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Generalized characteriza- tions of Generalized characterizations of semicom- putable semicomputable semimeasures semimeasures Tom Sterkenburg Motivation Tom Sterkenburg The semicom- putable semimeasures The universal semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Generalized characterizations of semicomputable semimeasures

Tom Sterkenburg

CCR 2015 June 24, 2015 Heidelberg University

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Solomonoff’s theory of prediction

◮ How to predict the continuation of a given finite string of bits?

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Solomonoff’s theory of prediction

◮ How to predict the continuation of a given finite string of bits? ⊲ Devise an “a priori” probability distribution on 2<ω, and predict by conditionalization.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Solomonoff’s theory of prediction

◮ How to predict the continuation of a given finite string of bits? ⊲ Devise an “a priori” probability distribution on 2<ω, and predict by conditionalization. A priori probabilities are assigned to strings of symbols by examining the manner in which these strings might be produced by a universal Turing machine. Strings with short (...) “descriptions” (...) are assigned high a priori probabilities.

(Solomonoff, A formal theory of inductive inference, Inform. Contr. 7, 1964) Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Solomonoff’s theory of prediction

◮ How to predict the continuation of a given finite string of bits? ⊲ Devise an “a priori” probability distribution on 2<ω, and predict by conditionalization. A priori probabilities are assigned to strings of symbols by examining the manner in which these strings might be produced by a universal Turing machine. Strings with short (...) “descriptions” (...) are assigned high a priori probabilities.

(Solomonoff, A formal theory of inductive inference, Inform. Contr. 7, 1964)

◮ The algorithmic probability distribution QU via universal machine U is given by QU(σ) :=

  • ρ∈DU,σ

2−|ρ|, with DU,σ the set of minimal U-descriptions of σ.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

More precisely

Definition (Levin, 1973) A monotone machine is a c.e. set M ⊆ 2<ω × 2<ω of pairs of strings such that if (ρ1, σ1), (ρ2, σ2) ∈ M and ρ1 ρ2 then σ1 ∼ σ2.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

More precisely

Definition (Levin, 1973) A monotone machine is a c.e. set M ⊆ 2<ω × 2<ω of pairs of strings such that if (ρ1, σ1), (ρ2, σ2) ∈ M and ρ1 ρ2 then σ1 ∼ σ2. ⊲ Monotone machine U is universal if (ρeρ, σ) ∈ U ⇔ (ρ, σ) ∈ Me for some encoding {ρe}e∈N of the class {Me}e∈N of all monotone machines.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

More precisely

Definition (Levin, 1973) A monotone machine is a c.e. set M ⊆ 2<ω × 2<ω of pairs of strings such that if (ρ1, σ1), (ρ2, σ2) ∈ M and ρ1 ρ2 then σ1 ∼ σ2. ⊲ Monotone machine U is universal if (ρeρ, σ) ∈ U ⇔ (ρ, σ) ∈ Me for some encoding {ρe}e∈N of the class {Me}e∈N of all monotone machines. Definition A continuous a priori semimeasure is defined by λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) for universal monotone machine U.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Foundational principles

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Foundational principles

◮ The principle of Occam’s razor. That [this model] might be valid is suggested by “Occam’s razor,” (...) that the more “simple” or “economical” of several hypotheses is the more likely. Turing machines are then used to explicate the concepts of “simplicity” or “economy”—the more “simple” hypothesis being that with the shortest “description.” (Solomonoff, 1964, p. 3)

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Foundational principles

◮ The principle of Occam’s razor. That [this model] might be valid is suggested by “Occam’s razor,” (...) that the more “simple” or “economical” of several hypotheses is the more likely. Turing machines are then used to explicate the concepts of “simplicity” or “economy”—the more “simple” hypothesis being that with the shortest “description.” (Solomonoff, 1964, p. 3) ◮ The principle of indifference. Another suggested point of support is the principle of

  • indifference. If all inputs to a Turing machine that are of

a given fixed length, are assigned “indifferently equal a priori” likelihoods, then the probability distribution on the

