Ger & HarSciPop LawsHierarchy The Incomputable · 1Departamento de Matem atica, Instituto Superior T ecnico [email protected] 2CMAF { Centro de Matem atica e Aplica˘c~oes Fundamentais

Ger & Har Sci Pop O Laws Hierarchy

The Incomputable

Classifying the Theories of Physics

Jose Felix Costa1,2,3

Sponsored by Templeton Foundation

1Departamento de Matematica, Instituto Superior Tecnico

[email protected] – Centro de Matematica e Aplicacoes Fundamentais

3CFCUL – Centro de Filosofia das Ciencias da Universidade de Lisboa

June 21, 2012

Jose Felix Costa Beamer


Overview

1 The Incomputable in Physics

2 The Scientist conceptThe Scientist conceptEXplain and Behavioural Correct classesUnification of theories

3 Popper’s Refutation Principle

4 Omniscient and Trivial OraclesOmniscient oraclesBounded queries

5 Identifying the Non-computableScale functionsHierarchies of scalesPREDπ generalises BCDegree of regularity of a physical law

6 A New Non-collapsing HierarchyParticular casesAnother non-collapsing hierarchyConclusion



Introduction: Newton-Bentley’s correspondence

Newton-Bentley’s correspondence led Newton to abandon theStoic Cosmos of a finite distribution of matter in infinite spaceand to adopt the Atomist Universe in which matter isdistributed throughout infinite space.



Introduction: Newton-Bentley’s correspondence

Newton-Bentley’s correspondence led Newton to abandon theStoic Cosmos of a finite distribution of matter in infinite spaceand to adopt the Atomist Universe in which matter isdistributed throughout infinite space.



Letter 1

If the distribution of matter were finite, then the matteron the outside of this space would by its gravity tendtoward the matter on the inside, and by consequence, falldown into the middle of the whole space, and therecompose one great spherical mass... But if the matter wasevenly diffused through an infinite space, it would neverconvene into one mass but some of it into one mass andsome of it into another so as to make an infinite number ofgreat masses scattered at great distances from one toanother throughout all of infinite space. And thus mightthe Sun and fixed stars be formed.



Letter 1

If the distribution of matter were finite, then the matteron the outside of this space would by its gravity tendtoward the matter on the inside, and by consequence, falldown into the middle of the whole space, and therecompose one great spherical mass... But if the matter wasevenly diffused through an infinite space, it would neverconvene into one mass but some of it into one mass andsome of it into another so as to make an infinite number ofgreat masses scattered at great distances from one toanother throughout all of infinite space. And thus mightthe Sun and fixed stars be formed.



Letter 2

Newton had fully agreed with Bentley that gravitymeant providence had created a universe of greatprecision.

The hypothesis of deriving the frame of the world bymechanical principles from matter evenly spread throughthe heavens being inconsistent with my system, I hadconsidered it very little before your letters put me upon it,and therefore trouble you with a line or two more about it...



Letter 2





Letter 2





Letter 3

Newton elaborated earlier arguments that a divinepower was essential in the design of initial conditions.

... this frame of things could not always subsist withouta divine power to conserve it.



Letter 3





Letter 3





Letter 3

Figure: Sensorium Dei in Newton’s metaphysics.



Introduction: Newton’s system of the world

Newton, equipped with his system of the world, was facingincomputability in Nature...



Introduction: Newton’s system of the world

Newton, equipped with his system of the world, was facingincomputability in Nature...



Overview









The Incomputable in Physics



Geroch and Hartle I [GH86]

Regard number w as measurable if there exists a finiteset of instructions for performing an experiment such thata technician, given an abundance of unprepared rawmaterials and an allowed error ε, is able by following thoseinstructions to perform the experiment, yielding ultimatelya rational number within ε of w.



Geroch and Hartle I [GH86]

Regard number w as measurable if there exists a finiteset of instructions for performing an experiment such thata technician, given an abundance of unprepared rawmaterials and an allowed error ε, is able by following thoseinstructions to perform the experiment, yielding ultimatelya rational number within ε of w.



Geroch and Hartle II [GH86]

Every computable number is measurable...

