02 10 Doyle complexity

Collaborators and contributors(partial list): 

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

For more details: 

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

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Finance Transportation Energy Information Consumer Utilities Manufacturing Commerce Health Our lives are run by/with networks Emergency

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Convergent Networks Transportation Energy Information Consumer Manufacturing Commerce Health Environment Emergency Finance Utilities

Convergent networking: the promise: 

Convergent networking: the promise Ubiquitous computing, communications, and control that is embedded and intertwined via sensors and actuators in complex networks of networks, with layers of protocols and feedback. Resulting in: Seamless integration and automation of everything Efficient and economic operation Robust and reliable services Convergent Networks Transportation Energy Information Consumer Manufacturing Commerce Health Environment Emergency Finance Utilities

Convergent networking: the promise: 

Convergent networking: the promise Ubiquitous computing, communications, and control that is embedded and intertwined via sensors and actuators in complex networks of networks, with layers of protocols and feedback. Resulting in: Seamless integration and automation of everything Efficient and economic operation Robust and reliable services Convergent Networks Transportation Energy Information Consumer Manufacturing Commerce Health Environment Emergency Finance Utilities

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When Start to Stink John Doyle Stink Things Things

Design objectives: 

Robustness to uncertainty in environment and components Scarce resources are used efficiently Scalable to large numbers (in order to do 1-2, it may be necessary to have high internal complexity (complicated), but we want simple, robust, verifiable external behavior, so…) Verifiable with short proofs Design objectives

Robustness, evolvability/scalability, verifiability: 

Robustness, evolvability/scalability, verifiability Ideal performance Robustness Verifiability?

Robustness, evolvability/scalability, verifiability: 

Robustness, evolvability/scalability, verifiability Ideal performance Robustness Verifiability in forward engineering translates into comprehensibility in reverse engineering of biological systems This research direction may be good news for understanding complex biological processes

Robustness and verifiability: 

Robustness and verifiability Ideal performance Robustness How do you prove “nothing bad can happen?” Need both new modeling techniques and new proof techniques.

Robust hybrid/nonlinear systems theory (of embedded networks)?: 

Robust hybrid/nonlinear systems theory (of embedded networks)? “Theory” without scalable algorithms. Linear theory plus bounds, with scalable algorithms. Hacking. (Scalable algorithms without theory.) Theory with scalable algorithms? Most research: Not scalable, no theory.

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Hacking. Theory with scalable algorithms. Robustness verification of embedded control software/hardware. Provably robust, scalable Internet protocols.

Complexity lessons review: 

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

Complexity lesson #3: 

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.

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z y x flow position Shear flow turbulence

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 Quick review of computational complexity Assume you already know: P/NP and NP complete SAT and 3-SAT Phase transitions (at least in percolation lattices) …but not necessarily NP vs coNP Duality and relaxations

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 Typically NP hard.

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Modeling Analysis Set of bad system behaviors Set of possible system behaviors Problem: Not robust NP: exhibit a point

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 Typically coNP hard. More important problem. Short proofs may not exist. Fundamental asymmetries* Between P and NP Between NP and coNP * Unless they’re the same…

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Modeling Analysis Set of possible system behaviors Set of bad system behaviors coNP: Give a proof

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 What makes a problem “harder”?

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Modeling Analysis Set of possible system behaviors Set of bad system behaviors Problem: Robust but no short proof Key idea: Complexity implies fragility of model

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 Easy to find solutions? Satisfiable or feasible

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Modeling Analysis Set of bad system behaviors Set of possible system behaviors Problem: Not robust NP: easy to find a point

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 Easy to find proofs? Unsatisfiable or infeasible

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Modeling Analysis Set of possible system behaviors Set of bad system behaviors coNP: easy to find a proof

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0 Complexity?

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Modeling Analysis Set of possible system behaviors Set of bad system behaviors Problem: Robust but no short proof Key idea: Complexity implies fragility of model

Example: Satisfiability: 

Example: Satisfiability SAT: Given a formula in propositional calculus, is there an assignment to its variables making it true? We consider clausal form, e.g.: (a OR (NOT b) OR c) AND (b OR d) AND (b OR (NOT d) OR a) a, b, c, and d are Boolean (True/False) variables. Problem is NP-Complete. (Cook 1971) Shows surprising “power” of SAT for encoding computational problems.

