Automata and Formal Languages - CM0081 Hypercomputation

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1 Automata and Formal Languages - CM0081 Hypercomputation Andrés Sicard-Ramírez Universidad EAFIT Semester

2 Motivation Absolute computability Gödel [(1946) 1990, p. 150]: the great importance of the concept of general recursiveness (or Turing s computability) is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. Hypercomputation 2/57

3 Motivation Absolute computability Davis [1958, p. 11]: For how can we ever exclude the possibility of our being presented, some day (perhaps by some extraterrestrial visitor), with a (perhaps extremely complex) device or oracle that computes a noncomputable function? However, there are fairly convincing reasons for believing that this will never happen. Hypercomputation 3/57

4 Motivation Relative computability Sylvan and Copeland [2000, p. 190]: Troubles with absolutism are deeper and more extensive than these cracks (analogue procedures and newer physics) reveal. For one thing, computability is relative not simply to physics, but more generally to systems of frameworks, which include or contain underlying logics. Hypercomputation 4/57

5 Hypercomputers Definition A hypercomputer is any machine (theoretical or real) that compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine [Copeland 2002b]. Hypercomputation 5/57

6 Hypercomputers Definition A hypercomputer is any machine (theoretical or real) that compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine [Copeland 2002b]. Super Turing Machines and Non Turing Machines L Σ Super-TM L Σ TM TM non-tm Hypercomputation 6/57

7 Possible Sources of Hypercomputation Computability Mathematics Logic Hypercomputation Model (HM) Biology Physics? Hypercomputation 7/57

8 First Hypercomputation Model: Oracle Turing Machines Definition A oracle Turing machine (OTM) is a Turing machine equipped with an oracle that is capable of answering questions about the membership of a specific set of natural numbers [Turing 1939]. Hypercomputation 8/57

9 First Hypercomputation Model: Oracle Turing Machines Definition A oracle Turing machine (OTM) is a Turing machine equipped with an oracle that is capable of answering questions about the membership of a specific set of natural numbers [Turing 1939]. Hypercomputability features If the oracle is a recursive set then OTM TM. If the oracle is a non-recursive set then OTM HM. Hypercomputation 9/57

10 About the Hypercomputation Term Copeland and Proudfoot [1999]: Right Hypercomputation Wrong Super-Turing computation Computing beyond Turing s limit Breaking the Turing barrier Etc. Hypercomputation 10/57

11 Hypercomputation Model: Accelerated Turing Machines Definition An accelerated Turing machine (ATM) is a Turing machine that performs its first step in one unit of time and each subsequent step in half the time of the step before [Copeland 1998, 2002a]. Hypercomputation 11/57

12 Hypercomputation Model: Accelerated Turing Machines Definition An accelerated Turing machine (ATM) is a Turing machine that performs its first step in one unit of time and each subsequent step in half the time of the step before [Copeland 1998, 2002a]. Hypercomputability features Since 1 + 1/2 + 1/4 + 1/8 + = i=0 1 = 2, 2i an ATM could complete an infinity of steps in two time units. Hypercomputation 12/57

13 Hypercomputation Model: Analog Recurrent Neural Network (ARNN) Description [Siegelmann 1999] u 1 b 11 x 1 b 12 a 14 u 2 b 13 b 22 b 23 a22 a x 24 2 x 4 a 23 a 34 c 3 x 3 X(t + 1) = σ( A X(t) + B U(t) + C) Hypercomputation 13/57

14 Hypercomputation Model: Analog Recurrent Neural Network (ARNN) Hypercomputability features a ij {N, Q, R} ARNN {DFA, TM, HM}. Hypercomputation 14/57

15 Standard Quantum Computation (SQC) Models Quantum Turing machines (QTM) [Deutsch 1985] and quantum circuits [Deutsch 1989]. Hypercomputation 15/57

16 Standard Quantum Computation (SQC) Models Quantum Turing machines (QTM) [Deutsch 1985] and quantum circuits [Deutsch 1989]. Relation between the models TMs Probabilistic TMs Reversible TMs QTMs Hypercomputation 16/57

