Transcript Predicate Logic (Resolution)

Propositional Logic Resolution
Dr. Rogelio Dávila Pérez
Profesor-Investigador
División de Posgrado
Universidad Autónoma Guadalajara
[email protected]
Propositional logic resolution
Conjunctive normal form: any formula of the predicate calculus
can be transformed into a conjunctive normal form.
Def. A formula is said to be in conjunctive normal form if it
consists in the conjunction of clauses.
A1  A2  …  An where Ai is a clause.
Def. A formula is said to be a clause if it consists in a disjunction of
literals. A clause has the following form:
L1 v L2 v … v Lm where Li is a literal.
Def. A literal is an atomic formula or the negation of an atomic
formula.
Def. A formula is said to be in clausal form if it can be expressed as a set
of clauses:
{C1 , … , Cn,}
where Ci is a clause
Propositional logic resolution
Transforming into clausal form
1. Eliminate implication symbols (), using the identity:
  v
2. Introduce negation: reduce scopes of negation symbols by repeatedly
applying the De Morgan rules:
(i)  ( v )      
(ii)  (  )    v  
3. Put matrix in conjunctive normal form by repeatedly applying the
distributive laws:
(i)  v (  )  ( v )  ( v )
(ii)   ( v )  (  ) v (  )
4. Eliminate conjunction () symbols separating the expression in clauses.
Propositional logic resolution
Resolution refutation procedure
In general a resolution refutation for proving an arbitrary wff
from a set of wffs , 
, proceeds as follows:

1. Convert the wffs in  to clausal form.
2. Negate the formula  to be proved and convert the result to
clausal form.
3. Combine the clauses resulting form steps 1 and 2 into a single
set, .
4. Iteratively apply resolution to the clauses in  and add the
results to  either until there are no more resolvents that can
be added or until the empty clause is produced.
Propositional logic resolution
Important results

Completeness of resolution refutation: the empty clause will be
produced by the resolution refutation procedure if  |= 
thus we say that propositional resolution is refutation complete.

Decidibility of propositional calculus by resolution refutation: if
 is a finite set of clauses and if  |  then the resolution
refutation procedure will terminate without producing the empty
clause.
Propositional logic resolution
Exercises
1. Transform into clausal form the following wff:
~[((p v ~q) r)  (p q)]
2. Prove using resolution refutation the axioms of the propositional
logic.
a. Implication introduction:
p  (q  p)
b. Implication distribution:
(p  (q  r))  ((p  q)  (p  r))
c. Contradiction realization:
(q  ~p)  ((q  p)  ~q)