exist a substitution f such that
• X /Z , Y / B X / A , Z / A, Y / B f
• Notice that X / A , Z / A, Y / B X /Z , Y /
  BZ / A

9
Unification

• Let S be a finite set of simple expressions.
• The Disagreement set of S is defined as follows.
• Locate the leftmost symbol position at which not
  all members of S have the same symbol, and
  extract from each expression in S the
  subexpression begining at that symbol position.
• The set of all these subexpressions is the
  disagreement set.
• Example let S be the set P(x, y) , P (x,
  f(g(a)))
• then
• D y , f(g(a))

10
Unification Algorithm

• Input a finite set S of simple expressions
• Output a mgu for S (if S is unifiable)
• 1. Set k 0 and ?0 ?.
• 2. If S?k is a singleton, then stop ?k is an
  mgu for S.
• Else find the disagreement set Dk of S?k .
• 3. If there exists X and t in Dk such that
• X is a variable not occurring in t,
• then set ?k1 ?k x / t, increment k by 1 and
  go to step 2.
• Else report that S is not unifiable, and stop.