# CS代考计算机代写 Java Context Free Languages python algorithm BU CS 332 – Theory of Computation

BU CS 332 – Theory of Computation

Lecture 13:

• Mid‐Semester Feedback • Enumerators

• Decidable Languages

Reading: Sipser Ch 4.1

Mark Bun March 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”

3/16/2020 CS332 ‐ Theory of Computation 3

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

3/16/2020 CS332 ‐ Theory of Computation 4

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

3/16/2020 CS332 ‐ Theory of Computation 5

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

3/16/2020 CS332 ‐ Theory of Computation 6

Enumerators

3/16/2020 CS332 ‐ Theory of Computation 7

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…)

…

3/16/2020 CS332 ‐ Theory of Computation 8

Enumerators

Finite control

Work tape “Printer”

• Starts with two blank tapes • Prints strings to printer

strings eventually printed by

• May never terminate (even if language is finite) • May print the same string many times

3/16/2020 CS332 ‐ Theory of Computation 9

Enumerator Example

1. Initialize

2.

Repeat forever:

• Calculate (in binary) • Send to printer

• Increment

What language does this enumerator enumerate?

3/16/2020 CS332 ‐ Theory of Computation 10

Enumerable = Turing‐Recognizable

Theorem: A language is Turing‐recognizable some enumerator enumerates it

Start with an enumerator for and give a TM

3/16/2020 CS332 ‐ Theory of Computation

11

Enumerable = Turing‐Recognizable

Theorem: A language is Turing‐recognizable some enumerator enumerates it

Start with a TM for and give an enumerator

3/16/2020 CS332 ‐ Theory of Computation 12

Decidable Languages

3/16/2020 CS332 ‐ Theory of Computation 13

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?

3/16/2020 CS332 ‐ Theory of Computation 14

Questions about regular languages

Design a TM which takes as input a DFA and a string , and determines whether accepts

How should the input to this TM be represented?

Let

separated by ;

. List each component of the tuple

• Represent

by ,‐separated binary strings by ,‐separated binary strings

• Represent

• Represent ,…

by a ,‐separated list of triples

Denote the encoding of by

3/16/2020 CS332 ‐ Theory of Computation 15

Representation independence

Computability (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”

3/16/2020 CS332 ‐ Theory of Computation 16

A “universal” algorithm for recognizing regular languages

Theorem: is decidable

Proof: Define a 3‐tape TM on input

1. 2.

Check if is a valid encoding (reject if not)

3.

Accept iff ends in an accept state

Simulate on , i.e.,

• Tape 2: Maintain 𝑤 and head location of 𝐷

• Tape 3: Maintain state of 𝐷, update according to 𝛿

3/16/2020 CS332 ‐ Theory of Computation 17

Other decidable languages

3/16/2020 CS332 ‐ Theory of Computation 18

CFG Generation

Theorem: recognizable

is Turing‐

Proof idea: Define a TM

recognizing

On input :

1. Enumerate all strings that can be generated from

(i.e., all length‐1 derivations, all length‐2 derivations, …) 2. If any of these strings equal , accept

3/16/2020 CS332 ‐ Theory of Computation 19

CFG Generation

Theorem: is decidable

Chomsky Normal Form for CFGs:

• Canhavearule𝑆 → 𝜀

• Allremainingrulesoftheform𝐴→𝐵𝐶or𝐴→𝑎 • Cannot have 𝑆 on the RHS of any rule

Lemma: Any CFG can be converted into an equivalent CFG in Chomsky Normal Form

Lemma: If is in Chomsky Normal Form, any nonempty string w that can be derived from has a derivation with at most steps

3/16/2020

CS332 ‐ Theory of Computation 20

CFG Generation

Theorem:

Proof idea: Define a TM recognizing

is decidable

On input :

1. Convert into Chomsky Normal Form 2. Enumerate all strings derivable in

3. If any of these strings equal , accept

steps

3/16/2020 CS332 ‐ Theory of Computation

21

Context Free Languages are Decidable

Theorem: Every CFL is decidable

Proof: Let be a CFG generating . The following TM

decides

On input :

1. Run the decider for on input

2. Accept if the decider accepts; reject otherwise

3/16/2020 CS332 ‐ Theory of Computation 22

Classes of Languages

recognizable

decidable context free

regular

3/16/2020 CS332 ‐ Theory of Computation 23

More Examples

3/16/2020 CS332 ‐ Theory of Computation 24

Decidability of

Theorem: decidable

is

Proof: The following TM decides

On input , where is a DFA with

1. 2.

Perform steps of breadth‐first search on state diagram of to determine if an accept state is reachable from the start state

Accept if an accept state reachable; reject otherwise

3/16/2020 CS332 ‐ Theory of Computation 25

states:

3/16/2020 CS332 ‐ Theory of Computation 26

Decidability of

Theorem: decidable

is

Proof: The following TM decides

On input , where is a CFG with

1. 2.

states: Repeat until no new variable is marked:

3.

every symbol 𝑈, … , 𝑈 is marked

Accept if the start variable is unmarked; else reject

Mark all terminal symbols in

Mark any variable 𝐴 where 𝐺 has a rule 𝐴 → 𝑈𝑈 …𝑈 and

3/16/2020 CS332 ‐ Theory of Computation 27

New Deciders from Old

Theorem: is decidable

Proof: The following TM decides

On input , where are DFAs:

1. Construct a DFA that recognizes the symmetric

difference

2. Run the decider for on and return its output 3/16/2020 CS332 ‐ Theory of Computation 28

Symmetric Difference

3/16/2020 CS332 ‐ Theory of Computation 29