1/18 Machines running on random tapes and the probabilities of events by George Barmpalias joint work with Cenzer/Porter and Lewis-Pye February 2017, Dagstuhl Victoria University of Wellington Chinese Academy of Sciences
2/18 Run a universal Turing machine on an arbitrary tape X. Becher et.al. showed that some of these are (highly) random. Can we characterize them in terms of algorithmic randomness? What is the probability that it will ▶ halt? compute a total function? ▶ enumerate a computable set? enumerate a co-fjnite set? ▶ enumerate a set which computes the halting problem? ▶ compute an (in)computable function? ▶ halt with an output inside a certain set A � ∅ ? These are reals in ( 0 , 1 ) .
3/18 References probabilities. JSL 2006. ▶ Becher/Grigoriefg. Random reals and possibly infjnite computations part I: Randomness in ∅ ′ . JSL 2005. ▶ Sureson. Random reals as measures of natural open sets. TCS 2005 ▶ Becher/Figueira/Grigoriefg/Miller. Randomness and halting ▶ Becher/Grigoriefg. Random reals à la Chaitin with or without prefjx-freeness. TCS 2007.
4/18 Universal halting probabilities Can we characterize all natural universal probabilities in terms Shown to be exactly the 1-random left-c.e. reals in ( 0 , 1 ) by ▶ Chaitin (1975) – Solovay (1975) ▶ Calude/Hertling/Khousainov/Wang (2001) ▶ Kučera/Slaman (2001) The Ω analysis. For any Y let Ω Y denote a Y-left-c.e. Y-random real in ( 0 , 1 ) . And let 1 − Ω Y denote a Y-right-c.e. Y-random real in ( 0 , 1 ) . of relativized Ω numbers?
5/18 Characterization of probabilities I Totality Enumeration of a computable set Enumeration of a co-fjnite set Universality probability 1 − Ω ∅ ′ Ω ∅ ( 2 ) Ω ∅ ( 2 ) Ω ∅ ( 3 ) Enumeration of a set which computes ∅ ′ 1 − Ω ∅ ( 3 ) ▶ Barmpalias/Cenzer/Porter TCS (2017) ▶ Barmpalias/Dowe Phi. Trans. R. Soc. (2012)
6/18 What about These questions are not subject to the previous analysis. Indeed these probabilities are do not need to be random. However the analysis is based on: ▶ computing a computable function? ▶ computing a co-fjnite set? ▶ recent and not-so-recent properties of omega numbers; ▶ some theory of lowness for randomness; ▶ additional constructions of universal machines.
7/18 Characterization of probabilities II Computing incomputable set Computing a computable set Computing cofjnite set Barmpalias/Cenzer/Porter Arxiv 1612.08537 (2017) 1 − Ω ∅ ′ ∅ ′ -d.c.e. reals in ( 0 , 1 ) ∅ ′ -d.c.e. reals in ( 0 , 1 )
8/18 Computing an (in)computable set Given machine M: i For every 2-random X we have …by the theory of lowness for randomness. Why the difgerence of two ∅ ′ -left-c.e. reals? ▶ TOT ( M ) is a Π 0 2 class ▶ INCTOT ( M ) is a Π 0 3 class. Let ( V i ) be a universal Martin-Löf test and let: INCTOT ∗ ( M ) = TOT ( M ) ∩ { X | X ∈ ∩ i V M ( X ) } . X ∈ INCTOT ( M ) ⇔ X ∈ INCTOT ∗ ( M ) .
9/18 Computing an (in)computable set Hence So µ ( INCTOT ( M )) = µ ( INCTOT ( M ) ∗ ) . Also INCTOT ( M ) ∗ is a Π 0 2 class. µ ( TOT ( M ) − INCTOT ( M ) ∗ ) is a ∅ ′ -d.c.e. real. The other direction relies on a recent fact about Ω numbers. The Ω derivation theorem.
10/18 is 1-random is 1-random. Barmpalias/Lewis Arxiv 1604.00216 (2016) s lim Given a left-c.e. approximation ( α s ) → α and ( Ω s ) → Ω , α − α s = r ∈ [ 0 , ∞ ) Ω − Ω s r � 0 ⇐⇒ α ⇐⇒ α − Ω r � 1 If α is 1-random then r ∈ ( 0 , 1 ) ⇐⇒ α − Ω is left-c.e. ⇐⇒ α − Ω is right-c.e. r > 1 r = 1 ⇐⇒ α − Ω is properly d.c.e.
11/18 Prescription machine theorems for every Martin-Löf random real X. Given a Σ 0 2 prefjx-free set of strings Q, there exist machines M 0 , M 1 such that ▶ M 0 ( X ) is computable ifg X ∈ ⟦ Q ⟧ ▶ M 1 ( X ) is computable ifg X � ⟦ Q ⟧
12/18 The harder direction Prescription machine theorems randomized universal machine has a computable output. Ω analysis Ω derivation theorem Every ∅ ′ -d.c.e real in ( 0 , 1 ) is the probability that a certain
13/18 Restricted halting probability Given the universal prefjx-free machine U and a set X let the probability that U halts with output in X. Grigoriefg (2002) asked if the arithmetical complexity of X is ∑ 2 −| σ | Ω ( X ) : = U ( σ ) ↓∈ X refmected on the randomness of Ω U ( X ) .
14/18 Becher/Figueira/Grigoriefg/Miller (2006) showed that n -complete X; n , n>1; …giving a negative answer to Grigoriefg’s question. ▶ Ω U ( X ) is rational for some X ≤ T ∅ ′ ; ▶ Ω U ( X ) is 1-random for Σ 0 ▶ Ω U ( X ) is not n-random for X ∈ Σ 0 1 then is Ω U ( X ) Martin-Löf random? If X � ∅ is Π 0
15/18 This question was discussed and/or attempted in probabilities. JSL 2006. J. Complexity 2006. Bul. Symb. Logic 2006. ▶ Becher/Grigoriefg. Random reals and possibly infjnite computations part I: Randomness in ∅ ′ . JSL 2005. ▶ Becher/Figueira/Grigoriefg/Miller. Randomness and halting ▶ Figueira/Stephan/Wu. Randomness and universal machines. ▶ Miller/Nies. Randomness and computability: open questions.
16/18 Overview of the argument Adding a random left-c.e. real to a non-random d.c.e. real gives a random c.e. real. is a Martin-Löf random left-c.e. real. 1 set and Ω U ( X ) is a right-c.e. real then If X is a Π 0 Ω U ( X ) is not Martin-Löf random. Ω derivation theorem 1 set, the number Ω U ( X ) If X is a nonempty Π 0
17/18 Decanter argument 1 1 / 2 1 / 3 1 / 4
18/18 Thanks! – and main references output from a random oracle. Arxiv:1612.08537 (2017) ▶ Barmpalias/Lewis. Difgerences of halting probabilities. Arxiv: 1604.00216 (2016) ▶ Barmpalias/Cenzer/Porter The probability of a computable ▶ Barmpalias/Cenzer/Porter Random numbers as probabilities of machine behaviour. TCS (2017)
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