COSC 4350 and 5350 Artificial Intelligence

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COSC 4350 and 5350Artificial Intelligence

• Propositional Logic and Resolution (Part II)
• Theorem Proving by Resolution
• 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
• As a mechanism for proving theorems implemented
  as a search algorithm

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Readings

• Section 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?
• 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 A Because we can pass all Math
• classes easily
• True A In some cases, we can automatically
  verify
• if a certain design is correct (e.g. a CPU)

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Proof System What is it?

• A proof system axioms rules of inference
• In other words, a proof system is a mathematical
  system that defines two things
• Axioms things assumed to be true from the very
  onset (e.g. 1 gt 0)
• Rules of inference how to derive new facts from
  existing facts and axioms (e.g. 1gt 0 gt 2 gt 1)
• So, the purpose of a proof system is to give us a
  methodology of deriving true statements that are
  guaranteed to be true that is, constructing
  proofs. ?

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Proof What is it?

• Since we know what a proof system is, defining
  the meaning of a proof is easy
• A proof is a finite list of true statements
• starting with the axioms or proven facts (aka
  theorems) and ending at the true statement that
  one wants to derive
• In such a way that a new true statement is
  derived from some set of previously proven true
  statements using a rule of inference

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Proof System An Example

• Axioms
• 1 gt 0 (1)
• If X gt Y, then X 1 gt Y 1 (2)
• Rules of Inference
• Modus Ponens (MP)
• Modus Tollens (the inverse of MP)
• Prove that 3 gt 2.
• Proof
• 1. 1 gt 0 (Axiom 1)
• 2. If 1 gt 0, then 2 gt 1 (Axiom 2, X1, Y0)
• 3. 2 gt 1 (By MP applied on fact 1, 2)
• 4. If 2 gt 1, then 3 gt 2 (Axiom 2, X2, Y1)
• 5. 3 gt 2 Done! ? (MP applied on fact 3, 4)

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Proof System In Real Life

• Actually, all the proofs that we did in our
  calculus classes implicitly relied on all the
  axioms in set theory and in Peano arithmetic (aka
  Peano axioms) AND the rule of inference used was
  Modus Ponens
• Just that your teacher might not have told you
  about all these things in the background
• One reason is probably because they were
  developed in the 19th century or earlier ?

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Soundness and Completeness of a Proof System

• Soundness if the system proves that something is
  true, then it really is true
• The system doesnt derive contradictions
• Completeness if something is really true, it can
  be proven using the system
• The system can be used to derive all the true
  mathematical statements one by one

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What is a Theorem Prover?

• A program that can decide the validity of a
  certain statement using a given proof system

Note Depending on the type of proof system used,
whether S is false can be undecidable. If so, the
proof system is semi-decidable and the theorem
prover will not stop running on a false statement
X.
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Soundness and Completeness of Theorem Provers

• Suppose we have a theorem prover P
• Soundness If P proved that some statement S is
  true, then S is really a true statement
• Completeness If a statement S is true, that S
  can be proven to be true by P

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Resolution Theorem Proving

• Resolution is a rule of inference sound and
  complete
• Developed in 1965 by J.A. Robinson
• This single rule of inference can be used to
  construct a sound and complete theorem prover
  when used with a complete search algorithm
  provided that the proof system is also sound and
  complete

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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
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Resolution Theorem Proving Block World Example

• Q Can we implement resolution theorem
• proving as a search procedure?
• A Yes, we started with a bunch of
• statements and ended in an empty set

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Treating Resolution Theorem Proving as a Search
Problem

• Initial state the set of clauses G K U s in
  CNF
• K the set of facts about a domain
• s negation of the conclusion s that we want to
  prove given K
• Goal State
• The empty clause contradiction
• Reminder deriving a contradiction in a proof by
  refutation means that s has to be true since
  otherwise we will have a contradiction
• Operator the resolution procedure that produces
  a resolvent R given two clauses C1, C2 from G
• G (G - C1, C2) U R

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Treating Resolution Theorem Proving as a Search
Problem (Contd)

• Since theorem proving can be viewed as a search
  problem, we need a search strategy!
• One naïve approach is to use randomized search
• Randomly picked a pair of clauses X, Y from the
  set G
• If resolve(X,Y) can produce a resolvent R, take
  away X, Y from G and add R to G. Otherwise,
  repeat 1) until a resolvent can be produced
• Repeat steps 1) to 2) until G

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Treating Resolution Theorem Proving as a Search
Problem (Contd)

• Since naïve approach doesnt work well, we need
• to ask two questions to make a feasible search
• algorithm
• Q1 Which pair of clauses should be resolved
  first?
• We need a search strategy
• Q2 What is the search tree? (Hint a tree
  implies there are levels in the tree)
• 0th level resolvents all the original clauses
  (the set K) plus the clausal form of the negation
  of the WFF to be proved
• (i1)th level resolvents any resolvent of an ith
  level resolvent, plus a jth level resolvent where
  j lt i

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Treating Resolution Theorem Proving as a Search
Problem (Contd)

• Uninformed search
• Breadth first search
• Generates one level of resolvents before another
  until the empty clause is found
• Depth first search
• Generates a resolvent at level i, then a
  resolvent at level i1, using resolvents from the
  current and/or previous levels backtrack if
  search is stuck. Stops when the empty clause is
  found.

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Treating Resolution Theorem Proving as a Search
Problem (Contd)

• Informed search use heuristic to guide the
• selection of a pair of clauses for resolution
• Unit-preference strategy
• We prefer to first resolve clauses with a single
  literal
• Linear-input strategy
• Only allows those resolutions in which one of the
  clauses being resolved is either
• a member of the original set of clauses (set K)
• the negation of the target clause to be proved
  (s)

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Example of Linear Input Strategy
S
K
resolvents
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Conclusion

• A proof system allows us to construct proofs in a
  systematic way
• A theorem prover can be used to decide if a
  statement is true or not if the proof system is
  decidable
• Resolution is a rule of inference we use it to
  derive new facts (just like Modus Ponens)
• A resolution theorem prover uses resolution as
  the inference procedure and it can be implemented
  as a search algorithm
• This implies that theorem proving can be viewed
  as a search problem