martin ziegler 1 heinz nixdorf institute university of paderborn algorithms and complexity...

Martin Ziegler 1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Physically-Relativized Church-Turing Hypotheses

Martin Ziegler

Theoretical Computer Science

University of Paderborn

33095 GERMANY

Martin Ziegler 2



„Does there exist a physical system of computational power strictly

exceeding that of a Turing machine?“• for example able to solve the Halting problem?• or in polynomial time some NPNP-complete problem?

Answer has tremendous effects on our conception of• nature (universe as a computer?, cf. eg. Seth

Lloyd)

• Turing machines: universal model of computation– in computer science (WHILE-programs, λ-calculus)– in mathematics (μ-recursive function class)– and in physics?

• engineering actual computing devices (Intel, AMD)

Martin Ziegler 3



„Does there exist a physical system of computational power strictly

exceeding that of a Turing machine?“(Physical/strong) Church-Turing Hypothesis: No!

Audience Poll:• Do you believe in this hypothesis?• Proof?

What is a physical system, anyway?

Martin Ziegler 4



Feynman, Shor, Deutsch, GroverAdamyan, Calude, Dinneen, Pavlov, Kieu

Martin Ziegler 5


Algorithms and ComplexityMalament, Hogarth, Nemeti …

Martin Ziegler 6


Algorithms and ComplexityA.C.-C.Yao, W.D.Smith, K.Svozil …

Martin Ziegler 7



Reif & Tate & Yoshida (1994), Oltean (2006ff), Woods

Martin Ziegler 8



Beggs&Tucker (2007): „[…] we should […] use a physical theory to define precisely the class of physical systems under investigation“

So, what is a Physical System?

Celestial Mechanics, Newtonian Mechanics, Continuum Mechanics, Magneto-statics, Electrostatics, Ray Optics, Gaussian Optics, Electrodynamics, Special Relativity, General Relativity, Quantum Mechanics, Quantum Field Theory

Ludwig: „Die Grundstrukturen einer physikalischen Theorie“, Springer (1990)Schröter: „Zur Meta-Theorie der Physik“, de Gruyter (1996)

„Reality“ described/covered by a patchwork of physical theories

A physical theory Φ consists of 3 parts:

• a mathematical theory MT

• a part WB of nature it aims to describe

• a correspondence AP from WB to MT

WB2WB1

WB3

Martin Ziegler 9



„Does there exist in Φ a physical system of compu-tational power exceeding that of a Turing machine?“„Does there exist in Φ a physical system of compu-tational power exceeding that of a Turing machine?“

Church-Turing Hypothesis relative to a Physical Theory

Principle 2.2 in Beggs&Tucker (2007):“Classifying computers in a physical theory.”

Principle 2.3 in Beggs&Tucker (2007): Mapping the border between computer and hyper-computer in physical theory.

→ Research Programme: For various physical theories Φ, investigate CTHΦ.

That is, fix a physical theory Φ and consider validity of the Church-Turing Hypothesis (CTH) relative to Φ.

Martin Ziegler 10


Algorithms and ComplexityResearch Programme

Ontological commitment; again Beggs&Tucker (2007):“It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false”

For various physical theories Φ, investigate CTHΦ.

For a fixed Φ, does there exist in Φ• a system able to solve the Halting problem?• or in polynomial time some NPNP-complete problem?

Compare Baker&Gill&Solovay (1975):„Relativizations of the PP=?NPNP Question”:For one oracle A, provably PPA=NPNPA;for another oracle B, provably PPB≠NPNPB.

Martin Ziegler 11


Algorithms and ComplexityOntological Commitment

• Is there a theory which is not „false“ somehow?

→ Grand Unified Theory/Grand Unified Theory/Theory of EverythingTheory of Everything→ dream, not science

• Pragmatic: each Φ describes some part of reality more or less accurately

It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false.

„Reality“ described/covered by a

patchwork of physical theories

WB2WB1

WB3

GUT/GUT/ToEToEGUT/GUT/ToEToE

Martin Ziegler 12


Algorithms and ComplexityOntological Commitment II

• Pragmatic: each Φ describes some part of reality more or less accurately

• Compare Models of ComputationModels of Computation in Theoretical Computer Science:

– Is a ZX81 more appropriately described by a TM or by a DFA?

