bu cs 332 – theory of computation · 2020. 3. 16. · bu cs 332 – theory of computation lecture...

BU CS 332 – Theory of Computation

Lecture 13:• Mid-Semester Feedback• Enumerators• Decidable Languages• Countability

Reading:Sipser Ch 4.1

Mark BunMarch 16, 2020

What aspects of the course help you learn best?

• Examples in class• Reviewing past homeworks/exams in class• Textbook• Posting materials online• Lecture, generally• Office hours• In-depth problem-solving in discussion section• Top Hat questions• Piazza discussions / instructor response

3/16/2020 CS332 - Theory of Computation 2

What in the class so far has hindered your learning?• Pace of information transmission / workload• Criteria for formality of proofs on homework and exams• Poor handwriting• Questions in class not fully answered• Lack of organization in discussion• Broad concepts

• “Bureaucratic descriptions”• “All materials concluded”


What specific changes can we make to improve your learning?• More examples• Post solutions / other materials online• Discussion solutions• More Top Hat questions• Go slower• More guidelines for how to solve each type of problem• Looser grading• Midterm too long• More detailed slides


Do you understand what is expected from you in this class?• Reading the book before vs. after class• Need to do every problem in the book to succeed?• Lack of coordination between readings and lectures• “I have to attend lectures, read the material in the book,

do some practice problems and then attempt the homework”

• Exam grading critical over formatting vs. looser standards on homework


How can you improve your own learning?• Read the book• Solve more practice problems• Review HW solutions• Come to office hours• Time management• Open mind to more abstract ways of thinking


Enumerators


TMs are equivalent to… • TMs with “stay put”• TMs with 2-way infinite tapes• Multi-tape TMs• Nondeterministic TMs• Random access TMs• Enumerators• Finite automata with access to an unbounded queue = 2-

stack PDAs• Primitive recursive functions• Cellular automata• “Turing-complete” programming languages (C, Python,

Java…)…


Enumerators

• Starts with two blank tapes• Prints strings to printer𝐿𝐿(𝐸𝐸) = {strings eventually printed by 𝐸𝐸}• May never terminate (even if language is finite)• May print the same string many times


Finite control

Work tape

“Printer”

Enumerator Example1. Initialize 𝑐𝑐 = 12. Repeat forever:

• Calculate 𝑠𝑠 = 𝑐𝑐2 (in binary)• Send 𝑠𝑠 to printer• Increment 𝑐𝑐

What language does this enumerator enumerate?


Enumerable = Turing-RecognizableTheorem: A language is Turing-recognizable ⇔ some enumerator enumerates it⇐ Start with an enumerator 𝐸𝐸 for 𝐴𝐴 and give a TM


Enumerable = Turing-RecognizableTheorem: A language is Turing-recognizable ⇔ some enumerator enumerates it⇒ Start with a TM 𝑀𝑀 for 𝐴𝐴 and give an enumerator


Decidable Languages


1928 – The Entscheidungsproblem


The “Decision Problem”

Is there an algorithm which takes as input a formula (in first-order logic) and decides whether it’s logically valid?

Questions about regular languagesDesign a TM which takes as input a DFA 𝐷𝐷 and a string 𝑤𝑤, and determines whether 𝐷𝐷 accepts 𝑤𝑤

How should the input to this TM be represented?Let 𝐷𝐷 = (𝑄𝑄, Σ, 𝛿𝛿, 𝑞𝑞0,𝐹𝐹). List each component of the tuple separated by ;• Represent 𝑄𝑄 by ,-separated binary strings • Represent Σ by ,-separated binary strings• Represent 𝛿𝛿 ∶ 𝑄𝑄 × Σ → 𝑄𝑄 by a ,-separated list of triples

(𝑝𝑝,𝑎𝑎, 𝑞𝑞), …

Denote the encoding of 𝐷𝐷,𝑤𝑤 by 𝐷𝐷,𝑤𝑤


Representation independenceComputability (i.e., decidability and recognizability) is not affected by the choice of encoding

Why? A TM can always convert between different encodings

For now, we can take to mean “any reasonable encoding”


A “universal” algorithm for recognizing regular languages𝐴𝐴DFA = 𝐷𝐷,𝑤𝑤 DFA 𝐷𝐷 accepts 𝑤𝑤}Theorem: 𝐴𝐴DFA is decidable

