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Automata and Formal Languages - CM0081 Hypercomputation

Andrés Sicard-Ramírez

Universidad EAFIT

Semester 2017-2

Motivation

Absolute computability “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 fjrst time succeeded in giving an absolute defjnition of an interesting epistemological notion, i.e., one not depending on the formalism chosen.”∗

∗Gödel, K. (1990). Remarks Before the Princeton Bicentennial Conference on

Problems in Mathematics (1946), p. 150.

Automata and Formal Languages - CM0081. Hypercomputation 2/44

Motivation

Absolute computability “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.”∗

∗Davis, M. (1958). Computability and Unsolvability, p. 11. Automata and Formal Languages - CM0081. Hypercomputation 3/44

Motivation

Relative computability “Troubles with absolutism are deeper and more extensive than these cracks (analogue procedures and newer physics) reveal. For one thing, comput- ability is relative not simply to physics, but more generally to systems of frameworks, which include or contain underlying logics.”∗

∗Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative, p. 190. Automata and Formal Languages - CM0081. Hypercomputation 4/44

Hypercomputers

Defjnition (Hypercomputer) A hypercomputer is any machine (theoretical or real) that compute functions

• r numbers, or more generally solve problems or carry out tasks, that cannot

be computed or solved by a Turing machine.∗ Super-TM and Non-TM TM Super-TM 𝑀 ⊆ Σ∗ non-TM TM 𝑀 ⊆ Σ∗

∗Copeland, B. J. (2002). Hypercomputation. Automata and Formal Languages - CM0081. Hypercomputation 5/44

Hypercomputers

Defjnition (Hypercomputer) A hypercomputer is any machine (theoretical or real) that compute functions

• r numbers, or more generally solve problems or carry out tasks, that cannot

be computed or solved by a Turing machine.∗ Super-TM and Non-TM TM Super-TM 𝑀 ⊆ Σ∗ non-TM TM 𝑀 ⊆ Σ∗

∗Copeland, B. J. (2002). Hypercomputation. Automata and Formal Languages - CM0081. Hypercomputation 6/44

Possible Sources of Hypercomputation

Mathematics Computability Logic Hypercomputation Model (HM) Biology Physics ?

Automata and Formal Languages - CM0081. Hypercomputation 7/44

First Hypercomputation Model: Oracle Turing Machines

Defjnition (Oracle Turing machines) A oracle TM (OTM) is a TM equipped with an oracle that is capable

• f answering questions about the membership of a specifjc set of natural

numbers.∗ Hypercomputability features If the oracle is a recursive set then OTM ≡ TM. If the oracle is a non-recursive set then OTM ≡ HM.

∗Turing, A. M. (1939). Systems of Logic Based on Ordinales. Automata and Formal Languages - CM0081. Hypercomputation 8/44

First Hypercomputation Model: Oracle Turing Machines

Defjnition (Oracle Turing machines) A oracle TM (OTM) is a TM equipped with an oracle that is capable

• f answering questions about the membership of a specifjc set of natural

numbers.∗ Hypercomputability features If the oracle is a recursive set then OTM ≡ TM. If the oracle is a non-recursive set then OTM ≡ HM.

∗Turing, A. M. (1939). Systems of Logic Based on Ordinales. Automata and Formal Languages - CM0081. Hypercomputation 9/44

About the ‘Hypercomputation’ Term

Right∗ Wrong Hypercomputation Super-Turing computation Computing beyond Turing’s limit Breaking the Turing barrier …

∗Copeland, B. J. and Proudfoot, D. (1999). Alan Turing’s Forgotten Ideas in

Computer Science.

Automata and Formal Languages - CM0081. Hypercomputation 10/44

Hypercomputation Model: Accelerated Turing Machines

Defjnition (Accelerated Turing Machines) An accelerated TM (ATM) is a TM that performs its fjrst step in one unit

• f time and each subsequent step in half the time of the step before.∗

Hypercomputability features Since 1 + 1/2 + 1/4 + 1/8 + … =

∞

∑

𝑗=0

1 2𝑗 = 2, an ATM could complete an infjnity of steps in two time units.

