Prepositional Logic

1
Prepositional Logic

• Lecture 5

2
Logical Equivalence

• A stronger form of implication is expressed by
  the statement
• B is TRUE if and only if A is TRUE. This is
  usually written as A ?? B and is called logical
  equivalence
• The double arrow signifies that A ?? B is the
  same as
• A?B AND B?A
• A??B (NOT A OR B) AND (NOT B OR A)

3
Proof

• The proposition that is to be proved Y, is called
  the conclusion and the propositions that are
  taken to be TRUE X1, X2, X3,.,XN are called
  premises
• In principle all you have to do to prove Y is to
  use the two rules of inference modus ponens and
  the chain rule, to deduce new prepositions from
  the premises until you produce the conclusion Y
• (A AND B) ? ((C AND D) ?(E AND F))
• (E AND F) ? (NOT F OR G)
• A AND B
• Prove that
• (C AND D) ? NOT (F AND NOT G)

4
Proof

• Using Modus Ponens with 1 and 3 we can deduce
• (C AND D) ?(E AND F)
• Which is a new true preposition not contained in
  the premises, and using the chain rule with 2 and
  4 we can deduce
• (C AND D) ?(NOT F OR G)
• And finally, using DeMorgans Law on
• (NOT F OR G) gives
• (C AND D) ? NOT(F AND NOT G) which is the desired
  conclusion

5
Human or Automated

• When a human tries to prove something using
  prepositional logic a variety of vague hunches
  and inspirations are used to guide the process of
  deduction
• This is clearly going to be a difficult process
  to automate unless we find or force some sort of
  regularity on the way the premises are used to
  deduce the conclusion

6
Resolution

• The use of two rules of inference, modus ponens
  and the chain rule, is a complicating factor in
  the automation of proof using prepositional
  logic.
• It is possible to combine modus ponens and the
  chain rule into a single inference rule
  resolution-that also suggests the use of a
  standard form for all compound prepositions

7
Resolution

• As
• A?B can be written as
• NOT A OR B
• And the chain rule
• From A?B and B?C deduce A?C can be written as
• From NOT A OR B
• AND NOT B OR C
• Deduce NOT A OR C
• And this can be thought of as cancelling the
  terms B and NOT B
• This cancellation of terms like B and NOT B
  between two compound prepositions to produce a
  third that does not involve B is called resolution

8
Conjunctive Normal Form

• Resolution is an inference rule that would be
  easy for a computer to use if all the
  prepositions that constituted the premises were
  simple prepositions or their negation Ored
  together
• That is each preposition should be something like
  
• (A OR B OR NOT C OR D)
• This can be achieved by using the Conjunctive
  Normal Form

9
CNF

• It can be proved, using Boolean Algebra, that any
  compound preposition can be written as the AND of
  a number of sub prepositions called clauses each
  one being the OR of a number of terms for example
  
• (A OR B) AND (NOT C OR D) AND (E OR NOT F OR G)

10
Rules that can be used to convert any preposition
to CNF

• 1. Remove equivalences of the form A??B by
  writing them as A ? B AND B?A
• 2. Remove implications of the form A ?B by
  writing them as NOT A OR B
• 3. Move NOTs inside brackets using De. Morgans
  Laws e.g
• Change NOT(A OR B) into (NOT A AND NOT B)
• 4. Distribute Ors over ANDs e.g change A OR (B
  AND C) into (A OR B) AND (A OR C)

11
Example

• (-P ?? Q) ?(P AND (Q OR R))
• (--P OR Q) AND (-Q OR P) ?(P AND(Q OR R))
• -((--P OR Q) AND (-Q OR P)) OR (P AND(Q OR R))
• -(P OR Q) OR (-Q OR P) OR (P AND (Q OR R))
• (-P AND -Q) OR (Q AND P) OR(P AND (Q OR R))
• (-P AND -Q) OR (Q AND P) OR (P AND Q) OR (P AND R)

12
Logic Normal Forms

• Automatic Theorem proving, logic programming and
  knowledge representation, are often the aims for
  maximum uniformity and standardization in the the
  syntax, avoiding the full syntactic variety of
  prepositional and predicate calculus
• The main reason for this is that the more variety
  there is in the syntax the more inference rules
  you need. If one can reduce the complexity of the
  syntax in terms of the number of connectives, the
  degree of embedding and the significance of
  order, then corresponding reductions in the
  complexity of the inference rules are possible.

