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Computer theory

创建日期：2024-06-28 17:25:51

所属分类：计算机computer science

标签： Computer

Computer theory

Problem 1 (TM with Undo, 15 points). A TM with undo is the same as a basic single-tape TM, except in one computational step it is also allowed to perform the following “undo” action: Return the head to where it was before the most recent transition, and replace the symbol under the head with whatever was there previously. The state of the TM’s ﬁnite control should remain the same.

For example, suppose a TM with undo starts in conﬁguration 8675p409. If its transition function speciﬁes(p,4) = (q,3,R), then the next conﬁguration will be 86753q09. If its transition function then speciﬁes(q,0) = U (where U stands for “undo”), then the next conﬁguration will be 8675q409.

Things get more interesting when the TM with undo is asked to do multiple undos in succession, or interleave them with normal transitions. In these cases, the machine must be able to undo all of the normal transitions in the correct order – from most recent to least recent. If there is no instruction left to undo, the machine will stay in the same conﬁguration.

For example, suppose the transition function of a TM with undo speciﬁes(p,0) = (p,1,R), (p,1) = (q,0,R),(q,0) = U,(q,1) = (r,0,R), and(r,0) =(r,1) = U. Then if the TM’s starting conﬁguration is p010, the next few conﬁgurations the TM with undo enters are

p010

1. (read 0, write 1, move right)

1p10

2. (read 1, write 0, move right)

10q0

3. (undo instruction 2) 1q10

4. (read 1, write 0, move right)

10r0

5. (undo instruction 4)

1r10

6. (undo instruction 1)

r010

7. (undo…but there is no instruction left to undo)

r010

Give a simulation argument, using an implementation-level description, to show that every language recognizable by a TM with undo is also Turing-recognizable.

Hint: It may be useful to maintain an explicit record of past actions in a stack. Every time the TM with undo performs a normal read/write/move transition, push a record of this action to the stack. And every time it performs an undo, pop the most recent action from the stack. How would you implement this behavior on, say, a multi-tape TM?

Problem 2 (Countability and Diagonalization, 18 points).

(a) An undirected graph is a ﬁnite set of vertices V ✓ N together with a ﬁnite set of edges

E = {{u,v}|u,v 2 V}. Show that the set G of all undirected graphs is countable. (8 points) (b) Let M be the set of all encodings of Turing machines. A TM property is a function f :

M!{0,1}. Use a diagonalization argument to show that the set of all TM properties,

P = {f | f is a TM property}, is uncountable. (10 points)

Problem 3 (Loopy TM, 17 points). Consider the following computational problem. Given a TM M, doesM loop forever on every input w?

(a) Formulate this problem as a language LTM. (3 points)

(b) Use a reduction to prove that LTM is undecidable. Make sure to explain (brieﬂy) why the TM computing your reduction is correct. (14 points)

Problem 4 (Bonus Problem). A function f : N!N is non-increasing if f(i) f(i + 1) for every

i 2 N. Sof(1) = 8,f(2) = 3,f(3) = 2,f(4) = 2,… are the ﬁrst few values of (what looks like)

a non-increasing function f. But the function g where g(1) = 5,g(2) = 6,g(3) = 4,… is not non-increasing.

Is the set NIof non-increasing functions countable or uncountable? Prove your answer.

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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 PHIL2617 PHYS3116 MIE250 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