Formal Languages and Automata in Programming Languages

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Formal Languages and Automata in Programming
Languages
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Motivation
The problem of parsing structured text is very
common Consider the structure of email addresses
(using a grammar) ltemailAddressgt ltpersongt _at_
lthostgt ltpersongt ltwordgt lthostgt ltwordgt
ltwordgt.lthostgt Describe and recognize email
addresses in arbitrary text.
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Languages and Automata in Programming Languages

• Regular languages
• Recognized(accepted) by finite automata
• Useful for tokenizing program text (lexical
  analysis)
• Context-free languages
• Recognized(accepted) by pushdown automata
• Useful for parsing the syntax of a program

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Deterministic Finite Automata (DFA)

• Q finite set of states
• S finite set of letters (alphabet)
• d QxS -gt Q (transition function)
• q0 start state (in Q)
• F set of accept states (subset of Q)
• Acceptance input consumed with the automaton in
  a final state.

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Example of DFA
1
0
1
q2
q1
0
Accepts all strings that end in 1
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Another Example of a DFA
S
b
a
b
a
r1
q1
a
b
a
b
q2
r2
a
b
Accepts all strings that start and end with a
OR start and end with b
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Non-deterministic Finite Automata (NFA)

• Transition function is different
• d QxSe -gt P(Q)
• P(Q) is the powerset of Q (set of all subsets)
• Se is the union of S and the special symbol e
  (denoting empty)

String is accepted if there is at least one path
leading to an accept state, and input consumed.
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Example of an NFA
0, 1
0, 1
0, e
1
1
q1
q2
q3
q4
What strings does this NFA accept?
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Regular Expressions

• R is a regular expression if R is.
• a for some a in S.
• e (the empty string).
• member of the empty language.
• the union of two regular expressions.
• the concatenation of two regular expr.
• R1 (Kleene closure zero or more repetitions of
  R1).

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Examples of Regular Expressions
0, 1 0 all strings that end in 0 0, 1
0 string that start with 1 or 0 followed
by zero or more 0s. 0, 1
all strings 0n1n, n gt0 not a regular
expression!!! The value of a regular expression
is a regular language (a set of
strings).Utilities such as AWK, GREP, PERL,
emacs, provide facilities for the description
and matchingof patterns using regular
expressions
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Important Theorems

• A language is regular if a regular expression
  describes it.
• A language is regular if a finite automaton
  recognizes it.
• DFAs and NFAs are equally powerful.

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Context-free Grammars
Context-free grammars are defined by substitution
rules Big Jim ate gree cheesegreen Jim
ate green cheese Jim ate cheese Cheese ate Jim
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Backus-Naur Form

• Context-free grammars are used to formally
  describe the syntax of programming languages.
• Every syntactically correct program is derived
  using the context-free grammar of the language.
• Parsing a program involves tracing such
  derivation, given the context-free grammar and
  the program.
• See SlonnegerKurz for more details.

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Context-free Grammars

• A context-free grammar consists of
• V a finite set of variables
• S a finite set of terminals
• R a finite set of rules of the formvariable -gt
  variable, terminal
• S the start variable

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Pushdown Automata (PDA)

• A pushdown automaton consists of
• Q a set of states
• S input alphabet (of terminals)
• G stack alphabet
• d a set of transition rulesQ x Se x Ge -gt P(Q
  x Ge)currentState, inputSymbol, headOfStack
  -gtnewState, pushSymbolOnStack
• q0 the start state
• F the set of accept states (subset of Q)
• Deterministic At most one move is possible from
  any configuration

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How does a PDA accept?

• By final state
• Consume all the input while
• Reaching a final state
• By empty stack
• Consume all the input while
• Having an empty stack
• Set of final states is irrelevant

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Example of a PDA
e, e -gt
0, e-gt0
q2
q1
1, 0-gte
e, -gte
q3
q4
1, 0-gte
Notation a, b-gtc when PDA reads a from input,
it replaces b at the top of stack with c.
What does this PDA accept?
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Important Theorems

• A language is context-free iff a pushdown
  automaton recognizes it
• Non-deterministic PDA are more powerful than
  deterministic ones

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Example of Context-free Language That Requires a
Non-deterministic PDA
w wR w belongs to 0, 1 i.e. wR is w
written backwards
Idea Non-deterministically guess the middle of
the input string
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The Solution
e, e -gt
0, e-gt0 1, e-gt1
q2
q1
e, e-gte
e, -gte
q3
q4
1, 1-gte0, 0-gte