Download presentation

Presentation is loading. Please wait.

Published byAaliyah Bailey Modified over 3 years ago

1
Some important properties Lectures of Prof. Doron Peled, Bar Ilan University

2
We repeat some notation and definitions. An assignment z satisfies a formula if M PL (, z)=1 |== means that logically implies, i.e., each assignment that satisfies all the formulas in also satisfies. |-- means that we can prove from, i.e., there is a proof sequence that uses assumptions from, the axioms A1, A2, A3 and the proof rule MP, and ends with.

3
Substitution of variables by formulas Let be a WFF. A function s:Var WFF is called a substitution function. We define inductively a function subst(,s) that replaces in all the variables p i by s(p i ). Basis: subst(p i,s)=s(p i ), subst(F,s)=F, subst(T,s)=T. Closure: subst(( 1 /\ 2 ),s)=(subst( 1,s)/\subst( 2,s)) subst(( 1 \/ 2 ),s)=(subst( 1,s) subst( 2,s)) Etc.

4
For example s={, } subst((p 1 \/p 2 ),s)=(subst(p 1,s)\/subst(p 2,s))= ((p 3 \/p 4 )\/(p 5 \/p 6 )) Finite dependency of substitution: Let be a WFF with all its propositions inside {p 1,p 2,…,p n }. Let s 1 and s 2 be two substitution functions such that for each i, s 1 (p i )=s 2 (p i ). Then subst(,s 1 )=subst(,s 2 ).

5
Some propositions If is not satisfiable and s is a substitution, then subst(,s) is not satisfiable. If is a tautology and s is a substitution, then subst(,s) is a tautology.

6
Provability and satisifiability Lemma(*): |== iff { F} is unsatisfiable. One direction: from |==, any assignment that satisfies must also satisfy, thus cannot satisfy { F}. Conversely, if { F} is unsatisfiable, any assignment that satisfies cannot satisfy { F}, hence it must satisfy.

7
Consistency A set of formulas is consistent if |-/-F. A set is maximally consistent if it is consistent and for every formula, either or ( F) belongs to the set.

8
Consistency and satisfiability inconsistent unsatisfiable (=soundness!) Can be proved from soundness: if we can prove F from then since our proof system is sound, F logically follows from. Thus there cannot be a satisfying assignment to our assumptions. This implies consistency, as will be shown later. consistent satisfiable (=completeness!) The contraposition is implied by completeness: If is unsatisfiable then |==F. By the completeness theorem, |--F, i.e., is inconsistent. This implies completeness as will be shown later.

9
Landscape Consistent Satisfiable Soundness Completeness

10
Proving that consistent satisfiable Same as completeness theorem: start with and extend into * by enumerating the formulas in some order and adding a formula if the set remains consistent. Then * behaves as a truth assignment and thus gives a satisfying assignment to.

11
consistent satisfiable implies completeness! We showed that every consistent set is satisfiable. Lemma: If |-/- then { F} is consistent Proof: We show the contraposition: If { F} is inconsistent, then |--. If so, then { F}|-- (since from F we can prove anything). But we can also prove from { }. Thus, by proof by cases, we have |--. Now assume |== but |-/-. Then according to the Lemma above { F} is consistent. But according to lemma (*), { F} is not satisfiable. Contradiction.

12
Compactness theorem Let be an infinite set of propositional formulas. Then is satisfiable iff every finite subset of is satisfiable. Proof: as shown before, is satisfiable iff is consistent. If is consistent, then every finite subset of it is consistent. Conversely, if is inconsistent, there is a finite subset of it that proves F, hence is inconsistent.

13
inconsistent unsatisfiable implies consistency Lemma: If |-- then { F} is inconsistent. Proof: If |-- it follows that from { F} one can prove both and { F}, hence one can prove F, which makes { F} inconsistent. Now suppose that |--. Then { F} is inconsistent. Thus, { F} is unsatisfiable. Therefore from Lemma (*) |==.

Similar presentations

OK

Rosen 1.6. Approaches to Proofs Membership tables (similar to truth tables) Convert to a problem in propositional logic, prove, then convert back Use.

Rosen 1.6. Approaches to Proofs Membership tables (similar to truth tables) Convert to a problem in propositional logic, prove, then convert back Use.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ethanol production process Business ppt on flipkart Ppt on forward rate agreement pricing Ppt on obstructive sleep apnea By appt only movie Stereoscopic image display ppt online Ppt on remote controlled screw jack Ppt on sentence completion test Ppt on human chromosomes 46 Ppt on principles of peace building