COSC 4355 and 5355 Expert Systems

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COSC 4355 and 5355Expert Systems

• Knowledge Representation and Reasoning (Part 2)
• Resolution Theorem Proving in Propositional Logic
• Dr. Lappoon R. Tang

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Overview

• What is automated theorem proving?
• Proof System What is it?
• Theorem Prover What is it?
• Resolution Theorem Proving
• Resolution
• a rule of inference
• a mechanism for deriving new facts

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Readings

• ESPP
• Ch 3.10 Ch 3.12
• AI
• Ch 7.5

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Revision The Rules of Inference (Proof by
Refutation)

• To prove X, assume X is NOT true (i.e. not(X) is
  true), and show that it leads to a contradiction
• Example prove that there is no greatest integer
• Proof
• Assume the greatest integer is N
• Since N1 is also an integer
• But N1 gt N
• !!

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What is Automated Theorem Proving?

• It is a branch of research in AI that concerns
  the following issues
• Can I create a system that can automatically
  prove if a statement is true or not?
• Reminder a statement is something that is either
  true or false (it has to be one or the other)
• If so, how can I make sure that the system will
  run efficiently?

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Why would Automated Theorem Proving be Desirable?

• Q Why would it be nice to have a theorem
• prover?
• False Because we can pass all Math
• classes easily
• True It has tremendous application values

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What is a Theorem Prover?

• Idea A program that can decide the validity of a
  certain statement under a certain system of
  logic.

S can be proven true given everything in K is true
Theorem Prover
A statement S
S cannot be proven true given everything in K is
true
A knowledge of true statements K
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Resolution Theorem Proving

• Resolution is a rule of inference
• Just like Modus Ponens, it can be used for
  deriving new statements that are true provided
  the premises are true.
• Developed in 1965 by J.A. Robinson
• This single rule of inference can be used to
  construct a theorem prover

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Resolution Theorem Proving Important definitions

• A literal is either an atomic sentence (or simply
  atom) or the negation of it
• Atoms p, q, r
• Negation of atoms p, q, s
• A clause is a disjunction of literals (or a set
  of literals)
• Example p v q p, q
• An empty clause is the empty set
• A WFF is in conjunctive normal form (CNF) if its
  expressed as a conjunction of clauses
• Example (p v q) (r v s)
• Any WFF in propositional logic can be converted
  to a logically equivalent CNF using laws about
  logical connectives (e.g. distributive law of
  over v)

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Resolution Theorem Proving The resolution rule
of inference

• The resolution rule of inference
• Given p U A and p U B where p is an
  atom, A and B are sets of literals
• Derived A U B
• p U A p U B
• A U B
• p is the atom that has been resolved
• A U B is the resolvent
• This rule can be implemented as a procedure
  called resolution

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Resolution An Example

• Suppose A r, B s
• p U r p U s
• r, s
• The atom p has been resolved
• r, s is the resolvent

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Resolution Theorem Proving The Intuition

• The idea behind resolution theorem proving is
  this
• To prove that a statement s is true, we assume
  that it is false we assume s is true
• If we can derive a contradiction at the end the
  empty clause, then we know s is true

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Resolution Theorem Proving Algorithm
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Resolution Theorem Proving Algorithm

• Given a set of WFF K, and a sentence s
• Task find out if s can be derived from K (Note
  this is the same as asking if the statement K gt
  s is true)
• Convert all the sentences in K into clauses in
  CNF
• Convert s to s (negation of s) Do you smell a
  proof by refutation here? ?
• Convert s into a clause in CNF
• Combine all the clauses obtained in 1) and 2)
  into a single set G (G s U K)
• Iteratively apply the resolution procedure on a
  pair of clauses in G and add the resolvents to G
  until
• There are no more new resolvents that can be
  added and return FALSE K gt s is false
• The empty clause is produced (a contradiction is
  found) and return TRUE Yes, K implies s

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Resolution Theorem Proving Block World Example

• Given set K (of facts)
• BAT_OK (battery is ok)
• CNF BAT_OK
• MOVES (robot arm is not moving)
• CNF MOVES
• BAT_OK LIFTABLE ? MOVES (if the battery is ok
  and the object is liftable, then the robot arm
  moves)
• What is the disjunction of literals?
• CNF BAT_OK, LIFTABLE, MOVES
• Prove LIFTABLE (the object is too heavy, thus
  not liftable by the robot arm)

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Resolution Theorem Proving Block World Example