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Prove this Padding Lemma.

Prove this Padding Lemma.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Need help with part (c). Thank you :)

3. This question is about the Padding Lemma for FPD Languages. Definitions A prefix of a string w is any substring that occurs at the start of w. In other words, a word p is a prefix of w if and only if there exists a word r such that w = px. (Note that w is a prefix of itself, in which case x = €.) A language L over an alphabet E is finitely prefix-determined (FPD) if there exists a finite set P of words such that, for all words w e E*, W E L w has a prefix in P. For example: • Consider the language of binary strings whose first three bits contain two 1s. This language is FPD, using the prefix set P = {011, 101, 110}. • Consider the language of all utterancesover the English alphabet that start with an Australia state capital city. This language is FPT, using the prefix set P = {Hobart, Melbourne, Sydney, Brisbane, Perth, Adelaide}. The Padding Lemma for FPD Languages states: Let L be any nonempty FPD language. Then there exists N such that, for all w e L with |w| > N and all r e E*, we have wx E L. (a) Write the conclusion of the Padding Lemma ... There exists N such that, for all w e L with w > N and all r e E*, we have wx e L. ... using quantifiers and appropriate logical and mathematical symbols. (b) Write the logical negation of (a) using quantifiers, ensuring that all your quantifiers are out the front (i.e., as far to the left as possible), and that all other logical and mathematical symbols you use are to the right of all the quantifiers. (c) Prove this Padding Lemma. 'so they don't have to be grammatical sentences 3 (d) Let EVEN BINARY be the language of binary representations of even nonnegative integers: EVEN BINARY = {0, 10, 100, 110, 1000, 1010,...}. Use the Padding Lemma to prove by contradiction that EVEN BINARY is not FPD.
Transcribed Image Text:3. This question is about the Padding Lemma for FPD Languages. Definitions A prefix of a string w is any substring that occurs at the start of w. In other words, a word p is a prefix of w if and only if there exists a word r such that w = px. (Note that w is a prefix of itself, in which case x = €.) A language L over an alphabet E is finitely prefix-determined (FPD) if there exists a finite set P of words such that, for all words w e E*, W E L w has a prefix in P. For example: • Consider the language of binary strings whose first three bits contain two 1s. This language is FPD, using the prefix set P = {011, 101, 110}. • Consider the language of all utterancesover the English alphabet that start with an Australia state capital city. This language is FPT, using the prefix set P = {Hobart, Melbourne, Sydney, Brisbane, Perth, Adelaide}. The Padding Lemma for FPD Languages states: Let L be any nonempty FPD language. Then there exists N such that, for all w e L with |w| > N and all r e E*, we have wx E L. (a) Write the conclusion of the Padding Lemma ... There exists N such that, for all w e L with w > N and all r e E*, we have wx e L. ... using quantifiers and appropriate logical and mathematical symbols. (b) Write the logical negation of (a) using quantifiers, ensuring that all your quantifiers are out the front (i.e., as far to the left as possible), and that all other logical and mathematical symbols you use are to the right of all the quantifiers. (c) Prove this Padding Lemma. 'so they don't have to be grammatical sentences 3 (d) Let EVEN BINARY be the language of binary representations of even nonnegative integers: EVEN BINARY = {0, 10, 100, 110, 1000, 1010,...}. Use the Padding Lemma to prove by contradiction that EVEN BINARY is not FPD.
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