Problem 75 - Discrete Mathematics with Applications

Question

Use the grammar in Exercise 74 to see if each is a valid sequence of parentheses. $$(())$$

Step-by-Step Solution

Answer

The given sequence is \((( ) \)). We can check its balance by walking through the sequence and keeping a count of open parentheses. After going through all characters, our open count is 0, which means the sequence is valid and properly balanced.

Step 1: Inspect the given sequence

The given sequence is (( )). We have two opening parentheses followed by two closing parentheses.

Step 2: Check the balance of the sequence

We will walk through the sequence and keep a count of open parentheses. Starting from the first character, our steps are as follows: - First character: ( - increment the open count to 1 - Second character: ( - increment the open count to 2 - Third character: ) - close one open parenthesis, decrement the open count to 1 - Fourth character: ) - close another open parenthesis, decrement the open count to 0

Step 3: Determine if the sequence is valid

Since the open count is 0 at the end, the sequence of parentheses is valid and properly balanced.

Key Concepts

Discrete Mathematics and Balanced Parentheses

The challenge of identifying whether a sequence of parentheses is balanced intersects the domain of discrete mathematics, a fundamental component of computer science and analysis of algorithms. In discrete mathematics, structures are studied that are fundamentally discrete rather than continuous, such as integers, graphs, and statements in logic.

When dealing with balanced parentheses, we are looking at a specific case of a well-formed formula, which is critical in logical expressions and programming languages. A sequence of parentheses is considered balanced if each opening parenthesis '(' has a corresponding closing parenthesis ')', and they are properly nested.

For example, the sequence \( (( )) \) is balanced because every opening bracket is matched with a closing bracket in the correct order. This concept is foundational in writing error-free code and creating functions that operate as intended. It also serves as a simple introduction into the much broader field of formal languages and automata theory within discrete mathematics.

Improving our understanding of this concept involves practicing with different combinations of parentheses and verifying their balance using a systematic approach, similar to the steps followed in the given problem.

Formal Grammar and Parentheses Structure

Formal grammar plays a crucial role in defining the syntax of programming languages and determining the correct sequence of symbols. It consists of a finite set of rules that describe which strings belong to the language and which do not. In context to our exercise, the grammar for parentheses would involve rules that dictate how parentheses can be properly opened and closed.

In deeper terms, formal grammar defines the rules for generating sequences in a formal language. For balanced parentheses, this typically involves a recursive definition where a sequence is valid if it's empty, or if it contains a valid sequence within a matching pair of parentheses, or if valid sequences are placed adjacent to each other.

An example of a recursive grammar that generates balanced parentheses is as follows:

Grammar for Balanced Parentheses

• S → (S)S | ε

Where S is a non-terminal symbol that represents a possible sequence of balanced parentheses and ε denotes an empty string. This simple rule can generate all possible strings of balanced parentheses and is a powerful tool for parsing.

Parsing Algorithms for Checking Parentheses

Parsing algorithms are computational processes used to analyze a string of symbols according to a given grammar. They serve as automated checks in compilers and interpreters to verify whether code syntax is correct or not, parsing being a fundamental step in compiling programs.

For validating balanced parentheses, a parsing algorithm processes the sequence, often using a stack data structure. Each time the algorithm encounters an opening parenthesis, it 'pushes' it onto the stack. Whenever a closing parenthesis appears, the algorithm 'pops' an item from the stack, checking for a corresponding opening parenthesis.

Let's examine the algorithm with our sequence \( (( )) \) again. The stack starts empty, and as the parser reads each symbol:

• First '(': push onto stack (stack now has one '(')
• Second '(': push onto stack (stack now has two '(')
• First ')': pop from stack (matches with an '(' , stack now has one '(')
• Second ')': pop from stack (matches with the remaining '(', stack is now empty)

Since the stack is empty at the end and all symbols were matched correctly, the sequence is validated as balanced.

Improving parsing algorithms for higher efficiency and accuracy continues to be an important task in computer science, allowing sophisticated analysis of complex programming languages and various forms of data.