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CS 245 Final Assessment
创建日期：2024-04-15 15:44:03
标签： CS 245
Final Assessment
CS 245 Final Assessment 
Question 1 (12 marks)
Consider the following assumptions.
If the dragon is mythical, then it is immortal, but if it is not mythical, then it is a mortal
mammal. If the dragon does not have scales, then it is mortal and not a mammal. The
dragon is magical if it has scales.
(a) Use the following proposition symbols, represent each sentence of the above as a propositional[3]
formula.
= The dragon is magical.
= The dragon is a mammal.
= The dragon is mortal.
= The dragon is mythical.
= The dragon has scales.
Convert each of the formulas into conjunctive normal form.
(b) For each of the following questions, determine whether or not its answer is implied by the
assumptions above. If so, give a resolution proof of the implication. If not, give sufficient
counter-example(s).
i. Is the dragon mythical?[3]
ii. Is the dragon magical?[3]
iii. Does the dragon have scales?[3]
Question 2 (8 marks)
(a) Give the parse tree for the formula ?(?() → ()).[2]
(b) Give a formal proof (using Natural Deduction) of ? ? ?(?() → ()).[6]
In your proof, you may use the 17 basic rules and also any of the following.
? (∈), (???), (?+), and/or (Tr).
? Excluded Middle.
? , ? ? , for any and . (Sometimes called “ex falso”.)
Question 3 (8 marks)
In this question, you will use resolution to determine whether or not the formula
??(() = ) ∧ ???((()) = )
is satisfiable.
When this requires you to make an arbitrary choice, specify what choice you have made.
(a) Convert the formula to an equivalent formula in prenex normal form. (Keep all of the quan-[2]
tifiers present at this step.)
(b) Modify the formula from the previous step to produce a set of clauses (without quantifiers)[2]
suitable to apply resolution.
(c) Show the steps of the resolution, until either the process stops or you have made four new[3]
clauses.
(d) From the outcome of the previous step, what can you conclude about the satisfiability of the[1]
original formula?
Question 4 (6 marks)
(a) Prove that[2]
{?(() → ()), ?()} ? ?().
(b) Use part 4a to prove that[2]
{?(() → ()), ?()} ? ?().
(c) Because of the result of part 4b, there must be at least one error in this “proof”:[2]
(1) ?(() → ()), () ? () (by (∈))
(2) ?(() → ()), () ? ?(() → ()) (by (∈))
(3) ?(() → ()), () ? () → () (by (??), (2))
(4) ?(() → ()), () ? () (by (→ ?), (1), (3))
(5) ?(() → ()), () ? ?() (by (?+), (4))
(6) ?(() → ()), ?() ? ?() (by (??), (5))
State which line number is the first one having en error. Briefly explain what the error is.
Question 5 (6 marks)
An enhanced Turing machine (ETM) has all the features of an ordinary Turing machine,
except that its transitions may leave the tape head unmoved (i.e., move neither left nor right).
Prove that an enhanced Turing machine is no more powerful than an ordinary Turing machine.
Explain, in detail, how to implement an ETM transition of the form
(1, 1) = (2, 2, ),
(where indicates that the tape head should stay in its current position), in an ordinary Turing
machine.
Question 6 (12 marks)
(a) Fill in the blanks with appropriate annotations to prove partial correctness, with the given[7]
pre- and post-conditions.
Note that you will first need to determine an invariant.
? = 0 ∧ 0 ≥ 0? precondition
? ?
s := 0 ;
? ?
x := 1 ;
? ?
while ( n > 0 ) {
? ?
? ?
n := n - 1 ;
? ?
s := s + x ;
? ?
x := x + 2 ;
? ?
}
? ?
? = 20 ?
Question 6, continued
(b) Justify (informally but carefully) any “implied” conditions required by the annotations, to[3]
complete the proof of partial correctness.
(c) Is the program totally correct? Justify your answer.[2]
Question 7 (6 marks)
Recall that the array-assignment rule: ?[{1 ← 2}/] ? A[e1]:= e2 ; ? ?.
Consider the following code and its annotation. and ′ are formulas that you will specify below.
?1? ? ? pre-condition
?2? ?′ ? implied
?3? A[ m ] := n ;
?4? ?[[]] = ? array assignment
(The line numbers in angle brackets are only for reference. The variables , and take arbitrary
integer values. The array can have any integer as index [i.e., it is infinite – which does not matter
here].)
(a) Write down the exact formula ′, so that the array-assignment rule is used correctly. (Do not[2]
simplify it for this part.)
(b) Determine a non-trivial pre-condition and prove (informally) the resulting implication.[4]
(To get full marks, your should be equivalent to ′ and also as simple as possible. You do
not need to prove either of these conditions, however.)
Question 8 (6 marks)
Prove that the following problem is undecidable:
Given two Turing machines 1 and 2,
Is the set of strings accepted by 1 a subset of the set of strings accepted by 2?
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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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