Implementation of Lexical Analysis

Outline

• Specify lexical structure using regular expressions

• Finite automata

  • Deterministic Finite Automata (DFA)

  • Non-deterministic Finite Automata (NFA)

• Implementation of regular expressions

  • Regular expression \(\rightarrow\) NFA \(\rightarrow\) DFA \(\rightarrow\) Tables

Regular Expressions in Lexical Specification

• A regular expression specifies a predicate \(s \in L(R)\), that is, is a string \(s\) a member of the language \(L(R)\)

• Testing for set membership is not enough, we need to partition the input into tokens

• We can adapt regular expressions to meet this goal

Regular Expressions to Lexical Specification

1. Select a set of tokens

   • Integer, Keyword, Identifier, \(\ldots\)

2. Write a regular expression (or rule) for the lexemes of each token

   • Integer = \([0123456789]+\)

   • Keyword = \((if \; | \; else \; | \; \ldots)\)

   • Identifiers: \([A-Za-z] \; ( \; [A-Za-z] \; | \; [0123456789] \; )*\)

Regular Expressions to Lexical Specification

3. Construct a regular expression that matches all lexemes for all tokens

   • \(R\) = Integer \(|\) Keyword \(|\) Identifier \(|\) \(\ldots\)

   • \(R\) = \(R_1\) \(|\) \(R_2\) \(|\) \(R_3\) \(|\) \(\ldots\)

• If \(s \in L(R)\) then \(s\) is a lexeme

  • Furthermore \(s \in L(R_i)\) for some \(i\)

  • This \(i\) determines the token that is reported

Regular Expressions to Lexical Specification

4. Let the input be \(x_1 \ldots x_n\)

   • \(x_1 \ldots x_n\) are characters

   • For \(1 \leq i \leq n\) check if \(x_1 \ldots x_i \in L(R)\)

   • If so, it must be that \(x_1 \ldots x_i \in L(R_j)\) for some \(j\)

   • Otherwise, \(s \notin L(R)\)

5. Remove \(x_1 \ldots x_i\) from the input and got to the previous step

Options for Handling Whitespace and Comments

1. We could create a token for whitespace or comments

   • Whitespace = \((\; \texttt{' '} \; | \; \texttt{'\n'} \; | \; \texttt{'\t'} \; )+\)

   • Comment = \(\ldots\)

   • An input of " \t\n 42 " is transformed to the token stream Whitespace Integer Whitespace

2. The lexer skips whitespace and comments

   • This is the preferred method because whitespace and comments are irrelevant to the parser (for most languages)

   • The lexer still needs to match a whitespace (or comment) regular expression, but a token is not output

Ambiguities

• How much input is used?

  • What if \(x_1 \ldots x_i \in L(R)\) and \(x_1 \ldots x_k \in L(R)\)?

  • Rule: choose the longest possible substring (“maximal munch”)

• Which token is used?

  • What if \(x_1 \ldots x_i \in L(R_j)\) and \(x_1 \ldots x_i \in L(R_k)\)?

  • Rule: choose the rule listed first (\(j\) if \(j < k\))

Error Handling

• What if no regular expression matches a prefix of the input?

• Problem: the algorithm needs to terminate

• Solution: write a rule matching all invalid strings and place it at the end of the rules

• Lexer tools allow you to write
  \(R = R_1 \; | \; \ldots \; | \; \texttt{Error}\)
  where the token Error matches if nothing else matches

Summary

• Regular expressions provide a concise notation for string patterns

• Adapting regular expressions to lexical analysis requires small extensions to resolve ambiguities and handle errors

• Good algorithms are known that

  • Require only a single pass over the input

  • Require few operations per character (table lookup)

Regular Languages and Finite Automata

• Result from formal language theory: regular expressions and finite automata both define the class of regular languages

• Thus, lexical analysis uses:

  • Regular expressions for specification

  • Finite automata for implementation (automatic generation of lexical analyzers)

Finite Automata

• A finite automata is a recognizer for the set of strings of a regular language

• A finite automaton consists of:

  • A finite input alphabet \(\Sigma\)

  • A set of states \(S\)

  • A start state \(n\)

  • A set of accepting states \(F \subseteq S\)

  • A set of transitions in \(S \rightarrow S\) (mappings from states to states)

Finite Automata

• Transition notation
  \(s_1 \; \rightarrow{^a} \; s_2\)
  is read: in state \(s_1\) on input \(a\) go to state \(s_2\)