  • utput strings is equivalent to that imposed by the (...)

model described. (Solomonoff, 1964, p. 4)

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The question

◮ Both associations rely on unique features of the uniform measure λ. ⊲ But is λ really essential in the definition? Couldn’t we equivalently define µU(σ) := µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}), for some other computable measure µ? ⊲ If the choice for λ is only circumstantial, this might be thought to undermine both associations.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The question

◮ Both associations rely on unique features of the uniform measure λ. ⊲ But is λ really essential in the definition? Couldn’t we equivalently define µU(σ) := µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}), for some other computable measure µ? ⊲ If the choice for λ is only circumstantial, this might be thought to undermine both associations. Question Can we replace the uniform measure λ in the definition by an element µ from some wider class of computable measures and still obtain the same class of a priori semimeasures? That is, does it hold for some wider class (and what class?) of computable µ that {µU}U = {λU}U?

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Overview

◮ Machine characterizations of the semicomputable semimeasures ⊲ The continuous semimeasures and monotone machines ⊲ The discrete semimeasures and prefix-free machines ◮ The a priori semicomputable semimeasures ⊲ Bayesian mixtures over all semicomputable semimeasures ⊲ The (positive) answer ◮ Wrapping up

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its characterization

Definition A continuous semimeasure ν : 2ω → R≥0 satisfies

1

ν(ǫ) ≤ 1;

2

ν(σ0) + ν(σ1) ≤ ν(σ) for all σ ∈ 2<ω.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its characterization

Definition A continuous semimeasure ν : 2ω → R≥0 satisfies

1

ν(ǫ) ≤ 1;

2

ν(σ0) + ν(σ1) ≤ ν(σ) for all σ ∈ 2<ω. Let M denote the class of all semicomputable continuous semimeasures. Theorem (Zvonkin & Levin, 1970) For every semicomputable continuous semimeasure ν, there is a monotone machine M such that ν = λM. Hence M = {λM}M.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its characterization

Definition A continuous semimeasure ν : 2ω → R≥0 satisfies

1

ν(ǫ) ≤ 1;

2

ν(σ0) + ν(σ1) ≤ ν(σ) for all σ ∈ 2<ω. Let M denote the class of all semicomputable continuous semimeasures. Theorem (Zvonkin & Levin, 1970) For every semicomputable continuous semimeasure ν, there is a monotone machine M such that ν = λM. Hence M = {λM}M. ◮ Full proof given by A.R. Day, Increasing the gap between descriptional complexity and

algorithmic probability, Trans. Amer. Math. Soc. 363, 2011. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its generalized characterization

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its generalized characterization

Definition Measure µ is nonatomic if there is no X ∈ 2ω with µ({X}) > 0.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its generalized characterization

Definition Measure µ is nonatomic if there is no X ∈ 2ω with µ({X}) > 0. Theorem For every computable continuous measure µ that is nonatomic, there is for every semicomputable continuous semimeasure ν a monotone machine M such that ν = µM. Hence M = {µM}M. ◮ Full proof by a generalization of the construction

  • f Day.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class M and its generalized characterization

Definition Measure µ is nonatomic if there is no X ∈ 2ω with µ({X}) > 0. Theorem (Zvonkin & Levin, 1970) For every computable continuous measure µ that is nonatomic, there is for every semicomputable continuous semimeasure ν a monotone machine M such that ν = µM. Hence M = {µM}M. ◮ Full proof by a generalization of the construction

  • f Day.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class P and its generalized characterization

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class P and its generalized characterization

Definition A discrete semimeasure P : 2<ω → R≥0 satisfies

σ∈2<ω P(σ) ≤ 1.

Let P denote the class of all semicomputable discrete semimeasures.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class P and its generalized characterization

Definition A discrete semimeasure P : 2<ω → R≥0 satisfies

σ∈2<ω P(σ) ≤ 1.

