/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 Use the grammar in Exercise 74 t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 78

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

Short Answer

Expert verified
The given sequence of parentheses is \(()()()\). Using the grammar from Exercise 74, we can generate this sequence through the following steps: 1. Start with the initial non-terminal: \(P\) 2. Apply production 1: \(P \rightarrow (P)P\) giving \((P)P\) 3. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) giving \(()P\) 4. Apply production 1: \(P \rightarrow (P)P\) giving \(()(P)P\) 5. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) giving \(()()P\) 6. Apply production 1: \(P \rightarrow (P)P\) giving \(()()(P)P\) 7. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) giving \(()()()\) Since we were able to generate the given sequence using the grammar, it is a valid sequence of parentheses.

Step by step solution

01

Introduce Exercise 74 Grammar

In Exercise 74, the grammar for generating valid sequences of parentheses is given by the following two productions: 1. \(P \rightarrow (P)P\) 2. \(P \rightarrow \epsilon\) A string of parentheses is considered valid if it can be generated by this grammar.
02

Demonstrate the Generation of the Test Sequence

We want to determine if the sequence \(()()()\), is valid using the grammar from Exercise 74. Let's attempt to generate the given sequence: 1. Start with the initial non-terminal: \(P\) 2. Apply production 1: \(P \rightarrow (P)P\) \((P)P\) 3. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) \(()P\) 4. Apply production 1: \(P \rightarrow (P)P\) \(()(P)P\) 5. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) \(()()P\) 6. Apply production 1: \(P \rightarrow (P)P\) \(()()(P)P\) 7. Apply production 2: \(P \rightarrow \epsilon\) to the first non-terminal \(P\) \(()()()\) Now that we have generated the given sequence, we can conclude that it is a valid sequence of parentheses.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Context-Free Grammar
In the world of formal languages, a context-free grammar (CFG) is a set of recursive rewriting rules, also known as productions, used to generate patterns of strings. A CFG is composed of a set of terminal symbols, which are the characters of the alphabet that appear in the strings generated by the grammar, non-terminal symbols, which can be replaced by groups of terminal and non-terminal symbols, a start symbol, and a set of production rules, which are transformations that replace a non-terminal symbol with a group of terminal and non-terminal symbols.

The power of CFGs lies in their ability to describe many natural syntactical forms, such as the valid sequences of parentheses in source code, mathematical expressions, or in the syntax of programming languages. For example, considering the grammar given in Exercise 74, the production rules enforce that every opening parenthesis must be matched with a closing parenthesis and that the parentheses must be properly nested.
Formal Languages
Formal languages are sets of strings formed from an alphabet and are governed by specific grammatical rules. They are essential for defining the syntax of programming languages, creating patterns in computer science, and even analyzing human languages. When a sequence of characters (or 'string') adheres to a given set of rules, it's said to be a 'word' in the formal language.

In discrete mathematics and computer science, the study of formal languages provides the theoretical underpinnings for the compilation and interpretation of programming languages. This understanding allows us to determine if a sequence of code is syntactically correct, which is crucial for both writing and understanding code.
Discrete Mathematics
Discrete mathematics lies at the heart of computer science. It encompasses a variety of topics that deal with discrete elements that assume distinct, separate values. This differs from continuous mathematics, which deals with objects that can vary smoothly. Discrete mathematics includes studies of algorithms, graphs, and mathematical logic, as well as the design of computer systems.

Understanding the concepts of discrete mathematics is essential for solving problems in algorithm design and system optimization. In the case of parentheses matching, the discrete structure of the sequences allows for mathematical proofs that can verify the correctness of an algorithm designed to recognize valid sequences of parentheses. It provides a solid foundation for addressing problems related to computability, complexity, and formal grammar construction.
Parentheses Generation
Parentheses generation refers to creating all possible combinations of well-formed parentheses. A valid sequence of parentheses is one that can be correctly matched, meaning that each left parenthesis has a corresponding right parenthesis and that the pairs are properly nested. This is a common exercise in computer science, particularly in algorithm design and programming languages that require syntax parsing.

The process can be algorithmically implemented using a context-free grammar, as it was shown in Exercise 74, or by employing recursive functions and backtracking techniques. In practical applications, these algorithms make it possible to parse expressions in programming languages, validate input in web forms, and even format code in text editors. Understanding the generation of valid parentheses sequences is a good example of intersecting knowledge of formal languages, context-free grammars, and discrete mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Create a NDFSA that accepts the regular language over \(\\{\mathrm{a}, \mathrm{b}\\}\) of strings that: Begin with \(a a\) , but not end in \(b b.\)

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. $$0.23$$

Let \(m\) denote the number of \(a^{\prime} s\) in a string. Design an FSA that accepts strings over \(\\{a, b\\}\) which: Contain baab as a substring.

By making a DFSA, define a regular grammar \(G=(N, T, P, \sigma)\) that generates the language consisting of strings over \(\\{a, b\\}\) that: Contain \(a a a\) as a substring.

The production rules of a grammar for simple arithmetic expressions are: $$\langle\text { expression }\rangle :=\langle\text { digit })(\langle\text { expression })) |+(\langle\text { expression }\rangle) |$$ $$-(\langle\text { expression }\rangle) | \langle\text { expression }\rangle \langle\text { operator }\langle\text { expression }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\text { operator }\rangle : :=+|-| / | \uparrow$$ Use this grammar to answer Exercises \(52-59\). Determine if each is a valid arithmetic expression. $$2 * 3+4$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.