Problem 24 - Discrete Mathematics with Applications

Question

Consider the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{\sigma\\}, T=\\{\mathrm{a}, \mathrm{b}\\},\) and \(P=\\{\sigma \rightarrow \mathbf{a} \sigma \mathbf{b}, \sigma \rightarrow \mathbf{a b}\\} .\) Determine if each word belongs to \(L(G).\) abba

Step-by-Step Solution

Answer

The word "abba" belongs to the language L(G) generated by the grammar G, as it can be derived from the start symbol σ using the production rules P: σ → aσb → aabb.

Step 1: Understand the given grammar

We have a grammar G with: - Non-terminal symbols N = {σ} - Terminal symbols T = {a, b} - Production rules P = {σ → aσb, σ → ab} - Start symbol σ

Step 2: Apply production rules

We will try to derive the word "abba" from the start symbol σ using the production rules in P. Let's see if it's possible: 1. σ → aσb (using the first production rule) 2. aσb → aabb (using the second production rule) We have successfully derived the word "abba" from the start symbol σ using the production rules in P.

Step 3: Conclusion

Since we can derive the word "abba" from the start symbol σ using the production rules P, the word "abba" belongs to the language L(G) generated by the grammar G.

Key Concepts

Production Rules

In the context of a context-free grammar, production rules are the guidelines that allow you to create strings within a language. These rules outline how non-terminal symbols can be replaced with combinations of terminal and non-terminal symbols. For instance, the grammar provided, denoted as \(G=(N, T, P, \sigma)\), has the following production rules:

• \(\sigma \rightarrow \mathbf{a} \sigma \mathbf{b}\)
• \(\sigma \rightarrow \mathbf{ab}\)

These rules serve as blueprints for generating strings in the grammar's language. In our example, the string "abba" can be constructed by applying these production rules starting from the symbol \(\sigma\). Initially, \(\sigma\) is replaced by \(a\sigma b\), and then \(\sigma\) is finally substituted with "ab" resulting in "aabb".

Terminal Symbols

Terminal symbols are like the building blocks of strings in a language generated by a grammar. In simple terms, these are the characters that appear in the final strings and cannot be further replaced or transformed. In our given grammar, the terminal symbols are \(\{a, b\}\).These characters are fixed and signify the end of a production process when forming a string. They form the actual content of a language's words. As you perform derivations using production rules, the resulting strings should only consist of these terminal symbols. For instance, when deriving "abba" from the start symbol, you end up with a string consisting entirely of the terminal symbols \(a\) and \(b\). This confirms that a string is part of the language defined by the grammar.

Non-Terminal Symbols

Non-terminal symbols are placeholders or variables within a grammar that help guide the transformation process from start to completion. They can't appear in the final language strings themselves, but control the structure and generation of the strings. In the given grammar, the only non-terminal symbol is \(\sigma\).The role of non-terminal symbols is crucial because they kickstart the derivation process. They are replaced by other symbols (both terminal and non-terminal) through production rules until a string entirely made of terminal symbols is created. For example, during our derivation of "abba", the non-terminal symbol \(\sigma\) is transformed first into \(a\sigma b\), and subsequently revised into the terminal sequence "aabb", demonstrating their importance in deriving the desired output.

Language Generation

Language generation in the context of a context-free grammar involves creating a set of strings from a given start symbol using production rules. This set of strings makes up what we know as the language of the grammar, denoted as \(L(G)\).For the grammar \(G=(N, T, P, \sigma)\), the aim is to determine if a string "abba" can be constructed using the provided production rules. By sequentially applying these rules, starting from the non-terminal symbol \(\sigma\), you either transform it directly to "ab" or to "a\sigma b" and then to "aabb".Successfully deriving "abba" implies it belongs to the grammar's language \(L(G)\). This process of transforming non-terminal symbols into terminal strings using production rules highlights the power and role of grammar in language generation. In essence, it captures how new words or sentences that belong to the language can be systematically created.