Generating Hard Random Formulas: 

Generating Hard Random Formulas Key: Use fixed-clause-length model. (Mitchell, Selman, and Levesque 1992) Critical parameter: ratio of the number of clauses to the number of variables. Hardest 3SAT problems at ratio = 4.3

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Hardness of 3SAT 0 2 3 4 5 Ratio of Clauses-to-Variables 6 7 8 1000 3000 DP Calls 2000 4000 50 var 40 var 20 var Easy Easy Hard

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The 4.3 Point 0.0 2 3 4 5 Ratio of Clauses-to-Variables 6 7 8 0.2 0.6 Probability 0.4 50% sat Mitchell, Selman, and Levesque 1991 0.8 1.0 At low ratios: few clauses (constraints) many assignments easily found At high ratios: many clauses inconsistencies easily detected

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Refer to as a SAT transition Complexity transition Is SAT transition either necessary or sufficient for complexity transition? Connections with phase transitions in statistical physics? Are transitions “sharp” in large size limit?

Theoretical Status Of Threshold: 

Theoretical Status Of Threshold Very challenging problem ... Current status: 3SAT threshold lies between 3.45 and 4.6 (Motwani et al. 1994, Achlioptas et al. 2001, Kirousis 2002, Broder and Suen 1993, Dubois 2000; Achlioptas and Beame 2001, Friedgut 1997, etc.) Other problems better characterized (NPP)

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? ? SAT Phase transitions Complexity

Quasigroups or Latin Squares: 

Quasigroups or Latin Squares Quasigroup or Latin Square (Order 4) 32% preassignment Gomes and Selman 96 A quasigroup is an n-by-n matrix such that each row and column is a permutation of the same n colors

Quasigroup with Holes (QWH): 

Quasigroup with Holes (QWH) Given a full quasigroup, “punch” holes into it Always completable (satisfiable), so no SAT transition. Appears to have a complexity transition (easy-hard-easy).

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? ? SAT Phase transitions Complexity

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Lots of problems with statistical physics story.

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SAT transition Complexity transition Since P=NP question and status of this “phase transition” are unresolved… Can we find other settings in which both are clearer, yet reveal the essential features of the general problem? And suggest the right research directions to pursue.

Short term strategy: 

Short term strategy Abandon 3SAT temporarily Phase transition is not fully understood Computational complexity not fully understood Both should hopefully be resolved soon Focus on other problems where both phase transition and/or computational complexity are well-understood P: Percolation lattices, linear programming, convex quadratic programming coNP: polynomial optimization

Lattice models?: 

What can we do with lattices that will be easy to understand, yet relevant to the “real” computational complexity problems that we most care about? Key abstractions: Robustness/Fragility Verifiability Complexity Resource scarcity Lattice models?

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.2 .4 .6 .8 Density = fraction of occupied sites (black) Focus on “horizontal” paths.

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Some (nonstandard) definitions “Vertical” paths in empty sites are allowed to connect through corners or edges. (8 neighbors) “Horizontal” paths connect only on edges. (4 neighbors.Ordinary square site percolation.) Focus on “horizontal” paths.

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Critical phase transition at density = .59…

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.2 .4 .6 .8 Density = fraction of occupied sites (black) Focus on “horizontal” paths.

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Robustness is provided by barriers in some state space. These prevent cascading failure events. Lattices offer a crude abstraction, in that paths can be thought of as barriers, with robustness to perturbations in the lattice. Verifiability complexity is measured in the length of the proof required to verify robustness. Lattices can offer a variety of crude abstractions to this as well. The length of minimal paths would be a simple measure of “proof length.”

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Caution: potential source of confusion. Very special features: Dual and primal problems are “essentially” the same. There is no duality gap.

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Barriers in 1d lattices are 0d cuts. Barriers in 3d lattices are 2d cuts. path fragments barrier In general, barriers are d-1 dimensional (dual) cuts stopping 1-dim (primal) paths in a d-dim lattice.

