John Doyle

1
Turbulence and a new physics?

• John Doyle
• Control and Dynamical Systems, Electrical
  Engineering, Bioengineering
• Caltech

2
Collaborators and contributors(partial list)

• Turbulence Bamieh, Dahleh, Bobba, Gharib,
  Marsden,
• Theory Parrilo, Carlson, Paganini,
  Papachristodoulo, Prajna, Goncalves, Fazel, Lall,
  DAndrea, Jadbabaie, many current and former
  students,
• Biology Csete,Yi, Arkin, Simon, AfCS, Borisuk,
  Bolouri, Kitano, Kurata, Khammash, El-Samad,
  Gross, Endelman, Sauro, Hucka, Finney,
• Web/Internet Low, Willinger, Vinnicombe,Kelly,
  Zhu,Yu, Wang, Chandy, Effros,
• Physics Mabuchi, Doherty, Barahona, Reynolds,
  Asimakapoulos,
• Engineering CAD Ortiz, Murray, Schroder,
  Burdick,
• Disturbance ecology Moritz, Carlson, Robert,
• Finance Martinez, Primbs, Yamada, Giannelli,

Caltech faculty
Other Caltech
Other
3
Universal features of complex systems

• Commonly shared goal
• Consistent, coherent view of the fundamental,
  universal characteristics of complex systems that
  transcend specific domains
• Details are complicated and easy to get wildly
  wrong.
• Initial attempts at unified theories of complex
  systems have been spectacular (i.e. popular)
  failures, but instructive ones.

4
Universal features of complex systems

• New story is completely different, but far more
  powerful.
• Builds on 3 sources of experience, insight, and
  theory
• Advanced engineering and technology
• Molecular and systems biology
• Mathematics of complex engineering systems
• Big surprise? Multiscale physics!
• What can we say in an hour?

5
Outline for today

• A thin slice through a thick collection of
  subjects
• Concentrate on one of many persistent mysteries
  at the foundations of physics Turbulence
• More specifically the origin and nature of
  coherent structures in shear flow turbulence
• Thus this will be a convenient domain in which to
  introduce new concepts
• You know much more about turbulence than I do

6
Outline for today

• If we have a truly universal story, this will
  necessarily reveal essential features of the
  whole
• The details do, however, matter enormously
• Illustrates universal features of complex systems
• Suggest a new, unifying conceptual framework plus
  math tools for complex multiscale physics,
  networks, and biology
• Math details tomorrow?

7
Using our imagination

• Think of me as a theoretical biologist
• Imagine that we have successfully developed a
  theoretical foundation for systems biology
• (Actually, were optimistic, but it remains to be
  seen whether well be successful in biology. But
  were using our imagination, so)
• Were interested in whether the mathematical
  tools that have been so successful in biology
  will apply to physics
• So lets look at the age-old problem of turbulence

8
Assumed background

• Familiarity with basics of fluid mechanics (you
  easily know much more than me)
• Will use cartoons and stories and minimal math to
  tell the story
• Proofs of the turbulence results obviously
  require math, but this part isnt too bad and can
  be handcrafted with elementary mathematics
• The general story involves much more
  sophisticated mathematics. More fun but less
  accessible.

9
Topics skipped today

• Math details. Tomorrow?
• Other related physics problems Origin of power
  laws, dissipation and entropy, quantum
  measurement and quantum/classical transition
• Much work involving control of turbulence,
  quantum systems, etc (e.g. Speyer, Kim,
  Cortellezi, Bewley, Burns, King, Krstic, )
• Related multiscale problems in networking
  protocols (Low, Paganini, Vinnicombe,)
• Biological regulatory networks (Khammash, El
  Samad, Yi, Csete)

10
Caveats

• Neither thorough nor scholarly
• Leaves out the heart of the subject, which
  still lacks an accessible introduction for a
  general audience (but its getting better)
• Im here for awhile and eager to tell you more

