Proving more specific problems are not recursive

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Lecture 10

• Proving more specific problems are not recursive
• Reduction technique
• Use subroutine theme to show that if one problem
  is unsolvable, so is a second problem
• Need to clearly differentiate between
• use of program as a subroutine and
• a program being an input to another program

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Question 1

• What can we conclude from the following scenario?
• We prove/know language L2 is recursive
• We show that we can construct a program P1 which
  decides L1 using any program P2 which decides
  language L2 as a subroutine.
• We conclude that
• Does this occur in real-life programming?

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Question 2

• What can we conclude from the following scenario?
• We prove/know language L1 is not recursive
• We show that we can construct a program P1 which
  decides L1 using any program P2 which decides
  language L2 as a subroutine.
• We conclude that

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Rephrasing key step

• We show that we can construct a program P1 which
  decides L1 using any program P2 which decides
  language L2 as a subroutine.
• This can be rephrased in the following 2 ways
• If L2 is recursive, then L1 is recursive
• Scenario in question 1
• If L1 is not recursive, then L2 is not recursive
• Scenario in question 2

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Illustration of If L1 is not recursive, then L2
is not recursive
Set of all languages
REC
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What we will typically do
Set of all languages
REC
If L1 is not recursive, then L2 is not recursive
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Proving If L2 is recursive, then L1 is
recursive

• Assume L2 is recursive
• Let P2 be a program which decides L2
• Construct a program P1 which decides L1 using P2
  subroutine theme
• We have no idea how P2 decides L2
• We still treat P2 as a black box
• L1 and L2 are now SPECIFIC languages
• Thus, we can use properties of L1 and L2 in this
  construction
• Argue P1 decides L1

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Constructing P1 from P2

• Many ways to construct P1 from P2
• We focus on one method
• Construct a program P3 which computes a function
  f which we call a many-one reduction
• P3 (or f) transforms inputs to problem L1 into
  inputs to problem L2 so that
• yes inputs map to yes inputs
• no inputs map to no inputs

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Construction Overview

• Properties of the construction
• Input string x is transformed into a new string
  P3(x)
• x is an input to program P1 (problem L1)
• P3(x) is an input to program P2 (problem L2)
• We must give actual program P3
• P3 will use specific knowledge about L1 and L2
• We use P2 as a black box routine

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Many-one Reductions in Detail

• Properties of f and program P3

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Properties of P3 (f)

• P3 must compute a many-one reduction function
  fS --gt S
• For all x in S, P3(x) must be defined
• That is P3 must halt with some output P3(x)
• For all x in S, (x in L1) iff (P3(x) in L2)
• (x in L1) --gt (P3(x) in L2)
• Not (x in L1) --gt Not (P3(x) in L2)

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Yes-gtYes and No-gtNo
L1
L1
S
L2
L2
S
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Many-one Reduction

• If there is such a many-one reduction function f
  (and the corresponding program P3), we say that
  L1 many-one reduces to L2
• Notation
• L1 bm L2
• b denotes L1 is no harder than L2

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Example

• L1 is even length strings over 0,1
• L2 is odd length strings over 0,1
• Tasks
• Give a reduction function f which shows that L1bm
  L2
• Give a corresponding program P3 which computes f

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Example 2

• L1 is 0,1
• L2 is 0
• Tasks
• Give a reduction function f which shows that L1bm
  L2
• Give a corresponding program P3 which computes f

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Constructing an example P3
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Example

• L1 is H
• Input
• Program P
• input y to program P
• Yes/No Question
• Does P halt on y?

• L2
• Input
• Program P
• Yes/No question
• Does L(P) the set of even length strings?

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Construction overview again
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Keep this in mind

• Programs which are PART of program P1 and thus
  executed when P1 executes
• Program P3, an actual program we construct
• Program P2, an assumed program which solves
  problem L2
• Programs which are INPUTS/OUTPUTS of programs P1,
  P2, and P3 and which are not executed when P1
  executes
• Programs P, P, and P

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Analysis of L2

• L2
• Input
• Program P
• Yes/No question
• Does L(P) the set of even length strings?
• Program P2
• Solves L2
• We dont know how

• Consider a program P such that L(P)
• P2 rejects program P as input
• Consider the following program P
• If input z has even length then yes else no
• L(P) set of even length strings
• P2 accepts program P as input

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Specifying P3
x
P3(x)
Yes/No
P2
P3
P1

• Remember, we are solving H (L1), not L2
• Input x to P3 is an input to H
• Program P
• Input string y to program P
• Output P3(x) of P3 is an input to problem L2
• Program P
• Specification for P3
• P P3(x) P3(P,y)
• L(P) L(P) set of even length strings iff
  P halts on y

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How does P3 work?
P,y
P
Yes/No
P2
P3
P1

• Key
• P3 creates P by modifying program P to run y
  to test if P halts on y
• P3 then appends a program P where P has the
  right property
• Example

Input to P3 Program P Variables Integer array
A10 Integer max,i Instructions read input
string into array A max A0 for
(i1ilt10i) IF Ai gt max THEN max
Ai return max Input y 01532758210
If input z has even length THEN yes ELSE no
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Program P
P,y
P
Yes/No
P2
P3
P1

• Specification for program P
• L(P) even length strings if and only if P
  halts on y
• In actuality
• If P halts on y, L(P) L(P) set of even
  length strings
• If P does not halt on y, P will loop on all
  strings
• Thus L(P)

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Key Step

• If P loops on y, L(P)
• P begins by running P on y
• Suppose P does NOT halt on y
• P never even looks at input z it just loops
• Typically, L(P) means P is a no input
  to problem L2
• If P halts on y, L(P) L(P)
• Note, P could be ANY program we want
• CHOOSE P such that P is a yes input to
  problem L2
• or if L(P) means P is a yes input to
  problem L2, then choose P such that P is a no
  input to problem L2

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P3 illustrated
P,y
P
Yes/No
P2
P3
P1
Program P Variables Integer array A10 Integer
max,i Instructions read input string into array
A max A0 for (i1ilt10i) IF Ai gt max
THEN max Ai return max Input
y 01532758210
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P3 in more detail
P,y
P
Yes/No
P2
P3
P1

• Pseudocode for P3
• Make P a procedure of P
• Make the first line of the main procedure for
  P a call to P with y as the input
• P(y)
• Ignore what P returns only interested if P
  halts on y
• Make P a procedure of P
• Make the second line of the main procedure for
  P a call to P with the input z to P as
  the input
• P(z)
• Note P will return what P(z) returns

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Summary

• Many-one Reductions
• L1 bm L2
• Construct P1 from P2
• Similarities with closure property constructions
• Use P2 as a subroutine
• Differences with closure property constructions
• Use specific properties of L1 and L2
• P3 is inside P1 and it maps inputs of L1 to
  inputs of L2
• Often have programs as inputs and outputs