Chapter 10: Q8P (page 439)

Let A be a regular language over ${0, 1}$ . Show that A has size-depth complexity $(O (n) . O (l o g n))$ .

Short Answer

Expert verified

Using our knowledge of the parity function with n inputs can be computed by a branching program that has $O (n)$ nodes, we are going to solve the above problem.

Give an appropriate answer that shows the proof.

Step by step solution

Defining a circuit with O(n) size in two different ways

Let us take $a_{1}, a_{2}, . . . . . ., a_{n}$ as the input to the circuit.

One method to define a circuit with $O (n)$ size is to construct gates $g_{i}$ ,

Where, $g_{i} = a_{i}$ andlocalid="1657783611009" $g_{i} = a_{i} + g_{i - 1}$ for $i \leq n$ . The size and depth used by this construction is $O (n)$ .

Defining second way to get a circuit for parity function

We Know thatthe parity function with n inputs can be computed by a branching program that has $O (n)$ nodes.
Which can be achieved by building a binary tree of gates that compute XOR function, where XOR function is used as equivalent to the parity function. Each of XOR gate can be implemented by using two AND’s, two NOT’s and one OR gates.
This construction uses $O (n)$ and $O (logn)$ depth.

Hence, the size depth complexity of A is $(O (n) . O (logn))$ .

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Let A be a regular language over ${0, 1}$ . Show that A has size-depth complexity $(O (n) . O (l o g n))$ .

Short Answer

Step by step solution

Defining a circuit with O(n) size in two different ways

Defining second way to get a circuit for parity function

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Let A be a regular language over{0,1} . Show that A has size-depth complexity(O(n).O(logn)).

Short Answer

Step by step solution

Defining a circuit with O(n) size in two different ways

Defining second way to get a circuit for parity function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Algorithms in Computer Science

Data Representation in Computer Science

Computer Organisation and Architecture

Problem Solving Techniques

Functional Programming

Computer Programming

Study anywhere. Anytime. Across all devices.

Let A be a regular language over ${0, 1}$ . Show that A has size-depth complexity $(O (n) . O (l o g n))$ .