Proof: Define a 3-tape TM 𝑀𝑀 on input 𝐷𝐷,𝑤𝑤 :1. Check if 𝐷𝐷,𝑤𝑤 is a valid encoding (reject if not)2. Simulate 𝐷𝐷 on 𝑤𝑤, i.e.,

• Tape 2: Maintain 𝑤𝑤 and head location of 𝐷𝐷• Tape 3: Maintain state of 𝐷𝐷, update according to 𝛿𝛿

3. Accept iff 𝐷𝐷 ends in an accept state


Other decidable languages𝐴𝐴DFA = 𝐷𝐷,𝑤𝑤 DFA 𝐷𝐷 accepts 𝑤𝑤}

𝐴𝐴NFA = 𝑁𝑁,𝑤𝑤 NFA 𝑁𝑁 accepts 𝑤𝑤}

𝐴𝐴REX = 𝑅𝑅,𝑤𝑤 regular expression 𝑅𝑅 generates 𝑤𝑤}

𝐴𝐴CFG = 𝐺𝐺,𝑤𝑤 CFG 𝐺𝐺 generates 𝑤𝑤}


CFG GenerationTheorem: 𝐴𝐴CFG = 𝐺𝐺,𝑤𝑤 CFG 𝐺𝐺 generates 𝑤𝑤} is Turing-recognizableProof idea: Define a TM 𝑀𝑀 recognizing 𝐴𝐴CFG

On input 𝐺𝐺,𝑤𝑤 :1. Enumerate all strings that can be generated from 𝐺𝐺

(i.e., all length-1 derivations, all length-2 derivations, …)

2. If any of these strings equal 𝑤𝑤, accept


CFG GenerationTheorem: 𝐴𝐴CFG = 𝐺𝐺,𝑤𝑤 CFG 𝐺𝐺 generates 𝑤𝑤} is decidable

Chomsky Normal Form for CFGs:• Can have a rule 𝑆𝑆 → 𝜀𝜀• All remaining rules of the form 𝐴𝐴 → 𝐵𝐵𝐵𝐵 or 𝐴𝐴 → 𝑎𝑎• Cannot have 𝑆𝑆 on the RHS of any rule

Lemma: Any CFG can be converted into an equivalent CFG in Chomsky Normal FormLemma: If 𝐺𝐺 is in Chomsky Normal Form, any nonempty string w that can be derived from 𝐺𝐺 has a derivation with at most 2 𝑤𝑤 − 1 steps


CFG GenerationTheorem: 𝐴𝐴CFG = 𝐺𝐺,𝑤𝑤 CFG 𝐺𝐺 generates 𝑤𝑤} is decidableProof idea: Define a TM 𝑀𝑀 recognizing 𝐴𝐴CFG

On input 𝐺𝐺,𝑤𝑤 :1. Convert 𝐺𝐺 into Chomsky Normal Form2. Enumerate all strings derivable in ≤ 2 𝑤𝑤 − 1 steps3. If any of these strings equal 𝑤𝑤, accept


Context Free Languages are DecidableTheorem: Every CFL 𝐿𝐿 is decidableProof: Let 𝐺𝐺 be a CFG generating 𝐿𝐿. The following TM decides 𝐿𝐿.

On input 𝑤𝑤:1. Run the decider for 𝐴𝐴CFG on input 𝐺𝐺,𝑤𝑤2. Accept if the decider accepts; reject otherwise


Classes of Languages


context free

regular

recognizable

decidable

More Examples


𝐸𝐸DFA = 𝐷𝐷 𝐷𝐷 is a DFA that recognizes the empty language}

𝐸𝐸CFG = 𝐺𝐺 𝐺𝐺 is a CFG that recognizes the empty language}

Decidability of 𝐸𝐸𝐷𝐷𝐷𝐷𝐷𝐷Theorem: 𝐸𝐸DFA = 𝐷𝐷 𝐷𝐷 is a DFA that recognizes ∅} is decidableProof: The following TM decides 𝐸𝐸DFA