∗Copeland, B. J. (1998). Super Turing-Machines. Automata and Formal Languages - CM0081. Hypercomputation 11/44

Hypercomputation Model: Accelerated Turing Machines

Defjnition (Accelerated Turing Machines) An accelerated TM (ATM) is a TM that performs its fjrst step in one unit

• f time and each subsequent step in half the time of the step before.∗

Hypercomputability features Since 1 + 1/2 + 1/4 + 1/8 + … =

∞

∑

𝑗=0

1 2𝑗 = 2, an ATM could complete an infjnity of steps in two time units.

∗Copeland, B. J. (1998). Super Turing-Machines. Automata and Formal Languages - CM0081. Hypercomputation 12/44

Hypercomputation Model: Analog Recurrent Neural Network (ARNN)∗

𝑣1 𝑣2 𝑦1 𝑦2 𝑦3 𝑦4 𝑐11 𝑐12 𝑐13 𝑐22 𝑐23 𝑑3 𝑏14 𝑏22 𝑏24 𝑏23 𝑏34 ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢 + 1) = 𝜏( ⃗⃗⃗⃗⃗⃗ 𝐵 ⋅ ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐶 ⋅ ⃗⃗⃗⃗⃗⃗ 𝑉(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐷) Hypercomputability features 𝑏𝑗𝑘 ∈ {ℕ, ℚ, ℝ} ⇒ ARNN ≡ {DFA, TM, HM}.

∗Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the

Turing Limit.

Automata and Formal Languages - CM0081. Hypercomputation 13/44

Hypercomputation Model: Analog Recurrent Neural Network (ARNN)∗

𝑣1 𝑣2 𝑦1 𝑦2 𝑦3 𝑦4 𝑐11 𝑐12 𝑐13 𝑐22 𝑐23 𝑑3 𝑏14 𝑏22 𝑏24 𝑏23 𝑏34 ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢 + 1) = 𝜏( ⃗⃗⃗⃗⃗⃗ 𝐵 ⋅ ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐶 ⋅ ⃗⃗⃗⃗⃗⃗ 𝑉(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐷) Hypercomputability features 𝑏𝑗𝑘 ∈ {ℕ, ℚ, ℝ} ⇒ ARNN ≡ {DFA, TM, HM}.

∗Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the

Turing Limit.

Automata and Formal Languages - CM0081. Hypercomputation 14/44

Standard Quantum Computation (SQC)

Models Quantum Turing machines (QTM)∗ and quantum circuits.† Relation between the models TMs Probabilistic TMs Reversible TMs QTMs ≡ ≡ ≡ ≡

∗Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the

Universal Quantum Computer.

†Deutsch, D. (1989). Quantum Computational Networks. Automata and Formal Languages - CM0081. Hypercomputation 15/44

Standard Quantum Computation (SQC)

Models Quantum Turing machines (QTM)∗ and quantum circuits.† Relation between the models TMs Probabilistic TMs Reversible TMs QTMs ≡ ≡ ≡ ≡

∗Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the

Universal Quantum Computer.

†Deutsch, D. (1989). Quantum Computational Networks. Automata and Formal Languages - CM0081. Hypercomputation 16/44

“Weak” Hypercomputation Based on SQC

Generation of truly random numbers 𝑉𝐼 | 0⟩ = 1 √ 2 (| 0⟩ + | 1⟩) → 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.”∗

• 2. The problem: It is not clear how to use this property to solve a

non-computable TM problem.†

∗Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing,

• p. 160.

†Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. Automata and Formal Languages - CM0081. Hypercomputation 17/44

“Weak” Hypercomputation Based on SQC

Generation of truly random numbers 𝑉𝐼 | 0⟩ = 1 √ 2 (| 0⟩ + | 1⟩) → 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.”∗

• 2. The problem: It is not clear how to use this property to solve a

non-computable TM problem.†

∗Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing,

• p. 160.

†Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. Automata and Formal Languages - CM0081. Hypercomputation 18/44

“Weak” Hypercomputation Based on SQC

Generation of truly random numbers 𝑉𝐼 | 0⟩ = 1 √ 2 (| 0⟩ + | 1⟩) → 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.”∗

• 2. The problem: It is not clear how to use this property to solve a

non-computable TM problem.†

∗Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing,

• p. 160.

†Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. Automata and Formal Languages - CM0081. Hypercomputation 19/44

Others Quantum Computation Models

Common misunderstanding quantum computation ≡ SQC ≡ adiabatic quantum computation (AQC) The real situation∗ Kieu’s hypercomputational quantum algorithm (KHQA). fjnite AQC ≡ SQC infjnite AQC ≡ Kieu’s hypercomputational quantum algorithm

∗Kieu, T. D. (2003). Computing the Non-Computable. Automata and Formal Languages - CM0081. Hypercomputation 20/44

Others Quantum Computation Models

Common misunderstanding quantum computation ≡ SQC ≡ adiabatic quantum computation (AQC) The real situation∗ Kieu’s hypercomputational quantum algorithm (KHQA). fjnite AQC ≡ SQC infjnite AQC ≡ Kieu’s hypercomputational quantum algorithm

∗Kieu, T. D. (2003). Computing the Non-Computable. Automata and Formal Languages - CM0081. Hypercomputation 21/44

Hypercomputational Quantum Algorithm à la Kieu∗

Classically non- computable 𝑄 problem (Hilbert’s 10th problem) Hypercomputational quantum algorithm Physical referent (Infjnite square well) Simulation Dynamical algebra (Lie alg. 𝔱𝔳(1, 1)) Partial solution to 𝑄

∗Sicard, A., Ospina, J. and Vélez, M. (2006). Quantum Hypercomputation Based on

the Dynamical Algebra 𝔱𝔳(1, 1).

Automata and Formal Languages - CM0081. Hypercomputation 22/44

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.”∗ Thesis M “What can be calculated by a machine is Turing machine computable.”†

∗Copeland, B. J. and Sylvan, R. (1999). Beyond the Universal Turing Machine. †Gandy, R. (1980). Church’s Thesis and Principles for Mechanisms. Automata and Formal Languages - CM0081. Hypercomputation 23/44

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.”∗ Thesis M “What can be calculated by a machine is Turing machine computable.”†

∗Copeland, B. J. and Sylvan, R. (1999). Beyond the Universal Turing Machine. †Gandy, R. (1980). Church’s Thesis and Principles for Mechanisms. Automata and Formal Languages - CM0081. Hypercomputation 24/44

Physical Hypercomputation?

Open problem The refutation of a general/physical version of Gandy’s Thesis M. Based on quantum physics (infjnite adiabatic quantum computation)? Based on relativistic physics (cosmological phenomena)?∗ † ‡

∗Hogarth, M. (1992). Does General Relativity Allow an Observardor to View an

Eternity in a Finite Time?

†Etesi, G. and Németi, I. (2002). Non-Turing Computations Via Malament-Hogart

Space-Times.

‡Németi, I. and Dávid, G. (2006). Relativistic Computers and the Turing Barrier. Automata and Formal Languages - CM0081. Hypercomputation 25/44

Physical Hypercomputation?

Open problem The refutation of a general/physical version of Gandy’s Thesis M. Based on quantum physics (infjnite adiabatic quantum computation)? Based on relativistic physics (cosmological phenomena)?∗ † ‡

∗Hogarth, M. (1992). Does General Relativity Allow an Observardor to View an

Eternity in a Finite Time?

†Etesi, G. and Németi, I. (2002). Non-Turing Computations Via Malament-Hogart

Space-Times.

‡Németi, I. and Dávid, G. (2006). Relativistic Computers and the Turing Barrier. Automata and Formal Languages - CM0081. Hypercomputation 26/44

Physical Hypercomputation?

