Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion A mixture of computability and ordinals, the infjnite time Turing machines Sabrina Ouazzani Paris-Est Créteil University Avril 2017 1 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Computability Describe what is a computation. Describe what runs a computation. 2 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Ordinals: counting through the infjnite infjnite… We can carry on counting! 3 / 44 We denote ω the set of all natural numbers. But ω is not the only
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Ordinals: counting through the infjnite infjnite… We can carry on counting! 4 / 44 We denote ω the set of all natural numbers. But ω is not the only
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion 1 Compute? 2 Infjnite time Turing machines 3 Some particularities of infjnite time 4 Conclusion 5 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Compute Sequence of instructions: fjnite; not ambiguous; allows to solve a problem. Defjnition (Algorithm) . 6 / 44 ⇝
Compute? if m is even then Open question (conjecture, 1952): for all m , does we always reach end end Infjnite time Turing machines else Variables : counter k sequence (Collatz conjecture) Example: compute the n fjrst terms of the hailtstone Conclusion Some particularities of infjnite time 7 / 44 for k from 0 to n do m ← m /2 ; m ← m × 3 + 1 ; k ← k + 1 ; With n = 7 and m = 10 we obtain the sequence 10 , 5 , 16 , 8 , 4 , 2 , 1 . a 1 ?
Compute? Turing machine. H Infjnite time Turing machines start 8 / 44 1936 theory Compute Conclusion Some particularities of infjnite time q 0 0 | 1 ◀ 0 | 1 ▶ 1 | 1 ◀ q 1 1 | 1 ▶ q 0 0 . . . . . . 0 0 1 1 1 0 0
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Timeline theory architecture 1936 1945 9 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Timeline theory architecture computer 1936 1945 1949 10 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Timeline theory architecture computer 1936 1945 1949 2000 11 / 44 theory ∞
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion 2000 Solve the Collatz conjecture. For all the natural numbers, apply the algorithm. 12 / 44 theory ∞
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Timeline theory architecture computer 1936 1945 1949 2000 13 / 44 theory ∞
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Timeline theory architecture ordinateur ? 1936 1945 1949 2000 ? 14 / 44 theory ∞
Compute? ? ? ? 2000 1945 1949 1936 15 / 44 Infjnite time Turing machines computer architecture theory Timeline Conclusion Some particularities of infjnite time theory ∞ computer ∞ ?
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Motivations: build links between Computer Science and Logic Ordinals as time for computation. Peculiar ordinal properties. Proof of mathematical properties from an algorithmic point of view. 16 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion 1 Compute? 2 Infjnite time Turing machines 3 Some particularities of infjnite time 4 Conclusion 17 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Ordinals: counting through the infjnite infjnite… We can carry on counting! 18 / 44 We denote ω the set of all natural numbers. But ω is not the only
Compute? … limit ordinal. let A be a set of ordinal … Infjnite time Turing machines 19 / 44 Defjnition (Ordinal) . Transitive well-ordered set for the membership relation. Ordinals Conclusion Some particularities of infjnite time 0 := ∅ If α is an ordinal, then α ∪ { α } , 1 := { 0 } = {∅} denoted α + 1 is called successor of α and is an ordinal; ω := { 0 , 1 , 2 , 3 , · · · } ω + 1 := { 0 , 1 , 2 , 3 , · · · , ω } numbers, then α = ∪ β ∈ A β is a ω. 2 := { 0 , 1 , 2 , · · · , ω, ω + 1 , ω + 2 . . . }
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Encoding countable ordinals the i -th bit of r is 1 if and only if x < y . Codage 1 (Encoding countable ordinals by reals) . 20 / 44 Countable ordinal = well order on N . Let < be an order on the natural numbers. The real r is a code for the order-type of < if, for i = ⟨ x , y ⟩ , Example: ω. 2 = ω + ω ⇝ even integers lower than odd integers. 0 = ⟨ 0 , 0 ⟩ 1 = ⟨ 0 , 1 ⟩ · · · r = 0 0 1 1 0 2 0 3 0 4 1 5 0 6 1 7 1 8 1 9 1 10 · · ·
Compute? additional special limit state Confjguration indexed by ordinals Infjnite time Turing machines lim computation steps are a single head 3 right-infjnite tapes Structure of infjnite time Turing machines (ITTM) Conclusion Some particularities of infjnite time 21 / 44 binary alphabet { 0 , 1 } q 0 . . . input 0 1 0 0 1 0 0 . . . work 0 0 0 0 0 0 0 . . . output 0 0 0 0 0 0 0
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Operating an ITTM … 22 / 44 q 1 . . . t = 420 0 1 0 0 1 0 0 Confjguration at α + 1 . ⇝ Confjguration at α . q 3 . . . t = 007 0 0 1 1 1 0 0
Compute? each cell: lim sup … lim sup Infjnite time Turing machines before. of cell values lim state: lim ; Operating an ITTM Some particularities of infjnite time Conclusion position; 23 / 44 Confjguration limit: head: initial . . . t = ω 0 1 1 0 1 0 0 ↑ ↑ ↑ ↑ ↑ ↑ ↑ q 1 . . . t = 420 0 1 0 0 1 0 0 q 3 . . . t = 007 0 0 1 1 1 0 0
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Halting Machines halt when they reach the halting state. Either an ITTM halts in a countable numer of steps, either it begins looping in a countable number of steps . Theorem 1 (Hamkins, Lewis [HL00]) . 24 / 44 We consider the strong stabilisation of cells at 0 . We focus on the halting problem on 0 .
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion 1 Compute? 2 Infjnite time Turing machines 3 Some particularities of infjnite time 4 Conclusion 25 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Clockable and writable ordinals Two natural notions: Defjnition (Clockable ordinal) . Defjnition (Writable ordinal) . 26 / 44 α clockable: there exists an ITTM that halts on input 000 . . . in exactly α steps of computation. α writable: there exists an ITTM that writes a code for α on input 000 . . . and halts.
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Supremum The supremum of the clockable ordinals is equal to the supre- Theorem 2 (Welch [Wel09]) . 27 / 44 mum of the writable ordinals. It is called λ . λ is a rather large countable ordinal…
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Let’s count! Like an hourglass, execute operations while clocking the desired ordinal. Speed-up lemma (Hamkins, Lewis [HL00]) . It is about counting through the encoding of an ordinal. 28 / 44 Count with a clockable ordinal ⇝ Clock. If p halts on 0 in α + n steps, then there exists p ′ which halts on 0 in α steps (and computes the same). ⇝ limit ordinals Count with a writable ordinal ⇝ Empty an order.
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion … What about the particularities of these ordinals? 29 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Gap There exist writable ordinals that are not clockable such that: they form intervalles; these intervalles have limit sizes. Intervalles of not clockable ordinals. Defjnition (Gap) . 30 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion size beg 31 / 44 λ · · · · · · · · · 0
Compute? Infjnite time Turing machines … … … gap checking … Conclusion Proof of gap existence Some particularities of infjnite time 32 / 44 α β β + ω 0 p Simulation of all programs on input 0 . In blue: halting programs. In red: limit step, begins a gap?
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion Proof of gap existence But …does the algorithm halt? Halting of the algorithm, proof by contradiction: Contradiction . 33 / 44 Above λ , by defjnition, there are no clockable ordinals. If no gaps before λ , thus beginnning of gap detected at λ .
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion What can we say about gaps? 34 / 44
Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion regular structure 35 / 44 λ · · · β 0 β 0 } · · · α · · · ω 0
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