Theory of computation

 
 
The theory of computation is mathematically model a machine (for example a computer) and study the theory about it which means what are the problems which would be solved by this machine, what are the limitations of the machine etc.

More information about this subject can be found at: YouTube: Theory of Computation or Automata Theory by Gate Lectures by Ravindrababu Ravula

Ravindrababu Ravula has created more than 70 excellent YouTube movies about this subject.
Based on these YouTube movies, notes have been made which you can find on this page.






Compiler basics


Information on this page based on: YouTube: Theory Of Computation 1,Introduction to TOC and DFA


  • Compilers vs interperters

    Compilers vs interperters

  • Structure of a compiler
    The structure of a compiler consists of 5 major fases:

    1. Lexical analysis
      Lexical analysis divides program text into words also known as lexemes by reading the text from left to right.

      The output of the lexical analysis is a series of tokens where the token class for each lexeme is identified. A token consists of: <token class, lexeme>

      Lookahead is always needed to decide where one token ends and the next token begins.
      Minimize the amount of lookaheads to simplify the implementation of the lexical analyzer.

    2. Parsing
      Parsing is diagramming sentences to understand sentence structure.

    3. Semantic Analysis
      Understand meaning to sentences to catch inconsistencies.

    4. Optimization
      Automatically modify program so that they run faster, use less memory, reduce power usage, use less database/network connection etc.
      Modern compilers spend large amount of time in optimization compare to other fases.

    5. Code generation
      Produces usually assembly code (translate into another language).

  • Regular languages
    Regular languages are used to determine which set of strings is in a token class.

    To define regular languages, regular expressions are used. Each regular expression (=syntax) denotes a set of strings (=regular languages).
    There are two basic regular expressions:

    • A single character

      A single character 'c' is an expression which denotes a language containing a string with a single character:
      'c' = {"c"}

    • Epsilon

      ε is an expression which denotes a language containing an empty string:
      ε = {""}

    There are three compound regular expressions:

    • Union (∪)

      A + B is an expression which denotes a language containing a string:
      A + B = {a | a∈A} ∪ {b | b∈B}

    • Concatenation (∧)

      AB is an expression which denotes a language containing a string:
      AB = {ab | a∈A ∧ b | b∈B}

    • Iteration (*)

      A* is an expression which denotes a language containing a string:
      A* = ∪i>=0Ai
      Ai = A0, A1, A2, A3,...
      A0 = ε
      Note: The empty string (ε) is always a elements of A*.

    Definition: The regular expression over Σ (=alphabet) are the smallest set of expressions including:
    R = ε | 'c' | R+R | RR | R*
    Note: 'c' ∈ Σ

    If we want to know the regular language we need to know what the alphabet (Σ) is.

    Σ = {0,1}

    Question: The expression 1* what language does it denotes?
    Answer: ε + 1 + 11 + 111 + 1111 + ...
    All strings of 1's.

    Question: The expression (1+0)1 what language does it denotes?
    Answer: {11,01}

    Question: The expression 0* + 1* what language does it denotes?
    Answer: {0i | i>=0} ∪ {1i | i>=0}

    Question: The expression (0 + 1)* what language does it denotes?
    Answer: {"",(0 + 1),(0 + 1)(0 + 1),(0 + 1)(0 + 1)(0 + 1),...}
    This means ALL strings of 0's and 1's. Can also be indicated by Σ*

  • Formal languages

    A language over Σ is a set of strings of characters drawn from Σ.
    Formal language examples:

    Alphabet = English characters
    Language = English sentences

    Alphabet = ASCII
    Language = C programs

  • Meaning function (L)

    Meaning function L maps syntax to semantics. A meaning function is used:
    • It makes clear what is syntax and what is semantics.
    • Allow us to consider notation as a separate issue.
    • Because expressions and meanings are not 1-1. There are more expressions than meanings. Meaning is many to one and never one to many.

