CSC3403 Assignment 1 2007 Sample solution

CSC3403 Assignment 1 2007 Sample solution
All questions were allocated 4 marks each.
These answers are sample only. In some cases there are many possible correct solutions. Any
correct solution will receive full marks.
1. Consider the following grammar
assign → id := expr
id → A | B | C
expr → expr + term | term
term → term * factor | factor
factor → id | - id
Now consider the sentence of the grammar:
A := - A + B * C
Produce a rightmost derivation of this sentence.
Solution
assign ⇒ id := expr
⇒ id := expr + term
⇒ id := expr + term * factor
⇒ id := expr + term * id
⇒ id := expr + term * C
⇒ id := expr + factor * C
⇒ id := expr + id * C
⇒ id := expr + B * C
⇒ id := term + B * C
⇒ id := factor + B * C
⇒ id := - id + B * C
⇒ id := - A + B * C
⇒ A := - A + B * C
2. For the grammar and sentence in Question 1, show the parse tree for the sentence.
Solution
assign
/|\
/ | \
/ | \
id := expr
| /|\
| / | \
A
expr + term
| /|\
| / | \
term term * factor
| | |
| | |
factor factor id
/ | | |
/ | | |
- id id C
| |
| |
A B

3. For the grammar and sentence in Question 1, show the abstract syntax tree for the
sentence.
Solution
:=
/ \
A +
/ \
- *
| / \
A B C
4. Consider the following grammar
S → a S | X
X → b X | Y
Y → a Y | ǫ
Write a single EBNF rule that describes this grammar.
(That is, write a rule of the form: “S → EBNF string”.)
Solution Either of the following is acceptable
S → {a}{b}{a}
S → a ∗ b ∗ a ∗
5. Prove that the grammar in question 4 is ambiguous
Solution We choose a sentence of the language that can be produced by two different derivations
(that is, also, two different parse trees).
S ⇒ aS ⇒ aX ⇒ aY ⇒ a
S ⇒ X ⇒ Y ⇒ aY ⇒ a
6. Write an unambiguous grammar in BNF that describes the same language as grammar
of question 4.
Solution The key here is to determine first why the grammar is ambiguous. The problem
occurs with sentences like “aaaa”. It is possible to generate the ’a’ symbol from either the
existing S rule or the Y rule. A key insight is that sentences that contain at least one b can
only be produced in one way. So one solution is to write a grammar that produces either
sentences of the form a ∗ , or it produces sentences of the form a ∗ b + a ∗ . A possible solution is
as follows.
S → a S | b X | ǫ
X → b X | Y
Y → a Y | ǫ
Only the S rule has changed. Note S → b X ensures that if the Y rule is used to generate a
symbols, then there must be at least one b in the sentence.
7. Describe in English the strings that will be produced by the following grammar. Strings
consist of the terminal symbols a, b, c, and k. You must identify the number (e.g. one,
zero or more, zero or one, one or more) of occurrences of each symbol.
S → a S b b | X
X → c X | c Y
Y → d Y | k
Solution a n c + d ∗ kb m where n ≥ 0 and m = 2n and x n means that symbol x is repeated n
times.

8. Write the BNF grammar rules for the language whose sentences have the following
structure.
(a) Zero or more x followed by one or more y.
(b) One or more x followed by one or more y. There are the same number of x as there
are y.
(c) A single x, followed by optionally one y, followed by a single z.
Solution
(a) S → x S | y Y
Y → y Y | ǫ
(b) S → x S y | x y
(c) S → x y z | x z
9. Write the BNF grammar rules for the language whose sentences have the following
structure
[x](y + |z) +
Solution Examination of this EBNF description reveals that it is a little over-complex. Indeed
an equivalent description would be [x](y|z) + . (If you don’t believe this, try to find a sentence
that can be produced by the original specification, but which cannot be produced by the new
one.)
S → x Y | Y
Y → y Y | z Y | y | z
10. Draw a single syntax diagram to describe the following grammar fragment. Assume ident
and expression are defined elsewhere (you may simply refer to them in your diagram).
Terminal symbols are enclosed in boxes .
designator → ident
| ident slist
slist → suffix slist
| suffix
suffix → . ident
| [ explist ]
| ^
explist → expression , explist
| expression
Solution
designator
ident
.
ident
[
^
expression
,
]