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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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