Problem 67 - Discrete Mathematics with Applications

Question

A number in ALGOL (excluding the exponential form) is defined as follows: $$\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle$$ \(\langle\text { decimal number }\rangle : :=\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle |\) $$\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle$$ $$\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\operatorname{sign}\rangle : :=+|-$$ Use this grammar to answer Exercises \(60-67\). Draw a derivation tree for each ALGOL number. $$0.23$$

Step-by-Step Solution

Answer

To draw the derivation tree for the ALGOL number \(0.23\), we start by applying the grammar rules: 1. \(\langle\text { number }\rangle\) = \(\langle\text { decimal number }\rangle\) 2. \(\langle\text { decimal number }\rangle\) = \(\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle\) 3. Derive the unsigned integers: - \(\langle\text { unsigned integer }\rangle\) = \(\langle\text { digit }\rangle\) (for 0) - \(\langle\text { unsigned integer }\rangle\) = \(\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) (for 23) 4. Derive the digits: - \(\langle\text { digit }\rangle\) = 0 - \(\langle\text { digit }\rangle\) = 2 - \(\langle\text { digit }\rangle\) = 3 The derivation tree is: - \(\langle\text { number }\rangle\) - \(\langle\text { decimal number }\rangle\) - \(\langle\text { unsigned integer }\rangle\) - \(\langle\text { digit }\rangle\) - "0" - "." - \(\langle\text { unsigned integer }\rangle\) - \(\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) - \(\langle\text { digit }\rangle\) - "2" - \(\langle\text { digit }\rangle\) - "3"

Step 1: Apply the number rule

We look at the definition of a number: \(\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle\) Since our number doesn't include a sign, we can deduce that our number is simply a decimal number. Therefore, we can write: \(\langle\text { number }\rangle\) = \(\langle\text { decimal number }\rangle\) 2. Derive a decimal number

Step 2: Apply the decimal number rule

Now we look at the definition of a decimal number: \(\langle\text { decimal number }\rangle : =\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle | \langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle\) We can see that our given number (0.23) falls into the second category, so we can write: \(\langle\text { decimal number }\rangle\) = \(\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle\) 3. Derive the unsigned integers

Step 3: Apply the unsigned integer rules

We have two unsigned integers in our number, 0 and 23. Let's derive both of them using the definition of an unsigned integer: \(\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) For the first unsigned integer (0), we have a single digit, so we can write: \(\langle\text { unsigned integer }\rangle\) = \(\langle\text { digit }\rangle\) For the second unsigned integer (23), we have two digits, so we can write: \(\langle\text { unsigned integer }\rangle\) = \(\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) 4. Derive the digits

Step 4: Apply the digit rules

Now, we will replace each derived digit with its corresponding rule: \(\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9\) For the first digit (0), we have: \(\langle\text { digit }\rangle\) = 0 For the second and third digits (2 and 3), we have: \(\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) = \(\langle\text { digit }\rangle\langle\text { digit }\rangle\) \(\langle\text { digit }\rangle\) = 2 \(\langle\text { digit }\rangle\) = 3 Now we can construct the derivation tree: - \(\langle\text { number }\rangle\) - \(\langle\text { decimal number }\rangle\) - \(\langle\text { unsigned integer }\rangle\) - \(\langle\text { digit }\rangle\) - "0" - "." - \(\langle\text { unsigned integer }\rangle\) - \(\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle\) - \(\langle\text { digit }\rangle\) - "2" - \(\langle\text { digit }\rangle\) - "3"

Key Concepts

Understanding Decimal Numbers

In the world of programming and number systems, a **decimal number** is one based on the base-10 numeral system. This is the most common way humans represent numbers, as it uses the digits 0 through 9. In the context of ALGOL number grammar, a decimal number can be seen as either two **unsigned integers** separated by a decimal point (.), or as the concatenation of unsigned integers.

Here's what you should keep in mind about decimal numbers in ALGOL:

• Decimal numbers use a point "." to differentiate between integer and fractional parts.
• Any part of a decimal can be a single unsigned integer or multiple unsigned integers combined.

When dealing with an ALGOL number like 0.23, it is understood as an integer part "0" and a fractional part "23". Recognizing how these parts are represented in the grammar is crucial for understanding their derivation.

Defining Unsigned Integers

An **unsigned integer** in programming is any whole number that does not include a sign (positive or negative). This differs from signed integers, which can be both positive and negative.

In ALGOL, each unsigned integer is made up of one or more digits from the set **0 through 9**, according to its grammar rules. Specifically, an unsigned integer can be seen as a sequence initiated by a digit which may be followed by additional digits.

Key points include:

• An unsigned integer starts with at least one digit.
• Subsequent digits just extend this number, but they cannot introduce a sign.
• It's essential for forming more complex numbers like decimal numbers.

In our example, "0" and "23" are both unsigned integers. "0" is a simple unsigned integer, whereas "23" involves two digits following each other.

Exploring Digit Derivation

**Digit derivation** is the foundation of building numbers in ALGOL. Each digit from "0" to "9" is individually recognized and cataloged according to the digit rules, which then proceed to construct larger numeric values through combination.

In programming grammar, the derivation of digits follows a strict order:

• Each digit is represented as is unless part of a larger multi-digit number.
• An unsigned integer considers each digit sequentially to form the whole.
• Digit derivation ensures numbers are composed correctly as per the rule sets.

For ALGOL number 0.23, firstly, the digit "0" is acknowledged. Then, "2" and "3" are derived to form "23", showing how digits are systematically combined to represent numbers.

Navigating Derivation Trees

A **derivation tree** is a graphical representation of how a sentence in a language is derived based on grammar rules. In programming, and specifically in ALGOL, it shows how each part of a number or expression follows the defined grammar to form a complete construct.

Understanding derivation trees involves:

• Starting from the highest-level rule, like a or .
• Following branches to lower-level components like or .
• Ensuring each tree branch adheres to the grammar rules.

For 0.23, the tree starts with the rule, connects to the , then branches to and components, mapping the whole structure graphically to show the compilation of each grammatical rule into a tangible number.