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Critical phase transition at density = .59…

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Lattices offer pedagogically useful but potentially dangerously misleading simplifications, which are thus both strengths and weaknesses: Internal complexity Computational complexity Duality Focus on “horizontal” paths.

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Internal vs external complexity: Real biology and technology uses extremely complex (complicated) hierarchical organization in order to create robust and verifiably (simple) behavior. Lattices allow no distinction between complex organization and complex behavior. This can be very misleading. Computational complexity: Most lattice computational problems are in P and thus easily explored, but fail to illustrate the P/NP asymmetry. We will rely on notions of complexity that are good analogies, but not precisely comparable. Duality: Duality is greatly simplified and transparent. This makes exposition easy but hides the NP/coNP asymmetry which is central to the general problem.

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Lattices offer enormous (and potentially dangerous) simplifications: Robustness problem= existence of horizontal path Verification = prove existence of horizontal path Complexity = minimum horizontal path length (of proof) Model fragility = minimum number of site changes to break all horizontal paths (= create a vertical path) Focus on “horizontal” paths.

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Note: I’m going to draw small lattices and rely on your imagination for what large lattices would look like.

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.2 .4 .6 .8 Alternative definition of “complexity:” The “computer” is you, looking at the lattice and determining by inspection whether there is a path or not. This can be easy or hard, depending on the density. This is not exactly the same as minimal path length, but close enough for now. Do a very informal story, and then make it rigorous. Density = fraction of occupied sites (black)

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Exist horizontal path? No Yes Easy Hard For random lattices, there are 4 regimes, with all combinations of Easy/Hard and Yes/No. The hard cases correspond to lattices that are of intermediate density, near the critical point. Easy cases are either high or low densities, which always correspond to Yes or No, respectively.

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No Yes Easy Hard No Yes Easy Hard It is much easier to see with all the clusters colored. But that’s cheating, because determining the clusters is essentially the computational problem. Note that the complexity (minimum proof) can be much simpler than the complicated description of the lattice

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The orthodox story: No Yes Easy Hard Hard problems are associated in some way with the phase transition.

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The counter-examples No Yes Easy Hard Exactly the opposite of criticality Yes or no Easy or hard High or low density Robust or fragile (to perturbations)

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The counter-examples Exactly the opposite of criticality Yes or no Easy or hard Low or high density Robust or fragile (to perturbations) 16 different possible combinations

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The counter-examples Exactly the opposite of criticality Yes or no Easy or hard Low or high density Robust or fragile (to perturbations) 16 different possible combinations 8

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Low Density (but connected) Easy Hard Robust Fragile Easy Hard Robust Fragile High density Hard implies fragile (we’ll prove this later). So only 6 of the 8 possibilities exist, and the critical density is nothing special. We will prove that these and only these implications hold.

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Low Density Easy Hard Robust Fragile Easy Hard Robust Fragile High density Easy Hard Robust Fragile

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Low Density Easy Hard Robust Fragile Easy Hard Robust Fragile All interesting real world problems are in this regime, with efficient, highly structured, rare configurations, using scarce (limited) resources.

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Low Density Easy Hard Robust Fragile Easy Hard Robust Fragile Impossible. Improbable in random lattices.

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Low Density Easy Hard Robust Fragile Easy Hard Robust Fragile High density Proof to follow.

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Low Density High density Easy Hard Robust Fragile Random lattices are complex (and fragile) only at critical phase transition.

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Occupied Empty MinPath n = length of side r = density l = MinPath length Definitions. Assume there is a connected (horizontal) path of minimal length l . Typical “minimal” path

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Occupied Empty MinPath Typical “minimal” cut n = length of side r = density l = MinPath length b = MinCut barrier length b Definitions. Assume there is a connected path of minimal length l . Typical “minimal” path

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n = length of side r = density l = MinPath length b = MinCut barrier length b Definitions. Assume there is a connected path of minimal length l . Vertical path

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n = length of side r = density l = MinPath length b = MinCut barrier length Assume a path exists. (Otherwise L=F=.) Necessarily   1/n, n2  l  n and define

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l = MinPath length b = MinCut barrier length

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l = MinPath length b = MinCut barrier length Proof (Vinnicombe&sushi): To provide robustness to b changes, there must be at least b independent paths, which by assumption have minimum length l. Necessarily  n2  lb, or  n/b  l/n. Take log of both sides.

