浙江大学计算理论期末考试考博复习提纲和练习

Ars Digita University Theory of Computation

Topics

Languages

All computational problems can be reframed as questions of membership in a language (set). Drawing Deterministic Finite Automata (DFAs).

Drawing Nondeterministic Finite Automata (NFAs).

Converting NFAs to DFAs

Regular Languages, Closure properties of Regular languages.

Problems to work on

What is a language?

Draw a DFA that accepts the strings ending in 01.

Draw a DFA that accepts the strings that have an even number of zeros.

Draw a DFA that accepts the strings that have an even number of zeros or end in 01.

Draw a DFA that accepts the strings that have an even number of zeros and end in 01.

Draw a DFA that accepts the strings that have an even number of zeros but don't end in 01.

Draw an NFA that accepts the strings that end in 01. Try to make it as simple as possible.

Draw an NFA that accepts the strings that have an even number of zeros or end in 01 using six states.

Draw an NFA that accepts any string that is a concatenation of a string that has an even number of zeros with a string that ends in 01.

Draw an NFA that accepts the strings that have an even number of zeros and end in 01.

Draw an NFA that accepts the language 0, using only two states.

Convert the previous NFA into a DFA using the method described in class.

Draw an NFA with two states that corresponds to the language 0?2*.

Convert the previous NFA to a DFA.

What is a regular language?

Draw the DFA that corresponds to the language 10*10*.

What is the Prefix of the language corresponding to the language 10*10*.

Draw the DFA for the prefix of the language corresponding to the regular expression 10*10*. What is the MIN of the language 10*10*?

Draw the DFA corresponding to the MIN of the language 10*10*.

Main Topics

More practice with NFAs, Converting to DFAs, getting rid of epsilons.

Formal defintion of Automata

The math mumbo-jumbo formulation is: Some five-tuple (Q, Sigma, delta, q, F)....

But that's not as bad as it seems. This is just some object that contains four other objects and a method. Which one is the method?

Operations on Languages: Our friends PREFIX and MIN

An example Proof. Just so you have a reference, we'll show you how to "prove" that Regular languages are closed under difference, ie. L1-L2 is regular if L1 and L2 are.

? Let M1 = (Q1, Sigma, delta1, q1, F1) accept L1, and let

? Let M2 = (Q2, Sigma, delta2, q2, F2) accept L2

? Then

? M = (Q1xQ2, Sigma, delta, (q1,q2), F1 x(Q2-F2)) accepts L1-L2

? where delta(r1,r2,a) = (delta1(r1,a), delta2(r2,a))

At this point you should give a few words of justification about why you think this new machine would accept L1-L2. If the mathematical notation is just way to much for you, just try to explain things in intuitively.

Regular expressions

If we run out of things to talk about :-)

Generalized NFAs

NFA + reg. ex. transitions = GNFA

Converting NFAs to Regular expressions

Converting Regular expressions to NFAs

Problems to work on

Write out the NFA corresponding to the language and convert to a DFA: All strings that end in 0. Same for all strings whose penultimate character is 0.

If you like this sort of thing: same things for all strings that that have 0 as the third to the last character.

One more if you'd like the practice: All strings ending in {0,1}

Write regular expressions for all strings that end in aaba. Where Sigma is just the usual alphabet. Do the same for all strings that do not contain a.

Write a regular expression for the strings that contain the substring 0101, where Sigma = {0,1}.

Do the same for all strings that alternate between 0 and 1 and end in 0.

Do the same for strings that alternate between 0 and 1.

Hard problem: Write a regular expression for the strings over {0,1} that are divisible by 3.

Hard Problem: Build an automaton that recognizes the strings that have the same number of zeros and ones.

Topics

The Java Computability Toolkit.

More on Regular Expressions.

GNFAs and converting an NFA to a regular expression.

Converting a Regular Expression to an NFA.

Diagonalization.

The Pumping Lemma.

Problems to work on

(Warm up) Give one string that is in the language and one that is not:

a*b*

a(ba)*b

a* + b*

(b + aaa)*

aba+bab

Write a regular expression

That accepts the set of all strings.

That accepts the set of all strings not containing 00 as a substring.

Convert to regular expressions:

Convert to an NFA:

(001)*(1+e)

(01*0 + e + 00)

(001)*(1+e)(01*0 + e + 00)

How long does a string in the following language have to be before we are guaranteed to have a loop?

Is the set of all strings that are palindromes regular? Why or why not?