We now ask whether, conversely, every measurablenumber is computable — or, in more detail, whethercurrent physical theories are such that their measurablenumbers are computable. This question must be askedwith care.













Barry Cooper [Coo12]

So — there is a qualitatively different apparentbreakdown in computability of natural laws at the quantumlevel — the measurement problem challenges us to explainhow certain quantum mechanical probabilities are convertedinto a well-defined outcome following a measurement. Inthe absence of a plausible explanation, one is denied acomputable prediction. The physical significance of theTuring model depends upon its capacity for explaining whatis happening here. If the phenomenon is not composite, itdoes need to be related in a clear way to a Turing universedesigned to model computable causal structure. We lookmore closely at definability and invariance.



Barry Cooper [Coo12]

So — there is a qualitatively different apparentbreakdown in computability of natural laws at the quantumlevel — the measurement problem challenges us to explainhow certain quantum mechanical probabilities are convertedinto a well-defined outcome following a measurement. Inthe absence of a plausible explanation, one is denied acomputable prediction. The physical significance of theTuring model depends upon its capacity for explaining whatis happening here. If the phenomenon is not composite, itdoes need to be related in a clear way to a Turing universedesigned to model computable causal structure. We lookmore closely at definability and invariance.



Purpose

1 Barry Cooper has been emphasising that the Turing world canmodel the incomputability of the physical world...

2 Might well be that degree theory does not easily establish aclear link between Computability and Physics...

3 On the other side, the recursion theoretic model of identificationhas been developed with some connections with Physics...

4 However, their mentors do not accept that expected reality ispossibly non-computable...

5 On yet another side, Gregory Chaitin has been claiming that atheory of Physics can be seen as a compression of (potentialinfinite) physical observations...

6 In this work, we will try to unite all these different, butconfluent views of the world.



Purpose









Purpose









Purpose









Purpose









Purpose









Purpose









Purpose








Ger & Har Sci Pop O Laws Hierarchy Sci EXn and BCn Unification

Overview









The Scientist Concept



Boyle’s law

Example (Scientist Boyle)

Boyle’s law :

The pressure of an ideal gas inside a flexible con-tainer, maintained at a constant temperature dur-ing a process of expansion or contraction, is pro-portional the the inverse of its volume.

pV = const

The scientist ‘Boyle’, on inputting text like this〈5, 25〉〈10,

15〉〈20,

110〉 . . . , outputs the code e for the instance of

Boyle’s law.



Boyle’s law


Boyle’s law :


pV = const


15〉〈20,


Boyle’s law.



Boyle’s law


Boyle’s law :


pV = const


15〉〈20,


Boyle’s law.



Boyle’s law


Boyle’s law :


pV = const


15〉〈20,


Boyle’s law.



Scientists work with text!

Text et al.

1 A text T for a function is a map of type N→ N× N. By T [t]we denote the sequence of the first t elements of T .

2 INIT = T [t] : T is a text for a function and t ∈ N.

Definition (Scientist, Gold [Gol67])

A scientist is a function (possibly partial, not necessarilycomputable) of type INIT → N.




Text et al.








Text et al.








Text et al.








Text et al.







Depicting a scientist

SCIENTIST M

φe(x) = 2x

From some order p ∈ N on0#2#4#6# . . . Output stabilizes

. . . e

General Identification

Figure: For all t ≥ p, scientist M on input ψ[t] outputs code e of ψ.




SCIENTIST M

φe(x) = 2x


. . . e






SCIENTIST M

φe(x) = 2x


. . . e






SCIENTIST M

φe(x) = 2x


. . . e






SCIENTIST M

φe(x) = 2x


. . . e





Success for functions

Definition (Scientific success on a single function, Gold [Gol67])

Let ψ : N→ N a total function. We say that scientist M identifiesψ if there exists an e ∈ N and an order p such that, for t ≥ p,M(ψ[t]) = e and φe = ψ.

Definition (... on a collection of functions, Gold [Gol67])

Let S be a set of total functions. We say that scientist M identifiesS just in case she identifies every ψ ∈ S.

















