# CS代考计算机代写 algorithm compiler BU CS 332 – Theory of Computation

BU CS 332 – Theory of Computation

Lecture 9:

• Midterm I review

Reading: Sipser Ch 0-2.3

Mark Bun February 19, 2020

Midterm I Topics

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Deterministic FAs (1.1)

• Given an English or formal description of a language 𝐿, draw the state diagram of a DFA recognizing 𝐿 (and vice versa)

• Know the formal definition of a DFA (A DFA is a 5 tuple…) and convert between state diagram and formal description

• Know the formal definition of how a DFA computes

• Regular operations: Union, concatenation, star and closure of regular languages under regular operations, construction for closure under complement

• Cross-product construction for union/intersection

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Nondeterministic FAs (1.2)

• Given an English or formal description of a language 𝐿, draw the state diagram of an NFA recognizing 𝐿 (and vice versa)

• Know the formal definition of an NFA

• Know the power set construction for converting an NFA

to a DFA

• Proving closure properties: Know the constructions for union, concatenation, star

• Recall other closure properties: reverse, intersection, complement

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Regular Expressions (1.3)

• Given an English or formal description of a language 𝐿, construct a regex generating 𝐿 (and vice versa)

• Formal definition of a regex

• Know how to convert a regex to an NFA

• Know how to convert a DFA/NFA to a regex

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Non-regular Languages (1.4)

• Know the proof ideas for the pumping lemma for regular languages

• Understand the statement of the pumping lemma and how to apply it

• Beyond the pumping lemma: Showing languages are non-regular by combining pumping lemma with closure properties

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Context-free Grammars (2.1)

• Given an English or formal description of a language 𝐿, give a CFG (in Backus-Naur form) generating 𝐿 (and vice versa)

• Formal definition of a CFG (A CFG is a 4-tuple…), context-free languages

• Parse trees, derivations

• You are not responsible for the material on ambiguity in parsing and Chomsky Normal Form

But these are interesting! Read about them if you have time

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Pushdown Automata (2.2)

• Given an English or formal description of a language 𝐿, describe a PDA recognizing 𝐿 in terms of:

• An algorithmic description of the machine • A state diagram for the machine

• (and vice versa)

• Formal definition of a PDA

• Know that PDAs recognize the context-free languages.

You are not responsible for knowing the proof.

• Closure properties of CFLs: Regular operations and intersection with regular languages, but not complement or intersection

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Non-context-free Languages

• Know the proof ideas for the pumping lemma for CFLs • Understand the statement of the pumping lemma and

how to apply it

• Beyond the pumping lemma: Showing languages are non-context-free by combining pumping lemma with closure properties

You are not responsible for Chapter 2.4 on deterministic CFLs (But read this if you’re interested in how CFLs are parsed in real compilers, etc.)

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Exam Tips

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Study Tips

• Review problems from HW 0-3, discussion sections 1-3, solved exercises/problems in Sipser, and suggested exercises on the homework

• We will literally ask you a question from the homework exercises, so make sure you understand these

• You may bring a page of notes to the exam. Preparing this note sheet is a great way to study.

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Study Tips

• Make sure you know how to solve the problems on the practice midterm and are familiar with the format. The format/length of the real midterm will be very similar.

• If you need more practice, there are lots of problems in the book. We’re happy to talk about any of these problems in office hours.

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For the exam itself

• You may cite without proof any result…

Stated in lecture

Stated and proved in the main body of the text (Ch. 0-2.3)

These include worked-out examples of state diagrams, regexes, CFGs, non-regular/non-CF languages

• Not included above: homework problems, discussion problems, (solved) exercise/problems in the text

• Showing your work / explaining your answers will help us give you partial credit

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Practice Problems

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Regular Languages

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Name six operations under which the regular languages are closed

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Prove or disprove: The non-regular languages are closed under union

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Give the state diagram of an NFA recognizing the language (01 U 10)*

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Give an equivalent regular expression for the

following NFA

0,1

0

𝑞0

𝑞1

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Let 𝑅 be a regular expression with 𝑛 symbols. If we convert 𝑅 into an NFA using the procedure described in class, how many states could it have in the worst case?

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Is the following language regular?

{𝑎𝑛𝑎𝑛|𝑛 ≥ 0}

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Is the following language regular?

{0𝑛1𝑛|0 ≤ 𝑛 ≤ 2020}

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Let𝐿 = 𝑤∈ 0,1 ∗ 𝑤hasthesamenumberof0sand1s}. Let 𝑝 be a pumping length and 𝑠 = (01)𝑝.

Give a decomposition of 𝑠 = 𝑥𝑦𝑧 which can be pumped in 𝐿. Is 𝐿 regular?

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Context-Free Languages

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Name three operations under which the context-free languages are closed.

Name two operations under which the CFLs are not closed

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What language is generated by the CFG

𝑆→𝑎𝑆𝑏 𝑏𝑌 𝑌𝑎 Y→𝑏𝑌|𝑎𝑌|𝜀 ?

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What language is recognized by the following

PDA?

𝑞0

𝜀,𝜀→$ 𝑞1,0→𝜀 1

0,𝜀→0 1,0→ 𝜀

𝜀, 0 → 𝜀

𝑞2

𝑞𝑓

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Give a CFG for the language

𝑤#0𝑛 𝑛≥0,𝑤=𝑛}

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Give a PDA recognizing the language

0𝑛1𝑛 𝑛 ≥ 0}

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Prove that 𝑤 ∈ 0,1 ∗ 𝑤 is a palindrome with the same number of 0s and 1s} is not context-free

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