Show that the set of all strings of zeros that have length that is a perfect cube can not be described by a regular expression.

Topics

How to minimize DFA's and why one would.

Turning a DFA into a Grammar

Decision Problems.

Some applications of Regular expressions

Everything is a string / number.

Problems to work on

Why do we care about minimizing DFA's ?

Minimize the following DFA. What Language does it generate?

Give a regular grammar that generates the same language as the DFA in problem 2.

Give a regular grammars that generates the same language as the following regular expressions: 010*

a(ab + bc)*

Is the following a valid argument? Why or why not? The language L = {0m1m 2n | m, n >= 0 } is not regular, since we already know that M = { 0m 1m | m >= 0} is not regular by the pumping lemma, and since L contains M it cannot therefore be regular.

Prove that the language { x2n | n >= 0 } is not regular.

Give a decision algorithm which takes as input a regular language L and decides whether L contains the string 1000101.

Give a decision algorithm which takes as input a regular language L and decides whether L equals the language {1000101}

Give a decision algorithm which takes as input a regular language L and decides whether L contains the language given by the regular expression 10*10*1

Miscellaneous problem: Eliminate the espilon transitons of this NFA, without converting it to a DFA (ie, figure out an easier way)

Mildy Challenging and Surprising Problem: Let L be any language over the alphabet {0}, not necessarily regular. Show that L* is regular.

Wicked hard challenging problem: Describe a grammar that generates the language { 0p| p is a prime }

Topics

Decision Problems

Context free grammars.

Problems to work on

Decision Problems

Finish the decision problems form the last recitation handout.

CFG warmup

Give a context free grammar that generates the language of palindromes {w(0+1+epsilon)w R | w is in (0 + 1)*}

Give a context free grammar that generates every possible string over {0,1}

Give a context free grammar that generates the language 0n1 0n 111 0m 1r where n, m, r >= 0 Give a context free grammar that generates the language of all strings that contain 101.

Give a context free grammar that generates the language of all strings with an equal number of zeros and ones.

And/Or

Give a context free grammar that generates the language that has equal numbers of 0's and 1's or contains 101.

Give a context free grammar that generates the language that has equal numbers of 0's and 1's and contains 101.

Complements

Give a context free grammar that generates the language of all strings of the form 0m1n where n >= m >= 0.

Give a context free grammar that generates the language of all strings of the form 0m1n where n > m >= 0.

Give a context free grammar that generates the complement of the language 0*1*.

Give a context free grammar that generates the language { w | w is not equal to 0n1n for any choice of n}

Challenging problem: Give a context free grammar that generates the language { w | w is not a

palindrome}

Topics

Rigorous proofs using the pumping lemma.

More constructions with context free grammars.

Chomsky Normal Form.

Problems to work on

Finish working on the CFG problems from the lst handout.

(Warmup). Generate the grammar for 0*1(0+1)*.

Using the grammar above, What are the leftmost and rightmost derivation for the strings 1001, 0011? What are the parse trees?

(Text 2.13) Consider the following grammar.

2.S --> TT | U

3.T --> 0T | T0 | 1

4.U --> 0U00 | 1

Describe the language it generates.

Chomsky Normal Form

Put the following grammar in Chomsky Normal form. (Text 2.14)

6. A --> BAB | B | epsilon

7. B --> 00 | espilon

Put the following grammar in Chomsky Normal form.

10.A --> BAB | B | BC

11.B --> 00 | epsilon

12.C --> AA

13.

Rigorous pumping lemma proofs

Problems from previous handouts kindly reproduced here. Rigorously prove using the pumping lemma.

Prove that the language { x2n | n >= 0 } is not regular.

Show that the set of all strings of zeros that have length that is a perfect cube can not be described by a regular expression.

Topics

Using regular expressions and context free grammars with Lex and Yacc to build complex programs quickly. A calculator example.

Chomsky Normal Form (Finish problems on Chomsky Normal Form that we didn't get to last time, i.e. all of them ;-)

Push Down Automata.

Problems to work on

Give a grammar for the language {a i b j | i <= j <= 2i}.

Consider the grammar

1.S --> z A s

2. A --> xb B yz

3. B --> cd A q | epsilon

Play around with this grammar and try to guess what a pumping lemma for CFG's might look like. Construct a PDA for {0n1n | n >= 0} without looking at your notes.