11
Thanks to Mory Gharib Jerry Marsden Brian Farrell
Turbulence in shear flows
Kumar Bobba, Bassam Bamieh
wings
channels
Turbulence is the graveyard of theories. Hans
Liepmann Caltech
pipes
12
velocity
high
low
equilibrium
periodic
chaotic
13
random pipe
14
bifurcation
laminar
flow (average speed)
turbulent
pressure (drop)
15
Random pipes are like bluff bodies.
16
flow
Typical flow
Transition
pressure
17
Chaos and turbulence

• The orthodox view
• Adjusting 1 parameter (Reynolds number) leads to
  a bifurcation cascade (to chaos?)
• Turbulence transition is a bifurcation
• Turbulent flows are perhaps chaotic, certainly
  intrinsically a nonlinear phenomena
• There are certainly many situations where this
  view is useful.
• But many people believe there is much more to the
  story. See Farrell, et al, etc.

18
wings
Streamline
channels
pipes
19
theory
laminar
log(flow)
experiment
turbulent
Random pipe
log(pressure)
20
log(flow)
Random pipe
log(Re)
21
Reynolds number

• Dimensionless constant
• Key determinant of qualitative flow
  characteristics

22
Average flow speed

• For generic flows
• Geometry plus R determines qualitative flow
  behavior
• Assume throughout that geometry and nominal
  velocity are fixed
• Vary viscosity, and hence R

23
This transition is extremely delicate (and
controversial).
Random pipe
It can be promoted (or delayed!) with tiny
perturbations.
log(Re)
24
Transition to turbulence is promoted (occurs at
lower speeds) by
Surface roughness Inlet distortions Vibrations The
rmodynamic fluctuations? Non-Newtonian effects?
25
None of which makes much difference for random
pipes.
Random pipe
26
Shark skin delays transition to turbulence
27
log(flow)
It can be reduced with small amounts of polymers.
log(pressure)
28
HOT
log(flow)
random
log(pressure)
29
Robustness of shear flows
Fragile
Surface roughness Inlet distortions Vibrations The
rmodynamics Non-Newtonian
Viscosity
Everything else
Robust
30
Robustness is a conserved quantity?
Fragile
Random
Viscosity
Everything else
Robust
31
streamlined
flow
Turbulence lesson
HOT
random
pressure drop

• Through streamlined design
• High throughput
• Robust to bifurcation transition (Reynolds
  number)
• Yet fragile to small perturbations
• Which are irrelevant for more generic flows

32
Complexity lesson 1
evolved

• Highly evolved systems are completely opposite
  from generic versions
• Extreme robustness is possible
• But the price may be new fragilities

generic
Fragile
Robust
33
Complexity lesson 1
thru-put
generic
demand
Fragile
Robust yet fragile
Robust
Uncertainty
34
Diseases of complexity
Fragile

• Parasites
• Cancer
• Epidemics
• Auto-immune disease

Complex development Regeneration/renewal Complex
societies Immune response
Uncertainty
Robust
35
Flow direction
Viewed from below through clear wall.
36
Flow direction
Viewed from below through clear wall.
37
streamwise
Couette flow
38
(No Transcript)
39
(No Transcript)
40
spanwise
Couette flow
41
high-speed region
From Kline
42
high-speed region
y
flow
position
z
x
43
3d/3c Nonlinear NS
44
3d/3c Linear NS
3d/3c Nonlinear NS
Linearize
3d/3c Linear NS
y
v
flow
flow
position
velocity
z
u
w
x
3 components
3 dimensions
45
3d/3c Linear NS
y
v
flow
flow
position
velocity
z
u
w
x
3 components
3 dimensions
46
streamwise
The mystery.
Thm The first instabilities are spanwise
constant.
All observed flows are largely streamwise
constant.
47
Thm The first instabilities are spanwise
constant.
48
Thm The first instabilities are spanwise
constant.
This is as different as two flows can be.
49
Emergilence
Theory
Conventional explanation Turbulence is a highly
nonlinear phenomena.
Translation I dont understand it, but neither
do you.
Experiment
50

• Definitions
• Nonlinear
• Not linear
• A word used by a confused and intimidated speaker
  in an attempt to create the same state in the
  listener

51

• Nonlinear, emergent, complex, bifurcation, phase
  transition, feedback, etc, can return to having a
  technical meaning.
• Emergilent
• Confused or confusing, intimidate by obfuscation
• Chaocritiplexity
• Emergilent chaos, criticality, and complexity

52

• Nonlinear
• Not linear
• Emergilent
• Confused or confusing, intimidate by obfuscation
• Chaocritiplexity
• Emergilent chaos, criticality, and complexity

53
3d/3c Linear NS

• Linearized Navier-Stokes
• Stable for all Reynolds numbers R
• Orthodox wisdom transition must be an inherently
  nonlinear phenomena
• Experimentally no evidence for an attractor or
  subcritical bifurcations
• Theoretically no evidence for
• The mystery deepens.

