BU CS 332 Theory of Computation Lecture 16: Reading: Mapping - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation Lecture 16: Reading: • Mapping Reducibility Sipser Ch 5.3 Mark Bun March 25, 2020

Problems in language theory 𝐔𝐍 𝐄𝐆𝐁 𝐃𝐆𝐇 undecidable decidable decidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 decidable decidable undecidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 ? ? decidable 3/30/2020 CS332 ‐ Theory of Computation 2

Reductions A reduction from problem to problem is an algorithm for problem which uses an algorithm for problem as a subroutine If such a reduction exists, we say “ reduces to ” 3/30/2020 CS332 ‐ Theory of Computation 3

Reminder: Using reductions to prove undecidability If reduces to and is undecidable, then is also undecidable Proof template: 1. Suppose to the contrary that is decidable 2. Using as a subroutine, construct an algorithm deciding 3. But is undecidable. Contradiction! 3/30/2020 CS332 ‐ Theory of Computation 4

Equality Testing for TMs �� � � � � � � Theorem: �� is undecidable Proof: Suppose for contradiction that there exists a decider for �� . We construct a decider for �� as follows: On input : 1. Construct TMs � , � as follows: 2. Run on input � � 3. If accepts, accept. Otherwise, reject. This is a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 5

Equality Testing for TMs �� � � � � � � Theorem: �� is undecidable Proof: Suppose for contradiction that there exists a decider for �� . We construct a decider for �� as follows: On input : 1. Construct TMs � , � as follows: � = � = 2. Run on input � � 3. If accepts, accept. Otherwise, reject. This is a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 6

Problems in language theory 𝐔𝐍 𝐄𝐆𝐁 𝐃𝐆𝐇 undecidable decidable decidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 decidable decidable undecidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 ? undecidable decidable 3/30/2020 CS332 ‐ Theory of Computation 7

Context ‐ free language testing for TMs �� Theorem: �� is undecidable Proof: Suppose for contradiction that there exists a decider for �� . We construct a decider for �� as follows: On input : 1. Construct a TM as follows: 2. Run on input 3. If accepts, accept. Otherwise, reject This is a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 8

Context ‐ free language testing for TMs �� Theorem: �� is undecidable Proof: Suppose for contradiction that there exists a decider for �� . We construct a decider for �� as follows: On input : 1. Construct a TM as follows: = “On input , � � � 1. If accept 2. Run TM on input 3. If accepts, accept.” 2. Run on input 3. If accepts, accept. Otherwise, reject This is a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 9

Mapping Reductions 3/30/2020 CS332 ‐ Theory of Computation 10

🚪🚪 Warning 🚪🚪 What’s wrong with the following “proof”? Bogus “Theorem”: �� is not Turing ‐ recognizable Bogus “Proof”: Suppose for contradiction that there exists a recognizer for �� . We construct a recognizer for �� : On input : 1. Run on input 2. If accepts, reject. Otherwise, accept. This sure looks like a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 11

Mapping Reductions: Motivation 1. How do we formalize the notion of a reduction? 2. How do we use reductions to show that languages are unrecognizable? 3. How do we protect ourselves from accidentally “proving” bogus theorems about recognizability? 3/30/2020 CS332 ‐ Theory of Computation 12

Computable Functions Definition: ∗ is computable if there is a TM ∗ A function ∗ , halts with only which, given as input any on its tape. Example 1: Example 2: = where is a TM, is a string, and is a TM that ignores its input and simulates running on 3/30/2020 CS332 ‐ Theory of Computation 13

Mapping Reductions Definition: Language is mapping reducible to language , written � ∗ such that for ∗ if there is a computable function ∗ , we have all strings 3/30/2020 CS332 ‐ Theory of Computation 14

Decidability Theorem: If and is decidable, then is also � decidable ∗ be a ∗ Proof: Let be a decider for and let mapping reduction from to . Construct a decider for as follows: On input : 1. Compute 2. Run on input 3. If accepts, accept. Otherwise, reject. 3/30/2020 CS332 ‐ Theory of Computation 15