Scientist Boyle

SCIENTIST

Boyle

φe is an instance of Boyle’s law

From some order p ∈ N on〈5, 2

5〉〈10, 1

5〉〈20, 1

10〉 . . . Output stabilizes

. . . e

EX -identification

Figure: For all t ≥ p, scientist M on input Volume[Pressure] outputs the instance ofBoyle’s law for the particular ideal gas under consideration.



Scientist Boyle

SCIENTIST

Boyle



5〉〈10, 1

5〉〈20, 1


. . . e

EX -identification




Scientist Boyle

SCIENTIST

Boyle



5〉〈10, 1

5〉〈20, 1


. . . e

EX -identification




Scientist Boyle

SCIENTIST

Boyle



5〉〈10, 1

5〉〈20, 1


. . . e

EX -identification




Scientist Boyle

SCIENTIST

Boyle



5〉〈10, 1

5〉〈20, 1


. . . e

EX -identification




Popper on precision in [Pop35]

Assume that the consequences of two theories differ solittle in all fields of application that the very smalldifferences between the calculated observable eventscannot be detected, owing to the fact that the degree ofprecision attainable in our measurements is not sufficientlyhigh. It will then be impossible to decide by experimentsbetween the two theories, without first improving ourtechnique of measurements. This shows that the prevailingtechnique of measurement determines a certain range — aregion within which discrepancies between observations arepermitted by the theory.



Popper on precision in [Pop35]

Assume that the consequences of two theories differ solittle in all fields of application that the very smalldifferences between the calculated observable eventscannot be detected, owing to the fact that the degree ofprecision attainable in our measurements is not sufficientlyhigh. It will then be impossible to decide by experimentsbetween the two theories, without first improving ourtechnique of measurements. This shows that the prevailingtechnique of measurement determines a certain range — aregion within which discrepancies between observations arepermitted by the theory.



Scientist Van der Walls

SCIENTIST

Van der Walls

φe is an instance of Van der Walls law

From some order p ∈ N on〈p0, V0〉〈p1, V1〉 . . . 〈pk, Vk〉 . . . Output stabilizes

. . . e

EX -identification

Figure: For all t ≥ p, scientist M on input Volume[Pressure] outputs the instance ofVan der Walls’ law for the particular gas under consideration:

(p+ an2

V 2 )(V − nb) = const.




SCIENTIST

Van der Walls



. . . e

EX -identification


(p+ an2





SCIENTIST

Van der Walls



. . . e

EX -identification


(p+ an2





SCIENTIST

Van der Walls



. . . e

EX -identification


(p+ an2





SCIENTIST

Van der Walls



. . . e

EX -identification


(p+ an2




EXplain and Behavioural Correct classes

Definition (EX n-identification, Gold [Gol67], Blum and Blum[BB75], Case and Smith [CS78, CS83])

A set S of (total) recursive functions is said to belong to class EX n,if there exists a scientist M such that, for each ψ ∈ S, there existsan order p ∈ N such that, for all t ≥ p, M on input ψ[t] outputs thesame n-variant code for ψ.

Definition (BC n-identification, Barzdins [B74], Feldman [Fel72],Case and Smith [CS78, CS83])

We say that a scientist M BC n-identifies a function ψ ∈ R, if thereexists an order p ∈ N such that, for all t ≥ p, φM(ψ[t]) is a n-variantcode for ψ. We say that scientist M BC n-identifies a set offunctions S ⊆ R just in case she BC n-identifies every ψ ∈ S.
























Almost and Everywhere Zero

Example (AEZ)

Standard example: The set AEZ of (total) computable functions ψidentical to 0 almost everywhere.Just consider the scientist M that, on input

ψ(0)# . . .#ψ(t− 1)# ,

builds the canonical list µ of the pairs 〈t, ψ(t)〉, where the ψ(t) isnon-zero, and outputs the code of the programme

λx.If x ∈ dom(µ), Then µ(x), Else 0 .

Such a scientist EX -identifies AEZ.



Almost and Everywhere Zero

Example (AEZ)

Standard example: The set AEZ of (total) computable functions ψidentical to 0 almost everywhere.Just consider the scientist M that, on input

ψ(0)# . . .#ψ(t− 1)# ,

builds the canonical list µ of the pairs 〈t, ψ(t)〉, where the ψ(t) isnon-zero, and outputs the code of the programme

λx.If x ∈ dom(µ), Then µ(x), Else 0 .

