Question

a. Write a BNF grammar that describes the structure of a nonterminal called \(\langle\) number \(\rangle\). Assume that contains an optional + sign followed by exactly two decimal digits, the first of which cannot be a 0 . Thus \(23,+91\), and \(+40\) are legal, but \(9,+01\), and 123 are not. b. Using your grammar from Exercise \(5 a\), show a parse tree for the value \(+90\).

Short answer

BNF grammar for \( \langle number \rangle \) is correct, and the parse tree for '+90' demonstrates its application accurately.

Step-by-step solution

Understanding BNF (Backus-Naur Form)

BNF is a notation used to express the grammar of a language in a formal way. It defines a set of production rules that describe how sentences in a language are generated. Non-terminal symbols are typically enclosed in angle brackets \( \langle ... \rangle \).

Identifying Rules for \( \langle number \rangle \)

The problem states that \( \langle number \rangle \) consists of an optional '+' sign followed by two decimal digits, with the first digit not being '0'. This can be expressed in BNF as follows:\[ \langle number \rangle ::= [\texttt{+}] \langle non-zero-digit \rangle \langle digit \rangle \]Where \( \langle non-zero-digit \rangle \) can be any digit from 1 to 9, and \( \langle digit \rangle \) can be any digit from 0 to 9.

Defining Non-terminals for BNF

Define the non-terminal symbols for digits:\[ \langle non-zero-digit \rangle ::= 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9 \]\[ \langle digit \rangle ::= 0 \mid 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9 \]

Writing Complete BNF Grammar

Combine the rules into a complete BNF grammar:\[ \begin{align*}\langle number \rangle &::= [\texttt{+}] \langle non-zero-digit \rangle \langle digit \rangle\\langle non-zero-digit \rangle &::= 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9 \\langle digit \rangle &::= 0 \mid 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9 \end{align*} \]

Constructing the Parse Tree for '+90'

To parse '+90' using the defined grammar, follow these steps:1. Start with \( \langle number \rangle \).2. Choose the option with \texttt{+}, since '+90' starts with a plus.3. The first digit (non-zero) is '9', matching \( \langle non-zero-digit \rangle \).4. The second digit is '0', matching \( \langle digit \rangle \).The parse tree is structured as follows:```\( \langle number \rangle \) | \texttt{+} | \langle non-zero-digit \rangle / 9``````\( \langle number \rangle \) | \langle digit \rangle / 0```

Verify Parse Tree

Check that each part of '+90' is correctly interpreted by the grammar rules:- The optional plus sign is present.- The first digit '9' complies with \( \langle non-zero-digit \rangle \).- The second digit '0' complies with \( \langle digit \rangle \).The parse tree correctly represents '+90' following our BNF grammar.

Key concepts

Syntax Rules

When discussing BNF grammar, one of the first things to understand is its **syntax rules**. Syntax rules are like a set of instructions that outline how expressions in a language should be formed. In BNF (Backus-Naur Form), these rules are used to precisely define the syntactic structure of a language, such as computer programming languages or even parts of human language.
BNF rules break down complex expressions into simpler ones, using a series of productions or transformations. Each production rule typically consists of a non-terminal symbol and a string of terminal and/or other non-terminal symbols. This allows for the recursive definition of structures within the language.

• For example, in BNF, the syntax rule for a "number" might look like this: \( \langle number \rangle ::= [\texttt{+}] \langle non-zero-digit \rangle \langle digit \rangle \). This states that a number can optionally start with a "+" sign, followed by two digits, where the first is non-zero.

Syntax rules are crucial because they ensure that the expressions formed are correct and meaningful within the language, which makes them an essential part of grammars described using BNF.

Parse Tree

A **parse tree** is a graphical representation of the syntactic structure of a sentence or string as described by a grammar, such as the BNF grammar you might utilize. It visually breaks down the sentence into its component parts based on the applied production rules of the grammar.
Each node in a parse tree represents a construct that can be further divided or broken down into parts (nodes) until the terminal or leaf nodes are reached, which represent the actual input being parsed. In this way, a parse tree shows one way a string can be derived from a start symbol according to the grammar rules.

• For instance, the parse tree for "+90" based on our BNF grammar begins with the non-terminal symbol \( \langle number \rangle \) and shows branches for "\texttt{+}", "9" as a \( \langle non-zero-digit \rangle \), and "0" as a \( \langle digit \rangle \).

The parse tree provides a systematic way to visualize and verify how each part of the sentence fits into the defined grammar, ensuring correctness and understanding of how expressions are constructed.

Non-Terminal Symbols

**Non-terminal symbols** in BNF grammar play a pivotal role in defining the structure of expressions. These symbols act as placeholders in syntax rules, representing patterns or sequences that can be replaced by other sequences according to the grammar.
Non-terminal symbols are always enclosed within angle brackets \( \langle ... \rangle \), distinguishing them from terminal symbols, which are the basic elements of the language like digits or letters. Non-terminals help form the hierarchical structure needed to break down complex expressions into simpler ones.

• In our example with \( \langle number \rangle \), non-terminal symbols like \( \langle non-zero-digit \rangle \) and \( \langle digit \rangle \) are used to describe parts of a number format allowing grammar to be concise yet infinitely expandable.

These symbols are integral to allowing BNF grammar to be as expressive and powerful as needed for accurately capturing the constraints and formations within a language. Non-terminal symbols, in essence, are what give life and flexibility to a grammar's framework.