Introduction to the Theory of Computation Assignment 3

CSC236: Introduction to the Theory of Computation
Assignment 3
Question 1. [16 marks]
Given a list L, a contiguous sublist M of L is a sublist of L whose elements occur in immediate
succession in L. For instance, [4, 7, 2] is a contiguous sublist of [0, 4, 7, 2, 4] but [4, 7, 2] is not a
contiguous sublist of [0, 4, 7, 1, 2, 4].
We consider the problem of computing, for a list of integers L, a contiguous sublist M of L with
maximum possible sum.
Algorithm 1 M axSublist(L)
<precondition: L is a list of integers.
<postcondition: Return a contiguous sublist of L with maximum possible sum.
Part (1) [5 marks]
Using a divide-and-conquer approach, devise a recursive algorithm which meets the requirements
of M axSublist.
Part (2) [8 marks]
Give a complete proof of correctness for your algorithm. If you use an iterative subprocess, prove
the correctness of this also.
Part (3) [3 marks]
Analyze the running time of your algorithm.
Question 2. [18 marks]
For a point x ∈ Q and a closed interval I = [a, b], a, b ∈ Q, we say that I covers x if a ≤ x ≤ b.
Given a set of points S = {x1, . . . , xn} and a set of closed intervals Y = {I1, . . . , Ik} we say that Y
covers S if every point xi
in S is covered by some interval Ij in Y .
In the “Interval Point Cover” problem, we are given a set of points S and a set of closed intervals
Y . The goal is to produce a minimum-size subset Y
′ ⊆ Y such that Y
′
covers S.
Consider the following greedy strategy for the problem.
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CSC236: Introduction to the Theory of Computation Due: August 3
rd, 2018
Algorithm 2 Cover(S, Y )
<precondition:
S is a finite collection of points in Q. Y is finite set of closed intervals which covers S.
<postcondition:
Return a subset Z of Y such that Z is the smallest subset of Y which covers S.
1: L = {x1, . . . , xn} ← S sorted in nondecreasing order
2: Z ← ∅
3: i ← 0
4: while i < n do
5: if xi+1 is not covered by some interval in Z then
6: I ← interval [a, b] in Y which maximizes b subject to [a, b] containing xi+1
7: Z.append(I)
8: i ← i + 1
9: return Z
Give a complete proof of correctness for Cover subject to its precondition and postcondition.
Question 3. [10 marks]
The first three parts of this question deals with properties of regular expressions (this is question
4 from section 7.7 of Vassos’ textbook). Two regular expressions R and S are equivalent, written
R ≡ S if their underlying language is the same i.e. L(R) = L(S). Let R, S, and T be arbitrary
regular expression. For each assertion, state whether it is true or false and justify your answer.
Part (1) [2 marks]
If RS ≡ SR then R ≡ S.
Part (2) [2 marks]
If RS ≡ RT and R ̸≡ ∅ then S ≡ T.
Part (3) [2 marks]
(RS + R)
∗R ≡ R(SR + R)
∗
.
Part (4) [4 marks]
Prove or disprove the following statement: for every regular expression R, there exists a FA M such
that L(R) = L(M). Note: even if you find the proof of this somewhere else, please try to write up
the proof in your own words. Just citing the proof is NOT enough.
Question 4. [16 marks]
In the following, for each language L over the alphabet Σ = {0, 1} construct a regular expression
R and a DFA M such that L(R) = L(M) = L. Prove the correctness of your DFA.
2
CSC236: Introduction to the Theory of Computation Due: August 3
rd, 2018
Part (1) [8 marks]
Let L1 = {x ∈ {0, 1}
∗
: the first and last charactes of x are the same}. Note: ϵ /∈ L since ϵ does
not have a first or last character.
Part (2) [8 marks]
Let a block be a maximal sequence of identical characters in a finite string. For example, the
string 0010101111 can be broken up into blocks: 00, 1, 0, 1, 0, 1111. Let L2 = {x ∈ {0, 1}
∗
:
x only contains blocks of length at least three}.
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