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FIT2014 logic, regular expressions, induction
创建日期：2024-08-06 18:17:46
标签： FIT2014
logic, regular expressions, induction
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FIT2014
Assignment 1
Linux tools, logic, regular expressions, induction
Start work on this assignment early. Bring questions to Consultation and/or the Ed Forum.
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.
1

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 is 3-colourable.
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
2
for (i = 1; i <= n; i++)
{
hbody of loopi
}
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 di↵erent, 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++) { hbody of loopi }
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 trac
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 'G 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 'G to make it True.
3
⼀
12
3
4
5
6
7
Figure 1: A graph, H, which has a number of di↵erent triangle-free vertex covers.
1
2
3
4
5 6
7
Figure 2: A graph with no triangle-free vertex cover.
For each vertex i 2 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 'H 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 'H 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:
logical operation text representation
¬ ~
^ &
_ |
Put your answer in a one-line text file called prob1.txt.
4
ˊ
D
T A c
y y h7B
CHIiinaiiBaEaGNCBN NAnDN
wvdrk
IV.liV3⼋V6⼋V7ˋˇ
V2ˇˇˇˇ
CV xV2⼋V5 V7
We are now going to look at the general process of taking any graph G on n vertices, and
constructing a corresponding Boolean expression 'G with n variables, in such a way that:
• the variables of 'G correspond to vertices in G, and
• 'G 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.
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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