Construct a PDA for {0n1m0m1n | n,m >= 0}

Topics

Today we'll mainly catch up on all the problems from previous handouts we haven't done yet. Aisde: Languages as infinitely long streams of bits, complexity and compression

Leftmost and Rightmost derivations.

Parse Trees.

Chomsky Normal Form. What it is useful for.

Converting a grammar to Chomsky normal form.

Intro to the pumping lemma for context free languages.

Languages and streams o' bits

Here is a list (well, the beginning of a list) of all possible strings that can be made from the characters a and b. They are ordered so that all strings of length 1 come first, then strings of length 2 then strings of length 3, ... etc.

e a b aa ab ba bb aaa aab aba abb baa bab bba bbb aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb aaaaa aaaab aaaba aaabb aabaa aabab aabba aabbb abaaa abaab ababa ababb abbaa abbab abbba abbbb baaaa baaab baaba baabb babaa babab babba babbb bbaaa bbaab bbaba bbabb bbbaa bbbab bbbba bbbbb aaaaaa aaaaab aaaaba aaaabb aaabaa aaabab aaabba aaabbb aabaaa aabaab aababa aababb aabbaa aabbab aabbba aabbbb abaaaa abaaab abaaba abaabb ababaa ababab ababba ababbb abbaaa abbaab abbaba abbabb abbbaa abbbab abbbba abbbbb baaaaa baaaab baaaba baaabb baabaa baabab baabba baabbb babaaa babaab bababa bababb babbaa babbab babbba babbbb bbaaaa bbaaab bbaaba bbaabb bbabaa bbabab bbabba bbabbb bbbaaa bbbaab bbbaba bbbabb bbbbaa bbbbab bbbbba bbbbbb

A language is an infinite stream of bits. Example a(a+b)*

0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 Example (ab)*

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Problems to work on

Work on the problems from previous handouts

Topics

How many languages are there?

Nondeterministic Pushdown Automata.

Deterministic Pushdown Automata.

Converting Context Free Grammars to Pushdown Automata.

Problems to work on

Languages

Could I ever write a computer program that enumerated (listed) all the possible languages? What about one that enumerated all the regular languages? What about one that enumerated all the context-free languages?

Pushdown Automata Warmup

What is the difference between a Nondeterministic Pushdown Automaton and a Deterministic Pushdown Automaton. Do you think they generate the same languages?

(Warm up) Construct a PDA that accepts the language {0,1}*.

(Warm up from last time:) Construct a PDA that accepts the language {0n1n| n >= 0}. Is your answer deterministic or not?

Construct a PDA that accepts the language {0n1n | n >= 2}.

More Pushdown Automata

Construct a PDA that accepts the language { 0m1n | n > m >= 0}

(From last time:) Construct a PDA that accepts the language {0n1m0m1n | n,m >= 0}.

Construct a PDA that accepts the language { w | w is not a palindrome}

And/Or

Construct a PDA that accepts the language { w | w is not a palindrome and w ends with a zero} Give a PDA that accepts the language {a i b j | i <= j <= 2i}

Deterministic vs Nondeterministic PDA's

Construct a NPDA that accepts the language of strings with the same number of zeros and ones. Construct a DPDA that accepts the language of strings with the same number of zeros and ones. Converting Grammars to PDA's

Convert the following Grammar to a PDA.

1.S --> AB

2. A --> 0

3. B --> 1

Convert the following Grammar to a PDA.

