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Chapter 11: Problem 76

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

Short Answer

Expert verified
The given sequence is \( ( ) ( ( ) ) \). We can break it down into individual characters and analyze each one to match opening and closing parentheses. In doing so, we find that all opening parentheses have a corresponding closing one, making this sequence a valid sequence of parentheses.

Step by step solution

01

Break down the given sequence into individual characters

We will analyze the sequence \( ( ) ( ( ) ) \) one character at a time, counting the open and closed parentheses as we go.
02

Analyze each character

1. The first character is an opening parenthesis: \( ( \) 2. The second character is a closing parenthesis: \( ) \) - This matches the first opening parenthesis, so we know we have one valid pair. 3. The third character is an opening parenthesis: \( ( \) 4. The fourth character is an opening parenthesis: \( ( \) - Notice that this does not close the previous opened parenthesis. Keep this in mind. 5. The fifth character is a closing parenthesis: \( ) \) - This matches the second opening parenthesis (the one we noticed that did not close previously). Now we have another valid pair. 6. The sixth character is a closing parenthesis: \( ) \) - This matches the third opening parenthesis, giving us the final valid pair.
03

Verify the sequence

We've matched all opening parentheses with their corresponding closing ones and have found valid pairs. Therefore, the given sequence \( ( ) ( ( ) ) \) is a valid sequence of parentheses.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formal Grammar
When discussing formal grammar, we are referring to a set of rules for generating strings in a language. In this context, a language is simply a collection of strings made from an alphabet, which in the case of parenthesis matching, consists of just two symbols: an opening parenthesis '(' and a closing parenthesis ')'.

A formal grammar for parentheses defines the correct arrangements of these symbols, and this is crucial in programming languages and mathematical computation where expressions need to be unambiguous. The grammar rules ensure that each opening parenthesis has a corresponding closing parenthesis and that they are properly nested.

Why Formal Grammar Matters

Understanding formal grammar is essential for parsing expressions correctly. It helps us to construct compilers and interpreters that can read and execute code without errors. In parentheses matching, formal grammar serves as the backbone that provides a structured way to determine if a sequence of parentheses is properly balanced or not.
Parentheses Matching
The concept of parentheses matching is a practical application of formal grammar. It involves scanning a sequence of parentheses to see if each opening parenthesis is matched correctly with a closing one. An important aspect of this process is the concept of balance and nesting.

For a sequence to be valid, each opening parenthesis must be followed by a corresponding closing parenthesis in the correct order. The sequence must not have extra opening or closing parentheses, and they must be properly nested within each other, as is the case with the sequence in the original exercise.

How to Match Parentheses

One common method for checking balance is to use a stack data structure. Every time an opening parenthesis is encountered, it is 'pushed' onto the stack, and when a closing parenthesis appears, it is 'popped' from the stack, which should contain the matching opening parenthesis. If the stack is empty at the end of this process, the sequence is valid.
Discrete Mathematics
The field of discrete mathematics encompasses a variety of topics that are fundamental to computer science, one of which is the study of algorithms for parsing and validating structures, like balanced parentheses sequences.

Discrete mathematics provides the theoretical underpinnings for algorithms and data structures used in solving such problems. This field covers topics such as combinatorics, graph theory, and logic鈥攁ll of which contribute to a deeper understanding of the algorithms for parentheses matching.

Role of Discrete Mathematics in Parentheses Matching

In the context of parentheses matching, discrete mathematics would study the algorithmic approach for validating sequences, analyze its efficiency, and even explore the number of possible valid parentheses sequences of a certain length. Understanding these concepts can not only help in programming but also in enhancing one's logical reasoning skills applicable in a wide range of disciplines.

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Most popular questions from this chapter

Construct a NDFSA that accepts the language generated by the regular grammar \(G=(N, T, P, \sigma),\) where: $$\begin{aligned}&N=\\{\sigma, \mathrm{A}, \mathrm{B}\\}, T=\\{\mathrm{a}, \mathrm{b}\\}, \text { and } P=\\{\sigma \rightarrow \mathrm{a} \mathrm{A}, \sigma \rightarrow \mathrm{b} \mathrm{A}, \mathrm{A} \rightarrow \mathrm{aB}, \sigma \rightarrow\\\ &\mathbf{b}, \mathbf{B} \rightarrow \mathbf{b}\\} \end{aligned}$$

Mark each as true or false, where \(A\) and \(B\) are arbitrary finite languages. \(|A B|=|B A|\)

Define the language \(L\) of all binary representations of non-negative integers recursively.

For Exercises \(68-73,\) use the following definition of a simple algebraic expression: $$\langle\text {expression}\rangle : :=\langle\text { term }\rangle |\langle\text { sign }\rangle\langle\text { term }\rangle |$$ $$\langle\text { expression }\rangle\langle\text { adding operator }\rangle\langle\text { term }\rangle$$ $$\langle\operatorname{sign}\rangle \therefore=+ 1-$$ $$\langle\text { adding operator}\rangle: :=+1-$$ $$\langle\text { term }\rangle : :=\langle\text { factor }\rangle |$$ $$\langle\text { term }\rangle\langle\text { multiplying operator }\rangle\langle\text { factor }\rangle$$ $$\langle\text { multiplying operator }\rangle := *| /$$ $$\langle\text { factor }\rangle : :=\langle\text { letter }|\rangle (\langle\text { expression }\rangle |\langle\text { expression }\rangle$$ $$\langle\text { letter }\rangle : := a|b| c | \ldots : z$$ Determine if each is a legal expression. $$a+b *(c / d)$$

Develop a grammar that generates each language over {0,1}. The set of words with prefix 00.
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