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Chapter 0: Q17P (page 1)

Let be the language of properly nested parentheses. For example, (()) and ()are in, but) (is not. Show that A is in L.

Short Answer

Expert verified

Prove that A is in L by counting the number of unmatched left parenthesis.

Step by step solution

01

Introduce nested parenthesis and L languages

Nested parentheses are parentheses that are contained within another parenthesis. L languages are decidable in logarithmic space using deterministic Turing Machine.

02

Show that A is in L. 

Count the number ofleft parenthesis (in binary) that aren't matched. Turing machine adds one to the counter, and when it comes across, it subtracts one from the number.

It rejects the language if the counter is zero. Also, rejects if the counter is not 1 when the TM completes counting.

There is no need to use 0 and 1 since they cannot be in the alphabet. This process takes log space to perform these operations.

Thus, A is in L.

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Most popular questions from this chapter

Question: Describe the error in the following 鈥減roof鈥� that 0*1*is not a regular language. (An error must exist because 0*1*is regular.) The proof is by contradiction. Assume that 0*1*is regular. Let p be the pumping length for localid="1662103472623" 0*1*given by the pumping lemma. Choose s to be the string 0p1p . You know that s is a member of 0*1*, but Example 1.73 shows that s cannot be pumped. Thus you have a contradiction. So 0*1* is not regular.

We generally believe that PATH is not NP-complete. Explain the reason behind this belief. Show that proving PATH is not NP-complete would prove P 鈮� NP

Myhill鈥揘erode theorem. Refer to Problem 1.51 . Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pair wise distinguishable by L . The index of L may be finite or infinite.

a. Show that if L is recognized by a DFA with k states, L has index at most k.

b. Show that if the index of L is a finite number K , it is recognized by a DFA with k states.

c. Conclude that L is regular iff it has finite index. Moreover, its index is the size of the smallest DFA recognizing it.

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