• Even ‘small‘ WB (=area of applica-bility) may have ‘large‘ applications!

– Ohm‘s Law & CM vs. QED


Martin Ziegler 13


Algorithms and ComplexityComputational Physics

Simulating of a (class of) physical systems Φ

Martin Ziegler 14


Algorithms and ComplexityComputational Physics

Computational complexity of simulating a (class of) physical systems Φ:

–complete if, in Φ, there exist systems implementing, e.g.,

• Boolean circuit evaluation• Travelling sales tour search• Universal Turing computation

Principle 2.2 in Beggs&Tucker (2007):“Classifying computers in a physical theory.”

→ CTHΦ as approach to dis-/prove optimality of algorithms in computational physics!

P, NP, PSPACE, RECP, NP, PSPACE, REC

Martin Ziegler 15


Algorithms and ComplexityResearch Programme

Principle 2.3 in Beggs&Tucker (2007): Mapping the border between computer and hyper-computer in physical theory.

For various physical theories Φ, investigate CTHΦ.Start with ‘simplest‘ theories!


Martin Ziegler 16


Algorithms and ComplexityExample: Celestial Mechanics

Various physical theories:

• full relativistic effects

• Newton gravitation

• Kepler ellipses around center of gravity, w/o interaction

• Copernican heliocentrism

• Ptolemaic geocentrism

• planar, circular rotation

Research Programme:For various physical theories Φ, investigate CTHΦ.It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false.

Martin Ziegler 17


Algorithms and ComplexityExample: Celestial Mechanics

Various physical theories

• Newton gravitation:PSPACEPSPACE-complete [Reif&Tate‘93]undecidable [W.D.Smith’06, K.Svozil‘07]

• planar, circular rotation NCNC1…#P#P-compl.

Martin Ziegler 18



1H 2H 3,4H

Example: Classical Mechanics

• An ideal solid can encode the Halting problem

• and may then be used to solve it by probing:

„Does there exist in CM a physical system of compu-tational power exceeding that of a Turing machine?“

Martin Ziegler 19


Algorithms and ComplexityExistence in Physical Theories

What makes CM unrealistic with respect to computability?

When is a mathematical object considered to exist?

a) If one can actually construct this object. („Constructivism“)

b) If its non-existence raises a contradiction. (indirect proof, e.g. Markov‘s Principle)

c) If the hypothesis of its existence does not raise a contradiction.(e.g. Zorn‘s Lemma is consistent with ZF)

1) Real bodies are not infinitely divisible.

But even if so (ontological commitment!):

2) In order to solve the Halting problem, does there exist a solid with it encoded?

CM should support only solidswhich can be ‚constructed‘

(e.g. cut/carved) from few basic ones

(e.g. cuboid)

Martin Ziegler 20


Algorithms and ComplexityConclusion• Current hot disputes on validity of Church-Turing hypothesis• mostly due to vagueness of the underlying notion of „nature“:

– ‚counterexamples‘ (=physical systems ‘solving‘ the Halting problem) exploit some physical theory Φ to its limits

• Better always speak of the CTH relative to a specific Φ.– independent of whether (and where) Φ is ‘realistic‘ or not.

• Investigate, for various Φ, the computational power of Φ• → Lower complexity bounds in computational physics

Realistic physical theory Φ=(MT,AB,WB) should• make the Church-Turing hypothesis a theorem

(meta-principle, like gauge-invariance or energy conservation)

• and employ some sort of constructivism in WB.

Martin Ziegler 21



Heinz Nixdorf Institute& Dept of Computer ScienceUniversity of PaderbornFürstenallee 1133095 Paderborn, Germany

Tel.: +49 (0) 52 51/60 30 67Fax: +49 (0) 52 51/62 64 82E-Mail: [email protected]://www.upb.de/cs/ziegler.html

Thanks for your attention!Thanks for your attention!

martin ziegler 1 heinz nixdorf institute university of paderborn algorithms and complexity...

Documents

physical theory principle

complexity beggstucker

patchwork of physical

various physical theories

class of physical systems

complexity feynman

complexity malament

amd slide