Problem 15 - 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 \(L(G)\) . $$\mathrm{ab}$$

Step-by-Step Solution

Answer

To derive the string "ab" using the given grammar G, we follow the steps: 1. Start with the initial symbol σ. 2. Apply the production rule σ → a A, resulting in the derivation "a A". 3. Apply the production rule A → b, resulting in the final derivation "ab". The derivation tree is: σ / \ a A \ b

Step 1: Start with the initial symbol σ

We begin with the initial symbol σ, which is given as a part of the grammar G. So we have: 1. σ

Step 2: Apply production rule σ → a A

Now, we need to apply a production rule to produce the given string "ab". We can apply the production rule σ → a A, as it gives us an "a" terminal followed by a non-terminal A. The derivation now looks like: 1. σ 2. a A

Step 3: Apply production rule A → b

The current derivation string is "a A" which has the non-terminal A at the end. We need to replace this non-terminal with the terminal "b" to obtain the given string "ab." We shall apply the production rule A → b: The final derivation becomes: 1. σ 2. a A 3. ab Now, we have derived the given string "ab" using the grammar G.

Step 4: Drawing the Derivation Tree

Finally, now that we have derived the string "ab" using the production rules, we can represent the derivation process as a tree: 1. σ is the root. 2. Then σ produced "a A" using rule σ → a A. So the two children of σ are "a" and "A." 3. Finally, A produced "b" using rule A → b. So, the child of A is "b." The derivation tree looks like: σ / \ a A \ b

Key Concepts

Derivation Tree

A derivation tree, also known as a parse tree, is a diagrammatic representation of how a string from a language is generated, using the grammar's production rules. This tree illustrates the steps needed to derive a string from the initial symbol to the string's completion.
For the given problem, the grammar allows us to derive the string "ab." In the derivation tree:

• The root node starts with the initial non-terminal symbol, \(\sigma\).
• Each branch represents the application of a production rule.
• The leaf nodes represent the terminal symbols, which in this case are 'a' and 'b'.

The tree structure clearly shows the order in which production rules are applied. This helps in understanding how complex strings can be progressively built up from simpler elements. A correct derivation tree allows students to visualize the hierarchical breakdown of the derivation process.

Production Rules

Production rules are the backbone of formal grammar, acting as instructions for replacing non-terminal symbols with other non-terminal and terminal symbols to form strings. The set of production rules defines the syntactic structure of the language generated by the grammar.
In our example, the production rules are:

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

These rules are applied sequentially to transform the start symbol \(\sigma\) into the desired terminal string "ab." Using the chosen production rules, you can gradually replace non-terminals until only terminals are left, forming the target string. Understanding these rules is crucial as they dictate how strings are formed from the grammar.

Non-Terminal Symbols

Non-terminal symbols are placeholders in a formal grammar that get replaced through production rules during the process of generating a string. They are not part of the final string output and serve as an intermediate step in the derivation.
In the given grammar, the non-terminal symbols are \(N = \{A, \sigma\}\).

• \(\sigma\) serves initially to start the derivation and transform into subsequent expressions.
• \(A\) is used as an intermediary which, through production rules, becomes the terminal symbol 'b'.

Non-terminal symbols are crucial because they provide flexibility and reusability of parts of the grammar, enabling the construction of complex strings by layering several derivations on top of each other.

Terminal Symbols

Terminal symbols are the basic characters or tokens of a language defined by a grammar, constituting the actual content of the language strings. They appear as the leaf nodes in a derivation tree and do not transform into any other symbols.
In our exercise, the terminal symbols are \(T = \{a, b\}\).

• "a" and "b" are used directly in forming the final string "ab."

These terminals are the end goal of a derivation, as the process is complete when only terminals remain. Recognizing and working with terminal symbols is fundamental, as they represent the final product of a language's grammar.