17 Weak Hypercomputation Based on SQC Generation of truly random numbers U H 0 = 1 2 ( ) measure Hypercomputation 17/57

18 Weak Hypercomputation Based on SQC Generation of truly random numbers U H 0 = 1 2 ( ) measure 1. We observe the superposition state: The act of observation causes the superposition to collapse into either 0 or the 1 state with equal probability. Hence you can exploit quantum mechanical superposition and indeterminism to simulate a perfectly fair coin toss. [Williams and Clearwater 1997, p. 160] Hypercomputation 18/57

19 Weak Hypercomputation Based on SQC Generation of truly random numbers U H 0 = 1 2 ( ) measure 1. We observe the superposition state: The act of observation causes the superposition to collapse into either 0 or the 1 state with equal probability. Hence you can exploit quantum mechanical superposition and indeterminism to simulate a perfectly fair coin toss. [Williams and Clearwater 1997, p. 160] 2. The problem: It is not clear how to use this property to solve a non-computable Turing machine problem [Ord and Kieu 2009]. Hypercomputation 19/57

20 Others Quantum Computation Models Common misunderstanding quantum computation SQC adiabatic quantum computation (AQC) Hypercomputation 20/57

21 Others Quantum Computation Models Common misunderstanding quantum computation SQC adiabatic quantum computation (AQC) The real situation Kieu s hypercomputational quantum algorithm (KHQA) [Kieu 2003]: finite AQC SQC infinite AQC KHQA Hypercomputation 21/57

22 Hypercomputational Quantum Algorithm à la Kieu Sicard, Ospina and Vélez [2006]: Classically noncomputable P problem (Hilbert s 10th problem) Hypercomputational quantum algorithm Simulation Partial solution to P Physical referent (Infinite square well) Dynamical algebra (Lie alg. su(1, 1)) Hypercomputation 22/57

23 Hypercomputation Model: Infinite Time Turing Machines Definition An infinite time Turing machine is a Turing machine working on a time clocked by transfinite ordinals [Hamkins and Lewis 2000; Hamkins 2002, 2007]. Figure from Hamkins [2002, Fig. 1]. Hypercomputation 23/57

24 Hypercomputation Model: Infinite Time Turing Machines Definition An infinite time Turing machine is a Turing machine working on a time clocked by transfinite ordinals [Hamkins and Lewis 2000; Hamkins 2002, 2007]. Description For convenience, the machines have three tapes: start input: scratch: output: Figure from Hamkins [2002, Fig. 1]. Hypercomputation 24/57

25 Hypercomputation Model: Infinite Time Turing Machines Description (cont.) In stage α + 1 the machine works as usual. Hypercomputation 25/57

26 Hypercomputation Model: Infinite Time Turing Machines Description (cont.) In stage α + 1 the machine works as usual. In limit ordinal stages the machine works as follows [Hamkins and Lewis 2000, p ]: To set up such a limit ordinal configuration, the head is plucked from wherever it might have been racing towards, and placed on top of the first cell. And it is placed in a special distinguished limit state. Now we need to take a limit of the cell values on the tape. And we will do this cell by cell according to the following rule: if the values appearing in a cell have converged, that is, if they are either eventually 0 or eventually 1 before the limit stage, then the cell retains the limiting value at the limit stage. Otherwise, in the case that the cell values have alternated from 0 to 1 and back again unboundedly often, we make the limit cell value 1. Hypercomputation 26/57

27 Hypercomputation Model: Infinite Time Turing Machines Description (cont.) This completely describes the configuration of the machine at any limit ordinal stage β, and the machine can go on computing, β + 1, β + 2, and so on, eventually taking another limit at β + ω co and so on through the ordinals. Hypercomputation 27/57

28 Hypercomputation Model: Infinite Time Turing Machines Description (cont.) This completely describes the configuration of the machine at any limit ordinal stage β, and the machine can go on computing, β + 1, β + 2, and so on, eventually taking another limit at β + ω co and so on through the ordinals. Hypercomputability features The halting problem is decidable in ω many steps by infinite time Turing machines [Hamkins and Lewis 2000]. Hypercomputation 28/57

29 Hypercomputation Model: Infinite Time Turing Machines Description (cont.) This completely describes the configuration of the machine at any limit ordinal stage β, and the machine can go on computing, β + 1, β + 2, and so on, eventually taking another limit at β + ω co and so on through the ordinals. Hypercomputability features The halting problem is decidable in ω many steps by infinite time Turing machines [Hamkins and Lewis 2000]. Remark Although not related with computability but algorithmic complexity, P NP for infinite time Turing machines [Schindler 2003]. Hypercomputation 29/57

