BU CS 332 – Theory of Computation Lecture 13: Reading: • Mid ‐ Semester Feedback Sipser Ch 4.1 • Enumerators • Decidable Languages 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 Work tape control “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 . 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 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. Check if is a valid encoding (reject if not) 2. Simulate on , 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 3/16/2020 CS332 ‐ Theory of Computation 17
Other decidable languages ��� ��� ��� ��� 3/16/2020 CS332 ‐ Theory of Computation 18
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 3/16/2020 CS332 ‐ Theory of Computation 19
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 3/16/2020 CS332 ‐ Theory of Computation 20
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 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: 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 3/16/2020 CS332 ‐ Theory of Computation 25
3/16/2020 CS332 ‐ Theory of Computation 26
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 → 𝑉 � 𝑉 � … 𝑉 � and every symbol 𝑉 � , … , 𝑉 � is marked 3. Accept if the start variable is unmarked; else reject 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
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