Context-Free Grammars and Their Notation

In the field of computer science and formal language theory, context-free grammars (CFGs) play a crucial role in designing and constructing compilers. A CFG is a powerful formalism that describes the syntax or structure of a programming language. It helps in parsing and understanding the source code, enabling the compiler to generate machine code or execute the program.


A context-free grammar consists of a set of production rules, non-terminal symbols, terminal symbols, and a start symbol. The rules define a language by specifying how the non-terminal symbols can be replaced by a sequence of terminal and/or non-terminal symbols. The generation of strings within the language starts from the start symbol and expands using the production rules until only terminal symbols remain.

The notation used to describe a CFG follows a standard format:

  • Non-Terminal Symbols: Represented by uppercase letters (e.g., A, B, S), they are variables that can be replaced by another sequence of symbols.
  • Terminal Symbols: Represented by lowercase letters (e.g., a, b, c), they are the basic units or tokens of the language.
  • Production Rules: Represented by "->", they describe the transformation of non-terminal symbols using a sequence of terminal and/or non-terminal symbols.
  • Start Symbol: The initial symbol from which the generation of strings begins.


Let's consider a simple example of a CFG for a basic arithmetic expression language:

E -> E + T | T
T -> T * F | F
F -> ( E ) | id

Here, E, T, and F are non-terminal symbols, and the symbols +, *, (, ), and id are terminal symbols representing addition, multiplication, opening and closing parentheses, and identifiers respectively. The production rules specify the allowed transformations and combinations.


A derivation in a CFG involves applying the production rules iteratively to generate strings within the language. Starting with the start symbol, each step replaces a non-terminal symbol according to the selected production rule. The process continues until only terminal symbols remain.

For example, considering the CFG mentioned earlier, the derivation of the expression id * id + id would be as follows:

E => E + T => T + T => F + T => id + T => id + T * F => id + F * F => id + id * F => id + id * id

Ambiguity and Parsing

Sometimes, a CFG can be ambiguous, meaning that a given string can be derived using multiple sequences of production rules. Ambiguity can arise when there are overlapping or conflicting rules. In the case of programming languages, ambiguity can lead to ambiguity in the meaning of the code.

To parse a string according to a CFG, we need techniques such as top-down parsing or bottom-up parsing. These parsing algorithms use the production rules and a parsing table to construct a parse tree, which represents the syntactic structure of the string.


Context-free grammars provide a formal and systematic way to describe the syntax of a programming language. The notation used to represent CFGs allows designers and compiler developers to define the rules and transformations necessary for parsing and understanding the source code. Understanding CFGs and their notation is essential for anyone involved in compiler design and programming language development.

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