54
Complexity lesson 2

• The orthodox view usually has some truth and
  applies to a limited domain
• complexity arises between order and disorder
• at a bifurcation or phase transition
• in otherwise homogeneous, generic configurations
  only weakly coupled to environment
• This is a complete red herring in general
• The mainstream view of complexity is most wrong
  in highly evolved systems
• In turbulence, highly evolved means highly
  streamlined
• (and theres usually some expert whos it all
  gotten right all along and has been ignored, e.g.
  Farrell)

55
Can we build an airplane without streamlining?
Yes
But it wont fly, so
not really.
56
Can we build an airplane, internet, organism or
ecosystem that is critical or at the edge of
chaos?
57
Can we build an airplane, internet, or organism
that is critical or at the edge of chaos?
Not really.
58
If Darwin were a physicist, he would have
proven that Life was emergilence at the edge of
chaocritiplexity Theoretical biologist
"If Galileo were a biologist, he would have
written a big fat tome on the details of how
different objects fall at different rates.
Geoffrey West Santa Fe Institute   Theoretical
Division, Los Alamos National Laboratory
59
Complexity definition

• Nothing controversial here
• Complicated ? long description
• Complex ? complicated proof or explanation
• These better not be the same or were in big
  trouble
• Complexity here is essentially proof length from
  computational complexity
• Seems to require a proof infrastructure to be
  assumed in order for complexity to be
  well-defined
• Actually seems to be the right way generally to
  look at complexity (coNP vs. NP), but thats
  another topic.

60

• Complicated long description
• Complex long proof or explanation

Complicated
Not
Chaos undecidability
Complex
Biology
Good engineering
Not
Physics
61

• Complicated long description
• Complex long proof or explanation

Complicated
Not
Turbulence is usually thought of as here.
Complex
We want to think of it as here.
Not
62
Complexity lessons review

• Highly evolved systems are robust yet fragile
• Orthodoxy of order-disorder transition is a red
  herring
• Complexity implies fragility
• complexity scarcity lt fragility

63
Two great abstractions of the 20th Century

• Separate systems from physical substrate
• Separate systems engineering into control,
  communications, and computing
• Theory
• Applications
• Facilitated massive, wildly successful, and
  explosive growth in both technology and
  mathematical theory
• but creating a new Tower of Babel where even the
  experts do not read papers or understand systems
  outside their subspecialty.

64
Biology and advanced technology

• Biology
• Integrates control, communications, computing
• Into distributed control systems
• Built at the molecular level
• Advanced technologies will do the same
• We need new theory and math, plus unprecedented
  connection between systems and devices
• Two challenges for greater integration
• Unified theory of systems
• Multiscale from devices to systems

65
Muddy water is not necessarily deep Nietzsche

• Failure of new science of complexity, complex
  adaptive systems, edge-of-chaos, self-organized
  criticality etc etc to get anything of this story
  right.
• Over reliance on superficial metaphors and
  simulation
• This failure has been very informative however.
  You learn as much from failure as success.
• Ideology Concepts from physics would resolve
  problems in complex technological and biological
  systems
• Reality Exactly the opposite (big surprise!)
• Persistent mysteries with powerful new theories
  power laws, turbulence, entropy, quantum
  entanglement and measurement, etc.

66
Highly Optimized Tolerance (HOT)(Jean Carlson,
Physics, UCSB)

• Complex systems in biology, ecology, technology,
  sociology, economics,
• are driven by design or evolution to
  high-performance states which are also tolerant
  to uncertainty in the environment and components.
• This leads to specialized, modular, hierarchical
  structures, often with enormous hidden
  complexity,
• with new sensitivities to unknown or neglected
  perturbations and design flaws.
• Robust, yet fragile!