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Lattices and paths can be: Resources: Scarce or rich Existence of path: Yes or no Complexity: Hard or easy Perturbations: Fragile or robust This is “maximally tight” in the sense that:

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Lattices and paths can be: Existence: Yes or no Resources: Scarce or rich Perturbations: Fragile or robust Complexity: Hard or easy Anything is possible, consistent with the theorem. We’ll just consider the 8 cases with paths.

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-S=log() Scarce Rich Robust Fragile Easy Hard

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Scarce Rich Robust Fragile Easy Hard

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Scarce Rich Robust Fragile Easy Hard

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Scarce Rich Robust Fragile Easy Hard

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Occupied Empty MinPath Easy F=S, L=0

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Easy F=S, L=0 Most robust possible.

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Easy and Fragile F=log(n)>S, L=0

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Scarce Rich Robust Fragile Easy Hard Occupied Empty MinPath F=S+L b d m

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To construct asymptotically tight cases where n2 = lb, consider the lattice below. b d r = density b = MinCut barrier length l = MinPath length n = length of side m = # of “cells” d = width of open regions m b d

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Now take limits: By constructing lattices as below, with n>>m>>1, it is possible to find lattices such that any n2  lb, with <1 is achievable.

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Scarce Rich Robust Fragile Easy Hard F=S+L

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Scarce Rich Robust Fragile Easy Hard The Fragile Face

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Scarce Rich Robust Fragile Easy Hard The Four Corners

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Scarce Robust Fragile Easy F=S+L Most Robust F=S Most Fragile F>>S

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Scarce Rich Robust Fragile Easy Hard Random

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Efficient and robust is far from random

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How general is this? Seems to hold in all theory where it has been investigated. Extensive literature on ill-conditioning in LPs and numerical linear algebra. Anecdotally, seems to capture essence of many complexity problems. Needs to be combine with laws constraining net system fragility.

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Polynomial functions: NP-hard problem.

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0 Complexity?

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Special case: Scalar QP

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Special case: Scalar QP Note to the overly pedantic: of course, you would never literally invert A.

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Polynomial functions: NP-hard problem. A “simple” relaxation (Shor): find the minimum γ such that γ- 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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A sufficient condition for nonnegativity: Sums of squares (SOS) Convex condition (Shor, 1987) Efficiently checked using SDP (Parrilo). Write: where z is a vector of monomials. Expanding and equating sides, obtain linear constraints among the Qij. Finding a PSD Q subject to these conditions is exactly a semidefinite program (LMI).

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Exactly as in QP case, “SAT 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 NP=co-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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Nested families of SOS (Parrilo) exhaust co-NP

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0 Conjectures on why such a boring “SAT transition:” One polynomial is generically robust, therefore no complexity. QPs capture the essence of this. Can make up other “SAT transitions” which create fragilities, and thus the possibility of complexity NPP: the problem the experts on phase transitions and computational complexity think is the best understood case. (It has a complete and beautiful story.)

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NPP ? Fragile = large changes in solution from small changes in data

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NPP

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Random f

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Very unlikely to be feasible. Contrast with random polynomial.

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Phase transitions Complexity

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A sufficient condition for nonnegativity: Sums of squares (SOS) Convex condition (Shor, 1987) Efficiently checked using SDP (Parrilo). Write: where z is a vector of monomials. Expanding and equating sides, obtain linear constraints among the Qij. Finding a PSD Q subject to these conditions is exactly a semidefinite program (LMI).

