Resolution Theorem Proving in Predicate Calculus

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Resolution Theorem Proving in Predicate Calculus

• Lecture No 10
• By Zahid Anwar

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Resolution

• Resolution is a technique for proving theorems in
  predicate calculus
• Resolution is a sound inference rule that, when
  used to produce a refutation, is also complete
• In an important practical application resolution
  theorem proving particularly the resolution
  refutation system, has made the current
  generation of Prolog interpreters possible

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Resolution Refutation

• The resolution principle, describes a way of
  finding contradictions in a data base of clauses
  with minimum substitution
• Resolution Refutation proves a theorem by
  negating the statement to be proved and adding
  the negated goal to the set of axioms that are
  known or have been assumed to be true
• It then uses the resolution rule of inference to
  show that this leads to a contradiction

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Resolution Refutation

• Once the theorem prover shows that the negated
  goal is inconsistent with the given set of
  axioms, it follows that the original goal must be
  consistent .
• This proves the theorem

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Steps in Resolution Refutation Proofs

1. Put the premises or axioms into clause form
2. Add the negations of what is to be proved in
   clause form, to the set of axioms
3. Resolve these clauses together, producing new
   clauses that logically follow from them
4. Produce a contradiction by generating the empty
   clause
5. The substitutions used to produce the empty
   clause are those under which the opposite of the
   negated goal is true

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Discussion about Steps

• Resolution Refutation proofs require that the
  axioms and the negation of the goal be placed in
  a normal form called the clause form
• Clausal form represents the logical database as a
  set of disjunctions of literals
• Resolution is applied to two clauses when one
  contains a literal and the other its negation

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Discussion about Steps

• If these literals contain variables, they must be
  unified to make them equivalent
• A new clause is then produced consisting of the
  disjunction of all the predicates in the two
  clauses minus the literal and its negative
  instance (which are said to have been resolved
  away)

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An Example

• We wish to prove that
• Fido will die from the statements that
• Fido is a dog and
• all dogs are animals and
• all animals will die

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Equivalent Reasoning by Resolution

• Convert these predicates to clause form

Predicate Form Clause Form
x dog(x)?animal(x) dog(x) V animal(x)
Dog(fido) Dog(fido)
yanimal(y) ?die(y) animal(y) V die(y)
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Apply Resolution

• Negate the conclusion that fido will die
  die(fido)
• dog(x) V animal(x) animal(y) V die(y)
• Y/X
• dog(fido) dog(y) V die(y)
• fido/Y
• die(fido) die(fido)
• a Null clause?Hence fido will die

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Another Example

• Anyone passing the Artificial Intelligence exam
  and winning the lottery is happy. But anyone who
  studies or is lucky can pass all their exams. Ali
  did not study be he is lucky. Anyone who is lucky
  wins the lottery. Is Ali happy?

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Another Example

• Anyone passing the AI Exam and winning the
  lottery is happy
• Xpass(x,AI) ? win(x, lottery) ?happy(x)
• Anyone who studies or is lucky can pass all their
  exams
• X Y studies(x) V lucky(x) ?pass(x,y)
• Ali did not study but he is lucky
• study(ali) ? lucky(ali)
• Anyone who is lucky wins the lottery
• X lucky(x) ?win(x,lottery)

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Change to clause Form

1. pass(X,AI) V win(X,lottery) V happy(X)
2. study(Y) V pass(Y,Z)
3. lucky(W) V pass(W,V)
4. study(ali)
5. Lucky(ali)
6. lucky(u) V win(u,lottery)
7. Add negation of the conclusion happy(ali)

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• pass(X,AI) V win(X,lottery) V happy(X)
• win(u,lottery) V lucky(u)
• u/v
• pass(u,AI) V happy(u) V lucky(u)
• ali/u happy(ali)
• pass(ali,AI) V lucky(ali)
• lucky(ali)
• pass(ali,AI)
• Ali/v,AI/w lucky(v) V pass(V,W)
• lucky(ali) lucky(ali)