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Assignment 1
Linux tools, logic, regular expressions, induction
DUE: 11:55pm, Friday 16 August 2024 (Week 4)
Instructions
Please read these instructions carefully before you attempt the assessment:
To begin working on the assignment, download the workbench asgn1 .zip from Moodle. Cre- ate a new Ed Workspace and upload this file, letting Ed automatically extract it. Edit the student-id file to contain your name and student ID. Refer to Lab 0 for a reminder on how to do these tasks.
The workbench provides locations and names for all solution files. These will be empty, needing replacement. Do not add or remove files from the workbench.
Solutions to written questions must be submitted as PDF documents. You can create a PDF file by scanning your legible (use a pen, write carefully, etc.) hand-written solutions, or by directly typing up your solutions on a computer. If you type your solutions, be sure to create a PDF file. There will be a penalty if you submit any other file format (such as a Word document). Refer to Lab 0 for a reminder on how to upload your PDF to the Ed workspace and replace the placeholder that was supplied with the workbench.
Before you attempt any problem—or seek help on how to do it—be sure to read and understand the question, as well as any accompanying code.
When you have finished your work, download the Ed workspace as a zip file by clicking on “Download All” in the file manager panel. You must submit this zip file to Moodle by the deadline given above. To aid the marking process, you must adhere to all naming conventions that appear in the assignment materials, including files, directories, code, and mathematics. Not doing so will cause your submission to incur a one-day late-penalty (in addition to any other late-penalties you might have). Be sure to check your work carefully.
Your submission must include:
a one-line text file, prob1.txt, with your solution to Problem 1;
an awk script, prob2.awk, for Problem 2;
a PDF file prob3.pdf with your solution to Problem 3;
an awk script, prob4.awk, for Problem 4;
a file prob5.pdf with your solution to Problem 5.
Introduction to the Assignment
In Lab 0, you met the stream editor sed, which detects and replaces certain types of patterns in text, processing one line at a time. These patterns are actually specified by regular expressions.
In this assignment, you will use awk which does some similar things and a lot more. It is a simple programming language that is widely used in Unix/Linux systems and also uses regular expressions.
In Problems 1–4, you will construct an awk program to construct, for any graph, a logical expression that describes the conditions under which the graph contains a triangle-free vertex cover.
Finally, Problem 5 is about applying induction to a problem about structure and satisfiability of some Boolean expressions in Conjunctive Normal Form.
Introduction to awk
In an awk program, each line has the form.
/pattern / { action }
where the pattern is a regular expression (or certain other special patterns) and the action is an instruction that specifies what to do with any line that contains a match for the pattern. The action (and the {... } around it) can be omitted, in which case any line that matches the pattern is printed. Once you have written your program, it does not need to be compiled. It can be executed di-rectly, by using the awk command in Linux:
$ awk -f programName inputFileName
Your program is then executed on an input file in the following way.
// Initially, we’re at the start of the input file, and haven’t read any of it yet. If the program has a line with the special pattern BEGIN, then
do the action specified for this pattern. Main loop, going through the input file: {
inputLine := next line of input file Go to the start of the program.
Inner loop, going through the program: {
programLine := next line of program (but ignore any BEGIN and END lines) if inputLine contains a string that matches the pattern in programLine, then
if there is an action specified in the programLine, then {
do this action }
else
just print inputLine // it goes to standard output
} }
If the program has a line with the special pattern END, then do the action specified for this pattern.
Any output is sent to standard output.
You should read about the basics of awk, including
? the way it represents regular expressions,
? the variables $1, $2, etc., and NR,
? the function printf( ··· ),
? for loops. For these, you will only need simple loops like
for (i = 1; i <= n; i++) {
(body of loop〉 }
This particular loop executes the body of the loop once for each i = 1, 2, . . . , n (unless the body of the loop changes the variable i somehow, in which case the behaviour may be different, but you should not need to do that). You can nest loops, so the body of the loop can be another loop. It’s also ok to write loops more compactly,
for (i = 1; i <= n; i++) { (body of loop〉 }
although you should ensure it’s still clearly readable.
Any of the following sources should be a reasonable place to start:
? A. V. Aho, B. W. Kernighan and P. J. Weinberger, The AWK Programming Language,
Addison-Wesley, New York, 1988.
(The first few sections of Chapter 1 should have most of what you need, but be aware also of the regular expression specification on p28.)
https://www.grymoire.com/Unix/Awk.html
the Wikipedia article looks ok
the awk manpage
the GNU Awk User’s Guide.
Introduction to Problems 1–4
Many systems and structures can be modelled as graphs (abstract networks). Many problems on these structures require allocation of some scarce resource to the objects in a network in such a way that there is suitable “coverage” of that resource throughout the network. For example, in a city road network, it may be necessary to place cameras at various road intersections, for traffic management or security, in such a way that every road throughout the network can be viewed by at least one camera. However, cost factors mean that there are constraints on the total number of cameras that can be placed.
This can be modelled abstractly by the vertex cover problem, which is a well-known graph theory problem that we will consider here.
Throughout, we use G to denote a graph, n denotes its number of vertices and m denotes its number of edges. V (G) denotes the set of vertices of G, and E(G) denotes the set of edges of G.
A vertex cover of a graph G is a set of vertices X in V (G) such that every edge in E(G) has at least one endpoint in X .
In this assignment, we are interested in a particular variant of the vertex cover problem in which triangles cannot be included in the vertex cover. A triangle in G is a set of three vertices, all of which are pairwise-adjacent (i.e., every pair in the set is joined by an edge). A vertex cover of G that contains no triangles is called a triangle-free vertex cover.
