two big questions: p vs. np and p vs. bppcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf ·...

Two Big Questions Derandomization Hierarchy Theorems

Two Big Questions:P vs. NP and P vs. BPP

Jeff Kinne

University of Wisconsin-Madison

Indiana State University, March 26, 2010

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Computational Complexity Theory

How much time, memory space, etc. are needed to solveproblems?

Is nondeterminism powerful? P vs. NP

Conjecture: P 6= NP

Techniques: hierarchy theorems, others

Is randomness powerful? P vs. BPP, L vs. BPL

Conjecture: P=BPP, L=BPL

My work: derandomization, hierarchy theorems

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Is nondeterminism powerful?

P vs. NP

Conjecture: P 6= NP

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Conjecture: P 6= NP

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Conjecture: P 6= NP

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Conjecture: P 6= NP

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Conjecture: P 6= NP

Is randomness powerful?

P vs. BPP, L vs. BPL

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Conjecture: P 6= NP

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Conjecture: P 6= NP

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Conjecture: P 6= NP

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Introducing two Big Questions

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Time Complexity

Decision Problem: yes/no questions

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Time Complexity

Poly Time (P)

Exp Time (EXP)

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Time Complexity

Poly Time (P)

Exp Time (EXP)

factoring,NP-complete sorting

shortest path

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Time Complexity

Poly Time (P)

Exp Time (EXP)

sorting?

factoring,NP-complete

shortest path

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Time Complexity

Poly Time (P)

Exp Time (EXP)

sorting?

factoring,NP-complete

shortest path

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Memory Space Complexity

Amount of working space memory needed

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Log Space (L)

Poly Space (PSPACE)

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Log Space (L)

Poly Space (PSPACE)

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Log Space (L)

Poly Space (PSPACE)

factoring,NP-complete arithmetic

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Log Space (L)

Poly Space (PSPACE)

arithmetic?factoring,

NP-complete

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Randomized Algorithm

Bounded error: Correct with probability > 99%

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01 10 10 1011 01 10

yes/no

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Random Strings

01 10 10 1011 01 10

yes/no

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Random Strings

01 10 10 1011 01 10

yes/no

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Polynomial Identity Testing

p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3

Do all terms cancel?

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p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3

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p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3

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p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3

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p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3

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Multi-variate Polynomial Identity Testing

p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

Pick point (x1, ..., x4), each xi ∈R S

p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

p non-zero, degree d

⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |

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p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)

30 + (x4 + x3 − x1)15 ·

(x3 − x2 + x1)20 − (x2 − x3 + x4)

20 · (x2 + x1)15

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

BPP: Bounded-error Probabilistic Poly time , BPL: log space

P?= BPP

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

BPP: Bounded-error Probabilistic Poly time

, BPL: log space

P?= BPP

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

P?= BPP

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

P?= BPP

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Nondeterministic Algorithm

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Graph 3-Coloring

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Graph 3-Coloring

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NP: Nondeterministic Polynomial time

“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...

P?= NP

NP?= coNP

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01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP

NP?= coNP

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Potential Proofs

01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP

NP?= coNP

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Potential Proofs

01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP

NP?= coNP

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Potential Proofs

01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP

NP?= coNP

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Potential Proofs

01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP

NP?= coNP

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Potential Proofs

01 10 10 1011 01 10

yes/no

Proof/Certificate

coloring

P?= NP NP

?= coNP

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Two Big Questions

P?= NP

Is finding proofs as easy as verifying them?

Is 3-coloring in Polynomial Time?

P?= BPP

Does randomness truly add power?

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Two Big Questions

P?= NP

Is finding proofs as easy as verifying them?

Is 3-coloring in Polynomial Time?

P?= BPP

Does randomness truly add power?

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Conjecture: P 6= NP

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Derandomization

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Naive Derandomization

Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

Try all possible random bit strings – exponentially many

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

Try all possible random bit strings

– exponentially many

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Derandomization – the Standard PRG Approach

Poly many strings to try

⇒ O(log n) seed, exp stretch

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Random Strings

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

Poly many strings to try ⇒ O(log n) seed, exp stretch

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

PRGhard

functionh

p(x1, x2, x3, x4)

x1, ..., x4

Poly many strings to try ⇒ O(log n) seed, exp stretch

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A New Approach – Typically-Correct Derandomization

Seed length n, poly stretch

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

11 01 10

p(x1, x2, x3, x4)

x1, ..., x4

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Random Strings

pseudo–random

01 10 10 1011 01 10

yes/no

11 01 10

p(x1, x2, x3, x4)

x1, ..., x4

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Standard Use of PRG’s vs. Typ-Correct

Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]

• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:

fast parallel time, streaming,communication protocols

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• Always correct • Small # mistakes

• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:

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• Always correct • Small # mistakes• Run PRG many times • Run PRG only once

• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:

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• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch

• Conditional results • Unconditional results:fast parallel time, streaming,communication protocols

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Conjecture: P 6= NP

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Hierarchy Theorems

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Hierarchy Theorems

Fix a model of computing(deterministic, randomized, nondeterministic)

Can we achieve more given more resources?

