Download presentation

1
**Chapter 3 Propositional Logic**

3.6. Propositional Resolution

2
**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.

3
**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 \/ 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}}

4
**The empty clause {} is unsatisfiable. The empty database {} is valid. **

Empty Sets 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 {{}}?

5
**Conversion to Clausal Form**

Implications out: P1 -> P > ~P1 \/ P2 P1 <-> P > (~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 \/ P > {P1,P2} P1 /\ P > {P1} , {P2}

6
**Example 1: (g & (r -> f))**

Examples Example 1: (g & (r -> f)) I (g & (~r | f)) N D O { g} , {~r,f} Example 2: ~(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}

7
**Propositional Resolution**

General form: {p1,..., r,...,pm} {q1,...,~r,...,qn} {p1,...,pm,q1,...,qn} Example: {Office, Home} {~Home,Sick} {Office,Sick}

8
**Multiple Conclusions:**

Issues Collapse: {~p,q} {p,q} {q} Multiple Conclusions: {~p,~q} {p,~p} {q,~q} Single Application Only: {p,q} {~p,~q} {} NO!

9
**Special Cases of Propositional Resolution**

Modus Ponens: p => q {-p,q} p {p} q {q} Modus Tollens: -q {-q} -p {-p} Chaining: p => q {-p,q} q => r {-q,r} p => r {-p,r}

10
**Example: 1. {p,q} premise 2. {-p,q} premise 3. {p,-q} premise**

Unsatisfiability Example: 1. {p,q} premise 2. {-p,q} premise 3. {p,-q} premise 4. {-p,-q} premise 5. {q} ,2 6. {-q} 3,4 7. {} ,6

11
**True or False Questions**

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.

12
**{p -> q, q -> r} |= p -> r ?**

True or False Example {p -> q, q -> r} |= p -> r ? p -> q ==> {-p,q} q -> r ==> {-q,r} -(p -> r) ==> {p} , {-r} 1. {~p,q} premise 2. {~q,r} premise 3. {p} negated goal 4. {~r} negated goal 5. {q} ,3 6. {r} ,5 7. {} ,6

13
**{p -> q,m -> p \/ q} |= m -> q?**

True or False Example {p -> q,m -> p \/ q} |= m -> q? p -> q ==> {~p,q} m -> p \/ q ==> {~m,p,q} ~ (m -> q) ==> {m} , {~q} 1. {~p,q} premise 2. {~m,p,q} premise 3. {m} negated goal 4. {~q} negated goal 5. {p,q} 2,3 6. {q} ,5 7. {} ,6

14
Incompleteness? 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}.

15
**Resolution proof of {} |= p->(q->p)**

I ~(p -> (q -> p)) : ~A N ~(~p \/ (~q \/ p)) (~~p /\ ~(~q \/ p)) (~~p /\ (~~q /\ ~p)) (p /\ (q /\ ~p)) D O {p}, {q}, {-p} 1. {p} negated goal 2. {q} negated goal 3. {-p} negated goal 4. {} 1,3

16
**Resolution Provability**

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

17
**Two finger method: 1. {P,Q} Premise**

Termination Two finger method: 1. {P,Q} Premise 2. {-P,Q} Premise 3. {P,-Q} Premise 4. {-P,-Q} Premise 5. {Q} ,2 6. {P} ,3 7. {Q,-Q} 1,4; 2,3 8. {P,-P} 1,4; 2,3 9. {-P} ,4 10. {-Q} ,4 11. {} ,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.

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google