1. Martin Ziegler 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Physically-Relativized Church-Turing Hypotheses Martin Ziegler Theoretical Computer Science University of Paderborn 33095 GERMANY

2. Martin Ziegler 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity „Does there exist a physical system of computational power strictly exceeding that of a Turing machine?“ for example able to solve the Halting problem? NPor 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. Martin Ziegler 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity „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. Martin Ziegler 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Feynman, Shor, Deutsch, Grover Adamyan, Calude, Dinneen, Pavlov, Kieu

5. Martin Ziegler 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Malament, Hogarth, Nemeti …

6. Martin Ziegler 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A.C.-C.Yao, W.D.Smith, K.Svozil …

7. Martin Ziegler 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Reif & Tate & Yoshida (1994), Oltean (2006ff), Woods

8. Martin Ziegler 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 WB 2 WB 1 WB 3

9. Martin Ziegler 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity „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 CTH Φ. That is, fix a physical theory Φ and consider validity of the Church-Turing Hypothesis (CTH) relative to Φ.

10. Martin Ziegler 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Research 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 CTH Φ. For a fixed Φ, does there exist in Φ a system able to solve the Halting problem? NP or in polynomial time some NP -complete problem? PNP Compare Baker&Gill&Solovay (1975): „Relativizations of the P =? NP Question”: PNP For one oracle A, provably P A = NP A ; PNP for another oracle B, provably P B ≠ NP B.

11. Martin Ziegler 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Ontological Commitment Is there a theory which is not „false“ somehow? Grand Unified Theory/ Theory of Everything → Grand Unified Theory/ Theory 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 WB 2 WB 1 WB 3 GUT/ToE GUT/ ToE

12. Martin Ziegler 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Ontological Commitment II Pragmatic: each Φ describes some part of reality more or less accurately Models of ComputationCompare Models 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 It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false.

13. Martin Ziegler 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Computational Physics Simulating of a (class of) physical systems Φ

14. Martin Ziegler 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 Principle 2.2 in Beggs&Tucker (2007): “Classifying computers in a physical theory.” → CTH Φ as approach to dis-/prove optimality of algorithms in computational physics! P, NP, PSPACE, REC  P, NP, PSPACE, REC

15. Martin Ziegler 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 CTH Φ. 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. Martin Ziegler 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Example: 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 CTH Φ. It does not matter whether we think of the theory Φ as true, or roughly applicable, or know it to be false.

17. Martin Ziegler 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Example: Celestial Mechanics Various physical theories Newton gravitation: PSPACE PSPACE -complete [Reif&Tate‘93] undecidable [W.D.Smith’06, K.Svozil‘07] planar, circular rotation NC#P  NC 1 … #P -compl.

18. Martin Ziegler 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 1H1H 2H2H 3,4  H 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?“

19. Martin Ziegler 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Existence 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 solids which can be ‚constructed‘ (e.g. cut/carved) from few basic ones (e.g. cuboid)

20. Martin Ziegler 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 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.

21. Martin Ziegler 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 Thanks for your attention!