Problem 18 - Discrete Mathematics with Applications

Question

Use the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{A, \sigma\\}, T=\\{\mathrm{a}, \mathrm{b}\\},\) and \(P=\\{\sigma \rightarrow \mathbf{a} \sigma, \sigma \rightarrow \mathbf{a} \mathbf{A}, \mathbf{A} \rightarrow \mathbf{b}\\}.\) Draw a derivation tree for each word in \(L(G).\) $$a^{4} b$$

Step-by-Step Solution

Answer

Step 2: Apply production rules to generate the word #tag_title# Apply production rules to generate the word #tag_content# To generate the word \(a^4b\), we will use the given production rules in the following sequence: 1. \(\sigma \rightarrow a\sigma\) gives \(a\sigma\) 2. \(\sigma \rightarrow a\sigma\) gives \(aa\sigma\) 3. \(\sigma \rightarrow a\sigma\) gives \(aaa\sigma\) 4. \(\sigma \rightarrow aA\) gives \(aaaaA\) 5. \(A \rightarrow b\) gives \(a^4b\) Step 3: Draw the derivation tree #tag_title# Draw the derivation tree #tag_content# The derivation tree for the word \(a^4b\) using the grammar \(G=(N, T, P, \sigma)\) is: ``` σ / \ a σ / \ a σ / \ a σ / \ a A \ b ``` Thus, the derivation tree for the given word demonstrates how to apply the production rules of the given grammar to generate the word \(a^4b\).

Step 1: Understand the Problem

We are given:
Use the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{A, \sigma\\},
T=\\{\mathrm{a}, \mathrm{b}\\},\) and \(P=\\{\sigma \rightarrow \mathbf{a}
\sigma, \sigma \rightarrow \mathbf{a} \mathbf{A}, \mathbf{A} \rightarrow
\mathbf{b}\\}.\) Draw a derivation tree for each word in \(L(G).\)
$$a^{4} b$$

Step 2: Apply Relevant Concepts

We apply mathematical definitions, properties, and theorems.

Step 3: Solution

Step 2: Apply production rules to generate the word #tag_title# Apply production rules to generate the word #tag_content# To generate the word \(a^4b\), we will use the given production rules in the following sequence: 1. \(\sigma \rightarrow a\sigma\) gives \(a\sigma\) 2. \(\sigma \rightarrow a\sigma\) gives \(aa\sigma\) 3. \(\sigma \rightarrow a\sigma\) gives \(aaa\sigma\) 4. \(\sigma \rightarrow aA\) gives \(aaaaA\) 5. \(A \rightarrow b\) gives \(a^4b\) Step 3: Draw the derivation tree #tag_title# Draw the derivation tree #tag_content# The derivation tree for the word \(a^4b\) using the grammar \(G=(N, T, P, \sigma)\) is: ``` σ / \ a σ / \ a σ / \ a σ / \ a A \ b ``` Thus, the derivation tree for the given word demonstrates how to apply the production rules of the given grammar to generate the word \(a^4b\).

Key Concepts

Derivation Tree

A derivation tree, also known as a parse tree, is a way to visually represent the derivation of a string in a formal grammar. In this tree, the root represents the initial non-terminal symbol from which derivation starts. Following this, each intermediate node is a non-terminal symbol, and each leaf node is a terminal symbol. For example, consider the grammar we've been given. We need to construct a derivation tree for the word \(a^4 b\). This tree will show step-by-step how to derive this particular word from the start symbol \(\sigma\), using the appropriate production rules. To begin constructing the tree:- We start with the root node \(\sigma\). - Apply the production rule \(\sigma \rightarrow a\sigma\) repeatedly to introduce the required number of \(a\)'s. - Finally, use \(\sigma \rightarrow aA\) and \(A \rightarrow b\) to generate the terminal \(b\) after the \(a\)'s.The derivation tree faithfully captures every step of this process, making it easier to understand how various components of the grammar work together.

Non-terminal Symbols

Non-terminal symbols are critical in grammar as they are the building blocks used to define the language. They are symbolic variables that can be replaced by other sequences of symbols, according to production rules, until terminal symbols are left. In this grammar:- The non-terminal symbols are \(N = \{A, \sigma\}\).- These symbols cannot appear in the final derived string, but they direct how the final string is formed.- \(\sigma\) acts as the starting node, which means every derivation process originates from it.The role of non-terminal symbols is crucial because they provide the structure and flexibility needed to create complex strings within a language.

Terminal Symbols

Terminal symbols are the actual characters of the language defined by the grammar. These symbols appear in the final strings and cannot be replaced by any other symbols. They represent the end products of the derivation process, as no further production rules can apply to them.In our grammar:- The terminal symbols set is \(T = \{a, b\}\).- These are the symbols you find in the language strings, such as \(a^4 b\).The transformation ends when only terminal symbols remain in the string, indicating that the derivation is complete. Terminal symbols are the stable endpoints in the derivation tree and show what components make up final derived words.

Production Rules

Production rules are the instructions that define how non-terminal symbols can be transformed into other non-terminal or terminal symbols. These rules are crucial as they outline the steps through which strings can be generated in the language.For this grammar, the production rules are:- \(\sigma \rightarrow a\sigma\)- \(\sigma \rightarrow aA\)- \(A \rightarrow b\)Each production rule indicates a possible transformation. - The first two rules explicitly describe how strings of one or more \(a\)'s can be generated by replacing \(\sigma\) with either \(a\sigma\) or \(aA\).- The last rule, \(A \rightarrow b\), allows the production of the \(b\) at the end of the string, completing the derivation when \(a^4b\) is formed.Thus, production rules are the essential directives that allow us to create strings step by step within the grammar's language.