Such a scientist EX -identifies AEZ.



Non-collapsing hierarchy

Proposition (The hierarchy of scientists, mainly from Case and Smith[CS78, CS83])

R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R

John Case writes in [Cas11]:

Hence, tolerating anomalies strictly increases the inferring power as doesrelaxing the restriction of (syntactic) convergence to single programmes.Physicists use of slightly faulty explanations is vindicated!





R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R







R /∈ EX = EX 0 ⊂ · · · ⊂ EX n ⊂ · · · ⊂ EX ? ⊂ BC 63 R

R /∈ BC = BC 0 ⊂ · · · ⊂ BC n ⊂ · · · ⊂ BC ? 3 R





Non-union theorem

Proposition (Blum and Blum [BB75], Case and Smith [CS78, CS83],Jain et al. [JORS99])

The classes EX n and EX ? are not closed under union.

Theorem (Unification of scientific laws)

Unification of scientific laws (as algorithms) is not always possiblewithin the paradigm EX .



Non-union theorem







Non-union theorem







Overview









Popper’s Refutation Principle



Falsifiability

[Popper states in the kernel of [Pop35]:] It has already beenbriefly indicated what role the basic statements play within theepistemological theory I advocate. We need them in order todecide whether a theory is to be called falsifiable, i.e. empirical[...] And we also need them for the corroboration of falsifyinghypothesis, and thus for the falsification of theories [...]

Basic statements must therefore satisfy the followingconditions: (a) From a universal statement without initialconditions, no basic statement can be deduced. On the otherhand, (b) a universal statement and a basic statement cancontradict each other. Condition (b) can only be satisfied if it ispossible to derive the negation of a basic statement from thetheory which it contradicts. From this and condition (a) itfollows that a basic statement must have a logical form suchthat its negation cannot be a basic statement in its turn.



Falsifiability





Falsifiability





Falsifiability





Falsifiability





Falsifiability





Falsifiability





Popper’s refutability principle

Popper’s refutability principle, Case and Smith [CS78, CS83]

The theory embedded in scientist M may not be refutable (!), for

1 It is not known if the instance φe is undefined on some y;

2 Programme e on input y does not halt, i.e., one can not prepareany experimental apparatus to refute “theory M on y”, given a basicstatement such as φe(y) 6= ψ(y), where φe(y) is the prediction andψ(y) is the observation, since it is not even known with generality ife(y) halts or not and, consequently, produce a prediction refutableby observation.






























Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded

Overview









Omniscient and Trivial Oracles



Omniscient oracles

Proposition (Adleman and Blum [AB91])

R ∈ OKEX .



Omniscient oracles

Definition (High sets)

A set A is high if and only if K ′ ≤T A′.


For all sets A, R ∈ OAEX if and only if A is high.



Omniscient oracles







Omniscient oracles







Bounded queries

Proposition

If A ≤T K, then OA,?EX = EX .

Proposition (Fortnow, Gasarch, Jain, Kinber, Kummer, Kurtz,Pleszkoch, Slaman, Solovay, and Stephan [FGJ+94])

For all sets A, A 6≤T K if and only if, for all n ∈ N,OA,nEX ⊂ OA,n+1EX .



Bounded queries

Proposition






Bounded queries

Proposition





Ger & Har Sci Pop O Laws Hierarchy Scale Ordinal scales PREDπ Degrees

Overview









Identifying the Non-computable Laws



Scale functions

Definition (Scale function)

We say that a (total) recursive function π : N→ N is a scalefunction if π is a non-decreasing surjective (linear or) sublinearfunction.

Definition (Ordering scale functions)

There is a relation between two total functions, ψ,ϕ : N→ N, bysaying that ψ ≺ ϕ if ψ ∈ o(ϕ). This relation can be generalised totwo classes of functions, F and G, by saying that F ≺ G if thereexists a function ϕ ∈ G, such that for all functions ψ ∈ F , ψ ≺ ϕ.