Open problem The refutation of a general/physical version of Gandy’s Thesis M. Based on quantum physics (infjnite adiabatic quantum computation)? Based on relativistic physics (cosmological phenomena)?∗ † ‡

∗Hogarth, M. (1992). Does General Relativity Allow an Observardor to View an

Eternity in a Finite Time?

†Etesi, G. and Németi, I. (2002). Non-Turing Computations Via Malament-Hogart

Space-Times.

‡Németi, I. and Dávid, G. (2006). Relativistic Computers and the Turing Barrier. Automata and Formal Languages - CM0081. Hypercomputation 27/44

An Interesting Project: Formal Verifjcation of Hypercomputation in Relativistic Physics∗ †

Goals Implement fjrst-order axiomatisations of theories of the relativity using the proof assistant Isabelle; Add a model of computation carried out by machines travelling along specifjc spacetime trajectories;

∗Stannett, M. (2015). Towards Formal Verifjcation of Computations and

Hypercomputations in Relativistic Physics.

†Stannett, M. and Németi, I. (2014). Using Isabelle/HOL to Verify First-Order

Relativity Theory.

Automata and Formal Languages - CM0081. Hypercomputation 28/44

An Interesting Project: Formal Verifjcation of Hypercomputation in Relativistic Physics∗ †

Goals Implement fjrst-order axiomatisations of theories of the relativity using the proof assistant Isabelle; Add a model of computation carried out by machines travelling along specifjc spacetime trajectories;

∗Stannett, M. (2015). Towards Formal Verifjcation of Computations and

Hypercomputations in Relativistic Physics.

†Stannett, M. and Németi, I. (2014). Using Isabelle/HOL to Verify First-Order

Relativity Theory.

Automata and Formal Languages - CM0081. Hypercomputation 29/44

An Interesting Project: Formal Verifjcation 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 𝑄 (for example, the Halting Problem) and machine-verify the following claims: in simpler relativistic settings, 𝑄 remains uncomputable; in some spacetimes, 𝑄 can be solved.

Automata and Formal Languages - CM0081. Hypercomputation 30/44

An Interesting Project: Formal Verifjcation 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 𝑄 (for example, the Halting Problem) and machine-verify the following claims: in simpler relativistic settings, 𝑄 remains uncomputable; in some spacetimes, 𝑄 can be solved.

Automata and Formal Languages - CM0081. Hypercomputation 31/44

Is Hypercomputation a Myth?

Davis’ refutations Davis, M. [2006]. Why There is no Such Discipline as

• Hypercomputation. Applied Mathematics and Computation 178.1,
• pp. 4–7.

Davis, M. [2004]. The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer,

• pp. 195–211.

Refutation to Davis Sundar, N. and Bringsjord, S. [2011]. The Myth of ‘The Myth of Hyper- computation’. In: Combined Pre-Proceedings of P&C 2011 and HyperNet

• 2011. Ed. by Stannett, M., pp. 185–196.

Automata and Formal Languages - CM0081. Hypercomputation 32/44

Is Hypercomputation a Myth?

Davis’ refutations Davis, M. [2006]. Why There is no Such Discipline as

• Hypercomputation. Applied Mathematics and Computation 178.1,
• pp. 4–7.

Davis, M. [2004]. The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer,

• pp. 195–211.

Refutation to Davis Sundar, N. and Bringsjord, S. [2011]. The Myth of ‘The Myth of Hyper- computation’. In: Combined Pre-Proceedings of P&C 2011 and HyperNet

• 2011. Ed. by Stannett, M., pp. 185–196.

Automata and Formal Languages - CM0081. Hypercomputation 33/44

Is Hypercomputation a Myth?

Davis’ refutations Davis, M. [2006]. Why There is no Such Discipline as

• Hypercomputation. Applied Mathematics and Computation 178.1,
• pp. 4–7.

Davis, M. [2004]. The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer,

• pp. 195–211.

