Backus-Naur Form (BNF): A Notation Technique for Context-Free Grammars

August 31, 2024 5 min read Computer Science Information Technology BNF Programming Languages Syntax Formal Language Grammar

Backus-Naur Form (BNF) is a notation technique used for describing the syntax of programming languages, pioneered in the specification of ALGOL 60.

Backus-Naur Form, often abbreviated as BNF, is a formal notation technique for defining the syntax of languages, particularly programming languages. It was first introduced by John Backus in his work on the ALGOL 60 programming language, and further refined by Peter Naur. This method was pioneering in formal language specification and has become an essential tool in the field of computer science.

Key Figures

John Backus: An American computer scientist known for developing the FORTRAN programming language and introducing BNF.
Peter Naur: A Danish computer scientist who contributed to refining BNF in the context of ALGOL 60.

Types/Categories of Grammar in BNF

BNF specifically defines context-free grammars which fall under Chomsky Hierarchy, classified as Type-2 grammars. These grammars are suitable for languages where the syntax can be described without needing to consider the context in which they appear.

Key Events

1959: Introduction of BNF by John Backus for the ALGOL 60 programming language.
1963: Peter Naur refined and popularized the use of BNF in ALGOL 60 report.

Detailed Explanation

BNF uses a series of rules (productions) to describe the structure of sentences in a language. Each rule specifies a single non-terminal symbol and the patterns of symbols that can replace it.

Basic Syntax of BNF

The fundamental components of BNF are:

Non-Terminal Symbols: Denoted by <angle brackets>, these represent abstract concepts.
Terminal Symbols: The actual characters or tokens of the language.
Production Rules: Descriptions of how non-terminal symbols can be expanded into sequences of non-terminals and terminals.
Derivation: The process of replacing non-terminals with terminals using the production rules.

Example of BNF for a simple arithmetic expression:

1<expression> ::= <term> "+" <term>
2<term> ::= <factor> "*" <factor> | <factor>
3<factor> ::= "(" <expression> ")" | <number>
4<number> ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

Chart/Diagram in Mermaid Format

Below is a Mermaid diagram showing a derivation in BNF:

    graph TD
	    A[<expression>] --> B[<term> + <term>]
	    B --> C[<factor> * <factor> + <factor> * <factor>]
	    C --> D["(" <expression> ")"]
	    D --> E[0]
	    D --> F[1]
	    D --> G[2]
	    D --> H[3]
	    D --> I[4]
	    D --> J[5]
	    D --> K[6]
	    D --> L[7]
	    D --> M[8]
	    D --> N[9]

Importance

BNF is critical for:

Language Design: Defining the syntax of new programming languages.
Compiler Construction: Providing a formal basis for parsing and interpreting code.
Documentation: Precisely describing the syntax of languages for educational and reference purposes.

Applicability

BNF is applied in numerous domains including:

Programming Language Development: For defining language grammar.
Compiler Design: For syntax analysis and parsing.
Software Engineering Education: Teaching formal methods and language theory.

Examples

Example: Simple Arithmetic Expression Grammar

1<expression> ::= <term> "+" <term> | <term>
2<term> ::= <factor> "*" <factor> | <factor>
3<factor> ::= "(" <expression> ")" | <number>
4<number> ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

Example: Defining a Simple Programming Language

1<program> ::= <statement> | <statement> ";" <program>
2<statement> ::= <assignment> | <if_statement> | <while_statement>
3<assignment> ::= <variable> "=" <expression>
4<if_statement> ::= "if" <condition> "then" <statement> "else" <statement>
5<while_statement> ::= "while" <condition> "do" <statement>

Considerations

When using BNF, it’s important to:

Maintain Consistency: Ensure rules are non-ambiguous.
Handle Recursion: Properly define and handle recursive rules to avoid infinite loops.
Optimize for Readability: Write clear and understandable rules.

Context-Free Grammar: A grammar where the productions replace non-terminals independently of the surrounding symbols.
Syntax Tree: A tree representation of the syntactic structure of a language, based on BNF rules.
Parsing: The process of analyzing a string of symbols using a formal grammar.

Comparisons

BNF vs. EBNF

Extended BNF (EBNF): An extension of BNF that includes additional notations for conciseness such as optionality, repetition, and grouping.

BNF vs. PEG

Parsing Expression Grammar (PEG): Another formal grammar system that is deterministic and easier to implement, but less expressive than BNF.

Interesting Facts

BNF not only influenced the development of programming languages but also impacted natural language processing and formal language theory.
Many modern programming languages, including Python and JavaScript, use derivatives of BNF for their syntax definitions.

Inspirational Stories

The Legacy of John Backus

John Backus’s contribution to computer science went beyond BNF. He was awarded the Turing Award in 1977 for his work on programming languages and for creating the first high-level language, FORTRAN. His innovative thinking laid the groundwork for modern software engineering.

Famous Quotes

“One accurate measurement is worth a thousand expert opinions.” – Grace Hopper, an advocate of formal methods in computing.

Proverbs and Clichés

“Syntax is the seatbelt of programming.”
“A grammar a day keeps the bugs away.”

Expressions, Jargon, and Slang

Parse: To analyze a string of symbols based on a formal grammar.
Grammar: The set of rules that define the structure of a language.
Token: The smallest unit in a language.

FAQs

What is BNF used for?

BNF is used to describe the syntax of programming languages and other formal languages in a precise, unambiguous way.

How is BNF different from regular expressions?

BNF can describe hierarchical structures and recursive patterns, which regular expressions cannot.

Can BNF describe all programming languages?

BNF can describe the syntax but not the semantics (meaning) of programming languages.

References

Backus, J. W. (1960). “The Syntax and Semantics of the Proposed International Algebraic Language of Zurich ACM-GAMM Conference”. Proceedings of the International Conference on Information Processing.
Naur, P. (1963). “Revised Report on the Algorithmic Language ALGOL 60”. Communications of the ACM.
Aho, A. V., Sethi, R., & Ullman, J. D. (1986). “Compilers: Principles, Techniques, and Tools”. Addison-Wesley.

Final Summary

Backus-Naur Form (BNF) is an essential notation technique for defining the syntax of programming languages. With its roots in the development of ALGOL 60, BNF has become a fundamental tool in compiler design, language specification, and software engineering education. By using a structured series of production rules, BNF enables precise and unambiguous description of language syntax, ensuring clear and accurate communication in the development and documentation of programming languages.