two big questions: p vs. np and p vs. bppcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf ·...
TRANSCRIPT
![Page 1: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/1.jpg)
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
1 / 35
![Page 2: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/2.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 3: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/3.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 4: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/4.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 5: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/5.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 6: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/6.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 7: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/7.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 8: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/8.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 9: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/9.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 10: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/10.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
2 / 35
![Page 11: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/11.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Introducing two Big Questions
3 / 35
![Page 12: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/12.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Decision Problem: yes/no questions
4 / 35
![Page 13: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/13.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
Decision Problem: yes/no questions
4 / 35
![Page 14: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/14.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
factoring,NP-complete sorting
shortest path
Decision Problem: yes/no questions
4 / 35
![Page 15: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/15.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
sorting?
factoring,NP-complete
shortest path
Decision Problem: yes/no questions
4 / 35
![Page 16: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/16.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
sorting?
factoring,NP-complete
shortest path
Decision Problem: yes/no questions
4 / 35
![Page 17: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/17.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Amount of working space memory needed
5 / 35
![Page 18: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/18.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
Amount of working space memory needed
5 / 35
![Page 19: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/19.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
Amount of working space memory needed
5 / 35
![Page 20: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/20.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
factoring,NP-complete arithmetic
Amount of working space memory needed
5 / 35
![Page 21: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/21.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
arithmetic?factoring,
NP-complete
Amount of working space memory needed
5 / 35
![Page 22: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/22.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Bounded error: Correct with probability > 99%
6 / 35
![Page 23: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/23.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
![Page 24: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/24.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
![Page 25: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/25.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
![Page 26: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/26.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
![Page 27: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/27.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
![Page 28: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/28.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
![Page 29: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/29.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
![Page 30: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/30.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
![Page 31: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/31.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 32: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/32.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 33: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/33.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 34: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/34.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 35: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/35.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 36: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/36.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d
⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 37: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/37.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
![Page 38: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/38.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
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
9 / 35
![Page 39: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/39.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
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
9 / 35
![Page 40: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/40.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
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
9 / 35
![Page 41: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/41.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
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
9 / 35
![Page 42: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/42.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
10 / 35
![Page 43: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/43.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Graph 3-Coloring
11 / 35
![Page 44: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/44.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Graph 3-Coloring
11 / 35
![Page 45: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/45.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 46: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/46.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 47: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/47.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 48: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/48.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 49: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/49.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 50: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/50.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
![Page 51: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/51.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP NP
?= coNP
12 / 35
![Page 52: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/52.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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?
13 / 35
![Page 53: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/53.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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?
13 / 35
![Page 54: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/54.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
14 / 35
![Page 55: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/55.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization
15 / 35
![Page 56: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/56.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
![Page 57: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/57.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
![Page 58: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/58.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings
– exponentially many
16 / 35
![Page 59: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/59.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
![Page 60: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/60.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
![Page 61: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/61.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
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
17 / 35
![Page 62: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/62.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
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
17 / 35
![Page 63: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/63.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
![Page 64: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/64.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
![Page 65: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/65.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try ⇒ O(log n) seed, exp stretch
17 / 35
![Page 66: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/66.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
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
17 / 35
![Page 67: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/67.jpg)
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Seed length n, poly stretch
18 / 35
![Page 68: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/68.jpg)
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
![Page 69: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/69.jpg)
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
11 01 10
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
![Page 70: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/70.jpg)
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
11 01 10
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
![Page 71: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/71.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 72: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/72.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 73: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/73.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 74: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/74.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 75: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/75.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 76: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/76.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
19 / 35
![Page 77: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/77.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
20 / 35
![Page 78: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/78.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
21 / 35
![Page 79: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/79.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 80: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/80.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing
(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 81: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/81.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 82: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/82.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 83: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/83.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 84: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/84.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 time
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 85: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/85.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 timen3 time
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 86: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/86.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 time
Poly Timen3 time
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 87: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/87.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n timePoly time =
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 88: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/88.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n space
log2 n space
log3 n space
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 89: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/89.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n spacelog3 n space =
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 90: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/90.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n space
log2 n space
log3 n space
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 91: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/91.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 92: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/92.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
![Page 93: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/93.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
23 / 35
![Page 94: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/94.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
24 / 35
![Page 95: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/95.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...A1
All Algorithms time n
24 / 35
![Page 96: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/96.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
time n
24 / 35
![Page 97: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/97.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time n
24 / 35
![Page 98: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/98.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ... D
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
![Page 99: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/99.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ... D
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
![Page 100: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/100.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic 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)
...
