Problem 12 - Discrete Mathematics with Applications

Question

Define each language \(L\) over the given alphabet recursively. \(L=\) set of words over \(\Sigma=\\{0,1\\}\) with suffix 11

Step-by-Step Solution

Answer

The language \(L\) over \(\Sigma = \{0,1\}\) with suffix \(11\) can be defined recursively as follows: - Base case: The word "11" is in \(L\). - Recursive case: Let \(w\) be a word in the language \(L\). Then, \(0w\) and \(1w\) are also in the language \(L\). This definition generates all words over the alphabet \(\Sigma\) that end with the suffix \(11\).

Step 1: Base words

We must find the base words in the language \(L\). The simplest words over the alphabet \(\Sigma = \{0,1\}\) that have a suffix of \(11\) are the words "11" themselves. These words will serve as the foundation for building all other words in \(L\).

Step 2: Concatenation

Given a word in the language \(L\), we can generate new words in \(L\) by concatenating either a \(0\) or a \(1\) to the beginning of the existing word. For example, if we have the word "11", we can create the new words "011" and "111" by concatenation. As this procedure is done recursively, these new words will serve as the base for generating even larger words in the language \(L\).

Step 3: Definition of the language \(L\)

Using the base words and concatenation rules established above, we can define the language \(L\) recursively as follows: - Base case: The word "11" is in \(L\). - Recursive case: Let \(w\) be a word in the language \(L\). Then, \(0w\) and \(1w\) are also in the language \(L\). Therefore, the language \(L\) consists of all words over the alphabet \(\Sigma = \{0,1\}\) that end with a suffix of \(11\), starting from the base word "11" and recursively generated based on the given rules from Step 3.

Key Concepts

Recursive Definitions

A recursive definition is a way to define an object in terms of itself. In the context of formal languages, it specifies how a language can be built starting from some basic elements and extending it using specific rules.

In our example concerning the language \(L\), the recursive definition begins with the simplest word that satisfies the condition, which is the word "11". This is our base case. Then, we expand the language by continuously applying a set of rules: namely, we add either a "0" or a "1" to the beginning of an existing word.

This method of construction illustrates how the language grows larger incrementally, with each new level of complexity based on preceding simpler elements.

Alphabet

In formal languages, an alphabet is a finite set of symbols from which strings are formed. It acts as the language's building blocks.

For the language \(L\), the alphabet \(\Sigma\) consists of only two symbols: \( \{0, 1\} \). These symbols are combined to form words or strings. Understanding the alphabet is crucial because it determines the possible characters that can appear in any string within the language.

• Alphabet size: Determines the complexity and variety of the language.
• Symbols: Provide the fundamental components used in defining words.

By harnessing this alphabet, we create various sequences, like those in \(L\), which have a defined pattern or property, such as ending with a particular suffix.

Concatenation

Concatenation is the operation of linking two strings end to end to form a new string. In formal languages, concatenation allows for the creation of more complex strings by joining simpler ones.

In our language \(L\), starting with the base word "11", we apply concatenation to generate new words. By attaching a "0" or a "1" to the start of any existing word, we extend the language. For instance, the word "11" becomes "011" or "111" after concatenation.

• Basic operation: Connects words or letters in sequence.
• Recursive process: Continually applies concatenation to expand the language.

This operation is essential for forming words in our defined recursive approach, acting as a cornerstone for the extension of language \(L\).

Base Case

In recursive definitions, the base case is the simplest, non-recursive part of the definition. It provides a starting point from which all other elements in the set are formed. It is crucial because it stops the recursion from continuing indefinitely.

In the case of language \(L\), the base case is the word "11". This initial word is the simplest form that fits the requirement of having a suffix "11". From this base word, more complex words are constructed by recursively applying the rules of concatenation as outlined in the recursive definition.

Having a clear and correct base case ensures the proper foundation for all subsequent steps in the recursive process and guides the entire generation of the language.

Suffix

A suffix in formal languages is a substring that appears at the end of a string. It's used to define certain properties or patterns within the language.

For language \(L\), every word must end with the suffix "11". This condition acts as a filter, ensuring that only strings meeting this requirement are included in the language.

• Suffix properties: Critical for maintaining language constraints.
• Consistency: Ensures all words comply with the language rule.

The suffix thus serves both as a defining requirement and a recognizable hallmark of the language \(L\), creating a boundary for word inclusion from the set of all possible strings over the alphabet.