Scale functions







Scale functions







Class of scale functions

Definition (Ordinal iterate of the logarithm)

For each ordinal α, we define inductively the class log(α): (a) if α is0, then log(0) = Os(λt.t), (b) if α is a successor ordinal, thenlog(α+1) = Os(λt. log(ψ(t)) : ψ ∈ log(α)), and (c) if α is a limitordinal, then log(α) = Os(∩γ∈αlog(γ)).

Proposition

For all α ∈ ωω, log(α+1) ≺ log(α).






Proposition







Proposition







Proposition







Proposition







Proposition




A first hierarchy of scales

Proposition (A first hierarchy of scales)

log(ω) ≺ · · · ≺ log(3) ≺ log(2) ≺ log(1) ≺ poly.

Proposition (Non-triviality of log(ω))

log(ω) 6= ∅.

Proof: Consider log?, defined by (a) log?(t) = 0, for t = 0, and(b) log?(t) = mink : log(k)(t) ≤ 1, for t > 0.







log(ω) 6= ∅.








log(ω) 6= ∅.








log(ω) 6= ∅.




Proposition (A second hierarchy of scales)

log(2ω) ≺ · · · ≺ log(ω+1) ≺ log(ω) ≺ · · · ≺ log(2) ≺ log(1) ≺ poly.

Proposition (Non-triviality of log(2ω))

log(2ω) 6= ∅.

Proof: Consider log?? = log? log?.

Example (Non-emptyness of limit classes)

We can continue descending by setting log(2ω+k) to be the classgenerated by log(k) log??...






log(2ω) 6= ∅.









log(2ω) 6= ∅.









log(2ω) 6= ∅.









log(2ω) 6= ∅.






Charles Sanders Peirce [Pei09]

Changing laws of Physics, Peirce [Pei09]

Now the only possible way of accounting for the laws of Nature andfor uniformity in general is to suppose them result of evolution. Thissupposes them not to be absolute, not to be obeyed precisely. Itmakes an element of indeterminacy, spontaneity, or absolute chancein nature.

Example (How to define a law?)

We assume that a law of physics corresponds to the extraction ofalgorithmic content of a possible non-computable phenomenon, i.e.,the processes in our brains, the instruments we use, extract frommore or less random behaviour some algorithmic content that allowsto predict Nature to some extent.

















PRED-identification

Definition (Predictive identification PRED)

Let π : N→ N be a scale function and β : N→ 0, 1 be an advice

sequence. We say that a scientist M PREDβπ -identifies the (total,

possibly non-computable) function ψ if there exists an order p ∈ Nsuch that, for all t ≥ p, M on input ψ[π(t)], and using at most theprefix β[π(t)] ⊂ β of the advice, outputs the programme code for ψsuch that, for every i ≤ t, we have ψ(i) = ψ(i).



PRED-identification

Definition (Predictive identification PRED)

Let π : N→ N be a scale function and β : N→ 0, 1 be an advice

sequence. We say that a scientist M PREDβπ -identifies the (total,

possibly non-computable) function ψ if there exists an order p ∈ Nsuch that, for all t ≥ p, M on input ψ[π(t)], and using at most theprefix β[π(t)] ⊂ β of the advice, outputs the programme code for ψsuch that, for every i ≤ t, we have ψ(i) = ψ(i).



Depicting a PRED-scientist

SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .

S ⊆countable FUNψ ∈ S

PREDπ -identification

β(0)β(0) . . . β(π(t)− 1)

From some order p = π(t)− 1 on

ψ(0)# . . .#ψ(π(t)− 1)# . . .

Figure: For all t ≥ p, scientist M on input ψ[π(t)] outputs code e of ψsuch that ψ and ψ coincide in the interval [0, t[.




SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .





SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .





SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .





SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .





SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .





SCIENTIST M

φe[t] = ψ[t]

Output stabilizes

ε0 . . . εp−1e0 . . . ek . . .



β(0)β(0) . . . β(π(t)− 1)


ψ(0)# . . .#ψ(π(t)− 1)# . . .




Full predictive classes

Definition (Success on a collection of functions)

We say that a scientist M PREDβπ -identifies the set S of total

functions if M PREDβπ -identifies each function of S. By PREDβ

π

we designate the class of sets PREDβπ -identifiable and by PREDπ

we designate the class of sets PREDβπ -identifiable with some advice

function β.