My work: hierarchy theorems for randomized algorithms

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Hierarchy Theorems

Fix a model of computing

(deterministic, randomized, nondeterministic)

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Hierarchy Theorems

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Hierarchy Theorems

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Hierarchy Theorems

n time

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Hierarchy Theorems

n time

n2 time

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Hierarchy Theorems

n time

n2 timen3 time

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Hierarchy Theorems

n time

n2 time

Poly Timen3 time

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Hierarchy Theorems

n timePoly time =

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Hierarchy Theorems

log n space

log2 n space

log3 n space

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Hierarchy Theorems

log n spacelog3 n space =

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Hierarchy Theorems

log n space

log2 n space

log3 n space

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Hierarchy Theorems

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Hierarchy Theorems

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Conjecture: P 6= NP

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Hierarchy Theorems for Deterministic Algorithms

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A2 A3 ...A1

All Algorithms time n

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A2 A3 ...

All Algorithms

time n

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A2 A3 ...

A1(x1) A2(x1) A3(x1)

A1(x2) A2(x2) A3(x2) ...

A1(x3) A2(x3) A3(x3)

All Algorithms

time n

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A2 A3 ... D

A1(x1) A2(x1) A3(x1)

A1(x2) A2(x2) A3(x2) ...

A1(x3) A2(x3) A3(x3)

All Algorithms

time ntime n2

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A2 A3 ... D

A1(x1) A2(x1) A3(x1)

A1(x2) A2(x2) A3(x2) ...

A1(x3) A2(x3) A3(x3)

All Algorithms

time ntime n2

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A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

All Algorithms

time ntime n2

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n time

n2 time

Poly Timen3 time

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log n space

log2 n space

log3 n space

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Hierarchy Theorems for Nondeterministic Algorithms?

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Hierarchy Theorems for Nondeterministic Algorithms?

A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

All Nondet. Algorithms

time ntime n2

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Hierarchy Theorems for Nondeterministic Algorithms

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A1 Dtime ntime n2

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0`−1x1

0`−2x1

` ≈ 2|x1|

time ntime n2

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A1(x1)

A1(0x1)A1(00x1)

A1(0`−1x1)

0`−1x1

A1(0`x1)0`x1

A1(0`−2x1)0`−2x1

` ≈ 2|x1|

time ntime n2

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A1(x1)

A1(0x1)A1(00x1)

A1(0`−1x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1)0`−2x1

6=` ≈ 2|x1|

time ntime n2

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

=========

` ≈ 2|x1|

time ntime n2

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

Assume A1 the same

=========

as D on all inputs

` ≈ 2|x1|

time ntime n2

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

Assume A1 the same

=========

as D on all inputs

` ≈ 2|x1|

time ntime n2

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¬A1(x1)

... ...

0`−1x1

¬A1(x1)0`x1

0`−2x1

Assume A1 the same

=========

as D on all inputs

` ≈ 2|x1|

¬A1(x1)¬A1(x1) ¬A1(x1)

¬A1(x1)¬A1(x1)

¬A1(x1)¬A1(x1)¬A1(x1)

¬A1(x1)¬A1(x1)

time ntime n2

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¬A1(x1)

... ...

0`−1x1

¬A1(x1)0`x1

0`−2x1

Assume A1 the same

=========

as D on all inputs

` ≈ 2|x1|

¬A1(x1)¬A1(x1) ¬A1(x1)

¬A1(x1)¬A1(x1)

¬A1(x1)¬A1(x1)¬A1(x1)

¬A1(x1)¬A1(x1)

time ntime n2

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n time

n2 time

Poly Timen3 time

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log n space

log2 n space

log3 n space

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Hierarchy Theorems for Randomized Algorithms?

What if Pr[A1(x1) = “yes”] ≈ .5?

Then D does not have bounded error, not valid

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A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

All Randomized Algorithms

time ntime n2

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A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

time ntime n2

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Random Strings

01 10 10 1011 01 10

yes/no

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A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

time ntime n2

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A2 A3 ... D

A1(x1) A2(x1) A3(x1) ¬A1(x1)

A1(x2) A2(x2) A3(x2) ... ¬A2(x2)

A1(x3) A2(x3) A3(x3) ¬A3(x3)

time ntime n2

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

=========

` ≈ 2|x1|

time ntime n2

Make sure D has bounded error – 1 bit of advice

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

=========

` ≈ 2|x1|

time ntime n2

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

=========

` ≈ 2|x1|

time ntime n2

Make sure D has bounded error

– 1 bit of advice

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A1(x1) A1(0x1)

A1(0x1) A1(02x1)A1(00x1) A1(03x1)

... ...A1(0`−1x1) A1(0`x1)

0`−1x1

A1(0`x1) ¬A1(x1)0`x1

A1(0`−2x1) A1(0`−1x1)0`−2x1

=========

` ≈ 2|x1|

time ntime n2

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Yes, for algorithms with 1 bit of advice!

My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...

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n time

n2 time

Poly Timen3 time

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log n space

log2 n space

log3 n space

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log n space

log2 n space

log3 n space

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log n space

log2 n space

log3 n space

My work [ Kinne, Van Melkebeek ]

Memory Space hierarchies: randomized, quantum, ...

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log n space

log2 n space

log3 n space

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Conjecture: P 6= NP

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The End, Thank You!

Slides available at:http://www.kinnejeff.com/GoSycamores/

(or E-mail me)

More on my research (slides, papers, etc.) at:http://www.kinnejeff.com/

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The End, Thank You!

Slides available at:http://www.kinnejeff.com/GoSycamores/

(or E-mail me)

More on my research (slides, papers, etc.) at:http://www.kinnejeff.com/

35 / 35

two big questions: p vs. np and p vs. bppcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf ·...

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