Context Free Grammars

1
Chapter 3

• Context Free Grammars
• Comparison to RE Notation
• Specification of Context Free Grammar Rules
• Derivations Language Defined by a Grammar
• Examples

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Parse Trees Abstract Syntax Trees

• Parse Trees
• Abstract Syntax Tree
• Examples

3
Ambiguity

• Ambiguous Grammars
• Precedence and Associativity
• Dangling Else Problem
• Inessential Ambiguity

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Extended Notations

• EBNF Notation
• Syntax Diagrams
• Examples

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Formal Properties of CFG

• Grammar Rles as Equations
• Chomsky Hierarchy
• Limits of Syntax as Context Free Rules

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FA from Grammar

• One state for each non-terminal
• A rule of the form
• Nt1 terminal, generates transition from a
  state to final state
• A rule of the form
• Nt1 terminal Nt2
• Generates transition from state 1 to state 2 on
  an arc labeled by the terminal

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Graphic representation of FA

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FA from RE

• Each RE corresponds to a grammar
• For all REs
• A natural translation to FSM exists
• Alternation often leads to non-deterministic
  machines

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Deterministic Finite Automata (DFA)

• For all states S
• For all characters C
• There is at most one arc from any state S that is
  labeled with C
• Easier to implement
• No backtracking
• Conventions for DFA
• Error transitions are not explicitly shown
• Input symbols that result in the same transition
  are grouped together (this set can even be given
  a name)
• Still not displayed stopping conditions and
  actions

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Non-Deterministic Finite Automata (NFA)

• A non-deterministic FA
• Has at least one state
• With two arcs to two distinct states
• Labeled with the same character
• Example from start state, a digit can begin an
  integer literal or a real literal
• Implementation requires backtracking

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Lookahead Backtracking in NFA
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Implementation of FA
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From RE to DFA RE to NFA
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NFA to DFA

• There is an algorithm for converting a
  non-deterministic machine to a deterministic one
• Result may have exponentially more states
• Intuitively need new states to express
  uncertainty about token int or real
• Other algorithms for minimizing number of states
  of FSM, for showing equivalence, etc.

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Example DFA
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Another view of the same DFA
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Yet another view of the same DFA
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State Minimization in DFA
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TINY DFA
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Lex for Scanner

• Lex Conventions for RE
• Format of a Lex Input File