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Nested families of SOS (Parrilo) exhaust co-NP

Semialgebraic Sets: 

Semialgebraic Sets Semialgebraic: finite number of polynomial equalities and inequalities. Continuous, discrete, or mixture of variables. Is a given semialgebraic set empty? Feasibility of polynomial equations: NP-hard… Search for bounded-complexity emptiness proofs, using SDP. (Parrilo 2000)

Positivstellensatz (Real Nullstellensatz): 

Stengle, 1974 Generalizes Hilbert’s Nullstellensatz and LP duality Infeasibility certificates of polynomial equations over the real field. Parrilo: Bounded degree solutions computed via SDP!  Nested family of polytime relaxations: for quadratics, the first level is the S-procedure… Positivstellensatz (Real Nullstellensatz) if and only if

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Special case: LP

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Special case: LP For LP case it is well-known that complexity implies fragility (long run times imply ill-conditioning).

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Search for counterexample Search for proof coNP NP Set of possible system behaviors Set of bad system behaviors coNP: Give a proof

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Search for counterexample Search for proof 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.

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Convex, but infinite dimensional. Efficient (P time) search subsets (relaxations) using SOS/SDP Guaranteed to converge Search for counterexample Seach for proof

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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 Search for short proof Failure to find short proof implies some relaxed model is nonempty (which is bad).

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First order relaxation Failure to find short proof implies some relaxed model is nonempty (which is bad). For NPP problem, first relaxation has simple interpretation! And it is not good. This problem is intrinsically fragile, thus complex. Known terrible test!

Entangled Quantum States(Doherty, Parrilo, Spedalieri 2001): 

Entangled Quantum States (Doherty, Parrilo, Spedalieri 2001) Entangled states are one of the most important distinguishing features of quantum physics. Bell inequalities: hidden variable theories must be non-local. Teleportation: entanglement + classical communication. Quantum computing: some computational problems may have lower complexity if entangled states are available. How to determine whether or not a given state is entangled ?

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QM state described by psd Hermitian matrices ρ States of multipartite systems are described by operators on the tensor product of vector spaces Product states: each system is in a definite state Separable states: a convex combination of product states. Entangled states: those that cannot be written as a convex combination of product states.

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Decision problem: find a decomposition of r as a convex combination of product states or prove that no such decomposition exists. (Hahn-Banach Theorem) Hard! Z is an “entanglement witness,” a generalization of Bell’s inequalities

First Relaxation: 

First Relaxation Restrict attention to a special type of Z: The bihermitian form Z is a sum of squared magnitudes.

First Relaxation: 

First Relaxation If minimum is less than zero, r is entangled Equivalent to known condition Peres-Horodecki Criterion, 1996 Known as PPT (Positive Partial Transpose) Exact in low dimensions Counterexamples in higher dimensions

Further relaxations : 

Further relaxations Broaden the class of allowed Z to those for which is a sum of squared magnitudes. Also a semidefinite program. Strictly stronger than PPT. Can directly generate a whole hierarchy of tests.

Second Relaxation: 

Second Relaxation If the minimum is less than zero then r is entangled. Detects all the non-PPT entangled states tried… minimize subject to

Quantum entanglement and Robust control: 

Quantum entanglement and Robust control

Quantum entanglement and Robust control: 

Quantum entanglement and Robust control

Higher order relaxations: 

Higher order relaxations Nested family of SDPs Necessary: Guaranteed to converge to true answer No uniform bound (or P=NP) Tighter tests for entanglement Improved upper bounds in robust control Special cases of general approach All of this is the work of Pablo Parrilo (PhD, Caltech, 2000, now Professor at ETHZ) My contribution: I kept out of his way.

Summary: 

Summary Single framework with substantial advances in Testing entanglement MaxCut Global continuous optimization Finding Lyapunov functions for nonlinear systems Improved robustness analysis upper bounds Many other applications This is just the tip of a big iceberg

Nested relaxations and SDP: 

Nested relaxations and SDP

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Standard tools of robust (linear) control Unmodeled dynamics, nonlinearities, and IQCs Noise and disturbances Real parameter variations D-K iteration for -synthesis Are all treated much better… And generalized to Nonlinear Hybrid DAEs Constrained

Caveats: 

Caveats Inherits difficulties from robust control High state dimension and large LMIs Must find ways to exploit structure, symmetries, sparseness Note: many researchers don’t want to get rid of the ad hoc, handcrafted core of their approaches to control (why take the fun out of it?)

