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# Homework 3

Due by Wednesday, 31st October, 10:15am sharp (right before the class).

## Problem 1

We want to construct an LL(1) parser for the following grammar.

```S -> ( L )
S -> a
L -> L , S
L -> S```

a) Can the grammar be used with a LL(1) parser as is? If not, modify it accordingly.

b) Compute the First and Follow sets for S and L for the new grammar.

c) Finally, construct the parsing table.

## Problem 2

A grammar has a cycle if there is a non-terminal such that , i.e. it is possible to derive the nonterminal A from A by a nonempty sequence of production rules.

a) Show that an LL(1) grammar is not allowed to have any cycles. Recall, that an LL(1) grammar is one for which a predictive parser needs just one look-ahead symbol.

b) Give an algorithm that eliminates cycles in a context-free grammar.

## Problem 3

Show that the regular languages can be recognized with LL(1) parsers. Describe a process that, given a regular expression, constructs an LL(1) parser for it.

## Problem 4

Put the following grammar into Chomsky normal form. Show your steps (i.e. don't just give the final grammar without explanation, or we won't be able/willing to check that it describes the same language).

```S -> A ( S ) B | ε
A -> S | S B | x | ε
B -> S B | y```

## Problem 5

Assume a grammar in Chomsky normal form has non-terminals. Show that if the grammar can generate a word with a derivation having at least steps, then the recognized language should be infinite.