Definition (Success with respect to a collection of scales)

For each class of scales F , we define PREDβF = ∪π∈F PREDβ

π . We

also define PREDF = S : S ∈ PREDβF for some advice β.







π



function β.



π . We








π



function β.



π . We








π



function β.



π . We








π



function β.



π . We








π



function β.



π . We




Depicting a PRED-function

Example

function fβ :

Function fβ(x : N) : N;Function β(x : N) : boolean;

Var i, j : N;Begin

i := 1;j := 1;While i ≤ x do begin

i := 2j+1 − 1;j := j + 1

End;Return β(j − 1)

End




Example

function fβ :


Var i, j : N;Begin


i := 2j+1 − 1;j := j + 1


End




Example

function fβ :


Var i, j : N;Begin


i := 2j+1 − 1;j := j + 1


End



Piecewise Self-Described

Function fβ has the form:

β(0)β(1)β(1)β(2)β(2)β(2)β(2)β(3)β(3) . . . β(3)β(4) . . . β(4) . . .

Functions in PSD βlog are such like the following:

β(0)e2β(1)e4β(2) . . . β(2)β(3) . . . β(3)e16β(4) . . . β(15)e65536β(16) . . . β(16) . . .

β(0)e2︸︷︷︸e2

β(1)e4

︸︷︷︸e4

β(2) . . . β(2)β(3) . . . β(3)e16

︸︷︷︸e16

β(4) . . . β(15)e65536

︸︷︷︸e65536

β(16) . . . β(16) . . .





β(0)β(1)β(1)β(2)β(2)β(2)β(2)β(3)β(3) . . . β(3)β(4) . . . β(4) . . .


β(0)e2β(1)e4β(2) . . . β(2)β(3) . . . β(3)e16β(4) . . . β(15)e65536β(16) . . . β(16) . . .

β(0)e2︸︷︷︸e2

β(1)e4

︸︷︷︸e4

β(2) . . . β(2)β(3) . . . β(3)e16

︸︷︷︸e16

β(4) . . . β(15)e65536

︸︷︷︸e65536

β(16) . . . β(16) . . .





β(0)β(1)β(1)β(2)β(2)β(2)β(2)β(3)β(3) . . . β(3)β(4) . . . β(4) . . .


β(0)e2β(1)e4β(2) . . . β(2)β(3) . . . β(3)e16β(4) . . . β(15)e65536β(16) . . . β(16) . . .

β(0)e2︸︷︷︸e2

β(1)e4

︸︷︷︸e4

β(2) . . . β(2)β(3) . . . β(3)e16

︸︷︷︸e16

β(4) . . . β(15)e65536

︸︷︷︸e65536

β(16) . . . β(16) . . .





β(0)β(1)β(1)β(2)β(2)β(2)β(2)β(3)β(3) . . . β(3)β(4) . . . β(4) . . .


β(0)e2β(1)e4β(2) . . . β(2)β(3) . . . β(3)e16β(4) . . . β(15)e65536β(16) . . . β(16) . . .

β(0)e2︸︷︷︸e2

β(1)e4

︸︷︷︸e4

β(2) . . . β(2)β(3) . . . β(3)e16

︸︷︷︸e16

β(4) . . . β(15)e65536

︸︷︷︸e65536

β(16) . . . β(16) . . .







Rβπ

Rβπ is the set of functions ψ containing the zero Z, the successor S, function β, and closed under composition,

primitive recursion and minimalisation, whenever the resulting function is total, and such that the values of ψ(0),

ψ(1), ..., ψ(t− 1) only depends on the values of β(0), β(1), ..., β(π(t)− 1).




