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COMP9020 Assignment
Creation date：2024-05-14 14:10:10
Belonging category：计算机computer science
Assignment

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COMP9020 Assignment 

Objectives and Outcomes
The purpose of this assignment is to build your mathematical maturity in the areas of Equivalence Relations,
Functions, Graph Theory, Recursion and Induction. Most questions are presented at a highly abstract level so
that the consequences are very general, and can be applied in a variety of situations: not just later on in the
course, but also beyond. The specific motivation for each problem can be summarised as follows:
Problem 1: This question shows another, very useful, characterization of equivalence relations, connecting
them with functions. This goes along with the characterization in terms of partitions (i.e. a connection
with sets) described in lectures. Most of the examples of equivalence relations given in lectures and
quizzes (e.g. Quiz 4, M1) fit this characterization; and we use the result again in Problem 2.
Problem 2: This question shows that, in certain circumstances, it is possible to “extend” addition and multi-
plication to equivalence classes. A variation of this result (for a different equivalence relation) establishes
the facts about how addtion and multiplication interact with =n stated in lectures. This particular re-
sult shows a connection between the matrix representation of relations and the adjacency matrix of their
graphical representation (the former corresponding to a matrix of equivalence classes). We will see the
semiring E again when we cover Boolean Logic.
Problem 3: The aim of this question is to model a concrete (“real-world”) problem abstractly using Graph
Theory. An abstract model lets us: identify the important aspects of a problem and ignore the extra-
neous; identify connections with other, seemingly unrelated problems; and make use of other, more
general, tools and results to solve the specific problem. We will revisit this problem in Assignment 3,
where we will model it with Propositional Logic.
Problem 4: The aim of this question is to show how you can prove and disprove whether or not a graph has
a particular graph property (in this case, planarity). While the specific technique of finding/excluding
subdivisions is not necessarily applicable to other graph properties, the process illustrates the general
principle of how you might prove a “negative” property.
Problem 5: The similarity between relational composition (;) and transitivity was remarked upon in lectures.
This problem clarifies that connection, showing how relational composition can be used to recursively
“add” transitivity to a non-transitive relation. This question ties together several topics: Relations,
Recursion, Induction (and, obliquely, Formal Languages), as well as results from the first Assignment;
and reinforces the basic concepts of those topics.
Problem 6: The binary tree data structure is commonly taught in introductory Computer Science and should
be familiar to most students taking COMP9020 (though it is not essential to know about them for the
purposes of this question). In this question we take an abstract model of a binary tree and use this
to create abstract, recursive “programs” (recursively defined functions) and then reason about these
programs. We will revisit this question in Assignment 3 when we use binary trees themselves as
abstract models (of logical formulas) and utilise the recursive definition to count these objects.
Problem 7: In Problems 5 and 6, the necessary recursive definition has been provided. The aim of this ques-
tion is to create your own recursive definition, providing a formal definition of the previously informally
defined lexicographic order. This provides a connection between recursion and partial orders, further
highlighting the ubiquity of recursion in many parts of Computer Science.
After completing this assignment, you will:
• Be able to make logical arguments about abstract concepts [Problems 1, 2, 4, 5 and 7]
• Be able to create and reason about abstract models of concrete (“real-world”) problems and objects
[Problems 3, 6 and 7]
• Understand several deeper connections between different topics of the course [Problems 1, 2, 5 and 7]
1
Due: Monday, 31st October, 12:00 (AEDST)
Submission is through WebCMS/give and should be a single pdf file, maximum size 2Mb. Prose
should be typed, not handwritten. Use of LATEX is encouraged, but not required.
Discussion of assignment material with others is permitted, but the work submitted must be your
own in line with the University’s plagiarism policy.
Problem 1 (12 marks)
Let S be a set.
(a) Show that for any set T and any function f : S! T, the relation Rf ✓ S⇥ S, defined as:
(s, s0) 2 Rf if and only if f (s) = f (s0)
is an equivalence relation. 6 marks
(b) Show that if R ✓ S⇥ S is an equivalence relation, then there exists a set T and a function fR : S ! T
such that:
(s, s0) 2 R if and only if fR(s) = fR(s0)
6 marks
Problem 2 (20 marks)
Let B = {0, 1} and consider the function f : N ! B given by
f (n) =
⇢
1 if n > 0,
0 otherwise.
(a) Show that for all a, b 2 N:
(i) f (a+ b) = max{ f (a), f (b)} 3 marks
(ii) f (ab) = min{ f (a), f (b)} 3 marks
From Problem 1, we know that Rf ✓ N⇥N, the relation given by:
(m, n) 2 Rf if and only if f (m) = f (n)
is an equivalence relation. Let E ✓ Pow(N) be the set of equivalence classes of Rf , and for n 2 N, let
[n] 2 E denote the equivalence class of n.
We would like to define binary operations, and , on E as follows:
[x] [y] := [x+ y]
[x] [y] := [xy].
2
The difficulty is that the operands [x] and [y] can have multiple representations (e.g. if z 2 [x] then
[x] = [z]), and so it is not clear that such a definition makes sense: if we take a different representation of
the operands, do we still end up with the same result? For example, suppose [1] = [2]. Then we would
want [1] [1] = [2] [2], but with the proposed definition above, we would have [1] [1] = [2], and