67
Robust, yet fragile

• Robust to uncertainties
• that are common,
• the system was designed for, or
• has evolved to handle,
• yet fragile otherwise
• This is the most important feature of complex
  systems (the essence of HOT).

68
Robust, yet fragile

• Robust to uncertainties
• that are common,
• the system was designed for, or
• has evolved to handle,
• yet fragile otherwise
• This is the most important feature of complex
  systems (the essence of HOT).

69
Complexity lesson 3

• Complexity implies fragility

complexity scarcity lt fragility
70
Complexity lesson 3

• Complexity implies fragility

If complexity seems overwhelming, look for some
hidden fragility.
71
3d/3c Linear NS
Forcing
Environment

• Mathematically
• External disturbances
• Initial conditions
• Unmodeled dynamics

• Physically
• Wall roughness
• Acoustics
• Thermo fluctuations
• NonNewtonian
• Upstream disturbances

72
3d/3c Linear NS
Forcing
energy
(Bamieh and Dahleh)
t
73
t
74
The predicted flows are robustly and strongly
streamwise constant.
y
flow
Consistent with experimental evidence.
z
x
75
3d/3c Nonlinear NS
3d/3c Linear NS
Linearize
Stable for all R.
y
flow
z
x
2d/3c Linear NS
76
3d/3c Nonlinear NS
3d/3c Linear NS
Linearize
Stable for all R.
y
flow
z
x
2d/3c LNS
2d/3c NLNS
Linearize
77

• 2d/3c NLNS solutions to 3d/3c NLNS for streamwise
  constant initial conditions
• 2d/3c NLNS has 3 velocity components depending on
  2 (spanwise) spatial variables

3d/3c NLNS
2d/3c NLNS
y
flow
z
x
78
3d/3c NLNS
Thm 2d/3c NLNS
Globally stable for all R.
2d/3c NLNS
Proof can rescale equations to be independent of
R!
79

• High gain, low rank operator
• Implications for
• Model reduction
• Computation
• Control

3d/3c Linear NS
Globally stable for all R.
2d/3c NLNS
Linearize
2d/3c LNS
80
The predicted flows are robustly and strongly
streamwise constant.
y
flow
z
x
Consistent with experimental evidence.
81
y
flow
z
x
82
2d/3c
83
Worst-case amplification is streamwise constant
(2d/3c). (Bamieh and Dahleh)
?
2d/3c
3d/3c
84
Theory
New explanation Shear flow turbulence is a
robustness problem with a simple explanation that
everyone can understand (except those stuck in
the conventional wisdom).
Complex mathematics is used to confirm that the
simple picture is robust to assumptions, such as
of linearity.
Experiment
85
Robustness of shear flows
Fragile
Viscosity
Everything else
Robust
86
Robustness is a conserved quantity?
Fragile
Random
Viscosity
Everything else
Robust
87
Complexity lesson 3

• Complexity implies fragility
• Complexity ? proof length
• Fragility ? ill-conditioning
• Fragile The answer changes a lot if the question
  changes a little.
• Complex The shortest explanation is long.

88
Complexity lesson 3

• Robust questions have simple explanations.
• Fragile questions may have complex explanations.
• Complex explanations imply fragility.
• (But fragility need not imply complexity and
  scarce resources aggravate all tradeoffs.)
• This is the heart of everything

89
Complexity lessons review

• Highly evolved systems are robust yet fragile
• Orthodoxy of order-disorder transition is a red
  herring
• Complexity implies fragility

90
Modeling complex systems
May need great detail here
Fragile
And much less detail here.
Uncertainty
Robust
91
Fragile
Robust (fragile) to perturbations in components
and environment ? Robust (fragile) to errors and
simplifications in modeling
More detail.
Required model complexity
Less detail.
Uncertainty
Robust
92

• Complexity of system evolution is driven by
  fragilities
• Complexity of experiments, modeling, and
  inference are also driven by fragilities

Fragile
More detail.
Required model complexity
Less detail.
Uncertainty
Robust
93
Topics skipped today

