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Logic in Computer Science
Chapter 3
Propositional Logic
3.6. Propositional Resolution
Transparency No. 3.3-1
Logic in Computer Science
Ch 3. Propositional Logic
Propositional Resolution (another Propositinal calculus)
 a single rule of inference:
P => Q , Q => R
-------------------------P => R
 Using propositional resolution alone (without axiom schemata
or other rules of inference), it is possible to build a theorem
prover that is sound and complete for All propositional
Calculus .
 The search space using propositional resolution is much
smaller than for standard propositional calculus (like H).
 works only on expressions in clausal form.
 General procedure to convert wffs into clausal forms
introduced before.
Transparency No. 3.1-2
Logic in Computer Science
Ch 3. Propositional Logic
Clausal Form revisited
 A literal is either an atomic sentence or a negation of an
atomic sentence.
p
-p
 A clause is either a literal or a disjunction of literals.
p
-p
p \/ q \/ r
 Clauses are usually written as sets of literals.
{ p} ,
{ -p} ,
{ p,q,r }
 A database in clausal form is a set of clauses.
{{-p,q},{r,-q}}
Transparency No. 3.1-3
Empty Sets
Logic in Computer Science
Ch 3. Propositional Logic
 The empty clause {} is unsatisfiable.
 Why: It is equivalent to an empty disjunction.
 The empty database {} is valid.

Why: It is equivalent to an empty conjunction.
 What about a database consisting of an empty clause {{}}?
Transparency No. 3.1-4
Conversion to Clausal Form
Logic in Computer Science
Ch 3. Propositional Logic
 Implications out:
P1 -> P2 ---> ~P1 \/ P2
P1 <-> P2 ---> (~P1 \/ P2) /\ (P1 \/ ~P2)
 Negations in:
~~P
---> P
~(P1 /\ P2) ---> ~P1 \/ ~P2
~(P1 \/ P2) ---> ~P1 /\ ~P2
 Disjunctions in:
P1 \/ (P2 /\ P3) ---> (P1 \/ P2) /\ (P1\/ P3)
 Operators out:
P1 \/ P2
P1 /\ P2
--->
--->
{P1,P2}
{P1} , {P2}
Transparency No. 3.1-5
Logic in Computer Science
Ch 3. Propositional Logic
Examples
 Example 1:




I (g & (~r | f))
N
D
O { g} , {~r,f}
 Example 2:






(g & (r -> f))
~(g & (r -> f))
I ~(g & (~r | f))
N (~g | ~(~r | f))
(~g | (~~r & ~f))
(~g | (r & ~f))
D (~g | r) & (~g | ~f)
O {~g, r} , {~g, ~f}
Transparency No. 3.1-6
Propositional Resolution
Logic in Computer Science
Ch 3. Propositional Logic
 General form:




{p1,..., r,...,pm}
{q1,...,~r,...,qn}
--------------------{p1,...,pm,q1,...,qn}
 Example:




{Office, Home}
{~Home,Sick}
---------------------{Office,Sick}
Transparency No. 3.1-7
Logic in Computer Science
Ch 3. Propositional Logic
Issues
 Collapse:




{~p,q}
{p,q}
-----------{q}
 Multiple Conclusions:





{p,q}
{~p,~q}
------------{p,~p}
{q,~q}
Single Application Only:
{p,q}
{~p,~q}
------------{}
NO!
Transparency No. 3.1-8
Special Cases of Propositional Resolution
Logic in Computer Science
Ch 3. Propositional Logic
 Modus Ponens:




p => q
p
---------q
{-p,q}
{p}
----------------{q}
 Modus Tollens:




