Physically-Relativized Church-Turing Hypotheses

1. Physically-Relativized Church-Turing Hypotheses Martin Ziegler Theoretical Computer Science University of Paderborn 33095 GERMANY

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 NP-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)

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?

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

5. Malament, Hogarth, Nemeti …

6. A.C.-C.Yao, W.D.Smith, K.Svozil …

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

8. So, what is a Physical System? WB2 WB1 WB3 • Beggs&Tucker (2007): „[…] we should […] use a physical theory to define precisely the class of physical systems under investigation“ 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) • A physical theory Φ consists of 3 parts: • a mathematical theory MT • a part WBof nature it aims to describe • a correspondence AP from WB to MT „Reality“ described/covered by a patchwork of physical theories

9. Church-Turing Hypothesis relative to a Physical Theory • Principle 2.2 in Beggs&Tucker (2007):“Classifying computers in a physical theory.” „Does there exist in Φa physical system of compu- tational power exceeding that of a Turing machine?“ „Does there exist in Φa physical system of compu- tational power exceeding that of a Turing machine?“ That is, fix a physical theory Φ and consider validity of the Church-Turing Hypothesis (CTH) relative to Φ. → Research Programme: For various physical theories Φ, investigate CTHΦ. Principle 2.3 in Beggs&Tucker (2007): Mapping the border between computer and hyper-computer in physical theory.

10. Research Programme 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 NP-complete problem? Compare Baker&Gill&Solovay (1975):„Relativizations of the P=?NP Question”: For one oracle A, provably PA=NPA; for another oracle B, provably PB≠NPB. 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”

11. Ontological Commitment GUT/ ToE GUT/ToE WB2 WB1 WB3 It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false. • Is there a theory which is not „false“ somehow? • → Grand Unified Theory/Theory of Everything • → dream, not science • Pragmatic: each Φ describes somepart of reality more or less accurately „Reality“ described/covered by a patchwork of physical theories

12. Ontological Commitment II It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false. • Pragmatic: each Φ describes somepart of reality more or less accurately • Compare Models of Computationin 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

13. Computational Physics • Simulating of a (class of) physical systems Φ

14. Computational 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 P, NP, PSPACE, REC 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!

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

16. Example: Celestial Mechanics 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. • Various physical theories: • full relativistic effects • Newton gravitation • Kepler ellipses around center of gravity, w/o interaction • Copernican heliocentrism • Ptolemaic geocentrism • planar, circular rotation

17. Example: Celestial Mechanics • Various physical theories • Newton gravitation: • PSPACE-complete [Reif&Tate‘93]undecidable [W.D.Smith’06, K.Svozil‘07] • planar, circular rotation NC1…#P-compl.

18. Example: Classical Mechanics 2H 1H 3,4H „Does there existin CMa physical system of compu- tational power exceeding that of a Turing machine?“ • An ideal solid can encode the Halting problem • and may then be used to solve it by probing:

19. Existence in Physical Theories CM should support only solidswhich can be ‚constructed‘ (e.g. cut/carved) from few basic ones (e.g. cuboid) What makes CM unrealistic with respect to computability? 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? • When is a mathematical object considered to exist? • If one can actually construct this object. („Constructivism“) • If its non-existence raises a contradiction. (indirect proof, e.g. Markov‘s Principle) • If the hypothesis of its existence does not raise a contradiction. (e.g. Zorn‘s Lemma is consistent with ZF)

20. Conclusion • 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 Φto 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.

21. Thanks for your attention! Heinz Nixdorf Institute & Dept of Computer Science University of Paderborn Fürstenallee 11 33095 Paderborn, Germany Tel.: +49 (0) 52 51/60 30 67 Fax: +49 (0) 52 51/62 64 82 E-Mail: ziegler@upb.de http://www.upb.de/cs/ziegler.html