Refutation to Davis Sundar, N. and Bringsjord, S. [2011]. The Myth of ‘The Myth of Hyper- computation’. In: Combined Pre-Proceedings of P&C 2011 and HyperNet

• 2011. Ed. by Stannett, M., pp. 185–196.

Automata and Formal Languages - CM0081. Hypercomputation 34/44

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), 2006. Burgin, M. [2005]. Super-Recursive Algorithms. Springer. Theoretical Computer Science. Vol. 317(1-3), 2004. Mind and Machines. Vols. 12(4)/13(1), 2002/2003.

Automata and Formal Languages - CM0081. Hypercomputation 35/44

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), 2006. Burgin, M. [2005]. Super-Recursive Algorithms. Springer. Theoretical Computer Science. Vol. 317(1-3), 2004. Mind and Machines. Vols. 12(4)/13(1), 2002/2003.

Automata and Formal Languages - CM0081. Hypercomputation 36/44

Final Remarks

“Once upon on time, back in the golden age of the recursive function theory, computability was an absolute.”∗ “Via the great pioneers of electronic computing…the 1930s analysis of com- putation led to the modern computing era. Who knows where a 21st-century

• verhaul of that classical analysis might lead.”†

“Is there any limit to discrete computation, and more generally, to scientifjc knowledge?”‡ ““In breaking the Turing barrier, our knowledge of the world, and therefore

• ur control of it, would be altered forever,” Professor Cooper added.”§

∗Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative, p. 189. †Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect

the Foundations?, p. 38.

‡Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier, p. 1. §Computing a way through the Turing barrier. The Reporter. The University of Leeds

• Newsletter. No. 505. 21 February 2005.

Automata and Formal Languages - CM0081. Hypercomputation 37/44

Final Remarks

“Once upon on time, back in the golden age of the recursive function theory, computability was an absolute.”∗ “Via the great pioneers of electronic computing…the 1930s analysis of com- putation led to the modern computing era. Who knows where a 21st-century

• verhaul of that classical analysis might lead.”†

“Is there any limit to discrete computation, and more generally, to scientifjc knowledge?”‡ ““In breaking the Turing barrier, our knowledge of the world, and therefore

• ur control of it, would be altered forever,” Professor Cooper added.”§

∗Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative, p. 189. †Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect

the Foundations?, p. 38.

‡Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier, p. 1. §Computing a way through the Turing barrier. The Reporter. The University of Leeds

• Newsletter. No. 505. 21 February 2005.

Automata and Formal Languages - CM0081. Hypercomputation 38/44

Final Remarks

“Once upon on time, back in the golden age of the recursive function theory, computability was an absolute.”∗ “Via the great pioneers of electronic computing…the 1930s analysis of com- putation led to the modern computing era. Who knows where a 21st-century

• verhaul of that classical analysis might lead.”†

“Is there any limit to discrete computation, and more generally, to scientifjc knowledge?”‡ ““In breaking the Turing barrier, our knowledge of the world, and therefore

• ur control of it, would be altered forever,” Professor Cooper added.”§

∗Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative, p. 189. †Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect

the Foundations?, p. 38.

‡Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier, p. 1. §Computing a way through the Turing barrier. The Reporter. The University of Leeds

• Newsletter. No. 505. 21 February 2005.

Automata and Formal Languages - CM0081. Hypercomputation 39/44

Final Remarks

“Once upon on time, back in the golden age of the recursive function theory, computability was an absolute.”∗ “Via the great pioneers of electronic computing…the 1930s analysis of com- putation led to the modern computing era. Who knows where a 21st-century

• verhaul of that classical analysis might lead.”†

“Is there any limit to discrete computation, and more generally, to scientifjc knowledge?”‡ ““In breaking the Turing barrier, our knowledge of the world, and therefore

• ur control of it, would be altered forever,” Professor Cooper added.”§

∗Sylvan, R. and Copeland, J. (2000). Computability is Logic-Relative, p. 189. †Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect

the Foundations?, p. 38.

‡Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier, p. 1. §Computing a way through the Turing barrier. The Reporter. The University of Leeds

• Newsletter. No. 505. 21 February 2005.