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
![Page 101: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/101.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
n time
n2 time
Poly Timen3 time
25 / 35
![Page 102: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/102.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
log n space
log2 n space
log3 n space
25 / 35
![Page 103: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/103.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms?
26 / 35
![Page 104: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/104.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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)
...
x1
x2
x3
...
A1
All Nondet. Algorithms
AllI
nput
s
...
time ntime n2
26 / 35
![Page 105: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/105.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
27 / 35
![Page 106: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/106.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 Dtime ntime n2
27 / 35
![Page 107: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/107.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
x1
0x1
00x1
...
0`−1x1
0`x1
0`−2x1
inpu
ts
` ≈ 2|x1|
time ntime n2
27 / 35
![Page 108: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/108.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1)
A1(0x1)A1(00x1)
...
A1(0`−1x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1)0`x1
A1(0`−2x1)0`−2x1
inpu
ts
` ≈ 2|x1|
time ntime n2
27 / 35
![Page 109: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/109.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1)
A1(0x1)A1(00x1)
...
A1(0`−1x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1)0`−2x1
inpu
ts
6=` ≈ 2|x1|
time ntime n2
27 / 35
![Page 110: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/110.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
27 / 35
![Page 111: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/111.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
as D on all inputs
` ≈ 2|x1|
time ntime n2
27 / 35
![Page 112: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/112.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
as D on all inputs
` ≈ 2|x1|
time ntime n2
27 / 35
![Page 113: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/113.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
¬A1(x1)
A1 D
... ...
x1
0x1
00x1
...
0`−1x1
¬A1(x1)0`x1
0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
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
27 / 35
![Page 114: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/114.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
¬A1(x1)
A1 D
... ...
x1
0x1
00x1
...
0`−1x1
¬A1(x1)0`x1
0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
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
27 / 35
![Page 115: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/115.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
n time
n2 time
Poly Timen3 time
28 / 35
![Page 116: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/116.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
log n space
log2 n space
log3 n space
28 / 35
![Page 117: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/117.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
![Page 118: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/118.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized 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)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
![Page 119: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/119.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized 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)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
![Page 120: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/120.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
30 / 35
![Page 121: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/121.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized 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)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
31 / 35
![Page 122: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/122.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized 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)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
31 / 35
![Page 123: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/123.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
![Page 124: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/124.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
![Page 125: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/125.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error
– 1 bit of advice
32 / 35
![Page 126: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/126.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
![Page 127: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/127.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 128: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/128.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 129: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/129.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
n time
n2 time
Poly Timen3 time
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 130: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/130.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 131: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/131.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 132: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/132.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]
Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 133: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/133.jpg)
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
![Page 134: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/134.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
34 / 35
![Page 135: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/135.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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
![Page 136: Two Big Questions: P vs. NP and P vs. BPPcs.indstate.edu/~jkinne/talks/isu-job-talk.pdf · Conjecture: P6= NP Techniques: hierarchy theorems, others Is randomness powerful? P vs](https://reader034.vdocuments.site/reader034/viewer/2022050423/5f927e7b7829a541a03d65d1/html5/thumbnails/136.jpg)
Two Big Questions Derandomization Hierarchy Theorems
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