11.1 Languages and Grammars

Languages and grammars are the building blocks of computation. They define how we communicate with computers and how computers interpret our instructions. This topic introduces the fundamental concepts of alphabets, strings, and formal languages.

Grammars provide the rules for generating valid strings in a language. We'll explore different types of grammars, from simple regular grammars to more complex context-free grammars, and see how they're used in real-world applications like programming languages and natural language processing.

Fundamentals

Alphabets and Strings

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• Alphabet consists of a finite set of symbols used to construct strings
• Symbols can include letters, numbers, or special characters (A, B, C, 1, 2, 3, @, #, $)
• String represents a finite sequence of symbols from the alphabet
• Empty string, denoted by ε, contains no symbols and has length zero
• String length measures the number of symbols in a string
• Concatenation operation combines two strings by appending one to the end of the other

Languages and Their Properties

• Language defines a set of strings over a given alphabet
• Can be finite or infinite in size
• Formal language specifies strings according to precise rules
• Natural language encompasses human spoken or written communication (English, Spanish, Mandarin)
• Language operations include union, intersection, and complement
• Closure property ensures a language remains closed under certain operations

Grammar Fundamentals

• Grammar provides a formal description of the structure of a language
• Consists of a set of rules for generating valid strings in the language
• Formal grammar includes four components: terminals, non-terminals, production rules, and start symbol
• Terminals represent the basic symbols of the alphabet
• Non-terminals serve as placeholders for patterns of terminals
• Start symbol designates the initial non-terminal for deriving strings
• Derivation process applies production rules to generate strings in the language

Grammar Components

Production Rules and Their Application

• Production rules define how to replace non-terminals with combinations of terminals and non-terminals
• Left-hand side (LHS) of a rule contains a single non-terminal
• Right-hand side (RHS) consists of a string of terminals and non-terminals
• Rules written in the form A → α, where A is a non-terminal and α is a string of terminals and non-terminals
• Derivation applies production rules sequentially to generate strings
• Leftmost derivation replaces the leftmost non-terminal at each step
• Rightmost derivation replaces the rightmost non-terminal at each step

Chomsky Hierarchy and Language Classifications

• Chomsky hierarchy categorizes formal grammars into four types
• Type 0 (unrestricted) grammars generate recursively enumerable languages
• Type 1 (context-sensitive) grammars produce context-sensitive languages
• Type 2 (context-free) grammars generate context-free languages
• Type 3 (regular) grammars create regular languages
• Each grammar type is a proper subset of the types above it in the hierarchy
• Hierarchy reflects the increasing restrictions on production rules from Type 0 to Type 3

Regular and Context-Free Languages

• Regular languages represent the simplest class of formal languages
• Can be described by regular expressions or finite automata
• Production rules for regular grammars have the form A → aB or A → a
• Context-free languages form a broader class than regular languages
• Described by context-free grammars or pushdown automata
• Production rules for context-free grammars have the form A → α, where A is a non-terminal and α is any string of terminals and non-terminals
• Context-free languages include nested structures (balanced parentheses, HTML tags)

Language Processing

Parsing Techniques and Applications

• Parsing analyzes the structure of a string according to the rules of a formal grammar
• Top-down parsing starts with the start symbol and attempts to derive the input string
• Bottom-up parsing begins with the input string and tries to reduce it to the start symbol
• Recursive descent parsing implements top-down parsing using a set of mutually recursive procedures
• Shift-reduce parsing employs a bottom-up approach using a stack and input buffer
• Parse tree represents the hierarchical structure of a string according to the grammar
• Abstract syntax tree (AST) provides a more compact representation of the parsed structure
• Parsing techniques find applications in compiler design, natural language processing, and XML processing