Let P denote the class of all semicomputable discrete semimeasures. Definition A prefix-free machine is a p.c. function T : 2<ω → 2<ω with prefix-free domain. Let Qµ

T(σ) := µ({ρ : (ρ, σ) ∈ T}). Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The class P and its generalized characterization

Definition A discrete semimeasure P : 2<ω → R≥0 satisfies

σ∈2<ω P(σ) ≤ 1.

Let P denote the class of all semicomputable discrete semimeasures. Definition A prefix-free machine is a p.c. function T : 2<ω → 2<ω with prefix-free domain. Let Qµ

T(σ) := µ({ρ : (ρ, σ) ∈ T}).

Proposition For every computable continuous measure µ that is nonatomic, there is for every semicomputable discrete semimeasure P a prefix-free machine T such that P = Qµ

  • T. Hence P = {Qµ

T}T. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to our main question

Question Can we replace the uniform measure λ in the definition by any other computable measure µ that is nonatomic and still obtain the same class of a priori semimeasures? That is, do we have {µU}U = {λU}U?

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to our main question

Question Can we replace the uniform measure λ in the definition by any other computable measure µ that is nonatomic and still obtain the same class of a priori semimeasures? That is, do we have {µU}U = {λU}U?

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)})

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

λ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)})

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

λ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe)

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

λ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)})

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

λ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λMe(σ).

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The a priori semimeasure as a mixture over M

Let’s write out λU(σ) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

λ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

λ(ρe) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

λ(ρe) · λMe(σ). ⊲ Thus λU is of the form ξW (·) :=

e∈N W (e)νe(·) with {νe}e = M.

◮ Function ξW is a W -weighted Bayesian mixture distribution over all semicomputable continuous semimeasures.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ)

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ) Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ)

=

  • e,i∈N

λ(ρe,i)λMe,i(σ)

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ)

=

  • e,i∈N

λ(ρe,i)λMe,i(σ) =

  • d∈N

λ(ρd) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Md)})

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ)

=

  • e,i∈N

λ(ρe,i)λMe,i(σ) =

  • d∈N

λ(ρd) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Md)}) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)})

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ)

=

  • e,i∈N

λ(ρe,i)λMe,i(σ) =

  • d∈N

λ(ρd) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Md)}) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) = λU(σ).

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Conversely

ξW (σ) =

  • e∈N

W (e)νe(σ) =

  • e∈N

Qλ

T(ρe)νe(σ)

=

  • e,i∈N

λ(ρe,i)λMe,i(σ) =

  • d∈N

λ(ρd) · λ({ρ : ∃σ′ σ((ρ, σ′) ∈ Md)}) = λ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) = λU(σ). ⊲ See Wood, Sunehag & Hutter, (Non-)Equivalence of Universal Priors, Proc.

Solomonoff Mem. Conf., 2011. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The big picture

M = {λM}M = {µM}M U {ξW }W = {λU}U

?

= {µU}U

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) = ...

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

The general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) = ... ⊲ Define µe(·) := µ(· | ρe). ◮ Do the semimeasures µe

Me for all e cover all elements in M? Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

A fixed-point lemma

Lemma Given encoding {ρe}e∈N of the class {Me}e∈N of all monotone machines. For every computable continuous measure µ that is nonatomic, {µe

Me}e = M. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

A fixed-point lemma

Lemma Given encoding {ρe}e∈N of the class {Me}e∈N of all monotone machines. For every computable continuous measure µ that is nonatomic, {µe

Me}e = M.

  • Proof. Let ν ∈ M.

For every given e, we can obtain a machine M such that µe

M = ν. Indeed, there is a total computable function g

that for given e retrieves index g(e) in given enumeration {Me}e∈N such that µe

Mg(e) = ν.

