Question

a. Write a BNF grammar that describes the structure of U.S. telephone numbers, which can be either \((x x x) x x x-x x x x\) or \(x x x-x x x x\), where \(x\) can be any digit from 0 to 9 . b. Modify your grammar from Exercise 6a so that (1) the middle digit of an area code must be either a 0 or a \(1,(2)\) the first digit of an area code cannot be a 0 or a 1 , and (3) the first digit of the seven-digit phone number cannot be a 0 or a 1 . c. Using your grammar from either Exercise \(6 a\) or \(6 b\), show a parse tree for the phone number (612)555-1212.

Short answer

Write a BNF grammar to define the phone number format, modify it as per given rules, and create a parse tree.

Step-by-step solution

Understand BNF Grammar

BNF, or Backus-Naur Form, is a notation technique for context-free grammars, often used to describe the syntax of languages. It consists of a set of production rules that define how the elements of the language are constructed.

Define BNF Grammar for US Telephone Numbers

We need to define the structure of a U.S. phone number. In BNF, this can be written as:
::= |
::= '(' ')'
::=
::=
::=
::= '-'
::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'

Modify BNF Grammar with Area Code and Number Conditions

Adjust your grammar to incorporate new conditions. Now the BNF becomes:
::= |
::= '(' ')'
::=
::=
::=
::= '-'
::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
::= '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
::= '0' | '1'

Create Parse Tree for (612)555-1212

Using the modified grammar, create a parse tree: - - - '(' - : '6' - : '1' - : '2' - ')' - : '-' - - - : '5' - : '5' - : '5' - : '-' - - : '1' - : '2' - : '1' - : '2'

Key concepts

Syntax Representation

BNF, which stands for Backus-Naur Form, is a method used to represent the syntax of a language. It helps in defining the structural composition of languages using context-free grammar. With BNF, we use a collection of production rules. These rules are guidelines about how to form valid sentences or expressions. The left side of a production rule indicates a non-terminal symbol. These are categories or groups within the structure. The right side shows the series of symbols (both terminal and non-terminal) that comprise a member of that category.
For instance, when creating a BNF grammar for U.S. telephone numbers, you define different components such as area codes and the digits. Each component follows specific rules, combining them to form the complete structure. This clear and methodical approach using BNF ensures linguistics are consistent and easily understandable.

U.S. Telephone Numbers

In the context of syntax representation through BNF, U.S. telephone numbers present a unique structure. These numbers can appear in two primary formats:

• Area code, then phone number (like (xxx) xxx-xxxx)
• Phone number only, starting with three digits (xxx-xxxx)

An area code is usually presented in brackets and consists of three digits. It must follow specific rules: the first digit must be between 2 and 9, and the middle digit can only be 0 or 1. The seven-digit phone number is split into two segments:

• The first segment (or prefix) must start with a digit from 2 to 9.
• A four-digit second segment completes the number.

These clearly defined formats and restrictions ensure that U.S. phone numbers have a standardized appearance.

Parse Tree

A parse tree is a visual representation commonly used in programming and linguistics to reflect the syntactic structure of a string as defined by a grammatical rule set, such as BNF. Each node in the tree represents a construct occurring in the grammar. This concept is helpful in verifying that the strings or sentences you create adhere to the established rules.
In creating a parse tree for a valid U.S. phone number like (612)555-1212, each section of the number (area code and phone number) is broken down to its components:

• The root node is the phone number itself.
• Branch off to nodes for the area code, separating each digit of '612' according to our conditions.
• Continue with nodes for the seven-digit number '555-1212'.

This segmentation into hierarchical levels simplifies the verification process by making it easy to see if each piece of the phone number string follows the rules of the grammar. Thus, it serves as an essential tool for grammarians and programmers alike.