1. A basis for computer theory and A means of specifying languages
Grammars
A basis for computer theory
and
A means of specifying languages

2. Languages defined by Grammars
<statement>
A grammar has
terminals
non-terminals
rules
start-symbol
<var> = <expression>
A <var> + <var>
B C

3. Example A grammar has terminals {A,B,C,==,+} non-terminals
{<statement>,<var>,<expression>}
rules
{<statement> -> <var> = <expression>
<expression> -> <var>+<var>
<var> -> { A | B | C }
start-symbol
<statement>

4. A sentence A derivation from a start symbol
Using the previous example:
<statement> -> <var> = <expression>
A = <var> + <var>
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
<statement> -> <var> = <expression>
A = <expression>
A = <var> + <var>
A = B + <var>
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
<A> -> a | a <A>
<A> -> a or
<A> -> a <A>
a a <A>
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
<A> = a | a <A> 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
<assign> -> <id> := <expr>
<id> -> A | B | C
<expr> -> <id> + <expr> |
<id> * <expr> |
( <expr> ) |
<id>

9. Parse Tree shows derivation
<assign>
<id> := <expr>
A <id> * <expr>
B ( <expr> )
<id> + <expr>
A <id> 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 that’s it
EBNF has other shorthands
{ , <id> }
represents an optional ,<id>
repeated indefinitely or not at all (zero or more)
[ , <id> ]
represents zero or one
….. ( <a> | <b> )
occurs in middle of expression giving choice (ex 3.5)

13. Be sure to practice problems
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:
a+b
a b+
a*bc*
Be sure your grammar generates ALL sentences in the language but not extra sentences which don’t belong
Typical errors are too few or too many!