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Phase transitions Complexity

Bad news and good news: 

Bad news and good news Bad news? Some hoped-for connections between phase transitions and complexity are not there. Good news?: Ideas still interesting. Lots more really good news! The alternative is much richer and useful, and connects in interesting ways with phase transitions New algorithms, new mathematics, new practical applications,… And deep implications for physics.

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Phase transitions Complexity ? Internet traffic and topology Biological and ecological networks Evolution and extinction Earthquakes and forest fires Finance and economics Social and political systems Physics and emergilence at the edge of chaocritiplexity

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Phase transitions Complexity ? Internet traffic and topology Biological and ecological networks Evolution and extinction Earthquakes and forest fires Finance and economics Social and political systems Physics and the edge of chaocritiplexity Rich new unifying theory of complex control, communication, and computing systems

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Physics and the edge of chaocritiplexity Ubiquity of power laws Coherent structures in shear flow turbulence Macro dissipation and irreversibility vs. micro reversibility. Quantum entanglement, measurement, and the QM/Classical transition Growing group of physicists and experimentalists are enabling this effort (Carlson, Mabuchi, Doherty, Gharib,…) Rich new unifying theory of complex control, communication, and computing systems

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z y x flow position Shear flow turbulence

Today’s new word: Orthophysics: 

Today’s new word: Orthophysics The use of mathematics, rigor, and careful attention to experimentalists… …to support physics and to treat, correct, and prevent irregularities, errors, and weaknesses, … …in order to create a more accurate and less distorted image of reality. Typically involves methods that are orthogonal to accepted orthodoxy. If emergilence is the disease, then orthophysics is part of the cure.

Ortho: 

Ortho Straight; upright; vertical: orthotropous. Perpendicular: orthorhombic. Correct; correction: orthopsychiatry. ETYMOLOGY:Middle English, from Old French, from Latin, from Greek, from orthos, straight, correct, right.

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orthochromatic : Of, relating to, or accurately reproducing the colors of the subject. orthodontics: The dental specialty and practice of preventing and correcting irregularities of the teeth. orthodox: Adhering to the accepted or traditional and established faith, especially in religion. orthogonal: 1. Relating to or composed of right angles. 2. Mathematics a. Of or relating to a matrix whose transpose equals its inverse. b. Of or relating to a linear transformation that preserves the length of vectors. orthography: The art or study of correct spelling.

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orthopedics: The branch of medicine that deals with the prevention or correction of injuries or disorders of the skeletal system and associated muscles, joints, and ligaments. orthopsychiatry: The psychiatric study, treatment, and prevention of emotional and behavioral problems, especially of those that arise during early development.

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orthoptics: The evaluation and nonsurgical treatment of visual disorders caused by imbalance of the eye muscles, such as strabismus. orthoscopic: 1.Having normal vision; free from visual distortion. 2. Giving an undistorted image. Used of an optical instrument. orthostatic: Relating to or caused by standing upright. orthotics: The science that deals with the use of specialized mechanical devices to support or supplement weakened or abnormal joints or limbs. orthotropous: Botany Growing straight, so that the micropyle is at the end opposite the stalk.

Orthophysics: 

Orthophysics The use of mathematics, rigor, and careful attention to experimentalists… …to support physics and to treat, correct, and prevent irregularities, errors, and weaknesses, … …in order to create a more accurate and less distorted image of reality. Typically involves methods that are orthogonal to accepted orthodoxy. “An important goal of orthophysics is to cure emergilence.”

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Orthophysics: The use of mathematics, rigor, and careful attention to experimentalists to support physics and to treat, correct, and prevent irregularities, errors, and weaknesses, in order to create a more accurate and less distorted image of reality. Typically involves methods that are orthogonal to accepted orthodoxy. Why put effort into orthophysics? Most physicists think you are working on “solved” problems Many non-physicists take their word for it No one else cares

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Because… …physics is the “queen of the sciences!”

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Emergilence Orthophysics: Searching for the cure to emergilence. www.cds.caltech.edu/~doyle www.aut.ee.ethz.ch/~parrilo