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COMPSCI 350 Mathematical Foundations of Computer Science
创建日期：2024-04-12 16:44:57
标签： COMPSCI 350
Mathematical Foundations of Computer Science

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COMPSCI 350 

City COMPUTER SCIENCE 

Mathematical Foundations of Computer Science 

(Time allowed: TWO hours) NOTE: Attempt to answer Question 1 and three other Questions in your answer script book.

 You may use without proof any result proved in class; in such a case, indicate exactly the statement of the result (theorem, 

fact, example, corollary, etc.) you use. Page 1 of 3 COMPSCI 350 Question 1 For each of the following sentences 

write in your script book Y (YES) if the sentence is true or N (NO) if the sentence is false: (a) The set of Turing-recognisable languages over {0, 1} is uncountably infinite. (b) Some infinite Turing-recognisable languages are not Turing-decidable. (c) ATM is not Turing-recognisable. (d) There are problems that can be solved using the universal programming language Python that cannot be solved using standard Turing machines. (e) If language A over alphabet Σ is not decidable, then neither is any language B over Σ such that A ⊆ B. (f) The language {ε} is decidable. (g) The complement of every Turing-recognisable language is Turing-recognisable. (h) The latest D-Wave quantum computer is deterministic. (i) If A ≤m B and B is decidable, then A is Turing-recognisable. (j) The complements of some Turing-recognisable languages are Turing-recognisable. infinite. equivalent to [25 marks] Question 2: Answer the following: (a) Design a Turing machine that decides the following language: L = {w ∈ {0, 1}∗ | w contains contains at least one copy of a substring 02m12n+10, for some m ≥ 0, n ≥ 0}. Give a formal description of the machine (by specifying the transition function in state diagram form), and show that your machine accepts 10 but rejects 110. [13 marks] (b) Prove that your construction is correct, that is, that the machine does indeed decide L. [12 marks] Question 3: Answer the following: (a) Given a language L over alphabet Σ, let the prefix set of L be Prefix(L) = {x ∈ Σ∗ | x is a prefix of some y ∈ L}. Show that the class of Turing-recognisable languages is closed under the prefix operation. [13 marks] (b) Show that the class of Turing-recognisable languages is not closed under the operation ∩, where ∩(A,B) = A ∩B. [12 marks] Page 2 of 3 COMPSCI 350 Question 4: Show that the following languages are Turing-decidable: (a) L = {〈A〉 | A is a DFA over the alphabet{0, 1} and L(A) only contains strings even in length}. (b) L = {〈A〉 | A is an NFA and L(A) contains exactly k strings}, where k ≥ 0 is a fixed number. (c) L = {〈A〉 | A is a DFA over the alphabet{0, 1} and there exists k ≥ 0, L(A) contains exactly k strings}. [25 marks] Question 5: Let g be a computable function. For every natural n, define the language Lg,n = {〈M〉 | M has more than g(n) states}. a) What is the language Lg,3 when g(n) = n2? [5 marks] b) Is Lg,3 undecidable? Justify your answer. [20 marks] Question 6: Answer the following: a) State Rice’s Theorem. [5 marks] 

b) Is the language {〈M〉 | M is a Turing machine with less than 100 states} undecidable? Justify your answer. [10 marks] c)

 Can we apply Rice’s Theorem to the language {〈M〉 | M stops on the input 101}? Justify your answer. [10 marks] 

Question 7: (a) Define the complexity function K. [5 marks] (b) Let x ∈ {0, 1}∗. 

Give a condition equivalent to K(x) ≤ |x|− 1. Justify your answer. [10 marks] (c) Is the set {x ∈ {0, 1}∗ | K(x) ≤ |x|− 1} 

Turing recognisable? Justify your answer. [5 marks] (d) Is the set {x ∈ {0, 1}∗ | K(x) ≤ |x|− 1} Turing decidable?

 Justify your answer. [5 marks] Page 3 of 3

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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 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