Automata and Formal Languages - CM0081. Hypercomputation 40/44

References

Burgin, M. (2005). Super-Recursive Algorithms. Springer. Calude, C. S. and Dinneen, M. J. (1998). Breaking the Turing barrier. Tech. rep. CDMTCS. Copeland, B. J. (1998). Super Turing-Machines. Complexity 4.1, pp. 30–32. – (2002). Hypercomputation. Minds and Machines 12.4, pp. 461–502. Copeland, B. J. and Proudfoot, D. (1999). Alan Turing’s Forgotten Ideas in Computer Science. Scientifjc American 280.4, pp. 76–81. Copeland, B. J. and Sylvan, R. (1999). Beyond the Universal Turing Machine. Australasian Journal of Philosophy 77, pp. 44–66. Copeland, J., Dresner, E., Proudfoot, D. and Shagrir, O. (2016). Time to Reinspect the Foundations? Communications of the ACM 59.11, pp. 34–38. Davis, M. (1958). Computability and Unsolvability. McGraw-Hill. – (2004). The Myth of Hypercomputation. In: Alan Turing: Life and Legaly of a Great Thinker. Ed. by Teuscher, C. Springer, pp. 195–211. – (2006). Why There is no Such Discipline as Hypercomputation. Applied Mathematics and Computation 178.1, pp. 4–7.

Automata and Formal Languages - CM0081. Hypercomputation 41/44

References

Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proc. R. Soc. Lond. A 400, pp. 97–117. – (1989). Quantum Computational Networks. Proc. R. Soc. Lond. A 425,

• pp. 73–90.

Etesi, G. and Németi, I. (2002). Non-Turing Computations Via Malament-Hogart Space-Times. Int. J. Theor. Phys. 41.2, pp. 341–370. 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. 101. Studies

in Logic and the Foundations of Mathematics. North-Holland Publishing Company, pp. 123–148. Gödel, K. (1990). Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics (1946). In: Kurt Gödel. Collected works. Ed. by Feferman, S. et al. Vol. I. Publications 1938–1974. Oxford University Press,

• pp. 150–153.

Hogarth, M. (1992). Does General Relativity Allow an Observardor to View an Eternity in a Finite Time? Foundations on Physics Letters 5.2, pp. 173–181. Kieu, T. D. (2003). Computing the Non-Computable. Contemporary Physics 44.1, pp. 51–71.

Automata and Formal Languages - CM0081. Hypercomputation 42/44

References

Németi, I. and Dávid, G. (2006). Relativistic Computers and the Turing Barrier. Applied Mathematics and Computation 178.1, pp. 118–142. Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. International Journal of Unconventional Computing 5, pp. 253–265. Sicard, A., Ospina, J. and Vélez, M. (2006). Quantum Hypercomputation Based

• n the Dynamical Algebra 𝔱𝔳(1, 1). J. Phys. A: Math. Gen. 39.40,
• pp. 12539–12558.

Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the Turing Limit. Progress in Theorical Computer Science. Birkhäuser. Stannett, M. (2015). Towards Formal Verifjcation of Computations and Hypercomputations in Relativistic Physics. In: Machines, Computations, and Universality (MCU 2015). Ed. by Durand-Lose, J. and Nagy, B. Vol. 9288. Lecture Notes in Computer Science, pp. 17–27. Stannett, M. and Németi, I. (2014). Using Isabelle/HOL to Verify First-Order Relativity Theory. Journal of Automated Reasoning 52 (4), pp. 361–378. Sundar, N. and Bringsjord, S. (2011). The Myth of ‘The Myth of Hypercomputation’. In: Combined Pre-Proceedings of P&C 2011 and HyperNet

• 2011. Ed. by Stannett, M., pp. 185–196.

Automata and Formal Languages - CM0081. Hypercomputation 43/44

References

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. 189–199. Syropoulos, A. (2008). Hypercomputation. Computing Beyond the Church-Turing

• Barrier. Springer.

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