CS1502 Formal Methods in Computer Science

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CS1502 Formal Methods in Computer Science
Lecture Notes 10 Resolutionand Horn Sentences
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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Conversion to Clausal Form

• Use P ? Q equiv P v Q to remove ?
• Use P ?? Q equiv ((P v Q) (Q v P)) to
  remove ??
• New 1st Step! We hadnt done conditionals yet
• Use DeMorgans laws and Elim to move as far
  inward as possible (gives NNF)
• Use the distributive laws until the sentence is
  in CNF

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Clausal Form

• Given a sentence S written in CNF S ( ) ?( )
  ? . . . ? ( )
• Convert each ( ) into a clause - a set consisting
  of each of the literals in ( ).Example S (A)
  ? (?B ? C ? D) ? (?A? D)Clauses are A, ?
  B, C, D, ? A, D

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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Satisfying a Set of Clauses

• Assigning truth-values to the atomic sentences so
  the CNF sentence the set corresponds to is true.
• A, ?C, D, C,?A,A, D is satisfied by
  
• A False
  C True D True

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Empty Clause

• denoted by .
• The empty clause is not satisfiable
• We want A, ? A to lead to ()
• The basic resolution step involves canceling
  pos and neg matching literals from two clauses
• So, we derive () from A, ? A

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Example not satisfiable

• ?P ? Q
• P
• ?Q

?P, Q
P
?Q
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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Resolution Step
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Res. Step with Larger Clauses
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Resolvent

• Clause R is a resolvent of clauses C1 and C2 if
  there is a literal in C1 whose negation is in C2
  and R consists of all the remaining literals in
  either clause.Example A, C , ?D and B, ?C
  have resolvent A, B, ?D

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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Resolution Theorem

• For any set of clauses that are not satisfiable,
  it is possible to arrive at the empty clause by
  using successive resolutions.

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Resolution Theorem Proving

• Method for searching for proofs automatically
• Sentences are translated into CNF, and then to
  sets of clauses
• At each step, a new clause is derived from two
  clauses you already have
• Proof steps all use the same rule
• If reach (__), sentences were not satisfiable
• Start premises negation of goal
• End if reach , premises goal

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Using Resolution to Determine Validity of
Arguments

• Q is a logical consequence of P1, P2, , Pn iff
  P1 P2 Pn Q is not satisfiable
• For sentences in clausal form Q
  is a logical consequence of a set of clauses S
  iff S ? ?Q is not satisfiable.
• So, convert premises negation of goal to
  clausal form, and do resolution steps, trying to
  reach ()

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Is this Argument Valid?
B v C C v D A v D B v D A
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Example

• Show ?A ? (B ? C) ? (?C ? ?D) ? (A ? D) ? (?B ?
  ?D) is not satisfiable.Clauses ? A, B, C,
  ?C, ?D, A, D, ?B, ?D. ?A A,D
  B,C ?C, ?D D
  B, ?D ?B,
  ?D
  ?D

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Example

• Modus PonesP?Q ?P?Q or
  ?P,QP
  PQ negate to get ?QApply
  resolution ?P,Q P
  Q
  ?Q
  ?

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What is a Horn sentence?

• A positive literal is any literal that is not
  preceded with a ?. For example, Cube(b) and P are
  positive literals.
• A sentence S is a Horn sentence if and only if it
  is in CNF and every conjunct has at most one
  positive literal.

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Examples

• (A ? ?B ? ?C) ? (?A ? ?B)
• (A ? ?B ? C) ? (D)
• (A ? B) ? (?C ? ?D)

Horn sentence
Not Horn sentence
Horn sentence
(A ? ?C) ? (B ? ?C) ? (A ? ?D) ? (B ? ?D)
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Alternate Form of Horn Conjunct

• ?A1? ?A2?...? ?An? B

?(A1 ? A2 ?... ? An) ? B
(A1 ? A2 ?... ? An) ? B
B - A1, A2, , An. In PrologRule
Conditional Form of Horn sentence
B if A1, A2, , An
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Special Cases

• No positive literal?A1? ?A2?...? ?An
• No negative literalsB

In Prolog, this is a query!
(A1 ? A2 ?... ? An) ? False
In Prolog, this is a fact!
True ? B
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Why are Horn sentences important?

• Very efficient algorithms exist for determining
  if a set of Horn sentences is satisfiable

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CNF to PROLOG

• ?A ? B ? ?C is ?(A?C) ? B is (A ?
  C)?B B - A, C.A.C.Query - B.Answer
  Yes.

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Example Prolog Program

• grandfather(X,Y) - father(X,Z),
  father(Z,Y).grandfather(X,Y) -
  father(X,Z), mother(Z,Y).
  mother(ann,bill).father(carl,ed).father(nick,a
  nn).father(ed,sam).