For example, in the graph H in Figure 1, the vertex sets {1, 3, 6, 7}, {2, 3, 4, 7}, and {1, 2, 5, 7} all form triangle-free vertex covers. But {1, 2, 3, 7} and {1, 3, 5, 7} are not triangle-free vertex covers, the first because it contains the subset {1, 2, 3}, which forms a triangle in H, and the second because it does not meet every edge in H. (Specifically, it misses the edge from 2 to 6.)
Although every graph has a vertex cover, not all graphs have a triangle-free vertex cover. An example of a graph with no triangle-free vertex cover is shown in Figure 2.
In Problems 1–4, you will write a program in awk that constructs, for any graph G, a Boolean expression φG in Conjunctive Normal Form that captures, in logical form, the statement that G has a triangle-free vertex cover. This assertion might be True or False, depending on G, but the requirement is that
G has a triangle-free vertex cover ←→ you can assign truth values to the variables of φG to make it True.
Figure 1: A graph, H, which has a number of different triangle-free vertex covers.
Figure 2: A graph with no triangle-free vertex cover.
For each vertex i ∈ V (G) we introduce a Boolean variable vi. It is our intention that each such variable represents the statement that “vertex i is included in the triangle-free vertex cover” (which might be True or False). When we write code, each variable vi is represented by a name in the form of a text string vi formed by concatenating v and i. So, for example, v2 is represented by the name v2.
But if all we have is all these variables, then there is nothing to control whether they are each True or False. We will need to build some logic, in the form of a CNF expression, to make this interpretation work. So, we need to encode the definition of a triangle-free vertex cover into a CNF expression using these variables. The expression we construct must depend on the graph. Furthermore, it must depend only on the graph.
We begin with a specific example (Problem 1) and then move on to general graphs (Problems 2–4).
Problem 1. [2 marks]
For the graph H shown in Figure 1, construct a Boolean expression φH using the variables vi which is True if and only if the assignment of truth values to the variables represents a triangle-free vertex cover of H.
Now type this expression φH into a one-line text file, using our text names for the variables (i.e., v1, v2, v3, etc.), with the usual logical operations replaced by text versions as follows:
Put your answer in a one-line text file called prob1.txt.
We are now going to look at the general process of taking any graph G on n vertices, and constructing a corresponding Boolean expression φG with n variables, in such a way that:
the variables of φG correspond to vertices in G, and
φG evaluates to True if and only if the assignment of truth values to those variables corresponds to a triangle-free vertex cover in G.
The next part of the assignment requires you to write an awk script to automate this construction. For the purposes of this task, every graph is encoded as an edge list in the following format:
The first line specifies the number of vertices, as a single positive integer. Let n be the number of vertices. Then the vertices of the graph are assumed to be numbered 1 , 2, . . . ,n.
Each subsequent line contains a single edge, represented as i -- j
where i and j are positive integers representing the two vertices linked by the edge (with 1 ≤ i ≤ n and 1 ≤ j ≤ n).
We allow any number of spaces before, between, and after the numbers on each line, subject to the requirement that there be at least one space on either side of the double-hyphen --.
For example, the graph in Figure 1 could be represented by the ten-line file on the left below, or (less neatly) by the one on the right.
Each graph is represented in a file of its own. Each input file contains exactly one graph represented in this way.
Positive integers, for n and the vertex numbers, can have any number of digits. They must have no decimal point and no leading 0.
Problem 2. [5 marks]
Complete the partial awk script. in Figure 3 so that when it takes, as input, a graph G represented in the specified format in a text file, it should produce, as output, a one-line file containing the text representation of a Boolean expression φG in CNF which uses the variables we have specified and which is True if and only if the variables describe a triangle-free vertex cover of G.
The text representation is the same as that described on p4 and used for Problem 1.
Put your answer in a file called prob2.awk.
BEGIN {
}
# The next line of code gets the number of vertices from the first line
# and stores it in the variable n
NR == 1 { n = $1; }
# Write code below to generate clauses ensuring that
# each edge must be included in the vertex cover
Your code goes here
# The line below should work in conjunction with what you wrote
# above, to add each edge into an associative array for checking later
a[$1, $3] = 1;
a[$3, $1] = 1;
}
END {
# Looping through triples of vertices in the array we created
for (i = 1; i <= n-2; i++)
{
for (j = i+1; j <= n-1; j++)
{
if (a[i, j] == 1) # (i,j) edge found, now check for triangles
{
for (k = j+1; k <= n; k++)
{
if (a[i, k] == 1 && a[j, k] == 1)
{
# Triangle found! Write code here to ensure
# the triangle is excluded from the vertex cover
Your code goes here
}
}
}
}
}
}
Figure 3: A partial awk script to use for Problem 2
Problem 3. [4 marks]
(a) Give, with brief justification, an expression for the number of clauses of φG in terms of m (the number of edges of G) and t (the number of triangles in G). For full marks, the number of clauses produced by your awk program (prob2.awk) must be correct as well as equalling the expression you give here.
(b) Give, with brief justification, an expression for an upper bound on the number of triangles t in G in terms of m. (You don’t need to find the tightest bound possible, but it needs to be tight enough to support your argument in part (c).)
(c) Based on your answers to (a) and (b), give with brief justification an upper bound on the number of clauses of φG, expressed only in terms of m.
Put your answers in a PDF file called prob3.pdf.
We are now going to modify the awk script so that the output it produces can be taken as input by a program for testing satisfiability.
SageMath is software for doing mathematics. It is powerful, widely used, free, open-source, and based on Python. It is already available in your Ed Workspace. You don’t need to learn SageMath for this assignment (although it’s good to be aware of what it is); you only need to follow the instructions below on how to use a specific function in SageMath for a specific task.
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