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# Lab for Automated Reasoning and Analysis LARA

## Homework 02

**Due Monday, 17th October, 10:15am. Please hand in before the lecture.**

For this homework, recall that a parse tree is a tree that represents the structure of the string to be parsed according to some grammar.

It can be defined recursively as follows:

- If t is a terminal, then t() is a parse tree.

- Given the parse trees t1(…), t2(..), …, t3(…) and a production rule
Y → t1 t2 … t3, then Y(t1(…), t2(…), …, t3(…)) is also a parse tree.

## Problem 1

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 2

A grammar is right regular if and only if the production rules are of the following form:

X -> a X -> aY X -> ε

where a is a terminal, X, Y are non-terminals. A left-regular grammar is defined analogously, with the second rule being X → Za.

Consider the following right regular grammar over :

S -> #A | !# | ! A -> ! | #B | ε B -> #B | !

a) Determine how the input string '###!' is derived.

b) Find a regular expression that accepts the same language as this grammar.

c) Show that a language is regular if and only if there exists a regular grammar that generates it. I.e. show that for each regular expression there is a regular grammar that defines the same language, and vice versa. (A regular grammar is a grammar that is either right or left regular.)

## Problem 3

The following grammar is motivated by type definitions in Scala.

TypeDefinition := "type" Identifier "=" Type Type := Type "," Type Type := Type "=>" Type Type := "Int" | "Boolean"

a) By giving at least one input which has two different parse trees prove that the grammar is ambiguous.

b) Given the fact that ”=>” has a lower precedence than ”,” resolve the ambiguity of the grammar. That is, write a new grammar that determines the same language but is not ambiguous, such that the Type “Int ⇒ Int, Int” is parsed as function from Int to Int,Int.

## Problem 4

The following grammar generates all the regular expressions over .

S -> S S S -> S '+' S S -> S '*' S -> '(' S ')' S -> 'a' | 'b'

a) Compute the **First** and **Follow** sets for S.

b) By giving two different parse trees for an input, show that the grammar is ambiguous.

c) Assuming that the '*' has the highest and '+' has the lowest precedence, find a grammar that accepts the same language but is unambiguous.

d) Compute the **First** and **Follow** sets for the non-terminals of the new grammar.

e) Is the new grammar LL(1)?

## Problem 5

A grammar is called LL(k) if it needs k tokens of lookahead when parsing a sentence. Does the following grammar belong to LL(k)? Justify your answer and find the smallest k you can, if applicable.

S -> AB S -> BC A -> a A -> Ba B -> Cc C -> c B -> a