Automata Theory and NFA Quiz

Chapter Two: Lexical Analysis

• Lexical analysis is the first phase of a compiler
• It reads the source program's characters
• It groups them into lexemes
• It outputs tokens for each lexeme
• The parser utilizes these tokens for syntax analysis
• This phase constructs tables, including a symbol table (identifiers) and a table of numeric constants
• Separating lexical analysis simplifies compiler design and implementation, improving portability

The Role of the Lexical Analyzer

• Acts as the front-end of a compiler
• Reads and parses the source program
• Groups characters into lexemes
• Produces a sequence of tokens for each lexeme
• The parser uses these tokens for syntax analysis
• Constructs tables of identifiers (symbol table) and numeric constants
• Key reasons for separating the phase:
  • Simplifies compiler design
  • Improves efficiency
  • Enhances portability

The Interaction of the Lexical Analyzer with the Parser

• The lexical analyzer sends tokens to the parser
• The parser receives tokens to perform syntax analysis
• Error messages can be sent back to the lexical analyzer for error handling

Tokens, Patterns, and Lexemes

• A token is a classification of lexical units. A token consists of:
  • A token name
  • An optional attribute value
• Examples: keyword, identifiers, and numbers.
• Token names are input symbols that the parser will process
• A pattern is a description of the form of lexemes within a token
• Example: "letter followed by letters and digits"
• A lexeme is a sequence of characters in the code that matches the token's pattern, and is identified by the lexical analyzer
• Examples of tokens, their descriptions, and sample lexemes are shown

Specification of Tokens

• Regular expressions are used to specify patterns for lexemes
• They are effective in specifying patterns needed for tokens
• These expressions are used by lexical analyzer generators
• These expressions are converted to automata for recognizing specified tokens

Alphabets

• In discrete math, a set is a collection of unique objects.
• Each element is listed only once and can be ordered.
• A set can have an infinite number of elements
• The empty set contains no elements, denoted as {} or Φ

Alphabets (continued)

• An alphabet is a finite, non-empty set of symbols.
• Examples:
  • Set of decimal numbers: Σ = {0, 1, 2, ..., 9}
  • Set of binary numbers: Σ = {0, 1}
  • Set of upper-case English letters: Σ = {A, B, ..., Z}
  • Set of lower-case English letters: Σ = {a, b, ..., z}

Strings

• A string (or word) is a finite sequence of symbols from a given alphabet.
• Examples:
  • If Sigma(Σ) = {a, b}, then abba is a string over the alphabet Σ.
• The order of elements in a string matters
• The length of a string (|w|) is the number of its symbols
• The empty string (ɛ or λ or γ) has zero occurrences of symbols

Strings (continued)

• The set of all strings over an alphabet Σ, including the empty string, is denoted by Σ*.
• Examples:
  • If Σ = {0, 1}, Σ* = {ɛ, 0, 1, 00, 01, 10, 11, 000, ...}
• A non-empty string set from Σ is denoted by Σ+
• Concatenation of two strings w₁ w₂ is the string composed by copying w₁ and copying w₂ after it
• Exponentiation of string w is defined for any integer i ≥ 0
• The reverse of a string w is denoted as wR
• Common string-related terms include prefix, suffix, substring, and subsequence.

Languages

• A language is a set of strings chosen from a particular alphabet.
• Σ* is a notation for a language in general.
• The empty language is denoted by {} or ∅.
• The language containing only the empty string is denoted by {ε}.