13
Normal Forms

• Such reductions can also lead to a reduction in
  the size of the resultant search space
• The three main syntactic schemes employed are
  conjunctive normal form (CNF), full clausal form
  and the Horn clause
• In the three it is customary to reduce the
  nesting of parenthesis by making AND and OR of
  variable arity I.e allowing AND and OR to govern
  any number of operands e.g
• (P OR (Q OR R)) (P OR Q OR R)
• (P AND (Q AND R)) (P AND Q AND R)

14
CNF An other Example

• -(p OR Q) ?(-P AND Q)
• (p?(q?r)) ?((P AND S) ?R)
• Clausal Form is very similar to CNF, except that
  the positive and negative literals in each
  disjunction are grouped together on different
  side of an arrow and the negation is dropped.
  Thus

15
Clausal Form

• (-P OR (P OR Q )) AND (-Q OR (P OR Q))
• (-P, P, Q) , (-Q, P, Q)
• Will be
• PQ?P
• PQ?Q
• Atoms on the left hand side of the arrow are
  implicitly disjoined while on the right hand side
  are implicitly conjoined

16
The Horn Clause Subset

• The Horn clause subset is just like the full
  clausal form, except that only one atom (at most)
  is allowed on the left hand side.
• Thus the Horn Clause equivalent of a rule will
  have the general form
• P?Q1 . . Qn
• Where Q are implicitly conjoined
• Write this as
• P- Q1,,Qn and you have the syntax of the PROLOG
  Programming language

17
Proof using resolution

• Using conjunctive normal form and resolution we
  have the beginnings of a proof strategy that can
  be automated.
• All we have to do is to convert all the
  prepositions in the premises to clause form and
  then apply resolution until we generate the
  conclusion
• There is still the possibility that it could take
  a long time to generate the conclusion by wild
  clause generation but if the conclusion can be
  deduced from the premises then this method will
  eventually find it

18
Reduction and Absurdum

• There is one slight improvement that we can make
  to this proof by resolution procedure and that is
  to use reduction and absurdum
• This involves adding the NOT of the conclusion to
  the premises that is denying the conclusion and
  trying to deduce a contradiction.

19
Null Clause

• If you are trying to prove A then add NOT A to
  the premises
• If A can be deduced from the premises then at
  some point in the resolution we will obtain the
  clauses A and NOT A and these can be resolved
  together to produce a clause with nothing in it
  the Null Clause

20
Advantage of Null Clause

• The advantage of this method is that if you add
  the NOT of the conclusion to the premises then no
  matter how you arrive at the null clause this can
  be taken as an indication that one of the
  premises was in fact FALSE, and as the only
  premise that is, in any doubt is the NOT of the
  conclusion you can deduce that that is the one
  that is FALSE. So the conclusion is indeed TRUE.

21
Proof By Reduction Example

• As an Example of Proof by reduction Consider the
  following
• Given
• A AND B
• A?C
• B?D as Premises
• Prove C AND D

22
Solution

• The first job is to convert the premises into
  C.N.F and then into separate clauses.
• A AND B Is already in CNF form and gives us 2
  separate clauses A, B
• A?C NOT A OR C
• B?D NOT B OR D
• Which gives

23
Example

• 1. A
• 2. B
• 3. NOT A OR C
• 4. NOT B OR D (The Second step is to negate the
  conclusion, convert it into clause form and add
  it to the premises)
• 5. NOT C OR NOT D
• The final step is to resolve clauses together
  until the null clause is produced
• 6. NOT C OR NOT B by resolving 5,4
• 7. NOT A OR NOT B by resolving 6,3
• 8. NOT A by resolving 7,2
• 9. NULL by resolving 8,1