• Other related physics problems Origin of power
  laws, dissipation and entropy, quantum
  measurement and quantum/classical transition
• Much work involving control of turbulence,
  quantum systems, etc (e.g. Speyer, Kim,
  Cortellezi, Bewley, Burns, King, Krstic, )
• Related multiscale problems in networking
  protocols (Low, Paganini, Vinnicombe,)
• Biological regulatory networks (Khammash, El
  Samad, Yi, Csete)

94
Topics skipped today

• Math details. Tomorrow?
• Universal architectures and protocols in complex
  systems
• Phase transitions in computational complexity
• Volatility in financial markets
• Robustness verification of hybrid/nonlinear
  systems
• Scaling the scientific enterprise

95
Postscript

• Revisit the problem of scaling.
• How do we study large networks with many
  components?
• Data
• Modeling
• Inference
• Inference is the hardest part to scale to large
  problems. Science as an enterprise has largely
  given up on it, focusing on data and modeling.
• Progress and problems.

96
Why this is impossible to explain

• Fundamentally new way of thinking about science
• Combination of mathematics that has never had
  contact before (operator theory, robust control,
  real semi-algebraic geometry, semi-definite
  programming, duality, P/NP/coNP and computational
  complexity)
• Either one of these would be an obstacle,
  together they seem completely hopeless.

97
Even worse!!

• Forces a complete rethinking of the foundations
  of theoretical physics, because there is no
  foundation where theory is done correctly.
• Were redoing statistical physics, turbulence,
  quantum mechanics, almost from scratch using
  new mathematics
• Good news Its working and the best, most
  visionary physicists are getting involved.
• Bad news Everyone else thinks were nuts.
  Unpublishable. Unfundable. So were doing this
  on the side.

98
The scientific iteration

• Do experiments and gather data.
• Make assertions about the data (modeling).
• Reason about the assertions to make inferences
  about the systems under study.
• Current scientific infrastructure (physics,
  chemistry, biology)
• scales ok for 1, badly for 2
• is completely intractably for 3
• The technological community is struggling hard
  with 3.

99
The challenge of inference

• Example Is a model consistent with data? Does a
  model have some property?
• Much harder than taking data or extracting
  information or forming models.
• Much of the technological community has become
  dominated by this challenge.
• Example Verifying that software works.
• New research breakthroughs offer unprecedented
  promise. Will sketch the ideas today.
• The cost and challenges of scaling inference will
  dominate future biology, whether we like it or
  not.
• (Just as the cost of software has come to
  dominate all other costs in large projects. We
  have to do this better.)

100
Could this be intrinsically hopeless?
(Turing/Godel)
Robustness
Ideal performance
Verifiability?
101
Robustness, evolvability/scalability,
verifiability
Robustness
Ideal performance

• Verifiability in forward engineering translates
  into comprehensibility in reverse engineering of
  biological systems
• Thus this research direction may also be good
  news for understanding complex biological
  processes

102
Robustness and verifiability
Robustness
Ideal performance

• How do you prove nothing bad can happen?
• Need both new modeling techniques and new proof
  techniques.

103
Robust hybrid/nonlinear systems theory (of
embedded networks)?
Linear theory plus bounds, with scalable
algorithms.
Theory without scalable algorithms.
Hacking. (Scalable algorithms without theory.)
Theory with scalable algorithms?
Most research Not scalable, no theory.
104
Provably robust, scalable Internet protocols.
Robustness verification of embedded control
software/hardware.
Hacking.
Theory with scalable algorithms.
105
Key issues

• Robustness/Fragility Uncertainty in components,
  environment, and modeling, assumptions, and
  computational approximations
• Verifiability Short proofs of robustness
• Complexity Extreme, highly structured internal
  complexity (complicated) is typically needed to
  produce verifiably robust behavior
• Scarce resources All tradeoffs are aggravated by
  efficiency and scarce resources

106

• Key unifying theoretical insights (Parrilo et al)
  
