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Chapter 11: Problem 22

Define the language \(L\) of all binary representations of non-negative integers recursively.

Short Answer

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The language \(L\) of all binary representations of non-negative integers can be defined recursively as follows: \(L(0) = "0" \) \( L(n) = \begin{cases} L(n-1)01 & \text{if}\ n \equiv 1\ (\text{mod}\ 2) \\[2ex] 0\overline{L(n-1)} & \text{if}\ n \equiv 0\ (\text{mod}\ 2) \end{cases} \) where \(\overline{L(n-1)}\) represents the bits of \(L(n-1)\) flipped from the most significant bit of \(n\) up to the least significant bit of \(n-1\).

Step by step solution

01

Base Case

The base case for the recursive definition of the language L is the smallest non-negative integer, which is 0. Its binary representation is also "0", so the base case is: \(L(0) = "0" \)
02

Recursive Rule

In order to generate the rest of the elements in the language L, we need to create a recursive rule. For each integer "n", its binary representation in L can be obtained by comparing n to the previous non-negative integer, n-1: 1. If n is odd, then the binary representation of n will be the same as n-1, but the last digit will be changed from 0 to 1. 2. If n is even, then find the most significant bit of n; change all the bits to the opposite (0 to 1 and 1 to 0) from the most significant bit of n up to the least significant bit of n-1, and add "0" as a new most significant bit. In other words, the recursive rule can be described as: \( L(n) = \begin{cases} L(n-1)01 & \text{if}\ n \equiv 1\ (\text{mod}\ 2) \\[2ex] 0\overline{L(n-1)} & \text{if}\ n \equiv 0\ (\text{mod}\ 2) \end{cases} \) where \(\overline{L(n-1)}\) represents the bits of \(L(n-1)\) flipped from the most significant bit of \(n\) up to the least significant bit of \(n-1\). Using this recursive definition, we can generate the binary representations for any non-negative integer, and the language L will contain all such binary numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Numbers
Binary numbers are the foundation of how information is processed and stored in computers. They consist of only two digits: 0 and 1. These digits, also known as bits, represent the two possible states of electrical circuits 鈥� off and on.
In binary, each bit represents a power of two. The rightmost bit (least significant bit) is the units position or 2^0, the next is in the 2^1 position, and so on. For example, the binary number 1011 corresponds to the decimal number:
  • 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0) = 8 + 0 + 2 + 1 = 11.
Binary numbers are essential when defining a language for non-negative integers, as shown in the recursive approach discussed. This is because every integer can be expressed as a unique binary number.
Non-negative Integers
Non-negative integers are simply all whole numbers greater than or equal to zero. These numbers include 0, 1, 2, 3, 4, and so on. Importantly, non-negative integers do not include negative numbers.
In the context of binary numbers, each non-negative integer has a corresponding binary representation. For example:
  • The integer 0 is represented as '0' in binary.
  • The integer 2 is represented as '10' in binary.
Recursive definitions come in handy when you want to construct binary representations systematically for such integers. The base case and recursive rules help in addressing how each subsequent integer can be derived from its predecessor.
Base Case
The base case is the starting point for a recursive definition. It's the simplest instance of the problem that can be solved directly without recursion. In defining a set of binary numbers, we begin with the smallest non-negative integer, which is 0.
The binary representation of 0 is just "0". This forms the base case in the recursive definition of the language L. By establishing this base case, we provide the initial value necessary for building more complex numbers using recursive rules.
Without the base case, the recursive process would lack a defined starting point, making it impossible to generate subsequent numbers.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value, known as the modulus. It is similar to the concept of a clock wrapping back to 1 after reaching 12.
In the context of binary numbers, modular arithmetic helps determine if a number is even or odd. This is crucial for our recursive rule, which defines how to find the next binary number:
  • If the number is odd (n \(\equiv\) 1 (mod 2)), append '1' to the binary representation of the previous number.
  • If the number is even (n \(\equiv\) 0 (mod 2)), manipulate the most significant bits as explained in the solution.
Through understanding modular arithmetic, you gain insight into how binary operations relate to the properties of numbers and how they define binary representations recursively.

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Most popular questions from this chapter

By making a DFSA, define a regular grammar \(G=(N, T, P, \sigma)\) that generates the language consisting of strings over \(\\{a, b\\}\) that: Contain at least one \(a\).

Create a NDFSA that accepts the regular language over \(\\{a, b\\}\) of strings that: Contain \(a^{3}\) as a substring.

Create a NDFSA that accepts the regular language over \(\\{a, b\\}\) of strings that: Contain \(b a^{2} b\) as a substring.

The production rules of a grammar for simple arithmetic expressions are: $$\langle\text { expression }\rangle :=\langle\text { digit })(\langle\text { expression })) |+(\langle\text { expression }\rangle) |$$ $$-(\langle\text { expression }\rangle) | \langle\text { expression }\rangle \langle\text { operator }\langle\text { expression }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\text { operator }\rangle : :=+|-| / | \uparrow$$ Use this grammar to answer Exercises \(52-59\). Determine if each is a valid arithmetic expression. $$3+\uparrow 7$$

Develop a grammar that generates each language over {0,1}. The set of words with prefix 00.
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