Grammars

1
Grammars

• A basis for computer theory
• and
• A means of specifying languages

2
Languages defined by Grammars
ltstatementgt

• A grammar has
• terminals
• non-terminals
• rules
• start-symbol

ltvargt ltexpressiongt
A ltvargt ltvargt
B C
3
Example

• A grammar has
• terminals
• A,B,C,,
• non-terminals
• ltstatementgt,ltvargt,ltexpressiongt
• rules
• ltstatementgt -gt ltvargt ltexpressiongt
• ltexpressiongt -gt ltvargtltvargt
• ltvargt -gt A B C
• start-symbol
• ltstatementgt

4
A sentence

• A derivation from a start symbol
• Using the previous example

ltstatementgt -gt ltvargt ltexpressiongt
A ltvargt ltvargt A B C

• A valid sentence in this language is
• A B C
• Defining a language by listing all sentences is
  not usually practical

5
Derivations have classifications

• Leftmost derivations replace one non-terminal at
  a time, always replacing the leftmost non-terminal

ltstatementgt -gt ltvargt ltexpressiongt
A ltexpressiongt
A ltvargt ltvargt A B
ltvargt A B C

• Rightmost derivations replace one non-terminal at
  a time, always replacing the rightmost
  non-terminal .. Surprised?

6
Recursion

• Non-terminals can appear on both the left and
  right side of a rule

ltAgt -gt a a ltAgt ltAgt -gt a or ltAgt -gt a ltAgt
a a ltAgt a a a
One can represent the language with a regular
expression a means one or more means
zero or more
THIS IS AN EXAMPLE OF RIGHT RECURSION!
7
Regular grammars

• ltAgt a a ltAgt is an example of a regular
  grammar
• RHS is
• a terminal
• a terminal followed by a non-terminal
• a non-terminal followed by a terminal
• Regular languages (defined by regular grammars)
  can typically be expressed by regular expressions

8
Context-free Grammar

• Single non-terminal on LHS
• Most programming language constructs
• Most examples in the text

ltassigngt -gt ltidgt ltexprgt ltidgt -gt A B
C ltexprgt -gt ltidgt ltexprgt ltidgt
ltexprgt ( ltexprgt )
ltidgt
9
Parse Tree shows derivation
ltassigngt ltidgt ltexprgt A ltidgt
ltexprgt B ( ltexprgt )
ltidgt ltexprgt
A ltidgt C
A B ( A C )
10
What is ambiguity in a grammar?

• First observe the significance of the structure
• See Figure 3.2
• Tree structure defines associativity/precedence
• Highest in the tree, last operation done
• Lowest in tree, earliest operation done
• SO ? If the grammar is defined so that there
  are two different parse trees, it implies two
  different interpretations to the same statement

11
How do we avoid ambiguity?

• Grammars are usually tricky
• Introduction of term and factor removes the
  ambiguity in algebraic expressions
• Another classical example is that of if-then-else
  
• How do you determine which if the else gets
  paired with?
• Use of braces, etc can eliminate the confusion.

12
BNF and extended BNF

• Just a syntax for writing grammar
• BNF uses for OR but thats it
• EBNF has other shorthands
• , ltidgt
• represents an optional ,ltidgt
• repeated indefinitely or not at all (zero or
  more)
• , ltidgt
• represents zero or one
• .. ( ltagt ltbgt )
• occurs in middle of expression giving choice (ex
  3.5)

13
Syntax Graphs

• Not a new idea
• Another means of representing a grammar
• Easier to read
• Fig. 3.6

Be sure to practice problems in the chapter!
14
Can you write grammars for

• ab
• a b
• abc
• Be sure your grammar generates ALL sentences in
  the language but not extra sentences which dont
  belong
• Typical errors are too few or too many!