5. A --> BAB | B | epsilon

6. B --> 00 | espilon

Dimitri Kountourogiannis, dimitrik@https://www.360docs.net/doc/a617871411.html,

浙大考博英语真题部分考试

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2016年浙江大学博士生英语听力 PART A Welcome to Everglades National Park. The Everglades is a watery plain covered with saw grass that's home to numerous species of plants and wildlife. At one and a half million acres, it's too big to see it all today, but this tour will offer you a good sampling. Our tour bus will stop first at T aylor Slough. This is a good place to start because it's home to many of the plants and animals typically associated with the Everglades. You'll see many exotic birds and, of course, our world famous alligators. Don't worry, there's a boardwalk that goes across the marsh, so you can look down at the animals in the water from a safe distance. The boardwalk is high enough to give you a great view of the saw grass prairie. From there we'll head to some other marshy and even jungle like areas that feature wonderful tropical plant life. For those of you who'd like a closer view of the saw grass prairie, you might consider renting a canoe sometime during your visit here. However, don't do this unless you have a very good sense of direction and can negotiate your way through tall grass. We'd hate to have to come looking for you. You have the good fortune of being here in the winter—the best time of year to visit. During the spring and summer, the mosquitoes will just about eat you alive! Right now they're not so bothersome, but you'll still want to use an insect repellent. Welcome to Everglades National Park. The Everglades is a watery plain covered with saw grass that's home to numerous species of plants and wildlife. At one and a half million acres, it's too big to see it all today, but this tour will offer you a good sampling. Our tour bus will stop first at Taylor Slough. This is a good place to start because it's home to many of the plants and animals typically associated with the Everglades. You'll see many exotic birds and, of course, our world famous alligators. Don't worry, there's a boardwalk that goes across the marsh, so you can look down at the animals in the water from a safe distance. The boardwalk is high enough to give you a great view of the saw grass prairie. From there we'll head to some other marshy and even jungle like areas that feature wonderful tropical plant life. For those of you who'd like a closer view of the saw grass prairie, you might consider renting a canoe sometime during your visit here. However, don't do this unless you have a very good sense of direction and can negotiate your way through tall grass. We'd hate to have to come looking for you. You have the good fortune of being here in the winter—the best time of year to visit. During the spring and summer, the mosquitoes will just about eat you alive! Right now they're not so bothersome, but you'll still want to use an insect repellent.----- 文章来源托福听力听力是托福内容，完型填空是大学英语教程第四册里面的内容~~~

浙江大学2005-2006学年冬季学期《物理化学(乙)》课程期末考试试卷1

浙江大学2005-2006学年冬季学期《物理化学(乙)》课程期末考试试卷开课学院：理学院，考试形式：闭卷，允许带___计算器 _入场考试时间：2006年1月11日,所需时间： 120 分钟考生姓名： ___ __学号：专业： ________ 一、填空题（20分，每题2分） 1.一定量的理想气体从V1自由膨胀到V2后，其ΔU 0； ΔA（或ΔF）0（请选择填入>、<、＝、不确定）。 2.理想气体的焦耳－汤姆逊系数μJ-T0（请选择>、<、＝、不确定）。 3.如果要确定一个“组成和总量都已知的均相多组分体系”的状态，我们至少还必须知道该体系的其它个独立状态变量。 4.当隔离体系中发生某种变化（包括化学变化）后，该体系的ΔU 0（请选择>、<、＝、不确定）。 5.在恒温条件下，对理想气体进行恒压压缩时，该过程的体系熵变ΔS体系0， ΔS体系+ΔS环境0（请选择填入>、<、＝、不确定）。 6.以汞作为工作物质的可逆卡诺热机的热机效率为以理想气体作为工作物质的可逆卡诺热机的热机效率的％。 7.零下5℃的过冷水变成同温同压下的冰时，该过程的体系熵变ΔS 0（请选择填入>、<、＝、不确定）。 8.已知某二元溶液对拉乌尔定律产生正偏差。如果以x B →0，γ B ＝1为标准态时，其活度系数是（请选择填入：>1、<1、＝1、不确定）。 9.当反应体系的总压一定时，加入惰性气体有利于气体物质的量的反应。

（请选择填入：增大、减小、不变、不确定） 10.I2(g)溶于互不相溶的水和CCl4(l)中并达到平衡，则该体系的组分数C＝；自由度数F＝。二、选择题（20分，每题2分） 1. 已知H2临界温度t c= －239.9°C, 临界压力p c = 1.297×103 kPa。现有一氢气钢瓶，在298 K时瓶中H2的压力为98.0×103 kPa，则H2的状态一定是 (a)气态(b) 液态(c) 气－液两相平衡(d)无法确定 2. 在一个绝热良好、抽成真空的容器中，灌满压力为101.325 kPa、温度为373 K 的纯水(容器内无气体存在)，此时水的饱和蒸气压p*(H2O) (a) > 101.325 kPa (b) < 101.325 kPa (c)= 101.325 kPa (d)无法确定 3. 被绝热材料包围的房间内放有一电冰箱，将电冰箱门打开的同时向电冰箱供给电能而使其运行。室内的温度将( ). (a) 逐渐降低(b) 逐渐升高(c) 不变(d)无法确定 4. 在温度为T、压强为100 kPa时，反应(1) A = 2B，反应(2) 2A = C及反应(3) C = 4B的标准摩尔焓分别为?r H m?(1)、?r H m?(2)及?r H m?(3)，则?r H m?(3)等于 (a) 2?r H m?(1) + ?r H m?(2) (b) ?r H m?(2)－2?r H m?(1) (c) ?r H m?(2) + ?r H m?(1) (d) 2?r H m?(1)－?r H m?(2) 5. 一定量的某真实气体，经节流膨胀后使系统的温度下降，p、V之积变大，此过程的Q( );?H ( ); ?U( ); ?S( )。 (a)Q=0, ?H =0, ?U<0, ?S>0 (b) Q=0, ?H =0, ?U=0, ?S>0 (c) Q<0, ?H =0, ?U<0, ?S>0 (d) Q=0, ?H =0, ?U=0, ?S=0 6. 在273 K、100 kPa下，过冷的液态苯凝结成固态苯，则此过程的 (a) ?S(系) > 0 (b) ?S(环) < 0 (c)?S(系) + ?S(环) > 0 (d) ?S(系) + ?S(环) < 0 7. 在300K下，一个抽真空的容器中放入过量的A(s), 发生下列反应： A(s) B(s) + 3D(g) 达到平衡时D(g)的压力p D* = 1.02 kPa。此反应的标准平衡常数K?为 (a) 1.02 (b) 1.061×10－6 (c) 1.04×10－4(d) 3.06 8. 已知

