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 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 (analogue procedures and newer physics) reveal. For one thing, comput- ability is relative not simply to physics, but more generally to systems of
Hypercomputers Defjnition (Hypercomputer) A hypercomputer is any machine (theoretical or real) that compute functions or numbers, or more generally solve problems or carry out tasks, that cannot 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 be computed or solved by a Turing machine. ∗
Hypercomputers Defjnition (Hypercomputer) A hypercomputer is any machine (theoretical or real) that compute functions or numbers, or more generally solve problems or carry out tasks, that cannot 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 be computed or solved by a Turing machine. ∗
Possible Sources of Hypercomputation Mathematics Computability Logic Hypercomputation Model (HM) Biology Physics ? Automata and Formal Languages - CM0081. Hypercomputation 7/44
If the oracle is a non-recursive set then OTM ≡ HM. First Hypercomputation Model: Oracle Turing Machines Defjnition (Oracle Turing machines) A oracle TM (OTM) is a TM equipped with an oracle that is capable of answering questions about the membership of a specifjc set of natural numbers. ∗ Hypercomputability features If the oracle is a recursive set then OTM ≡ TM. ∗ 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 of 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
2 𝑗 = 2, an ATM could complete an infjnity of steps in two time units. Hypercomputation Model: Accelerated Turing Machines Defjnition (Accelerated Turing Machines) An accelerated TM (ATM) is a TM that performs its fjrst step in one unit of 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 ∗ 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 of 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 ∗ Copeland, B. J. (1998). Super Turing-Machines. Automata and Formal Languages - CM0081. Hypercomputation 12/44 2 𝑗 = 2, an ATM could complete an infjnity of steps in two time units.
𝑏 𝑗𝑘 ∈ {ℕ, ℚ, ℝ} ⇒ ARNN ≡ { DFA , TM , HM } . Hypercomputation Model: Analog Recurrent Neural 𝑏 14 Automata and Formal Languages - CM0081. Hypercomputation Turing Limit. ∗ Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the Hypercomputability features 𝐷) ⃗⃗⃗⃗⃗⃗⃗ 𝑏 34 𝑏 23 Network (ARNN) ∗ 𝑏 22 𝑏 24 𝑑 3 𝑐 23 𝑣 1 𝑣 2 𝑦 1 𝑦 2 𝑦 3 𝑦 4 13/44 𝑐 11 𝑐 12 𝑐 13 𝑐 22 𝑌(𝑢 + 1) = 𝜏( ⃗⃗⃗⃗⃗⃗ 𝐵 ⋅ ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐶 ⋅ ⃗⃗⃗⃗⃗⃗ 𝑉(𝑢) + ⃗⃗⃗⃗⃗⃗
Hypercomputation Model: Analog Recurrent Neural 𝑏 14 Automata and Formal Languages - CM0081. Hypercomputation Turing Limit. ∗ Siegelmann, H. T. (1999). Neural Networks and Analog Computation. Beyond the Hypercomputability features 𝐷) ⃗⃗⃗⃗⃗⃗⃗ 𝑏 34 𝑏 23 Network (ARNN) ∗ 𝑏 22 𝑏 24 𝑑 3 𝑐 23 𝑣 1 𝑣 2 𝑦 1 𝑦 2 𝑦 3 14/44 𝑦 4 𝑐 11 𝑐 12 𝑐 13 𝑐 22 𝑌(𝑢 + 1) = 𝜏( ⃗⃗⃗⃗⃗⃗ 𝐵 ⋅ ⃗⃗⃗⃗⃗⃗⃗ 𝑌(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝐶 ⋅ ⃗⃗⃗⃗⃗⃗ 𝑉(𝑢) + ⃗⃗⃗⃗⃗⃗ 𝑏 𝑗𝑘 ∈ {ℕ, ℚ, ℝ} ⇒ ARNN ≡ { DFA , TM , HM } .
Standard Quantum Computation (SQC) ≡ Automata and Formal Languages - CM0081. Hypercomputation † Deutsch, D. (1989). Quantum Computational Networks. Universal Quantum Computer. ∗ Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the ≡ ≡ ≡ Models QTMs Reversible TMs Probabilistic TMs TMs Relation between the models 15/44 Quantum Turing machines (QTM) ∗ and quantum circuits. †
Standard Quantum Computation (SQC) ≡ Automata and Formal Languages - CM0081. Hypercomputation † Deutsch, D. (1989). Quantum Computational Networks. Universal Quantum Computer. ∗ Deutsch, D. (1985). Quantum Theory, the Church-Turing Principle and the ≡ ≡ ≡ Models QTMs Reversible TMs Probabilistic TMs TMs Relation between the models 16/44 Quantum Turing machines (QTM) ∗ and quantum circuits. †
superposition and indeterminism to simulate a perfectly fair coin “Weak” Hypercomputation Based on SQC Automata and Formal Languages - CM0081. Hypercomputation † Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. p. 160. ∗ Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing, non-computable TM problem. † 2. The problem: It is not clear how to use this property to solve a toss. ” ∗ equal probability. Hence you can exploit quantum mechanical Generation of truly random numbers the superposition to collapse into either | 0⟩ or the | 1⟩ state with 1. We observe the superposition state: “ The act of observation causes (| 0⟩ + | 1⟩) → measure 2 √ 1 17/44 𝑉 𝐼 | 0⟩ =
“Weak” Hypercomputation Based on SQC toss. ” ∗ Automata and Formal Languages - CM0081. Hypercomputation † Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. p. 160. ∗ Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing, non-computable TM problem. † 2. The problem: It is not clear how to use this property to solve a equal probability. Hence you can exploit quantum mechanical Generation of truly random numbers the superposition to collapse into either | 0⟩ or the | 1⟩ state with 1. We observe the superposition state: “ The act of observation causes (| 0⟩ + | 1⟩) → measure 2 √ 1 18/44 𝑉 𝐼 | 0⟩ = superposition and indeterminism to simulate a perfectly fair coin
“Weak” Hypercomputation Based on SQC toss. ” ∗ Automata and Formal Languages - CM0081. Hypercomputation † Ord, T. and Kieu, T. D. (2009). Using Biased Coins as Oracles. p. 160. ∗ Williams, C. P. and Clearwater, S. H. (1997). Explorations in Quantum Computing, non-computable TM problem. † 2. The problem: It is not clear how to use this property to solve a equal probability. Hence you can exploit quantum mechanical Generation of truly random numbers the superposition to collapse into either | 0⟩ or the | 1⟩ state with 1. We observe the superposition state: “ The act of observation causes (| 0⟩ + | 1⟩) → measure 2 √ 1 19/44 𝑉 𝐼 | 0⟩ = superposition and indeterminism to simulate a perfectly fair coin
Recommend
More recommend