Undecidability Theorem: If and is decidable, then is also � decidable Corollary: If and is undecidable, then is also � undecidable 3/30/2020 CS332 ‐ Theory of Computation 16

Old Proof: Equality Testing for TMs �� � � � � � � Theorem: �� is undecidable Proof: Suppose for contradiction that there exists a decider for �� . We construct a decider for �� as follows: On input : 1. Construct TMs � , � as follows: � = “On input 𝑦 , � = “On input 𝑦 , Ignore 𝑦 1. Ignore 𝑦 and accept” 1. Run 𝑁 on input 𝑥 2. If 𝑁 accepts, accept. 3. Otherwise, reject.” 2. Run on input � � 3. If accepts, accept. Otherwise, reject. This is a reduction from �� to �� 3/30/2020 CS332 ‐ Theory of Computation 17

New Proof: Equality Testing for TMs �� � � � � � � �� � � Theorem: �� hence �� is undecidable Proof: The following TM computes the reduction: On input : 1. Construct TMs � , � as follows: � = “On input 𝑦 , � = “On input 𝑦 , Ignore 𝑦 1. Ignore 𝑦 and accept” 1. Run 𝑁 on input 𝑥 2. If 𝑁 accepts, accept. 3. Otherwise, reject.” 2. Output � � 3/30/2020 CS332 ‐ Theory of Computation 18

Mapping Reductions: Recognizability Theorem: If and is recognizable, then is also � recognizable ∗ be a ∗ Proof: Let be a recognizer for and let mapping reduction from to . Construct a recognizer for as follows: On input : 1. Compute 2. Run on input 3. If accepts, accept. Otherwise, reject. 3/30/2020 CS332 ‐ Theory of Computation 19

Unrecognizability Theorem: If and is recognizable, then is also � recognizable Corollary: If and is unrecognizable, then is � also unrecognizable Corollary: If , then is unrecognizable �� � 3/30/2020 CS332 ‐ Theory of Computation 20

Recognizability and 3/30/2020 CS332 ‐ Theory of Computation 21

Consequences of 1. Since �� is undecidable, �� is also undecidable 2. �� implies �� � �� � �� Since �� is unrecognizable, �� is unrecognizable 3/30/2020 CS332 ‐ Theory of Computation 22

itself is also unrecognizable �� � � � � � � Theorem: 𝐵 �� � � �� hence �� is unrecognizable Proof: The following TM computes the reduction: On input : 1. Construct TMs � , � as follows: � = “On input 𝑦 , � = “On input 𝑦 , Ignore 𝑦 1. Ignore 𝑦 and accept” 1. Run 𝑁 on input 𝑥 2. If 𝑁 accepts, accept. 3. Otherwise, reject.” 2. Output � � 3/30/2020 CS332 ‐ Theory of Computation 23

More on Reductions and Undecidability 3/30/2020 CS332 ‐ Theory of Computation 24

Problems in language theory 𝐔𝐍 𝐄𝐆𝐁 𝐃𝐆𝐇 undecidable decidable decidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 decidable decidable undecidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 ? undecidable decidable 3/30/2020 CS332 ‐ Theory of Computation 25

is undecidable ��� ∗ Theorem: ��� hence ��� is undecidable �� � Proof idea: “Computation history method” On input : 1. Construct a CFG such that: ∗ does not accept 2. Output 3/30/2020 CS332 ‐ Theory of Computation 26

Problems in language theory 𝐔𝐍 𝐄𝐆𝐁 𝐃𝐆𝐇 undecidable decidable decidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 decidable decidable undecidable 𝐄𝐆𝐁 𝐃𝐆𝐇 𝐔𝐍 undecidable undecidable decidable 3/30/2020 CS332 ‐ Theory of Computation 27

Undecidable problems outside language theory Post Correspondence Problem (PCP): � Domino: �� . Top and bottom are strings. Input: Collection of dominos. Match: List of some of the input dominos (repetitions allowed) where top = bottom Problem: Does a match exist? This is undecidable 3/30/2020 CS332 ‐ Theory of Computation 28