p => q
-q
----------p
Chaining:
{-p,q}
{-q}
-----------{-p}
p => q
q => r
---------p => r
{-p,q}
{-q,r}
------------{-p,r}
Transparency No. 3.1-9
Unsatisfiability
 Example:
1. {p,q}
2. {-p,q}
3. {p,-q}
4. {-p,-q}
5. {q}
6. {-q}
7. {}
Logic in Computer Science
Ch 3. Propositional Logic
premise
premise
premise
premise
1,2
3,4
5,6
Transparency No. 3.1-10
True or False Questions
Logic in Computer Science
Ch 3. Propositional Logic
 Theorem: T |= A iff T U {~A} is unsatisfiable.
 Application: To determine whether a set T of sentences
logically implies a sentence A, rewrite T U {-s} in clausal form
and try to derive the empty clause.
Transparency No. 3.1-11
True or False Example
Logic in Computer Science
Ch 3. Propositional Logic
 {p -> q, q -> r} |= p -> r ?



1.
2.
3.
4.
5.
6.
7.
p -> q ==> {-p,q}
q -> r ==> {-q,r}
-(p -> r) ==> {p} , {-r}
{~p,q} premise
{~q,r} premise
{p}
negated goal
{~r}
negated goal
{q}
1,3
{r}
2,5
{}
4,6
Transparency No. 3.1-12
Logic in Computer Science
Ch 3. Propositional Logic
True or False Example
 {p -> q,m -> p \/ q} |= m -> q?
 p -> q
==>
 m -> p \/ q ==>
 ~ (m -> q) ==>
1.
2.
3.
4.
5.
6.
7.
{~p,q}
{~m,p,q}
{m} , {~q}
{~p,q} premise
{~m,p,q} premise
{m}
negated goal
{~q}
negated goal
{p,q} 2,3
{q}
1,5
{}
4,6
Transparency No. 3.1-13
Incompleteness?
Logic in Computer Science
Ch 3. Propositional Logic
 Theorem: Propositional Resolution is not generatively
complete. I.e., the statement:
“If T |= A then there is a proof of A from T using Propositional
Resolution only.”
is not true.
Pf: Show that {} |= p => (q => p) using propositional resolution.
Note since there are no premises, there are no conclusions
that can be generated. QED
 However it can be show that Propositional Resolution is
refutation complete. Namely, if T is unsatisfiable, then there is
a resolution proof of {} (contradiction) from T. Hence
 T |= A iff T U {~A} is unsatisfiable iff there is a resolution proof
of {} from T U {~A}.
Transparency No. 3.1-14
Resolution proof of {} |= p->(q->p)
Logic in Computer Science
Ch 3. Propositional Logic
I ~(p -> (q -> p)) : ~A
N ~(~p \/ (~q \/ p))
(~~p /\ ~(~q \/ p))
(~~p /\ (~~q /\ ~p))
(p /\ (q /\ ~p))
D
O {p}, {q}, {-p}
1.
2.
3.
4.
{p} negated goal
{q} negated goal
{-p} negated goal
{} 1,3
Transparency No. 3.1-15
Resolution Provability
Logic in Computer Science
Ch 3. Propositional Logic
Definition: A sentence A is provable from a set of sentences T
by propositional resolution (written T |-p A) iff there is a
derivation of the empty clause from the clausal from of T U
{~A} using resolution rule only.
 Soundness and Completeness of Resolution:
 Soundness Theorem: Proposition resolution is sound.
T |-r A => T |= A
 Completeness Theorem: Our proof system is complete.
T |= A => T |-r A
Transparency No. 3.1-16
Logic in Computer Science
Ch 3. Propositional Logic
Termination
 Two finger method:











1. {P,Q}
2. {-P,Q}
3. {P,-Q}
4. {-P,-Q}
5. {Q}
6. {P}
7. {Q,-Q}
8. {P,-P}
9. {-P}
10. {-Q}
11. {}
Premise
Premise
Premise
Premise
1,2
1,3
1,4; 2,3
1,4; 2,3
2,4
3,4
6,9
Theorem: There is a resolution
derivation of a conclusion from a set
of premises if and only if there is a
derivation using the two finger
method.
Theorem: Propositional resolution
using the two-finger method always
terminates.
Proof: There are only finitely many
clauses that can be constructed from
a finite set of logical constants.
Transparency No. 3.1-17