On input 𝐷𝐷 , where 𝐷𝐷 is a DFA with 𝑛𝑛 states:1. Perform 𝑛𝑛 steps of breadth-first search on state

diagram of 𝐷𝐷 to determine if an accept state is reachable from the start state

2. Accept if an accept state reachable; reject otherwise


Decidability of 𝐸𝐸𝐶𝐶𝐷𝐷𝐶𝐶Theorem: 𝐸𝐸CFG = 𝐺𝐺 𝐺𝐺 is a CFG that recognizes ∅} is decidableProof: The following TM decides 𝐸𝐸CFG

On input 𝐺𝐺 , where 𝐺𝐺 is a CFG with 𝑛𝑛 states:1. Mark all terminal symbols in 𝐺𝐺2. Repeat until no new variable is marked:

Mark any variable 𝐴𝐴 where 𝐺𝐺 has a rule 𝐴𝐴 → 𝑈𝑈1𝑈𝑈2 …𝑈𝑈𝑘𝑘 and every symbol 𝑈𝑈1, … ,𝑈𝑈𝑘𝑘 is marked

3. Accept if the start variable is unmarked; else reject


New Deciders from Old𝐸𝐸𝑄𝑄DFA = ⟨𝐷𝐷1,𝐷𝐷2⟩ 𝐷𝐷1,𝐷𝐷2 are DFAs and 𝐿𝐿(𝐷𝐷1) = 𝐿𝐿(𝐷𝐷2)}Theorem: 𝐸𝐸𝑄𝑄DFA is decidableProof: The following TM decides 𝐸𝐸𝑄𝑄DFA

On input ⟨𝐷𝐷1,𝐷𝐷2⟩ , where 𝐷𝐷1,𝐷𝐷2 are DFAs:1. Construct a DFA 𝐷𝐷 that recognizes the symmetric

difference 𝐿𝐿(𝐷𝐷1) △ 𝐿𝐿(𝐷𝐷2)

2. Run the decider for 𝐸𝐸DFA on 𝐷𝐷 and return its output


Symmetric Difference𝐴𝐴 △ 𝐵𝐵 = 𝑤𝑤 𝑤𝑤 ∈ 𝐴𝐴 or 𝑤𝑤 ∈ 𝐵𝐵 but not both}


Countability


Set Theory ReviewA function 𝑓𝑓:𝐴𝐴 → 𝐵𝐵 is• 1-to-1 (injective) if 𝑓𝑓 𝑎𝑎 ≠𝑓𝑓 𝑎𝑎𝑎 for all 𝑎𝑎 ≠ 𝑎𝑎𝑎

• onto (surjective) if for all 𝑏𝑏 ∈ 𝐵𝐵,there exists 𝑎𝑎 ∈ 𝐴𝐴 such that 𝑓𝑓 𝑎𝑎 = 𝑏𝑏

• a correspondence (bijective) if it is 1-to-1 and onto, i.e., every 𝑏𝑏 ∈ 𝐵𝐵 has a unique 𝑎𝑎 ∈ 𝐴𝐴 with 𝑓𝑓 𝑎𝑎 = 𝑏𝑏


How can we compare sizes of infinite sets?Definition: Two sets have the same size if there is a bijection between them

A set is countable if• it is a finite set, or• it has the same size as ℕ, the set of natural numbers


Examples of countable sets• ∅• 0,1• 0, 1, 2, … 8675309• 𝐸𝐸 = {2, 4, 6, 8, … }• 𝑆𝑆𝑄𝑄𝑈𝑈𝐴𝐴𝑅𝑅𝐸𝐸𝑆𝑆 = 1, 4, 9, 16, 25, …• 𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 1, 2, 4, 8, 16, 32, …

𝐸𝐸 = 𝑆𝑆𝑄𝑄𝑈𝑈𝐴𝐴𝑅𝑅𝐸𝐸𝑆𝑆 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = |ℕ|


How to show that ℕ × ℕ is countable?


(1, 1) (2, 1) (3, 1) (4, 1) 5, 1

(1, 2) (2, 2) (3, 2) (4, 2) (5, 2)

(1, 3) (2, 3) (3, 3) (4, 3) (5, 3)

(1, 4) (2, 4) (3, 4) (4, 4) (5, 4)

(1, 5) (2, 5) (3, 5) (4, 5) (5, 5)

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bu cs 332 – theory of computation · 2020. 3. 16. · bu cs 332 – theory of computation lecture...

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