Question

Assume that we represent dollar amounts in the following way: \(\$$ number.numberCR The dollar sign and the dollar value must be present. The cents part (including both the decimal point and the number) and the CR (which stands for CRedit and is how businesspeople represent negative numbers) are both optional, and number is a variablelength sequence of one or more decimal digits. Examples of legal dollar amounts include \)\$ 995\(, \)\$ 99 \mathrm{CR}, \$ 199.95\(, and \)\$ 500.000 \mathrm{CR}\(. a. Write a BNF grammar for the dollar amount just described. b. Modify your grammar so that the cents part is no longer an arbitrarily long sequence of digits but is exactly two digits, no more and no less. c. Using your grammar from either Exercise \)8 \mathrm{a}\( or \)8 \mathrm{~b}\(, show a parse tree for \)\$ 19.95 \mathrm{CR}$.

Short answer

Construct the BNF, modify it for two cent digits, and show the parse tree for $19.95CR.

Step-by-step solution

Step 8a: Define the BNF Grammar for Dollar Amounts

To create a BNF grammar for the described dollar amount, we need rules for the structure: 1. The entry begins with a dollar sign (`$`). 2. It includes an integer part composed of one or more decimal digits. 3. The cents part, if present, starts with a decimal point (`.`) followed by one or more digits. 4. The credit indication (`CR`) is optional. The BNF grammar will look like this: ``` ::= "$" [] [] ::= | ::= "." ::= | ::= "CR" ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ```

Step 8b: Modify BNF for Exactly Two Cent Digits

Now, we need to adjust the grammar so that the cents part is precisely two digits: 1. We will change `` to ensure it is exactly two digits long. The modified BNF grammar: ``` ::= "$" [] [] ::= | ::= "." ::= ::= "CR" ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ```

Step 8c: Parse Tree for $19.95CR Using BNF

To illustrate a parse tree using the BNF grammar from 8b:- The parse tree for "\(19.95CR" follows these components:``` / | \ ""\)"" | / \ / \ / \ "1" "9" "." | | "9" "5" | "C" | "R"```- The tree starts with ``, branching into the dollar sign `$`, the integer `19`, the cents `.95`, and the credit `CR`.

Key concepts

Backus-Naur Form

Backus-Naur Form, often abbreviated as BNF, is a formal notation used to describe the syntax of programming languages, command sets, or domain-specific languages. It's akin to grammar for natural languages, providing a precise framework to specify how valid strings can be constructed. This syntax consists of a set of derivation rules. Each rule defines a non-terminal symbol with alternatives that may define terminal symbols or other non-terminals. Non-terminal symbols represent syntactic "variables," while terminal symbols are the actual characters or tokens in the syntax. For instance, in the context of our exercise involving dollar amounts:

• The non-terminal `` tries to match the overall structure of a dollar representation.
• It can encompass terminals like the `"$"` symbol or numerical digits.

Using BNF helps in establishing clear guidelines for what constitutes correct syntax for any given language or data format.

Parse Tree

A parse tree is a visual representation of the syntactic structure of a string according to a formal grammar. It showcases how a sequence of input symbols can be derived using the production rules of a grammar like BNF. This is beneficial for visualizing the parsing process in compiler design, making it essential for students grasping concepts in computer science education.For example, when constructing a parse tree for the dollar amount "\\(19.95CR", the tree root, ``, branches into different segments:

• First, the dollar sign `\)`.
• The integer part which is the sequence `19`.
• The optional cents part, represented by the decimal point followed by `95`.
• Finally, the `` optional part, showing `CR`.

Through the parse tree, you can effortlessly observe how each segment of the string conforms to the predefined syntactic rules.

Syntax Rules

Syntax rules refer to the guidelines that govern how symbols and expressions are formed in a programming language or any structured format. In the context of BNF and grammar, they are quite pivotal as they determine the way non-terminal elements can be transformed into terminal elements. In our exercise example, the syntax rules for dollar amounts describe:

• The obligatory presence of a dollar sign `$` at the beginning.
• A sequence of digits for the integer part, ensuring it's not empty.
• Optional components like the cents and credit markers (`CR`).

Such rules ensure that each dollar amount is consistently and correctly structured, preventing potential parsing errors in applications relying on this format.

Computer Science Education

In computer science education, understanding formal grammars such as BNF is fundamental. It's essential for students studying programming languages, compilers, or language processors. By mastering BNF, students can gain insight into designing syntax for new languages or adapting existing ones. BNF provides clarity on how computer languages are constructed at a foundational level. This clarity aids in various educational aspects like:

• Designing new programming languages or systems that simplify software development tasks.
• Improving language compilers that translate high-level code into machine-readable instructions.
• Creating domain-specific languages tailored for specific tasks, increasing productivity.

Overall, BNF and related concepts equip students with valuable skills to tackle real-world challenges in software development, thus playing a pivotal role in their educational journey.