Problem 17 - 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 \(L(G)\) . $$\mathrm{a}^{3} \mathrm{b}$$

Step-by-Step Solution

Answer

The derivation tree for the word \(a^3b\) using the given grammar \(G = (N, T, P, \sigma)\) can be obtained by applying the production rules as follows: 1. Start with the axiom \(\sigma\). 2. Apply the rule \(\sigma \rightarrow a \sigma\) twice: \(\sigma \Rightarrow a \sigma \Rightarrow a a \sigma\). 3. Apply the rule \(\sigma \rightarrow a A\): \(a a \sigma \Rightarrow a a a A\). 4. Apply the rule \(A \rightarrow b\): \(a^3A \Rightarrow a^3b\). The resulting derivation tree is: ``` σ / \ / \ a σ / \ / \ a σ / \ / \ a A \ \ b ```

Step 1: Write down the given grammar

The given grammar \(G = (N, T, P, \sigma)\) consists of the following elements: - Non-terminal symbols \(N = \{A, \sigma\}\) - Terminal symbols \(T = \{a, b\}\) - Production rules \(P = \{\sigma \rightarrow a \sigma, \sigma \rightarrow a A, A \rightarrow b\}\) - Start symbol (axiom) \(\sigma\) Our goal is to generate the word \(a^3b\) using these production rules and draw the corresponding derivation tree.

Step 2: Start with the axiom

Since \(\sigma\) is the start symbol (axiom), we begin with it. Current word: \(\sigma\)

Step 3: Apply production rules

Since our target word begins with three \(a\)s and we know that there is a production rule \(\sigma \rightarrow a \sigma\), we will apply this rule twice to generate two \(a\)s. \(\sigma \Rightarrow a \sigma \Rightarrow a a \sigma\) Current word: \(a^2\sigma\) Now, we need to generate the third \(a\) and introduce the non-terminal symbol \(A\), which will later be replaced with \(b\). To accomplish this, we will use the production rule \(\sigma \rightarrow a A\). \(aa\sigma \Rightarrow aa a A\) Current word: \(a^3A\) Finally, we will apply the production rule \(A \rightarrow b\) to replace \(A\) with \(b\). \(a^3 A \Rightarrow a^3 b\) Now, we have successfully generated the word \(a^3b\).

Step 4: Draw the derivation tree

Based on the application of production rules in steps 2 and 3, we can draw the derivation tree as follows: ``` σ / \ / \ a σ / \ / \ a σ / \ / \ a A \ \ b ``` In summary, the derivation tree for the word \(a^3b\) using the given grammar \(G = (N, T, P, \sigma)\) is depicted above.

Key Concepts

Derivation Tree

A derivation tree, also known as a parse tree, is a visual representation of the syntactic structure of a string according to a formal grammar. This tree helps to understand how a word is derived from the grammar's start symbol using its production rules.
Each node in the derivation tree corresponds to a symbol from the grammar — either terminal, non-terminal, or both.

In our example, the objective was to derive the word \(a^3b\) using the grammar \(G\). The tree begins with the initial symbol \(\sigma\). Applying the production rules, \(\sigma\) transformed step by step into \(a^3b\).
The derivation process followed was:

• \(\sigma \rightarrow a \sigma\)
• \(\sigma \rightarrow a \sigma\)
• \(\sigma \rightarrow a A\)
• \( A \rightarrow b \)

Each branch in this tree shows the choice of production rules applied until reaching the terminal string \(a^3b\).

Context-Free Grammar

Context-free grammar (CFG) is a type of formal grammar used to define a language's syntax. In CFG, any production rule allows a single non-terminal symbol to be replaced by a string of non-terminal and/or terminal symbols.
The given grammar, \(G=(N, T, P, \sigma)\), includes such productions. Here, \(N\) represents non-terminal symbols; \(T\) represents terminal symbols; \(P\) represents production rules; and \(\sigma\) is the start symbol.

This grammar is 'context-free' because these rules allow the symbol on the left to be replaced independently of surrounding symbols. In other words, no contextual information is needed to apply the rules.

• The rule \(\sigma \rightarrow a \sigma\) continuously generates terminals "a" while keeping the process active.
• The rule \(\sigma \rightarrow a A\) signifies the ending path to generate "b" once all "a's" are produced.

These rules collectively allow us to build strings of symbols that belong to the grammar's language.

Non-terminal Symbols

Non-terminal symbols are fundamental elements of a context-free grammar, representing abstract concepts that need further expansion into other symbols.
They can be thought of as placeholders that require additional applications of production rules to be fully resolved into terminal symbols.

In our grammar example, the non-terminal symbols are \(N = \{A, \sigma\}\).

• \(\sigma\) serves as the starting point from which all derivations commence.
• \(A\) plays a crucial role in eventually being replaced by the terminal symbol \(b\).

Non-terminal symbols help in incrementally building up a specific structure outlined by the string, making complex languages easier to describe via recursive definitions.

Terminal Symbols

Terminal symbols are the basic symbols from which strings in a language are formed. They are the alphabet of the language, and once reached during a derivation, they can't be altered further.
Terminal symbols are the final output result after all non-terminal symbols in a derivation tree are resolved. In the grammar provided, the terminal symbols \(T = \{a, b\}\) have straightforward roles.

These symbols represent the actual characters in the language that are visible at the end of the derivation:

• Symbol \(a\) is repeatedly produced as per the production rules.
• Symbol \(b\) concludes the derivation by replacing the non-terminal \(A\).

The combination of terminal symbols, like \(a^3b\), forms the words accepted by the grammar's language. Once all non-terminals have been replaced by terminals, the process of derivation is complete.