1. cs466(Prasad)L6Proof1 Induction Proofs Equivalent Statements in Proofs

2. cs466(Prasad)L6Proof2

3. cs466(Prasad)L6Proof3 q p

4. cs466(Prasad)L6Proof4 Language Equivalence Proofs

5. cs466(Prasad)L6Proof5 Soundness: All strings derivable in the grammar G have the “characteristic” property of strings in L. Thou shall not lie. Typically the proof proceeds by associating an invariant with each variable that characterizes the strings it derives.

6. cs466(Prasad)L6Proof6 Completeness: All strings in L are derivable in G. Thou shall speak the whole truth. Typically the proof presents an algorithm for generating a string template of L using the rules of the grammar G.

7. cs466(Prasad)L6Proof7 How do we generate Derivation Rule Application

8. cs466(Prasad)L6Proof8 Let #w(x) = Number of occurrences of x in . Prove that Proof : By induction on the length of the derivation of a sentence. –Basis: l = 4

9. cs466(Prasad)L6Proof9 Induction Hypothesis: Induction Step: By context-freeness of the grammar:

10. cs466(Prasad)L6Proof10 By induction hypothesis: (Note that Sudkamp’s text illustrates an alternative approach. Its proof uses a property of sentential forms and shows that the rules in the grammar preserve it.)

11. cs466(Prasad)L6Proof11 Example Two proofs for the same result.

12. cs466(Prasad)L6Proof12 Prove that if then

13. cs466(Prasad)L6Proof13 Informal Argument E E E EE + * xxx + x * x x + * xx x Parse tree Abstract syntax tree Strictly Binary Trees: # nodes = (2 * #leaves) – 1 # ops + # vars = (2 * #vars) – 1 # ops + 1 = #vars #(+) + #(*) +1 = #(x)

14. cs466(Prasad)L6Proof14 Proof Alternatives

15. cs466(Prasad)L6Proof15 Proof 1: By induction on the length of the derivation of a sentence. Basis: n = 1 s = x 0 + 0 + 1 = 1 Induction Hypothesis:

16. cs466(Prasad)L6Proof16 Induction step: k=n+1 Possible patterns for w

17. cs466(Prasad)L6Proof17 By induction hypothesis:

18. cs466(Prasad)L6Proof18 LHS RHS

19. cs466(Prasad)L6Proof19 By induction hypothesis:

20. cs466(Prasad)L6Proof20 LHS RHS

21. cs466(Prasad)L6Proof21 Proof 2: By induction on the length of the derivation of a sentential form. Prove the general result so that the desired result for sentences follows given that

22. cs466(Prasad)L6Proof22 Basis: n = 1 w = +EE 1 + 0 + 1 = 0 + 2 w = *EE 0 + 1 + 1 = 0 + 2 w = x 0 + 0 + 1 = 1 + 0 Induction Hypothesis:

23. cs466(Prasad)L6Proof23 Induction step: k=n+1 Changes in the counts due to one-step rewrites:

24. cs466(Prasad)L6Proof24 In each case, the relationship (to be proved) among the counts of +,*, x and E for a sentential form holds. In other words, each rule application to a sentential form preserves this relationship. The desired result is obtained by observing that sentences do not contain any variable (non-terminal).

25. GENERAL STRATEGY FOR SENTENCE-BASED PROOF Variable of Induction: Length of derivation of sentence. Induction Hypothesis: Usually the same as the result to be proved. Basis Case: Explicitly verify for sentences derivable in (length =) 1 step.  That is, verify for subset of start symbol (LHS) productions that have only terminals on the RHS. cs466(Prasad)L6Proof25

26. Induction Step: (n+1) steps is divided into 1 step + n steps to create the sub-problems. This ensures that sub-problems created using can use induction hypothesis about sentences. The first step must use E-productions with at least one non-terminal on the RHS. Thus, there are as many cases as there are E-productions with non- terminals on the RHS. cs466(Prasad)L6Proof26

27. GENERAL STRATEGY FOR SENTENTIAL FORM-BASED PROOF Variable of Induction: Length of derivation of sentential form. Induction Hypothesis: Usually a generalization of the result to be proved, involving non-terminals. (The result about the sentences is a special case when number of non-terminals is set to 0.) Basis Case: Explicitly verify for sentential forms derivable in (length =) 0 steps and 1 step.  That is, verify for start symbol S and all start symbol (LHS) productions. cs466(Prasad)L6Proof27

28. Induction Step: (n+1) steps is divided into n steps + 1 step to create sub-problems. The sub-problems created using can use induction hypothesis about sentential forms. The last step can use all productions applicable to non-terminals in w. So, we need to show that each rule application preserves the induction hypothesis / invariant, e.g., maintain balance of counts. cs466(Prasad)L6Proof28

29. cs466(Prasad)L6Proof29 Language of Palindromes A man, a plan, a canal, Panama! Able was I ere I saw Elba. Rats live on no evil star. He goddam mad dog, eh? Madam, I'm Adam. A Santa stops pots at NASA. A Toyota's a Toyota. Tarzan raised a Desi Arnaz rat. Aibohphobia (The fear of palindromes!)

30. cs466(Prasad)L6Proof30

31. cs466(Prasad)L6Proof31

32. cs466(Prasad)L6Proof32

33. cs466(Prasad)L6Proof33 n = 2 is redundant. Palindromic property + Pattern specified

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36. cs466(Prasad)L6Proof36

37. cs466(Prasad)L6Proof37