    L(e) = M
    e = regular expression
    M = set of strings

    Example:
    L(ε) = {""}
    L('c') = {"c"}
    L(A+B) = L(A) ∪ L(B)
    L(AB) = {ab | a∈L(A) ∧ b | b∈L(B)}
    L(A*) = ∪i>=0 L(Ai)

  • Regular expression shorthand notation
    • At least one: A+ ≡ AA*
    • Union: A | B ≡ A + B
    • Option: A? ≡ A + ε
    • Range: [a-z] ≡ 'a' + 'b' + 'c' + 'd' + ...
    • Excluded Range: [^a-z] ≡ complement of [a-z]

  • Lexical specifications

    Regular expression describe many useful languages.
    Regular languages are a language specification.
    Regular expressions is used as the specification language for lexical analysis.

    • Keywords
      "if" + "then" + "else"

    • Integer: a non-empty string of digits
      digit = '0' + '1' + '2' + '3' + '4' + '5' + '6' + '7' + '8'+ '9'
      Integer = digit digit* = digit+

      Note:
      * = 0 or more
      + = 1 or more

    • Identifier: string of letters or digits, starting with a letter
      letter = 'a' + 'b' + 'c' + 'd' + ... + 'A' + 'B' + 'C' + 'D' + ...
      letter = [a-zA-Z]
      Identifier = letter(letter+digit)*

    • Whitespace: a non-empty sequence of blanks, newlines and tabs.
      Whitespace = (' ' + '\t' + '\n')+

    Examples:

    Email address = username@company.com
    Email address = letter+ '@' letter+ '.' letter+

    number = 12 or 12.34 or 12E+8
    digit = '0' + '1' + '2' + '3' + '4' + '5' + '6' + '7' + '8'+ '9'
    digits = digit+
    opt_fraction = ('.'digits) + ε = ('.'digits)?
    opt_exponent = ('E'('+' + '-' + ε) digits) + ε = ('E'('+' + '-')? digits)?
    number = digits opt_fraction opt_exponent

    To design the lexical specification of the language the following procedure must be followed:

    1. Create regular expression for the lexemes of each token class
      • Number = digit+
      • Keyword = 'if' + 'else' + ...
      • Identifier = letter(letter+digit)*
      • OpenPar = '('
      • :

    2. Construct a gigantic regular expression R, matching all lexemes for all tokens

      R = Keyword + Identifier + Number + ...
      R = R1 + R2 + R3 + ...

    3. Loop thru the input and check it against the meaning function L(R)

      Let input be x1...2
      For 1 <= i <= n check
          x1...i ∈ L(R)

      Note:
      • Always choose the longest input ("maximal munch").
      • If there is more than one token class which the input might belong choose the one listed first. In the lexical specification, keywords are given a higher priority than identifiers.

    4. If success then we know that:
          x1...i ∈ L(Rj) for some j

      If no rule matches, create error strings for all strings not in the lexical specifications.
      Put these error strings last in the priority list.

    5. Remove x1...i from input and go to step 3

  • Finite Automata

    Finite automata is used as an implementation model for regular expressions. Regular expressions and finite automaton are very close related. They can specify exactly the same languages called the regular languages.

    A finite automaton consists of:

    • An input alphabet: Σ
    • A finite set of states: S
    • A start state: n
    • A set of accepting states: F ⊆ S
    • A set of transistion states: state ->input state

    The language of a finite automaton ≡ set of strings that the automaton accepts.

  • Deterministic Finite Automata (DFA)

    • One transistion per input state
    • No ε-moves

    A DFA takes only one path through the state graph per input.
    DFAs are faster to execute than NFAs because there are no choices to consider.
    DFAs and NFAs recognize the same set of regular languages.

  • Nondeterministic Finite Automata (NFA)

    • Can have multiple transistions in a given state
    • Can have ε-moves

    An NFA accepts if some choices lead to an accepting state.
    NFAs are, in general, smaller compared to DFAs.

  • Regular expressions into NFAs

    Lexical specification we want to implement -> write a set of regular expressions -> translate regex to NFAs -> translate NFAs to DFAs -> implement DFAs as a set of Lookup tables and code for traversing thos tables.

    For each kind of regex define an equivalant NFA
    Notation: NFA for regexp M