[2] [2] = [4], and it is by no means clear that [2] = [4].
Our next step is to show that such a definition makes sense.
(b) Define relations , ✓ E2 ⇥E as follows:
((X,Y),Z) 2 if and only if there is x, y 2 N such that X = [x], Y = [y] and Z = [x+ y]
((X,Y),Z) 2 if and only if there is x, y 2 N such that X = [x], Y = [y] and Z = [xy]
(i) Show that is a function. 3 marks
(ii) Show that is a function. 3 marks
Part (b) shows that the informal definition of and given earlier is well-defined, so from now we will
view and as binary operations on E, that is , : E⇥E ! E.
(c) Show that for all A, B,C 2 E:
(i) A [1] = A 3 marks
(ii) A B = B A 3 marks
(iii) A (B C) = (A B) (A C) 2 marks
Remark
Objects that have a concept of “addition” () and “multiplication” () where:
• addition and multiplication are associative,
• both operations have identities (see (c)(i)),
• addition is commutative (see (c)(ii)), and
• multiplication distributes over addition (see (c)(iii))
are known as semirings. We have already seen a number of semirings in this course:
• The natural numbers with usual addition and multiplication,
• Integers modulo n with addition and multiplication modulo n,
• Subsets of a set X with union and intersection,
• Languages with union and concatenation,
• Binary relations with union and relational composition (see Assignment 1),
• Matrices with matrix addition and matrix multiplication.
3
Problem 3 (12 marks)
Eight houses are lined up on a street, with four on each side of the road as shown:
Each house wants to set up its own wi-fi network, but the wireless networks of neighbouring houses – that
is, houses that are either next to each other (ignoring trees) or over the road from one another (directly
opposite) – can interfere, and must therefore be on different channels. Houses that are sufficiently far
away may use the same wi-fi channel. Your goal is to find the minimum number of different channels the
neighbourhood requires.
(a) Model this as a graph problem. Remember to:
(i) Clearly define the vertices and edges of your graph. 4 marks
(ii) State the associated graph problem that you need to solve. 2 marks
(b) Give the solution to the graph problem corresponding to this scenario; and determine the minimum
number of wi-fi channels required for the neighbourhood? 2 marks
(c) How do your answers to (a) and (b) change if a house’s wireless network can also interfere with those
of the houses to the left and right of the house over the road? 4 marks
Problem 4 (12 marks)
This is the Petersen graph:
0
1
2 3
4
5
6
7 8
9
(a) Give an argument to show that the Petersen graph does not contain a subdivision of K5. 6 marks
(b) Show that the Petersen graph contains a subdivision of K3,3. 6 marks
4
Problem 5 (20 marks)
Let R ✓ S⇥ S be any binary relation on a set S. Consider the sequence of relations R0,R1,R2, . . ., defined
as follows:
R0 := I = {(x, x) : x 2 S}, and
Rn+1 := Rn [ (R;Rn) for n 0
(a) Prove that for all i, j 2 N, if i  j then Ri ✓ Rj. Hint: Let Pi(j) be the proposition that Ri ✓ Rj and prove
that Pi(j) holds for all j i. 4 marks
(b) Let P(n) be the proposition that for all m 2 N: Rn;Rm = Rn+m. Prove that P(n) holds for all n 2 N.
Hint: Use results from Assignment 1 4 marks
(c) Prove that if there exists i 2 N such that Ri = Ri+1, then Rj = Ri for all j i. 4 marks
(d) If |S| = k, explain why Rk2 = Rk2+1. 2 marks
(e) If |S| = k, show that Rk2 is transitive. 2 marks
(f) If |S| = k show that Rk2 is the minimum (with respect to ✓) of all reflexive and transitive relations that
contain R. 4 marks
Remark
The relation at the limita as n tends to infinity, R⇤ = limn!• Ri, is known as the reflexive, transitive
closure of R, and is closely connected to the Kleene star operator.
aBecause Rj ✓ Ri ✓ S⇥ S for all j  i, the Knaster-Tarski theorem ensures this limit always exists, even for infinite S.
Problem 6 (20 marks)
A binary tree is a data structure where each node is linked to at most two successor nodes:
If we include empty binary trees (trees with no nodes) as part of the definition, then we can simplify the
description of the data structure. Rather than saying a node has 0, 1, or 2 successor nodes, we can instead
say that a node has exactly two children, where a child is a binary tree. That is, we can abstractly define
the structure of a binary tree as follows:
• (B): An empty tree, t
• (R): An ordered pair (Tleft, Tright) where Tleft and Tright are trees.
5
So, for example, the above tree would be defined as the tree T where:
T = (T1, T2), where
T1 = (T3, T4) and T2 = (T5, t), where
T3 = T4 = T5 = (t, t)
That is,
T =
⇣
(t, t), (t, t)

,

(t, t), t
⌘
A leaf in a binary tree is a node that has no successors (i.e. it is of the form (t, t)). A fully-internal node in
a binary tree is a node that has exactly two successors (i.e. it is of the form (T1, T2) where T1, T2 6= t). The
example above has 3 leaves (T3, T4, and T5) and 2 fully-internal nodes (T and T1). For technical reasons
(that will become apparent) we assume that an empty tree has 0 leaves and 1 fully-internal nodes.
(a) Based on the recursive definition above, recursively define a function count(T) that counts the number
of nodes in a binary tree T. 4 marks
(b) Based on the recursive definition above, recursively define a function leaves(T) that counts the number
of leaves in a binary tree T 4 marks
(c) Based on the recursive definition above, recursively define a function internal(T) that counts the num-
ber of fully-internal nodes in a binary tree T. 4 marks
(d) If T is a binary tree, let P(T) be the proposition that leaves(T) = internal(T)+ 1. Prove that P(T) holds
for all binary trees T. Your proof should be based on your answers given in (b) and (c). 8 marks
Problem 7 (4 marks)
Let S be a finite set, totally ordered by <. Give a formal, recursive definition of the lexicographic ordering
lex✓ S⇤ ⇥ S⇤. 4 marks
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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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