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comp2022/2922 CFLs and non-regularity
创建日期：2024-06-03 16:57:19
标签： comp2022
CFLs and non-regularity

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comp2022/2922 A2 (90 marks) – CFLs and non-regularity 

• This assignment is due in Week 8 on Sunday 24 September, 11:59pm.
• Submission is on GradeScope.
• All work must be done individually without consulting anyone else’s solutions in accor-
dance with the University’s “Academic Dishonesty and Plagiarism” policies.
• For clarifications and more details on all aspects of this assignment (e.g., level of justifica-
tion expected, late penalties, repeated submissions, what to do if you are stuck, etc.) you
are expected to regularly monitor the Ed Forum post “Assignment FAQ”.
Problem 1. (10 marks) Fix Σ = {a, b}. For each of the following grammars,
provide five strings that are in the language and five strings that are not (you
score max(0, 5− num errors) points per part).
1.
S→ aSbb | ϵ
2.
S→ aA
A→ aA | bB
B→ bB | ϵ
Problem 2. (10 marks) Consider the following context-free grammar G over
Σ = {a, b}.
S→ AA
A→ AAA | bA | Ab | a
1. Show the grammar is ambiguous by providing two leftmost derivations of
the same string.
2. Provide a grammar in Chomsky Normal Form equivalent to G.
Problem 3. (20 marks) Fix Σ = {a, b}. Design context-free grammars for the
following languages:
1. The set of strings of length 6 that have the same number of a and b.
2. The set of strings of the form uvw with u, v,w ∈ {anbn|n ∈ Z+}.
3. The set of palindromes whose length is a multiple of 20. Use at most 100
rules.
4. The set of strings not in the set {anb2n+3|n ∈ Z+}.
1
comp2022/2922 A2 (90 marks) – CFLs and non-regularity s2 2023
Problem 4. (20 marks) Fix Σ = {a, b}. A string is extreme if its amount of a is
more than twice its amount of b plus one, or the other way around. For example,
aaaab and babbabbbbba are both extreme, while bbbbbaa and ababab are not.
1. Prove that the set E of extreme strings is not regular (10 marks).
2. Prove that the set P of strings that have an extreme string as a prefix is not
regular. Recall that a string x is a prefix of a string y if y = xw for some
string w. Thus aaaabbbb is in P (since it has aaaab as a prefix), while ababab
is not in P (since no prefix of it is extreme). (10 marks)
Problem 5. (10 marks) Fix Σ = {a, b}. Write a program that takes in a context-
free grammar G in CNF and outputs a context-free grammar for the language of
strings that are in L(G) and whose number of as is a multiple of 5.
For full marks, your solution should be able to take an arbitrary CNF grammar
with 10 variables and output a grammar with at most 100 variables.
Problem 6. (20 marks) You’re honing your skills for the world championship
of popular card game Magic: the Gathering. Your goal is to draw through all
the cards in your deck, and to this end, you’ve included lots of cards that help
you draw more cards. However, if you get unlucky, with some deck orderings
you’ve found that this is impossible. You’re interested in determining which
deck orderings can be won, and which can’t.
At the start of the game, you draw seven cards from your deck into your hand.
The cards in your deck are of the following types:
• Ancestral Recall. When played, you draw three cards.
• Wheel of Fortune. When played, you discard your hand, and then draw seven
cards.
• Cruel Bargain. When played for the first time in a game, you draw four
cards. It cannot be played more than once in a game.
• Other, useless cards.
Aside: Unlike in the actual game, you can play any number of cards from your
hand in any order, without needing to pay mana.
Your goal is to draw all the cards in your deck, at which point you win. Note that
”overdraw” (drawing more cards than exist in your deck) meets this condition,
and so is also a win. We encode a possible ordering of your deck as a string
over the alphabet Σ = {a,w, c, x} where a,w, c correspond to the cards starting
with those letters, and x corresponds to a useless card. We are interested in the
language of strings for which it is possible for you to win.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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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