• All robustness analysis (controls, comms,
  computing, statistical physics) reduces to
  creating and verifying existence of barriers in
  the appropriate state space
• Generalized tools from robust control
  (perturbation theory of operator algebras) allow
  sets of polynomial inequalities to provide rich
  descriptions of the geometry of uncertain
  systems and objectives
• Real semi-algebraic geometry connects coNP
  geometric problem with dual NP algebraic problem
• Semidefinite programming (SDP) optimization
  methods give scalable proof search for NP
  algebraic problem
• Long proof complexity feeds back to imply
  modeling fragility (Also via duality)

107

• Key unifying insights (Parrilo et al)
• Robustness barriers
• Polynomial inequalities
• Real algebraic geometry and duality between coNP
  geometric problem and dual NP algebraic problem
• SDP/SOS for scalable proof search
• Proof complexity feeds back to modeling fragility
  (also exploits duality)?

108

• What we did in the turbulence problem
• Robustness barriers
• Polynomial inequalities
• The rest of this was not used
• Real algebraic geometry and duality between coNP
  geometric problem and dual NP algebraic problem
• SDP/SOS for scalable proof search
• Proof complexity feeds back to modeling fragility
  (also exploits duality)?

Hand crafted
109
Why it all works.
Modeling Robustness barriers Perturbation
theory Polynomial inequalities
Analysis Real algebraic geometry Duality SDP/SOS
Model fragility
Proof complexity
110
Why its hard.
Modeling Robust control theory Operator Banach
Algebras
Analysis Real algebraic geometry Duality Optimizat
ion
Model fragility
Proof complexity
Theoretical CS NP-coNP
111
The big picture
Modeling
Analysis
112
Set of bad system behaviors
Set of possible system behaviors
Proof of robustness
Modeling
Analysis
113
Set of experimental behaviors
Set of possible model behaviors
Proof that model doesnt work
Modeling
Analysis
114
Set of bad system behaviors

• Sources of uncertainty
• Variations in components and environment
• Modeling assumptions
• Computational approximations
• Incomplete or inadequate description of
  objectives
• Want to manage these in a systematic and
  integrated way.

Set of possible system behaviors
Proof of robustness
115
Set of bad system behaviors
Set of possible system behaviors
NP exhibit a point
Modeling
Analysis
Problem Not robust
116
Set of experimental behaviors
Set of possible model behaviors
NP exhibit a point
Exists a model consistent with data.
Modeling
Analysis
117
Set of bad system behaviors
Set of possible system behaviors
coNP Give a proof
Modeling
Analysis
118
Set of experimental behaviors
Set of possible model behaviors
NP exhibit a point
Exists a model consistent with data.
Modeling
Analysis
119
Set of bad system behaviors
Key idea Complexity implies fragility of model
Set of possible system behaviors
Modeling
Analysis
Problem Robust but no short proof
120
Set of bad system behaviors

• Needs
• Rich modeling methods (hybrid, nonlinear, DAE,
  PDE)
• Systematic uncertainty management
• Scalable, automated proof system
• Feedback from proof complexity to model fragility
• Enormous progress on 1-3, promising new insights
  for 4.

Set of possible system behaviors
Proof of robustness
121
Set of experimental behaviors
Set of possible model behaviors
NP exhibit a point
coNP exclude large regions that need not be
searched.
122

• Key unifying insights
• All robustness analysis (controls, comms,
  computing, statistical physics) reduces to
  creating and verifying existence of barriers in
  the appropriate state space
• Generalized tools from robust control allow
  polynomial inequalities to provide rich
  descriptions of the geometry of uncertain
  systems and objectives
• Real algebraic geometry (Positivstellensatz)
  connects coNP geometric problem with dual NP
  algebraic problem
• SDP/SOS optimization methods give scalable proof
  search for NP algebraic problem (Parrilo et al)
• Proof complexity feeds back to modeling
  fragility? (Also via duality)

123

• Key unifying insights (Parrilo et al)
• Robustness barriers
• Polynomial inequalities
• Real algebraic geometry and duality between coNP
  geometric problem and dual NP algebraic problem
• SDP/SOS for scalable proof search
• Proof complexity feeds back to modeling fragility
  (also exploits duality)?