2019年浙江大学硕士生英语机考及考博英语-听力部分模拟题(难度相当)

Test One（听力播放） Section A Directions: In this section you will hear 10 short conversations between two speakers. At the end of each conversation, a third voice will ask a question about what was said. You will hear the question only once. When you have heard the question, read the four possible answers marked A, B, C and D and decide which is the best answer. Mark your answer by blackening the corresponding letter on the Answer Sheet with a pencil. 1. A. He’s worried. B. It’s getting late. C. He heard a noise outside. D. He can’t see out of the window. 2. A. Soon B. In 20 minutes. C. They don’t care. D. They don’t know. 3. A. Here. B. A pass. C. An official. D.A letter of introduction. 4. A. Home. B. To town. C. To his friends. D. To a restaurant. 5. A. She uses coffee. B. She hates coffee. C. She never liked coffee. D. She liked coffee before. 6. A. The pay isn’t important. B. They don’t need the money. C. He wants to save the money. D. He doesn’t like waiting in line for his pay. 7. A. He wants to go to sleep. B. Doctors always tell him lies. C. He doesn’t believe in medicine. D. He needs a rest without being bothered. 8. A. He eats too much. B. He kills chickens. C. He only eats chickens. D. He married a greedy person. 9. A. Buying trees. B. Who owns the tree. C. Their family trees. D. How old the tree is. 10. A. A stranger. B. A ticket seller. C. A train attendant. D. Another passenger. Section B Directions: In this section of the test you will hear three brief talks. You will hear them only once. After each one you will hear some questions. You will hear each question only once. After you hear the question, you will have 15 seconds to choose the best answer from the four choices given. Mark your answer on the Answer Sheet by blackening the corresponding letter with a pencil.

浙江大学工程热力学期末考试试题

一、简答题（每小题?5?分，共?30?分） 1、未饱和湿空气经历绝热加湿过程，其干球温度、湿球温度和露点温度如何变化 2、定压、定温、绝热和定容四种典型的热力过程，其多变指数的值分别是多少 3、画出燃气轮机装置定压加热理想循环的?p-v?图和?T-s?图，并写出其用循环增压比表示的热效率公式。（假设工质为理想气体，比热取定值） 4、反映往复活塞式内燃机混合加热循环特性的设计参数有哪几个写出其定义式。 5、住宅用空调机当夏天环境温度升高时，其制冷系数和耗功量如何变化 6、为什么在湿蒸汽区域进行的绝热节流过程总是呈现节流冷效应二、计算题（共?70?分） 1?．（?18?分）?3kmol?温度?t?1?＝?100 ℃的氮气流与?1kmol?温度?t?2?＝?20 ℃的空气流在管道中绝热混合。已知混合前空气的摩尔分数为：?x?N 2 ?＝?0.79?、?x?O2＝?0.21?，若混合前后氮气、空气和混合物的压力都相等，试求： (1)?混合后气体的温度； (2)?混合气体中?N 2?和?O?2?的摩尔分数； (3)?对应于?1kmol?的混合气产物，混合过程的熵增。

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2011浙江大学考博英语听力原文

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浙大2011考博英语部分题目及答案

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