Languages (continued)

• Natural languages are spoken by people (English, Spanish)
• Programming languages are formal language (C, Java)
• Languages expressed with formal specifications are used in computers

Languages (continued)

• Common language operations:
  • Union
  • Concatenation
  • Kleene closure (also called star closure)
  • Positive closure

Regular Expressions

• An integer constant token can be defined as: "at least one decimal digit, followed by zero or more additional digits"
• Regular expressions are represented in a compact notation
• Use "*" to denote zero or more instances of a character (Kleene star)
• Use "|" to express alternation between character options within an expression

Regular Expressions (continued)

• Precedence of operators:
  • * (Kleene star) has highest precedence.
  • Concatenation has medium precedence.
  • Union (|) has lowest precedence.
  • All operators are left associative

Regular Expressions (continued)

• Regular expressions are a formal notation for language patterns.
• These expressions specify the formal grammar for tokens

Regular Definitions

• A regular definition is a sequence of definitions, where each definition assigns a new symbol to a regular expression.
  • Each new symbol can appear in subsequent expressions.

Regular Definitions (continued)

• Example definitions:
  • Examples include for identifiers and possible numeric formats

Regular Expressions (continued):

• Extended Regular Expressions:
  • One or more instances: (r)+
  • Zero or one instances: r?
  • Character classes: [abc]
• Examples of applications in practice are to improve the efficiency of the formal specifications of regular expressions

Recognition of Tokens

• Starting point: the grammar of the programming language
• Formalize the patterns using regular expressions
• Implement using transition diagrams (finite automata)

Recognition of Tokens (continued)

• Handle whitespace (ws) with suitable notation
• Summary of lexical analyzer goal using tokens, names, and attribute values

Finite Automata (FA)

• A finite automaton (FA) is a mathematical model for recognizing patterns in strings
• Can be deterministic or nondeterministic
• The finite state machine (FSM) is a theoretical model of computation
• Has a fixed number of states
• Moves through states based on input symbols

Finite Automata (FA) (continued)

• Input tape: contains the input string to be processed
• Finite automaton: processes the input string for recognition
• Output: indicates acceptance or rejection of the input

Finite Automata (FA) (continued)

• Types of finite automata:
  • Deterministic finite automata (DFA)
  • Nondeterministic finite automata (NFA)

Finite Automata (FA) (continued)

• Designing an FA:
  • Analyze the patterns/strings the automata should accept
  • Ensure an output state exists for each possible input symbol on each possible state
  • Make sure there is one starting state and at least one accepting state that the input gets to
  • Use a transition diagram (graph) to represent the possible transitions

Finite Automata (FA) (continued):

• An NFA can consume the empty string during a state transition

Finite Automata (FA) (continued):

• Representation:
  • Can be represented using a transition table
• Implementation details:
  • Use arrays to represent transition table contents

Finite Automata (FA):

Transition diagrams for specific tokens (reserved words, identifiers, numbers) and whitespace.

Regular Languages

• A language is regular if there exists a DFA that accepts the language
• Examples of regular languages are provided

Nondeterministic Finite Automata (NFA)

• An NFA is a variation of finite automata where a transition can happen in parallel, or multiple transitions on the same symbol are possible.
• NFAs may have a lambda transition, where no input symbol is consumed (transition happens without consuming input symbol)

NFA Procedures

• Follow NFA rules to determine when an NFA accepts input, otherwise reject
• String is accepted if an accepting state is encountered

Equivalence of Machines (NFA and DFA)

• If both machines (NFA and DFA) represent the same formal language (L) then the machines are said to be equivalent.

Conversion of NFA to equivalent DFA

• Step 1: Start by defining the initial state of the equivalent DFA based on the initial state of the NFA

• Step 2: Create new states from the set of states in the NFA. Calculate the 8* transition function for each state (e.g., δ* (q0, a) = {q1})

• Step 3: Repeat Step 2 for each state to generate an equivalent state transition table to the resulting DFA.

• Step 4: Determine the final states based on the NFA's accepting states. Any state with an NFA accepting state will be made an accepting state in the corresponding DFA

Test your understanding of automata theory concepts, particularly focusing on Non-deterministic Finite Automata (NFA). This quiz covers topics like language acceptance, state transitions, and the representation of languages. Dive into the fundamental aspects that define the behavior of automata.