• Each transition “consumes” a character from the input

• At the end of input (or no transition possible)

  • If in accepting state, accept (\(s \in L(R)\))

  • Otherwise, reject (\(s \notin L(R)\))

Finite Automata State Graphs

• A state:
  

• A start state:
  

• An accepting state:
  

• A transition:
  

A Simple Example

• A finite automaton that accepts only “1”;
  

Another Simple Example

• A finite automaton accepting any number of 1s followed by a single 0

• Alphabet: \(\{0, 1\}\)
  

Another Example

• Alphabet: \(\{0, 1\}\)
  

Epsilon Transitions

• Epsilon transitions:
  

• The automaton can move from state \(A\) to state \(B\) without consuming input

Deterministic and Non-Deterministic Automata

• Deterministic Finite Automata (DFA)

  • One transition per input per state

  • No epsilon transitions

• Non-deterministic Finite Automata (NFA)

  • Can have multiple transitions for one input in a given state

  • Can have epsilon transitions

• Finite automata have finite memory – only enough to encode the current state

Execution of Finite Automata

• A DFA can take only one path through the state graph

  • Completely determined by input

• NFAs can choose:

  • whether to make epsilon transitions

  • which of multiple transitions for a single input to take

Acceptance of NFAs

• An NFA can get into multiple states
  

• An NFA accepts an input if it can get in a final state

• Exampe input: 1 0 1

NFA versus DFA

• NFAs and DFAs recognize the same set of languages (regular languages)

• DFAs are easier to implement

• A DFA can be exponentially larger than an equivalent NFA

NFA versus DFA

• For a given language the NFA can be simpler than the DFA

  • NFA:
    

  • DFA:
    

Regular Expressions to Finite Automata

• The implementation of a lexical specification as a finite automata has the following transformations:

  1. Lexical specification

  2. Regular expressions

  3. NFA

  4. DFA

  5. Table driven implementation of DFA

Regular Expressions to NFA

• We can define an NFA for each basic regular expression and than connect the NFAs together based on the operators

• Basic regular expressions

  • \(\epsilon\) transition
    

  • Input character ‘0’
    

Regular Expressions to NFA

• \(AB\): make an \(\epsilon\) transition from the accepting state of \(A\) to start state of \(B\)
  

• \(A \vert B\): create a new start state and add \(\epsilon\) transitions from the new start state to the start states of \(A\) and \(B\), then create a new accepting state and add \(\epsilon\) transitions from the accepting states of \(A\) and \(B\) to the new accepting state
  

Regular Expressions to NFA

• \(A*\): create a new start state and accepting state and add an \(\epsilon\) transitions: from the new start state to the start state of \(A\), from the accepting state of \(A\) to the new start state, and from the new start state to the new accepting state.
  

Regular Expressions to NFA Example

• Consider the regular expression: \((1 \vert 0)*1\)

• The NFA is
  

NFA to DFA (The Trick)

• Simulate the NFA

• Each state of the DFA is a non-empty subset of states of the NFA

• The start state is the set of NFA states reachable through epsilon transitions from the NFA start state

• Add a transition \(S \rightarrow^a S'\) to the DFA if and only if \(S'\) is the set of NFA states reachable from any state in \(S\) after seeing the input \(a\) (considering epsilon transitions as well)

NFA to DFA Remark

• An NFA may be in many states at any time

• If there are \(N\) states, the NFA must be in some subset of those \(N\) states

• There are \(2^N - 1\) possible subsets (finitely many)

NFA to DFA Example

• NFA

• DFA

Implementation

• A DFA can be implemented by a 2D table \(T\)

  • One dimension is “states”

  • The other dimension is “input symbols”

  • For every transition \(S_i \rightarrow^a S_k\) define \(T[i,a] = k\)

• DFA “execution”

  • If in state \(S_i\) and input \(a\), then read \(T[i,a] k\) and skip to state \(S_k\)

  • This is efficient

Example: Table Implementation of a DFA

Implementation Continued

• The NFA to DFA conversion is the core operation of lexical analysis tools such as lex

• But, DFAs can be huge

• In practice, lex-like tools trade off speed for space in the choice of NFA and DFA representations

Theory versus Practice

• DFAs recognize lexemes. A lexer must return a type of acceptance (token type) rather than simply an accept/reject indication

• DFAs consume the complete string and accept or reject it. A lexer must find the end of the lexeme in the input stream and then find the next one, etc.