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A mixture of computability and ordinals, the infjnite time Turing - - PowerPoint PPT Presentation

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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 Crteil University Avril 2017 1 / 44 Compute?

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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

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Computability

Describe what is a computation. Describe what runs a computation.

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Ordinals: counting through the infjnite

We denote ω the set of all natural numbers. But ω is not the only infjnite… We can carry on counting!

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Ordinals: counting through the infjnite

We denote ω the set of all natural numbers. But ω is not the only infjnite… We can carry on counting!

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Some particularities of infjnite time

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Conclusion

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Compute

Sequence of instructions: fjnite; not ambiguous; allows to solve a problem. Defjnition (Algorithm). ⇝

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Example: compute the n fjrst terms of the hailtstone sequence (Collatz conjecture)

Variables: counter k for k from 0 to n do if m is even then m ← m/2 ; else m ← m × 3 + 1 ; end k ← k + 1 ; end With n = 7 and m = 10 we obtain the sequence 10, 5, 16, 8, 4, 2, 1. Open question (conjecture, 1952): for all m, does we always reach a 1?

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Compute

theory 1936 Turing machine. q0 start q1 H 0 | 1 ▶ 1 | 1 ◀ 0 | 1 ◀ 1 | 1 ▶ 0 . . . 0 0 1 1 1 0 0 . . . q0

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Timeline

theory architecture 1936 1945

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Timeline

theory architecture computer 1936 1945 1949

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Timeline

theory architecture computer theory∞ 1936 1945 1949 2000

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theory∞ 2000 Solve the Collatz conjecture. For all the natural numbers, apply the algorithm.

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Timeline

theory architecture computer theory∞ 1936 1945 1949 2000

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Timeline

theory architecture

  • rdinateur

theory∞ ? 1936 1945 1949 2000 ?

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Timeline

theory architecture computer theory∞ ? computer∞ ? 1936 1945 1949 2000 ? ?

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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.

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Ordinals: counting through the infjnite

We denote ω the set of all natural numbers. But ω is not the only infjnite… We can carry on counting!

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Ordinals

Transitive well-ordered set for the membership relation. Defjnition (Ordinal). 0 := ∅ 1 := {0} = {∅} … ω := {0, 1, 2, 3, · · · } ω + 1 := {0, 1, 2, 3, · · · , ω} … ω.2 := {0, 1, 2, · · · , ω, ω + 1, ω + 2 . . . } If α is an ordinal, then α ∪ {α}, denoted α + 1 is called successor of α and is an ordinal; let A be a set of ordinal numbers, then α = ∪

β∈A β is a

limit ordinal.

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Encoding countable ordinals

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⟩, the i-th bit of r is 1 if and only if x < y. Codage 1 (Encoding countable ordinals by reals). Example: ω.2 = ω + ω ⇝ even integers lower than odd integers. 0 = ⟨0, 0⟩ 1 = ⟨0, 1⟩ · · · r = 00110203041506171819110 · · ·

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Structure of infjnite time Turing machines (ITTM)

3 right-infjnite tapes a single head binary alphabet {0, 1} additional special limit state lim computation steps are indexed by ordinals

Confjguration

input 0 1 0 0 1 0 0 . . . q0 work 0 0 0 0 0 0 0 . . .

  • utput 0 0 0 0 0 0 0

. . .

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Operating an ITTM

Confjguration at α + 1. ⇝ Confjguration at α. t = 420 0 1 0 0 1 0 0 . . . q1 … t = 007 0 0 1 1 1 0 0 . . . q3

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Compute? Infjnite time Turing machines Some particularities of infjnite time Conclusion

Operating an ITTM

Confjguration limit: head: initial position; state: lim; each cell: lim sup

  • f cell values

before. t = ω 0 1 1 0 1 0 0 . . . lim ↑ ↑ ↑ ↑ ↑ ↑ ↑ lim sup t = 420 0 1 0 0 1 0 0 . . . q1 … t = 007 0 0 1 1 1 0 0 . . . q3

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Halting

Machines halt when they reach the halting state. We consider the strong stabilisation of cells at 0. 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]). We focus on the halting problem on 0.

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Clockable and writable ordinals

Two natural notions: α clockable: there exists an ITTM that halts on input 000 . . . in exactly α steps of computation. Defjnition (Clockable ordinal). α writable: there exists an ITTM that writes a code for α

  • n input 000 . . . and halts.

Defjnition (Writable ordinal).

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Supremum

The supremum of the clockable ordinals is equal to the supre- mum of the writable ordinals. It is called λ. Theorem 2 (Welch [Wel09]). λ is a rather large countable ordinal…

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Let’s count!

Count with a clockable ordinal ⇝ Clock. Like an hourglass, execute operations while clocking the desired

  • rdinal.

If p halts on 0 in α + n steps, then there exists p′ which halts

  • n 0 in α steps (and computes the same). ⇝ limit ordinals

Speed-up lemma (Hamkins, Lewis [HL00]). Count with a writable ordinal ⇝ Empty an order. It is about counting through the encoding of an ordinal.

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…

What about the particularities of these ordinals?

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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).

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size · · · · · · · · · beg λ

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Proof of gap existence

α β β + ω … gap checking p … … …

Simulation of all programs on input 0. In blue: halting programs. In red: limit step, begins a gap?

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Proof of gap existence

But …does the algorithm halt? Halting of the algorithm, proof by contradiction: Above λ, by defjnition, there are no clockable ordinals. If no gaps before λ, thus beginnning of gap detected at λ. Contradiction.

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What can we say about gaps?

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ω α β0 · · · · · · · · · β0 λ regular structure

}

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ω α β0 · · · · · · · · · · · · β0 λ admissible = beginning of gaps

}

gaps containing admissible ordinals

}

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infjnite time Turing machines = model for algorithms proving logical properties

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Thank you for your attention.

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Some references: Joel D. Hamkins and Andrew Lewis. Infjnite time turing machines. Journal of Symbolic Logic, 65(2):567–604, 2000. Philip D. Welch. Characteristics of discrete transfjnite time turing machine models: Halting times, stabilization times, and normal form theorems. Theoretical Computer Science, 410(4-5):426–442, 2009.

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Computational power of ITTM

decidable decidable∞ Σ1

1

Σ1

2

Π1

1

Π1

2

∆1

1

arithmetic

∆1

2

s.d.∞ co s.d.∞ Figure: Projective hierarchy

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Admissible ordinals

A limit ordinal α is admissible if and only if there doesn’t exist a function f from γ < α to α such that: f is unbounded (no greatest element in α) and f is Σ1-defjnable in Lα. Proposition 3.

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Constructible hierarchy

L0 = ∅; Lα+1 = def(Lα); if α is a limit ordinal, Lα = ∪

β<α Lβ;

Defjnition (Constructible hierarchy L). Application: reals of Lλ are the writable reals.

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Defjnability

Let M be a set and F be the set of the formulas of the language {∈}. X is defjnable on a model (M, ∈) if: there exists a formula ϕ ∈ F, there exists a1, . . . , an ∈ M such that X = {x ∈ M : ϕ(x, a1, . . . , an) is true in (M, ∈)}. Defjnition (Defjnability). def(M) = {X ⊂ M : X is defjnable on (M, ∈)}.

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Admissible ordinals

A limit ordinal α is admissible if and only if there doesn’t exist a function f from γ < α to α such that: f is unbounded (no greatest element in α) and f is Σ1-defjnable in Lα. Proposition 4.

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