30 Hypercomputation Model: Infinite Time Turing Machines Theorem (Hamkins and Lewis [2000, Theorem 1.1]) Every halting infinite time computation is countable. Hypercomputation 30/57

31 Hypercomputation Model: Infinite Time Turing Machines Theorem (Hamkins and Lewis [2000, Theorem 1.1]) Every halting infinite time computation is countable. Remark The infinite time Turing machines have been generalised by the ordinal computability models, which are models based on ordinal numbers. These models include infinite time Turing machines or Turing machines working on tapes of transfinite length. Seyfferth [2013] shows a brief overview of these models. Hypercomputation 31/57

32 Church-Turing Thesis and Thesis M The Church-Turing thesis Any procedure than can be carried out by an idealised human clerk working mechanically with paper and pencil can also be carried out by a Turing machine. [Copeland and Sylvan 1999] Hypercomputation 32/57

33 Church-Turing Thesis and Thesis M The Church-Turing thesis Any procedure than can be carried out by an idealised human clerk working mechanically with paper and pencil can also be carried out by a Turing machine. [Copeland and Sylvan 1999] Thesis M What can be calculated by a machine is Turing machine computable. [Gandy 1980] Hypercomputation 33/57

34 Physical Hypercomputation? Open problem The refutation of a general/physical version of Gandy s Thesis M. Hypercomputation 34/57

35 Physical Hypercomputation? Open problem The refutation of a general/physical version of Gandy s Thesis M. Based on quantum physics (infinite adiabatic quantum computation)? Hypercomputation 35/57

36 Physical Hypercomputation? Open problem The refutation of a general/physical version of Gandy s Thesis M. Based on quantum physics (infinite adiabatic quantum computation)? Based on relativistic physics (cosmological phenomena)? [Pitowsky 1990; Hogarth 1992, 1994; Etesi and Németi 2002; Németi and Dávid 2006] Hypercomputation 36/57

37 An Interesting Project: Formal Verification of Hypercomputation in Relativistic Physics Stannett and Németi [2014] and Stannett [2015]: Goals Implement first-order axiomatisations of theories of the relativity using the proof assistant Isabelle; Hypercomputation 37/57

38 An Interesting Project: Formal Verification of Hypercomputation in Relativistic Physics Stannett and Németi [2014] and Stannett [2015]: Goals Implement first-order axiomatisations of theories of the relativity using the proof assistant Isabelle; Add a model of computation carried out by machines travelling along specific spacetime trajectories; Hypercomputation 38/57

39 An Interesting Project: Formal Verification of Hypercomputation in Relativistic Physics Goals (cont.) Consider how the power of these computational systems changes according to the underlying topology of spacetime; Hypercomputation 39/57

40 An Interesting Project: Formal Verification of Hypercomputation in Relativistic Physics Goals (cont.) Consider how the power of these computational systems changes according to the underlying topology of spacetime; Select a recursively uncomputable problem P (for example, the Halting Problem) and machine-verify the following claims: in simpler relativistic settings, P remains uncomputable; in some spacetimes, P can be solved. Hypercomputation 40/57

41 Is Hypercomputation a Myth? Davis refutations Davis, M. [2006]. Why There is no Such Discipline as Hypercomputation. In: Applied Mathematics and Computation 178.1, pp doi: /j.amc Hypercomputation 41/57

42 Is Hypercomputation a Myth? Davis refutations Davis, M. [2006]. Why There is no Such Discipline as Hypercomputation. In: Applied Mathematics and Computation 178.1, pp doi: /j.amc Davis, M. [2004]. The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer, pp doi: / _8. Hypercomputation 42/57

43 Is Hypercomputation a Myth? Davis refutations Davis, M. [2006]. Why There is no Such Discipline as Hypercomputation. In: Applied Mathematics and Computation 178.1, pp doi: /j.amc Davis, M. [2004]. The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer, pp doi: / _8. Refutation to Davis Sundar, N. and Bringsjord, S. [2011]. The Myth of The Myth of Hypercomputation. In: Combined Pre-Proceedings of P&C 2011 and HyperNet Ed. by Stannett, M., pp Hypercomputation 43/57

