macario-trujillo
Uploaded by
40 SLIDES
530 VUES
400LIKES

CS621: Artificial Intelligence

DESCRIPTION

This lecture by Pushpak Bhattacharyya from IIT Bombay delves into the fundamental concepts of soundness, completeness, and consistency in propositional logic. It covers the definitions and proofs of soundness and completeness, illustrating their significance in the context of semantic and syntactic worlds. The discussion includes truth tables, tautologies, and essential theorems that demonstrate the relationship between provability and validity. This comprehensive exploration provides valuable insights for anyone studying artificial intelligence and formal logic.

1 / 40

Télécharger la présentation

CS621: Artificial Intelligence

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CS621: Artificial Intelligence Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture 11- Soundness and Completeness; proof of soundness; start of proof of completeness 12th august, 2010

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

  3. 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) Syntactic ‘false Semantic ‘false’

  4. 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

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

  6. Soundness • Provability Validity • Completeness • Validity Provability

  7. Soundness:Correctness of the System • Proved entities are indeed valid • Completeness:Power of the System • Valid things are indeed provable

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

  9. Examine the relation between Soundness & Consistency Soundness Consistency

  10. 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

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

  12. 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

  13. Problem (P Q)(P Q) Semantic Proof A B P Q P Q P Q AB T F F T T T T T T T F F F F T F T F T T

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

  15. 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

  16. 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

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

  18. Statement Given A1, A2, … ,An |- B V(B) is ‘T’ for all Vs for which V(Ai) = T

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

  20. Case 2 B is one of Ais if V(Ai) = T, so is V(B) statement is correct

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

  22. i.e. Ei/Ej 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

  23. Towards Completeness Proof

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

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

  26. Necessary results Statement: (pq)((~pq)q) Proof: If we can show that (pq), (~pq) |- q Or, (pq), (~pq), qF |- F Then we are done.

  27. Proof continued 1. (pq) H1 2. (~pq) H2 3. qF H3 4. (~pq) (~qp) theorem of contraposition 5. ~qp MP, 2, 4 6. P MP, 3,5 7. q MP, 6, 1 8. F MP,7,3 QED

  28. How to prove contraposition To show (pq)(~q~p) Proof: pq, ~q, p |- F Very obvious!

  29. An example to illustrate the completeness proof

  30. Running the completeness proof For every row of the truth table set up a proof: • p, ~q |- p(p V q) • p, q |- p(p V q) • ~p, q |- p(p V q) • ~p, ~q |- p(p V q)

  31. p, ~q |- p  (p V q) i.e. p, ~q, p |- p V q p, ~q, p, ~p |- q p, ~q, p, ~p |- F |- F  q |- q

  32. p, q |- p  (p V q) i.e. p, q, p, ~p |- q same as 1

  33. ~p, q |- p  (p V q) ~p, q, p, ~p |- q Same as 1, since F is derived 4. ~p, ~q |- p  (p V q) Same as 1, since F is derived

  34. Why all this? If we have shown p, q |- A and p, ~q |- A then we can show that p |- A

  35. p |- (q  A) also p |- (~q  A) But (q  A)  ((~q  A)  A) is a theorem by MP twice p |- A

  36. General Statement of the completeness proof If V(A) = T for all models then |- A

  37. Elaborating, If P1, P2, …, Pn are constituent propositions of A and if V(A) = T for every model V(Pi) = T/F then |- A

  38. We have a truth table with 2n rows P1 P2 P3 . . . Pn A F F F . . . F T F F F . . . T T . . . T T T . . . T T

  39. If we can show P1’, P2’, …, Pn’ |- A’ For every row where Pi’ = Pi if V(Pi) = T = ~Pi if V(Pi) = F And A’ = A if V(A) = T = ~A if V(A) = F

  40. Lemma If row has P1’, P2’, …, Pn’, A’ Then P1’, P2’, …, Pn’ |- A’ A very critical result linking syntax with semantics

More Related