Problem 18 - Discrete Mathematics with Applications

Question

Use the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{A, \sigma\\}, T=\\{a, b\\},\) and \(P=\\{\sigma \rightarrow a \sigma, \sigma \rightarrow a A, A \rightarrow b\\},\) to answer Exercises \(15-23\) . Draw a derivation tree for each word in \(L(G)\) . $$\mathrm{a}^{4} \mathrm{b}$$

Step-by-Step Solution

Answer

The derivation tree for the word \(a^4b\) using the grammar \(G=(N, T, P, \sigma)\) can be constructed using the following derivation sequence: 1. \(\sigma \Rightarrow a\sigma\) 2. \(a\sigma \Rightarrow aa\sigma\) 3. \(aa\sigma \Rightarrow aaa\sigma\) 4. \(aaa\sigma \Rightarrow aaaaA\) 5. \(aaaaA \Rightarrow a^4b\) The resulting tree is: ``` Σ | aΣ | a| aΣ | a| a| aΣ | a| a| a| aA | a| a| a| ab ```

Step 1: Understand the given grammar

The given grammar is \(G=(N, T, P, \sigma)\), where: - \(N=\{A, \sigma\}\): The set of non-terminal symbols - \(T=\{a, b\}\): The set of terminal symbols - \(P=\{\sigma \rightarrow a \sigma, \sigma \rightarrow a A, A \rightarrow b\}\): The set of production rules - \(\sigma\): The starting non-terminal symbol

Step 2: Apply the production rules to derive the word \(a^4b\)

We need to find a sequence of production rules that will result in the word \(a^4b\). Here is one such derivation sequence: 1. \(\sigma \Rightarrow a\sigma\) (applying \(\sigma \rightarrow a\sigma\)) 2. \(a\sigma \Rightarrow aa\sigma\) (applying \(\sigma \rightarrow a\sigma\)) 3. \(aa\sigma \Rightarrow aaa\sigma\) (applying \(\sigma \rightarrow a\sigma\)) 4. \(aaa\sigma \Rightarrow aaaaA\) (applying \(\sigma \rightarrow aA\)) 5. \(aaaaA \Rightarrow a^4b\) (applying \(A \rightarrow b\))

Step 3: Draw the derivation tree for \(a^4b\)

Now we will draw the derivation tree, starting with the non-terminal symbol \(\sigma\), then making branches for each of the production rules applied. The tree should look like this: ``` Σ | aΣ | a| aΣ | a| a| aΣ | a| a| a| aA | a| a| a| ab ``` The derivation tree for the word \(a^4b\) using the grammar \(G=(N, T, P, \sigma)\) is shown above.

Key Concepts

Formal Grammar

When we talk about formal grammar within the context of computer science and linguistics, we're discussing a system that describes how to form strings from a language's alphabet that are valid according to specific language rules.

This system is comprised of a finite set of rules, often referred to as a grammar, which we use for constructing strings. In the exercise, the provided formal grammar is denoted by the notation \(G = (N, T, P, \text{S})\). Here's what each element represents in this context:

• \(N\) is a set of non-terminal symbols that can be transformed into terminal symbols using the production rules. These are sort of placeholders that guide the formation of the strings.
• \(T\) is the set of terminal symbols. These are the basic characters that appear in the strings generated by the grammar and cannot be replaced any further.
• \(P\) is a set of production rules which define how to replace non-terminal symbols with a combination of non-terminal and terminal symbols.
• \(\text{S}\) is the start symbol from which we begin to construct strings.

In essence, formal grammar gives us the blueprint for generating all strings that belong to a particular language.

Production Rules

Production rules, also known as productions, are the transformative steps applied to non-terminal symbols that help generate terminal strings in a formal grammar. They serve as instructions, guiding the grammar from a starting symbol to the end result—a string comprised entirely of terminal symbols.

In our exercise, production rules are represented in the set \(P\) with elements like \(\sigma \rightarrow a \sigma\). Let's unpack what this means:

• The symbol on the left side of the arrow (\(\sigma\)) is the non-terminal symbol being replaced.
• The sequence on the right side of the arrow (\(a \sigma\)) represents the new string after the rule has been applied, which may contain both non-terminal and terminal symbols.

Using the production rules effectively is essential for constructing a valid string in the language defined by the grammar. For example, to derive the word \(a^4b\) from our exercise, specific production rules were applied in sequence, creating a chain of replacements that lead us from the start symbol to the final string.

Word Derivation

Word derivation in the context of formal grammar involves applying production rules to transform a start symbol into a string that is in the language of the grammar. It's a sequential process, with each step replacing a non-terminal symbol according to the production rules until only terminal symbols remain.

In our exercise, the word \(a^4b\) is derived from the initial non-terminal \(\sigma\) by repeatedly applying appropriate rules:

• \(\sigma \Rightarrow a\sigma\)
• \(a\sigma \Rightarrow aa\sigma\)
• \(aa\sigma \Rightarrow aaa\sigma\)
• \(aaa\sigma \Rightarrow aaaaA\)
• \(aaaaA \Rightarrow a^4b\)

Each step moves closer to the final word, ensuring that the process follows the formal grammar's guidelines. To visualize word derivation, derivation trees can be extremely helpful. They illustrate the structure of the derivation process and are valuable for both understanding and teaching the concept of word derivation in formal grammar.