Announcements

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Announcements

• Reading Assignment
• Nilsson chapters 15-16
• Announcements
• LISP and Extra Credit Project Assigned
• 2nd Exam Tentatively Thu 12/4
• Todays Handouts
• Outline for Class 35
• Web Site
• www.mil.ufl.edu/eel5840
• Software and Notes

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Todays Menu

• Canonical Forms DNF and CNF
• Soundness and Completeness of the PC and
  Resolution
• Validity and Satisfiability of Resolution
• Refutation Completeness
• Conversion to Clause Form

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Predicate Calculus

• Soundness and Completeness
• If ? ? w ? w then the rules used to
  derive (obtain) w from ? are said to be sound.
• Soundness If any theorem derived from a set of
  wffs ? also logically follows from the set ?.
• Completeness All wffs that logically follow from
  a set ? are also derivable from the set ?.
• If ? ? w then the set ?? w is satisfiable
  and the set ?? w is unsatisfiable
  (inconsistent).
• Example Intuitively in the blocks world
• On(A,B) ? On(B,C) ? Above(A,C) if block A is
  on block B and block B is on block C, then
  block A is above block C
• Resolution is sound, but it is not complete. (mp
  is sound and complete)
• Why? Because P ? Q ? P ? Q but we cannot perform
  any resolutions because
• there are no negative literals!

4
Predicate Calculus

• Soundness and Completeness of Resolution
• How do we use Resolution?
• Answer To prove w from ? we form the set ??w
  and if this set is inconsistent (false) then ?? w
  is satisfiable and ? ? w is true.
• Check let ?P ? Q ? wP ? Q then wP
  ? Q and form ??w
• P ? Q ? P ? Q ? false therefore since
  ??w is inconsistent ? ? w is true then we say
  that Resolution is Refutation complete!
• The goal becomes (1) form ??w, (2) try to find
  nil (false)
• General Resolution
• The Resolution principle applies a sound rule of
  inference to a pair of clauses (a clause is a
  disjunction of literals) to obtain (derive) a new
  clause, called the resolvent, that logically
  follows from the original pair of clauses.
  Mathematically,
• ? w1 ? w2 ? w1 ? w3 ? w2 ? w3
  Resolvent

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Predicate Calculus

• Canonical Forms
• Any wff in the FOPC can be expressed in
• Conjunctive Normal Form (CNF) A wff that has
  the form of a conjunction of clauses given by F
  F1 ? F2 ... ? Fn where each Fi is a disjunction
  of literals (i.e., a clause). Example F (P?Q) ?
  (R?S)
• Disjunctive Normal Form (DNF) A wff that has
  the form of a disjunction of product terms given
  by F F1? F2 ... ? Fn where each Fi is a
  conjunction of literals. Example F (P?Q) ?
  (R?S)
• Example
• Given the set ? (P?Q) ? (P ?Q) ? (Q ?P) we
  can convert to CNF
• ? (P?Q) ? (P?Q) ? (Q?P) using the
  equivalent wffs
• Question Can you derive the wff (P?Q) from
  this axiom set ??
• Hint It will iff ??(P?Q) is unsatisfiable!

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Predicate Calculus

• Steps to Perform Resolution
• Step 1 Put your axiom set ? in CNF, or clause
  form. These axioms are your belief set which we
  assume is true. (e.g., our assumptions are true)
• Step 2 Negate the theorem w to be proved,
  convert to clause form and add it to the belief
  set. (e.g., form the set ??w)
• Step 3 The theorem w is true w.r.t. the axiom
  set ? iff ??w is inconsistent therefore, until
  the empty clause (false or nil) is produced
  choose pairs of clauses ? and ? that resolve
  until no resolvable pairs exist or you derive
  nil.
• A. Resolve the clauses ? and ?
• B. Add the resolvent ? to the axioms
• C. Repeat
• Step 4 If the empty clause is produced, report
  success (i.e., w is true). If you run out of
  resolvable clauses, then w is false.
• BIG PROBLEM The procedure does not tell us how
  to select ? and ?, which
• might result in a lot of useless
  resolvents! Also, the
• resolvents that may produce nil are
  not unique.

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Predicate Calculus

• Example
• Given the set ? (P?Q) ? (P ?Q) ? (Q ?P) Can
  you derive the wff (P?Q) from this axiom set
  ??
• Step 1 we can convert to CNF to obtain
• ? (P?Q) ? (P?Q) ? (Q?P) using the
  equivalent wffs
• Step 2 Form the set ??w
• ??w (P?Q) ? (P?Q) ? (Q?P) ?
  (Q?P)
• Step 3 Obtain ?1 Q ? Q Q from (P?Q) ? (P?Q)
• Obtain ?2 Q ? Q Q from (Q?P) ?
  (Q?P)
• Obtain ?3 nil from ?1 ? ?2
• Step 4 Since nil was obtained then (P?Q)
  logically follows from ?
• Note An alternative sequence is obtained as
  follows
• Obtain ?1 Q ? Q Q from (P?Q) ? (P?Q)
• Obtain ?2 P from ?1 ? (Q?P)
• Obtain ?3 Q from ?1 ? (Q?P)
• Obtain ?4 nil from ?2 ? ?3

Not the same sequence!
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Predicate Calculus

• Conversion to Clause Form
• We have seen that in order to perform resolution
  we need the ? set to be in clause form. The
  following procedure converts any wff in the FOPC
  to clause form.
• Step 1. Eliminate Implications (w1?w2 ? w1? w2)
• Step 2. Move Negations down to the Atfs (?xP(x)
  ? ?xP(x)
• ?xP(x) ? ?xP(x) (w1? w2) ? w1 ? w2
  etc.)
• Step 3. Skolemize
• Step 4. Standardize Variables Apart
• Step 5. Move all ? to the left
• Step 6. Distribute - move disjunctions down to
  the literals
• Step 7. Rewrite in Matrix Form
• Step 8. Standardize Variables Apart
• Step 9. Purge ?

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• The End!