BU CS 332 Theory of Computation Lecture 13: Reading: Mid Semester - - PowerPoint PPT Presentation

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

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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”

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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

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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

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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

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Enumerators

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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…) …

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Enumerators

• 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

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Finite control Work tape “Printer”

Enumerator Example

• 1. Initialize
• 2. Repeat forever:
• Calculate
• (in binary)
• Send to printer
• Increment

What language does this enumerator enumerate?

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Enumerable = Turing‐Recognizable

Theorem: A language is Turing‐recognizable some enumerator enumerates it Start with an enumerator for and give a TM

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Enumerable = Turing‐Recognizable

Theorem: A language is Turing‐recognizable some enumerator enumerates it Start with a TM for and give an enumerator

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Decidable Languages

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1928 – The Entscheidungsproblem

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The “Decision Problem” Is there an algorithm which takes as input a formula (in first‐

• rder logic) and decides whether it’s logically valid?

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

• . List each component of the tuple

separated by ;

• Represent

by ,‐separated binary strings

• Represent by ,‐separated binary strings
• Represent

by a ,‐separated list of triples , … Denote the encoding of by

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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”

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A “universal” algorithm for recognizing regular languages

• Theorem:

is decidable

Proof: Define a 3‐tape TM

• n input
• 1. Check if

is a valid encoding (reject if not)

• 2. Simulate
• n , i.e.,
• Tape 2: Maintain 𝑥 and head location of 𝐸
• Tape 3: Maintain state of 𝐸, update according to 𝜀
• 3. Accept iff

ends in an accept state

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Other decidable languages

• 3/16/2020

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CFG Generation

Theorem:

• is Turing‐

recognizable 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

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CFG Generation

Theorem:

• is decidable

Chomsky Normal Form for CFGs:

• Can have a rule 𝑇 → 𝜁
• All remaining rules of the form 𝐵 → 𝐶𝐷 or 𝐵 → 𝑏
• Cannot have 𝑇 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

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CFG Generation

Theorem:

• is decidable

Proof idea: Define a TM recognizing

• On input

:

• 1. Convert

into Chomsky Normal Form

• 2. Enumerate all strings derivable in

steps

• 3. If any of these strings equal , accept

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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
• n input
• 2. Accept if the decider accepts; reject otherwise

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Classes of Languages

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context free regular recognizable decidable

More Examples

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Decidability of

Theorem:

• is

decidable Proof: The following TM decides On input , where is a DFA with states:

• 1. Perform

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

• 2. Accept if an accept state reachable; reject otherwise

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Decidability of

Theorem:

• is

decidable Proof: The following TM decides

• On input

, where is a CFG with states:

• 1. Mark all terminal symbols in
• 2. Repeat until no new variable is marked:

Mark any variable 𝐵 where 𝐻 has a rule 𝐵 → 𝑉𝑉 … 𝑉 and every symbol 𝑉, … , 𝑉 is marked

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

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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

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Symmetric Difference

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