124
Too good to be true?
Modeling Robustness barriers Polynomial
inequalities
Analysis Real algebraic geometry Duality SDP/SOS

• Has proven to be astonishingly effective in a
  wide variety of areas
• Can we begin to understand why this is so?

125
Why may it be reasonable that mathematics is so
effective?

• Robust systems are verifiably so?
• Do only robust systems persist as coherent,
  structured objects of study (universes, solar
  systems, planets, life forms, protocols, )?
• If so, then mostly robust (and verifiably so)
  systems are around for us to study.

126
Focus for today
Modeling Robustness barriers Polynomial
inequalities
Analysis Real algebraic geometry Duality SDP/SOS
127
Quick review of computational complexity
?

• Assume you already know
• P/NP and NP complete
• SAT and 3-SAT
• but not necessarily
• NP vs coNP
• Duality and relaxations

128
Typically NP hard.
?
129
Set of bad system behaviors
Set of possible system behaviors
NP exhibit a point
Modeling
Analysis
Problem Not robust
130
Typically coNP hard.

• Fundamental asymmetries
• Between P and NP
• Between NP and coNP

?

• More important problem.
• Short proofs may not exist.

Unless theyre the same
131
Set of bad system behaviors
Set of possible system behaviors
coNP Give a proof
Modeling
Analysis
132
?
What makes a problem harder?
133
Set of bad system behaviors
Key idea Complexity implies fragility of model
Set of possible system behaviors
Modeling
Analysis
Problem Robust but no short proof
134
Easy to find solutions?
?
Satisfiable or feasible
135
Set of bad system behaviors
Set of possible system behaviors
NP easy to find a point
Modeling
Analysis
Problem Not robust
136
?
Easy to find proofs?
Unsatisfiable or infeasible
137
Set of bad system behaviors
coNP easy to find a proof
Set of possible system behaviors
Modeling
Analysis
138
?0
Complexity?
139
Set of bad system behaviors
Key idea Complexity implies fragility of model
Set of possible system behaviors
Modeling
Analysis
Problem Robust but no short proof
140
Search for counterexample

• Models describe sets of possible (uncertain)
  behaviors intersected with sets of unacceptable
  behaviors (failures)
• Thus verification of robustness (of protocols,
  embedded, dynamics, etc) involves showing that a
  set is empty.
• Searching for an element x ?M is in NP, since
  checking whether a given x ?M is typically in P.
• Proving that M is empty is in coNP and there may
  not be short proofs.

Search for proof
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Search for counterexample
Seach for proof

• Convex, but infinite dimensional.
• Efficient (P time) search subsets (relaxations)
  using SOS/SDP
• Guaranteed to converge

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Search for simple counterexample
Search for short proof
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Special case LP
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Search for simple counterexample
Search for short proof
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Search for simple counterexample
Failure to find short proof implies some relaxed
model is nonempty (which is bad).
Search for short proof
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?0
Complexity?
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Special case Scalar QP
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Special case Scalar QP
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• Polynomial functions NP-hard problem.
• A simple relaxation (Shor) find the minimum
  ?such that ?- F(x) is a sum of squares (SOS).
• Upper bound on the global maximum.
• Solvable using SDP, in polynomial time.
• A concise proof of nonnegativity.
• Surprisingly effective (Parrilo Sturmfels 2001).

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• Exactly as in QP case, SAT phase transition
  does not imply complexity.
• SOS/SDP relaxations much faster than standard
  algebraic methods (QE,GB, etc.).
• Before SOS/SDP, might have conjectured that this
  was an example of phase transition induced
  complexity.
• SOS/SDP gives certified upper bound in polynomial
  time.
• If exact, can recover an optimal feasible point.
• Surprisingly effective
• In more than 10000 random problems, always the
  correct solution
• Bad examples do exist (otherwise NPco-NP), but
  rare.
• Variations of the Motzkin polynomial.
• Reductions of hard problems (e.g. NPP is nice)
• None could be found using random search

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• e.g see Science, March 1 special issue on
  Systems Biology
• Csete and Doyle, Reverse Engineering of
  Biological Complexity

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For more details
www.cds.caltech.edu/doyle/IFAC www.aut.ee.ethz.ch
/parrilo