44 Academic Community Communities Hypercomputation Research Network Computability in Europe (CiE) Hypercomputation 44/57

45 Academic Community Communities Hypercomputation Research Network Computability in Europe (CiE) Books and dedicated journal issues Syropoulos, A. [2008]. Hypercomputation. Computing Beyond the Church-Turing Barrier. Springer. Applied Mathematics and Computation. Vol. 178(1), Burgin, M. [2005]. Super-Recursive Algorithms. Springer. Theoretical Computer Science. Vol. 317(1-3), Mind and Machines. Vols. 12(4)/13(1), 2002/2003. Hypercomputation 45/57

46 Final Remarks Once upon on time, back in the golden age of the recursive function theory, computability was an absolute. [Sylvan and Copeland 2000, p. 189] Computing a way through the Turing barrier. The Reporter. The University of Leeds Newsletter. No February Hypercomputation 46/57

47 Final Remarks Once upon on time, back in the golden age of the recursive function theory, computability was an absolute. [Sylvan and Copeland 2000, p. 189] Via the great pioneers of electronic computing the 1930s analysis of computation led to the modern computing era. Who knows where a 21st-century overhaul of that classical analysis might lead. [Copeland, Dresner, Proudfoot and Shagrir 2016, p. 38] Computing a way through the Turing barrier. The Reporter. The University of Leeds Newsletter. No February Hypercomputation 47/57

48 Final Remarks Once upon on time, back in the golden age of the recursive function theory, computability was an absolute. [Sylvan and Copeland 2000, p. 189] Via the great pioneers of electronic computing the 1930s analysis of computation led to the modern computing era. Who knows where a 21st-century overhaul of that classical analysis might lead. [Copeland, Dresner, Proudfoot and Shagrir 2016, p. 38] Is there any limit to discrete computation, and more generally, to scientific knowledge? [Calude and Dinneen 1998, p. 1] Computing a way through the Turing barrier. The Reporter. The University of Leeds Newsletter. No February Hypercomputation 48/57

49 Final Remarks Once upon on time, back in the golden age of the recursive function theory, computability was an absolute. [Sylvan and Copeland 2000, p. 189] Via the great pioneers of electronic computing the 1930s analysis of computation led to the modern computing era. Who knows where a 21st-century overhaul of that classical analysis might lead. [Copeland, Dresner, Proudfoot and Shagrir 2016, p. 38] Is there any limit to discrete computation, and more generally, to scientific knowledge? [Calude and Dinneen 1998, p. 1] In breaking the Turing barrier, our knowledge of the world, and therefore our control of it, would be altered forever, Professor Cooper added. Computing a way through the Turing barrier. The Reporter. The University of Leeds Newsletter. No February Hypercomputation 49/57

50 References Burgin, M. (2005). Super-Recursive Algorithms. Springer (cit. on pp. 44, 45). Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier. Tech. rep. CDMTCS (cit. on pp ). Copeland, B. J. (1998). Super Turing-Machines. In: Complexity 4.1, pp doi: /(SICI) (199809/10)4:1<30::AID-CPLX9>3.0.CO;2-8 (cit. on pp. 11, 12). (2002a). Accelerating Turing Machines. In: Minds and Machines 12.2, pp doi: /A: (cit. on pp. 11, 12). (2002b). Hypercomputation. In: Minds and Machines 12.4, pp doi: /A: (cit. on pp. 5, 6). Copeland, B. J. and Proudfoot, D. (1999). Alan Turing s Forgotten Ideas in Computer Science. In: Scientific American 280.4, pp doi: /scientificamerican (cit. on p. 10). Hypercomputation 50/57

51 References Copeland, B. J. and Sylvan, R. (1999). Beyond the Universal Turing Machine. In: Australasian Journal of Philosophy 77.1, pp doi: / (cit. on pp. 32, 33). Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect the Foundations? In: Communications of the ACM 59.11, pp doi: / (cit. on pp ). Davis, M. (1958). Computability and Unsolvability. McGraw-Hill (cit. on p. 3). (2004). The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer, pp doi: / _8 (cit. on pp ). (2006). Why There is no Such Discipline as Hypercomputation. In: Applied Mathematics and Computation 178.1, pp doi: /j.amc (cit. on pp ). Hypercomputation 51/57

