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comp2022/2922 Turing Machines
创建日期：2024-07-05 16:23:16
标签： comp2022
Turing Machines
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comp2022/2922  Turing Machines
– For problem 4, you will submit files named run.sh, build.sh and other files you
require. The first argument of run.sh, either tm to ptm or ptm to tm, should choose
the solution that is run. A skeleton will be provided on Ed.
– For problem 5, you will submit your 2-tape TM in one file named problem 5.txt.
• All work must be done individually without consulting anyone else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism” policies.
• For clarifications and more details on all aspects of this assignment (e.g., level of justification 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”.
DTM stands for ”Deterministic Turing Machine”, and NTM stands for ”Nondeterministic Turing
Machine”.
All tapes in this assignment are doubly-infinite. When asked to give a low-level description use
Morphett’s format, and its extension to 2 tapes as follows: the instruction
q a b c d L R s
means that if the TM is in state q and reads a on the first tape and b on the second tape then it
writes c on the first tape and d on the second, and moves the first tape left and the second tape
right, and changes to state s.
For instance, suppose the question asked you to provide a 2-tape DTM that checks if a string has
the same number of 0s as 1s. Here is an answer:
; copy0: copies 0’s from tape#1 to tape#2
copy0 0 _ 0 0 R R copy0
copy0 1 _ * * R * copy0
copy0 _ _ * * L L compare
; compare: matches 1s on tape#1 with 0s on tape#2
compare 1 0 * * L L compare
compare 0 * * * L * compare
; same number
compare _ _ * * * * halt-accept
; more 1s than 0s
compare * _ * * * * halt-reject
; more 0s than 1s
compare _ * * * * * halt-reject
1
comp2022/2922 A3 (90 marks) – Turing Machines s2 2023
NB. The Morphett’s simulator only correctly simulates deterministic TMs. For nondeterministic
TMs it resolves nondeterminism randomly, which is not what TMs do! Thus, you cannot rely on
the simulator showing you all possible runs on a given input.
Problem 1. (20 marks)
1. (5 marks) Is the complement of every context-free language Turingdecidable? Give a brief justification of your answer.
2. (5 marks) Suppose that L is the intersection of a context-free language L1
and a regular language L2. Argue that L is in P. You may use any facts
proven in the lecture slides.
3. (5 marks) Fix Σ = {0, 1}. Consider the operation dub that maps a string
u to the string in which every 0 is replaced by 00. For a language L let
dub(L) = {dub(u) : u ∈ L}.
For instance, if L = {10, 1001, 11, ϵ} then dub(L) = {100, 100001, 11, ϵ}, and
if L = {0
n1
n
: n ≥ 0} then dub(L) = {0
2n1
n
: n ≥ 0}.
Suppose that L is decidable. Give a high-level description of a decider for
dub(L).
4. (5 marks) Consider the language L consisting of the set of strings
(SourceM1
, SourceM2
, SourceM3
) such that L(M1) is not regular, L(M2) is
not regular, and L(M1) ∪ L(M2) = L(M3).
Show that L is undecidable. You may only use facts from the lecture slides
to do so, e.g., that the acceptance problem TMs is undecidable.
Problem 2. (10 marks) Below is a NTM over input alphabet {a, b}.
1. (5 marks) Describe in one sentence the language that it accepts. Briefly
justify your answer. No marks will be awarded for describing how the TM
operates.
2. (5 marks) What is the asymptotic time complexity (aka running time) of
the NTM? Give your answer in big-Theta notation, and briefly explain your
answer.
; initial state is 0
; input alphabet is {a,b}
0 a a R 0
0 b b R 0
0 a A L 1
2
comp2022/2922 A3 (90 marks) – Turing Machines s2 2023
1 * * L 1
1 _ _ R 2
2 A A R 4
2 a _ R 2a
2a * * R 2a
2a A A R 2A
2A A A R 2A
2A a A L 1
2 b _ R 2b
2b * * R 2b
2b A A R 2B
2B A A R 2B
2B b A L 1
4 A A R 4
4 _ * * halt-accept
Problem 3. (20 marks) Fix Σ = {0, 1}.
1. (5 marks) Give a low-level implementation (in Morphett’s format) of a 1-
tape DTM that decides the language consisting of strings of the form x1y
where |x| = |y|. For instance, 1, 011, 10100 should be accepted, while
ϵ, 0, 00, 000, 11, 11000 should be rejected.
2. (5 marks) Give a low-level implementation (in Morphett’s format) of a 1-
tape DTM that decides the language consisting of strings of the form x1x.
For instance, 1, 010, 10110 should be accepted, while ϵ, 0, 00, 000, 11, 10101
should be rejected.
3. (10 marks) Give a low-level implementation (in Morphett’s format) of a 1-
tape DTM that decides the language consisting of strings of the form xyxxz
where x is non-empty. For instance, 1011, 000, 01110110110, 1001000010101
should be accepted, while ϵ, 0, 1101, 01101101, 0011101 should be rejected.
Problem 4. (20 marks) Following recent advances in interdimensional technology, a new type of Turing Machine has been developed, the portal Turing Machine
(PTM). These machines have the remarkable power to use 0 symbols on the tape
as portals, allowing them to traverse unbounded distances in a single step. (0
looks like a portal, you see.)
The semantics of a PTM are that when a machine is in a cell with a 0 and moves
left or right, instead of moving one cell, it moves to the next 0 in that direction.
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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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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