Understanding Context-Free Grammars (CFGs) in Computer Science

Posted on Sep 4, 2024 in Computers

Context-Free Grammar

Early Computers and the Need for Higher-Level Languages

  • Early computers required instructions to be spelled out in their own machine language.
  • These primitive instructions were cumbersome and numerous, even for simple tasks.
  • This led to the need for higher-level languages that could translate complex mathematical expressions into single computer instructions.

The Role of Compilers and the Need for Rules

  • Compilers convert high-level language code into assembler language code.
  • Rules are necessary for compilers to process and understand expressions, ensuring valid syntax.
  • Invalid strings of symbols, like “((9)+”, should be rejected.

Limitations of Finite Automata (FA)

  • Finite Automata cannot effectively translate expressions into instructions due to their left-to-right scanning nature.
  • They may generate code before realizing the entire expression is invalid.

Rules for Valid Arithmetic Expressions

Rule 1: Any number is in the set AE.

Rule 2: If x and y are in AE, then so are:

  • (x), (x + y), (x – y), (x*y), (x/y), (x**y)

Example:

((3+4)*(6+7))

This expression can be derived from the rules as follows:

3 is in AE

4 is in AE

(3 + 4) is in AE

6 is in AE

7 is in AE

(6 + 7) is in AE

((3 + 4) * (6 + 7)) is in AE

Conversion to Assembly Language:

LOAD 3 in Register 1

LOAD 4 in Register 2

ADD the contents of Register 2 into Register 1

LOAD 6 in Register 3

LOAD 7 in Register 4

ADD the contents of Register 3 into Register 4

MULTIPLY Register 1 by Register 4

The Analogy to Human Language and Grammar

  • The problem of deciphering computer language expressions is analogous to understanding human language sentences.
  • Grammar, the study of language rules, helps us understand and generate valid sentences.

Semantics vs. Syntax

  • Semantics: Rules involving the meaning of words.
  • Syntax: Rules not involving the meaning of words.
  • In computer languages, syntax is primarily concerned with the structure and arrangement of symbols, while semantics deals with their meaning.

Context-Free Grammar (CFG) Definition

A context-free grammar is a collection of three things:

  1. An alphabet Σ of letters called terminals, used to form the words of a language.
  2. A set of symbols called nonterminals, including the start symbol S.
  3. A finite set of productions of the form: one nonterminal → finite string of terminals and/or nonterminals.

Productions can consist of terminals, nonterminals, or a mixture of both, including the empty string.

At least one production must have the nonterminal S on its left side.

Example of CFG for Arithmetic Expressions

Start → (AE)

AE → (AE + AE)

AE → (AE – AE)

AE → (AE * AE)

AE → (AE / AE)

AE → (AE ** AE)

AE → (AE)

AE → – (AE)

AE → ANY-NUMBER

  • Start initiates the process.
  • AE is the nonterminal.
  • Terminals include “any number” and the symbols: + – * / ** ( )

Defining “ANY-NUMBER”

We can further define “ANY-NUMBER” using additional rules:

Rule 1: ANY-NUMBER → FIRST-DIGIT

Rule 2: FIRST-DIGIT → FIRST-DIGIT OTHER-DIGIT

Rule 3: FIRST-DIGIT → 1 2 3 4 5 6 7 8 9

Rule 4: OTHER-DIGIT → 0 1 2 3 4 5 6 7 8 9

  • Rules 3 and 4 offer choices of terminals.

Example: Generating the Number 1066

ANY-NUMBER → FIRST-DIGIT (Rule 1)

→ FIRST-DIGIT OTHER-DIGIT (Rule 2)

→ FIRST-DIGIT OTHER-DIGIT OTHER-DIGIT (Rule 2)

→ FIRST-DIGIT OTHER-DIGIT OTHER-DIGIT OTHER-DIGIT (Rule 2)

→ 1 0 6 6 (Rules 3 and 4)

Conclusion

Context-Free Grammars provide a formal way to define the syntax of programming languages and other formal systems. They are essential for compiler design and understanding the structure of languages.