CS621: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–10: Soundness of Propositional Calculus 12 th August, 2010

Soundness, Completeness & Consistency Syntactic World Theorems, Proofs Semantic World Valuation, Tautology Soundness Completeness * *

Soundness Provability Truth Completeness TruthProvability

Soundness: Correctness of the System Proved entities are indeed true/valid Completeness: Power of the System True things are indeed provable

TRUE Expression s System Outside Knowledge Validation

Consistency The System should not be able to prove both P and ~P, i.e., should not be able to derive F

Examine the relation between Soundness & Consistency SoundnessConsistency

If a System is inconsistent, i.e., can derive F, it can prove any expression to be a theorem. Because F  P is a theorem

Inconsistency  Unsoundness To show that F  P is a theorem Observe that F, P  F ⊢ F By D.T. F ⊢ (P  F )  F; A3 ⊢ P i.e. ⊢ F  P Thus, inconsistency implies unsoundness

Unsoundness  Inconsistency Suppose we make the Hilbert System of propositional calculus unsound by introducing (A /\ B) as an axiom Now AND can be written as (A  (B  F ))  F If we assign F to A, we have ( F  (B  F ))  F But ( F  (B  F )) is an axiom (A1) Hence F is derived

Inconsistency is a Serious issue. Informal Statement of Godel Theorem: If a sufficiently powerful system is complete it is inconsistent. Sufficiently powerful: Can capture at least Peano Arithmetic

Introduce Semantics in Propositional logic Valuation Function V Definition of V V( F ) = F Where F is called ‘false’ and is one of the two symbols (T, F) Semantic ‘false’ Syntactic ‘false

V( F ) = F V(A  B) is defined through what is called the truth table V(A)V(B)V(A  B) T F F T T T F F T F T T

Tautology An expression ‘E’ is a tautology if V(E) = T for all valuations of constituent propositions Each ‘valuation’ is called a ‘model’.

To see that ( F  P) is a tautology two models V(P) = T V(P) = F V( F  P) = T for both

F  P is a theorem F  P is a tautology SoundnessCompleteness

If a system is Sound & Complete, it does not matter how you “Prove” or “show the validity” Take the Syntactic Path or the Semantic Path

Syntax vs. Semantics issue Refers to FORM VS. CONTENT Tea (Content) Form

Form & Content Godel, Escher, Bach By D. Hofstadter logician painter musician

Problem (PQ)  (PQ) Semantic Proof A B PQP QP QA  B TF F T T TT T T T FF F F T FT F T T

To show syntactically (PQ)(PQ) i.e. [(P (Q F )) F ] [(P F ) Q]

If we can establish (P(Q F )) F, (P F ), Q F ⊢ F This is shown as Q F hypothesis (Q F ) (P (Q F)) A1

Q  F; hypothesis (Q  F)  (P  (Q  F)); A1 P  (Q  F); MP F; MP Thus we have a proof of the line we started with

Soundness Proof Hilbert Formalization of Propositional Calculus is sound. “Whatever is provable is valid”

Statement Given A 1, A 2, …,A n |- B V(B) is ‘T’ for all V s for which V(A i ) = T

Proof Case 1B is an axiom V(B) = T by actual observation Statement is correct

Case 2B is one of A i s if V(A i ) = T, so is V(B) statement is correct

Case 3B is the result of MP on E i & E j E j is E i B Suppose V(B) = F Then either V(E i ) = F or V(E j ) = F...Ei...Ej...B...Ei...Ej...B

i.e.E i /E j is result of MP of two expressions coming before them Thus we progressively deal with shorter and shorter proof body. Ultimately we hit an axiom/hypothesis. Hence V(B) = T Soundness proved

A puzzle (Zohar Manna, Mathematical Theory of Computation, 1974) From Propositional Calculus

Tourist in a country of truth- sayers and liers Facts and Rules: In a certain country, people either always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions. Question: What single yes/no question can T ask of S, so that the direction of the capital is revealed?

Diagrammatic representation S (either always says the truth Or always lies) T (tourist) Capital

Deciding the Propositions: a very difficult step- needs human intelligence P: Left road leads to capital Q: S always speaks the truth

Meta Question: What question should the tourist ask The form of the question Very difficult: needs human intelligence The tourist should ask Is R true? The answer is “yes” if and only if the left road leads to the capital The structure of R to be found as a function of P and Q

A more mechanical part: use of truth table PQS’s Answer R TTYesT TF F FTNoF FF T

Get form of R: quite mechanical From the truth table R is of the form (P x-nor Q) or (P ≡ Q)

Get R in English/Hindi/Hebrew… Natural Language Generation: non-trivial The question the tourist will ask is Is it true that the left road leads to the capital if and only if you speak the truth? Exercise: A more well known form of this question asked by the tourist uses the X-OR operator instead of the X-Nor. What changes do you have to incorporate to the solution, to get that answer?