But by the Recursion Theorem, there is an ˆ e such that Mg(ˆ

e) = Mˆ

  • e. Hence µˆ

e Mˆ

e = ν.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to the general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

µ(ρe) · µe

Me(σ) Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to the general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

µ(ρe) · µe

Me(σ)

=

  • e∈N

Q ¯

µ T(ρe) · µe Me(σ) Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to the general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

µ(ρe) · µe

Me(σ)

=

  • e∈N

Q ¯

µ T(ρe) · µe Me(σ)

=

  • e,i∈N

¯ µ(¯ ρe,i) · ¯ µ

¯ e,i ¯ Me,i(σ) Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

slide-49
SLIDE 49

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Back to the general case

µU(σ) = µ({ρ : ∃σ′ σ((ρ, σ′) ∈ U)}) =

  • e∈N

µ({ρeρ : ∃σ′ σ((ρ, σ′) ∈ Me)}) =

  • e∈N

µ(ρe) · µ({ρ : ∃σ′ σ((ρ, σ′) ∈ Me)} | ρe) =

  • e∈N

µ(ρe) · µe

Me(σ)

=

  • e∈N

Q ¯

µ T(ρe) · µe Me(σ)

=

  • e,i∈N

¯ µ(¯ ρe,i) · ¯ µ

¯ e,i ¯ Me,i(σ)

= ¯ µ¯

U(σ). Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

A positive answer

Theorem For every nonatomic computable continuous measures µ, and U in the left-hand side ranging over those universal machines that are compatible with µ, {µU}U = {λU}U.

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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SLIDE 51

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

A positive answer

Theorem For every nonatomic computable continuous measures µ, and U in the left-hand side ranging over those universal machines that are compatible with µ, {µU}U = {λU}U. Theorem For every nonatomic computable continuous measures µ, and U in the left-hand side ranging over those universal machines that are compatible with µ, {Qµ

U}U = {Qλ U}U. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

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SLIDE 52

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Further results and questions

◮ Prefix-free Kolmogorov complexity is defined by KT(σ) := min{|ρ| : (ρ, σ) ∈ T} = − log max{λ(ρ) : (ρ, σ) ∈ T}. ⊲ Similarly, we can define for any computable measure µ K µ

T (σ) := − log max{µ(ρ) : (ρ, σ) ∈ T}. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

slide-53
SLIDE 53

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Further results and questions

◮ Prefix-free Kolmogorov complexity is defined by KT(σ) := min{|ρ| : (ρ, σ) ∈ T} = − log max{λ(ρ) : (ρ, σ) ∈ T}. ⊲ Similarly, we can define for any computable measure µ K µ

T (σ) := − log max{µ(ρ) : (ρ, σ) ∈ T}.

⊲ And we can prove a generalized Coding Theorem: Theorem For every computable continuous measures µ conditionally bounded away from zero, K µ

U =+ − log Qµ U =+ − log Qλ U =+ KU. Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

slide-54
SLIDE 54

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Further results and questions

◮ Prefix-free Kolmogorov complexity is defined by KT(σ) := min{|ρ| : (ρ, σ) ∈ T} = − log max{λ(ρ) : (ρ, σ) ∈ T}. ⊲ Similarly, we can define for any computable measure µ K µ

T (σ) := − log max{µ(ρ) : (ρ, σ) ∈ T}.

⊲ And we can prove a generalized Coding Theorem: Theorem For every computable continuous measures µ conditionally bounded away from zero, K µ

U =+ − log Qµ U =+ − log Qλ U =+ KU.

◮ What about monotonic Kolmogorov complexity?

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures

slide-55
SLIDE 55

Generalized characteriza- tions of semicom- putable semimeasures Tom Sterkenburg Motivation The semicom- putable semimeasures The universal semimeasures Conclusion

Further results and questions

◮ Prefix-free Kolmogorov complexity is defined by KT(σ) := min{|ρ| : (ρ, σ) ∈ T} = − log max{λ(ρ) : (ρ, σ) ∈ T}. ⊲ Similarly, we can define for any computable measure µ K µ

T (σ) := − log max{µ(ρ) : (ρ, σ) ∈ T}.

⊲ And we can prove a generalized Coding Theorem: Theorem For every computable continuous measures µ conditionally bounded away from zero, K µ

U =+ − log Qµ U =+ − log Qλ U =+ KU.

◮ What about monotonic Kolmogorov complexity? www.cwi.nl/˜tom

Tom Sterkenburg Generalized characterizations of semicomputable semimeasures