52 References Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. In: Proc. R. Soc. Lond. A 400, pp doi: /rspa (cit. on pp. 15, 16). (1989). Quantum Computational Networks. In: Proc. R. Soc. Lond. A 425, pp doi: /rspa (cit. on pp. 15, 16). Etesi, G. and Németi, I. (2002). Non-Turing Computations Via Malament-Hogart Space-Times. In: Int. J. Theor. Phys. 41.2, pp doi: /A: (cit. on pp ). Gandy, R. (1980). Church s Thesis and Principles for Mechanisms. In: The Kleene Symposium. Ed. by Barwise, J., Keisler, H. J. and Kunen, K. Vol Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, pp doi: /S X(08) (cit. on pp. 32, 33). Hypercomputation 52/57

53 References Gödel, K. [1946] (1990). Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics (1946). In: Kurt Gödel. Collected works. Ed. by Feferman, S. et al. Vol. II. Publications Oxford University Press, pp (cit. on p. 2). Hamkins, J. D. (2002). Infinite Time Turing machines. In: Minds and Machines 12.4, pp doi: /A: (cit. on pp. 23, 24). (2007). A Survey of Infinite time Turing machines. In: Machines, Computations, and Universality (MCU 2007). Ed. by Durand-Lose, J. and Margenstern, M. Vol Lecture Notes in Computer Science, pp doi: / _5 (cit. on pp. 23, 24). Hamkins, J. D. and Lewis, A. (2000). Infinite Time Turing machines. In: The Journal of Symbolic Logic 65.2, pp doi: / (cit. on pp ). Hypercomputation 53/57

54 References Hogarth, M. (1992). Does General Relativity Allow an Observer to View an Eternity in a Finite Time? In: Foundations on Physics Letters 5.2, pp doi: /BF (cit. on pp ). (1994). Non-Turing Computers and Non-Turing Computability. In: PSA 1994, pp (cit. on pp ). Kieu, T. D. (2003). Computing the Non-Computable. In: Contemporary Physics 44.1, pp doi: / (cit. on pp. 20, 21). Németi, I. and Dávid, G. (2006). Relativistic Computers and the Turing Barrier. In: Applied Mathematics and Computation 178.1, pp doi: /j.amc (cit. on pp ). Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. In: International Journal of Unconventional Computing 5, pp (cit. on pp ). Pitowsky, I. (1990). The Physical Church Thesis and Physical Computational Complexity. In: Iyyun 39, pp url: (cit. on pp ). Hypercomputation 54/57

55 References Schindler, R. (2003). P NP for Infinite Time Turing Machines. In: Monatshefte für Mathematik 139.4, pp doi: /s (cit. on pp ). Seyfferth, B. (2013). Three Models of Ordinal Computability. PhD thesis. University of Bonn. url: (visited on 14/05/2018) (cit. on pp. 30, 31). Sicard, A., Ospina, J. and Vélez, M. (2006). Quantum Hypercomputation Based on the Dynamical Algebra su(1, 1). In: J. Phys. A: Math. Gen , pp doi: / /39/40/018 (cit. on p. 22). Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the Turing Limit. Progress in Theorical Computer Science. Birkhäuser (cit. on p. 13). Hypercomputation 55/57

56 References Stannett, M. (2015). Towards Formal Verification of Computations and Hypercomputations in Relativistic Physics. In: Machines, Computations, and Universality (MCU 2015). Ed. by Durand-Lose, J. and Nagy, B. Vol Lecture Notes in Computer Science, pp doi: / _2 (cit. on pp. 37, 38). Stannett, M. and Németi, I. (2014). Using Isabelle/HOL to Verify First-Order Relativity Theory. In: Journal of Automated Reasoning 52 (4), pp doi: /s (cit. on pp. 37, 38). Sundar, N. and Bringsjord, S. (2011). The Myth of The Myth of Hypercomputation. In: Combined Pre-Proceedings of P&C 2011 and HyperNet Ed. by Stannett, M., pp (cit. on pp ). Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative. In: Sociative Logics and Their Applications: Essays by the late Richard Sylvan. Ed. by Priest, G. and Hyde, D. Ashgate Publishing Company, pp (cit. on pp. 4, 46 49). Hypercomputation 56/57

57 References Syropoulos, A. (2008). Hypercomputation. Computing Beyond the Church-Turing Barrier. Springer (cit. on pp. 44, 45). Turing, A. M. (1939). Systems of Logic Based on Ordinales. In: Proc. London Math. Soc , pp doi: /plms/s (cit. on pp. 8, 9). Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing. Springer-Telos (cit. on pp ). Hypercomputation 57/57