Self-Described

SD = ψ ∈ Rβlog : φ βψ(0) = ψ




Self

S = ψ ∈ Rβlog : (At) φ βψ(t) = ψ




Tower

f(0) = 0; f(t+ 1) = 2f(t)




Tower

f(0) = 0; f(t+ 1) = 2f(t)

Definition (Piecewise Self-Described)

PSD βlog = ∩t∈N ψ ∈ R β

log : φ βψ(f(t))[f(t)] = ψ[f(t)]




Tower

f(0) = 0; f(t+ 1) = 2f(t)

Definition (Piecewise Self-Described)

PSD βlog = ∩t∈N ψ ∈ R β

log : φ βψ(f(t))[f(t)] = ψ[f(t)]

Non-emptiness of PSD βlog

S ⊂ PSD βlog




Proposition

PSD βlog is PRED β

log-identifiable.




Proposition

PSD βlog is PRED β

log-identifiable.



PREDπ generalises BC

Proposition (PREDπ generalises BC )

If a set S of total functions ψ of type N→ N is BC -identifiable,then S is PREDβ

π -identifiable, for all scale functions π and all advicesequences β.



Degree of regularity of a physical law

Definition (PREDα)

Let β be some advice sequence. We say that a set S of (total)functions of type N→ N is in PREDβ

α, for some ordinal α, if S isPRED β

log(α)-identifiable.

Definition (Degree of predictability of a physical law)

A physical law S is said to be of ordinal degree α if S ∈ PREDα butS /∈ PREDα+1.




Definition (PREDα)









Definition (PREDα)








From non-computable to computable

Example (Recursive β)

If β is PREDβπ -identifiable for some strictly sublinear π, having β as

advice, then β is total recursive.

Proposition (From non-computable to computable)

If a set S of total functions of type N→ N is such thatS ∈ PREDβ

π , for some recursive advice sequence β, then each ψ ∈ Sis a recursive function.

Example (From PREDβπ -identifiable to BC -identifiable)

Let S be a set of total functions ψ of type ψ : N→ N. If S isPREDβ

π -identifiable for some recursive advice sequence β andstrictly sub-linear π, then S is BC -identifiable.







































Proof:

Let β be a recursive advice sequence such that the computable scientistM PREDβπ -identifies S. Let ψ ∈ S and T

be a text for ψ. Applying the smn -theorem to the function ϕ specified in the figure below, we get, for some recursive

function s and sufficient large σ, ψ(x) = φs(σ)(x) = ϕ(σ, x).

function ϕ :

Function ϕ(σ : INIT ;x : N) : N;Function β(t : N) : boolean; % β is recursive

Var i, j, e : N;Begin

While i < x Do Beginj := |σ| − 1;e :=M(σ, β[|σ|]);For k := j + 1 To π−1(j)− 1 Do σ := σ♦〈k, φe(k)〉;i := π−1(j)− 1

End;Return σ(x)

End



Proof:




function ϕ :




End;Return σ(x)

End



Proof:




function ϕ :




End;Return σ(x)

End


Ger & Har Sci Pop O Laws Hierarchy Cases Hierarchy Ω

Overview









A New Hierarchy of Scientists



Particular cases

Proposition

Every set S ⊆ FUN is PREDβλt.t-identifiable.

Proposition (Based on Loveland [Lov69], Kobayashi [Kob81])

All the characteristic functions of recursively enumerable sets belongto PRED log2 .



Particular cases

Proposition






Particular cases

Proposition






Another non-collapsing hierarchy

Proposition (Non-collapsing hierarchy)

If F and G are classes of scales such that F ≺ G, then there is a setin PREDG that is not in PREDF .

Proposition (PRED-hierarchy)

For all ordinal α ∈ ωω,

EX ⊂ BC ⊂ · · · ⊂ PREDα+1 ⊂ PREDα ⊂ · · · ⊂ PRED0



















What about R?

Proposition (R is in the inductive limit)

R ∈ PREDα, for all ordinal α ∈ ωω.



EX ⊂ BC ⊂ BC ? ⊂ · · · ⊂ PREDα+1 ⊂ PREDα ⊂ · · · ⊂ PRED0



What about R?








What about R?








Proof:

Example (sketch)

Just consider the scientist M that, on input

ψ(0)# . . .#ψ(t− 1)# ,

tests for all of the codes of Turing machines between 0 and t− 1 which halton all inputs and outputs the least code e such that φe[t] = ψ[t].

For sufficient large t ≥ p the scientist finds the code e of ψ to be e ≤ p ≤ t.

For the task described above it is enough p bits of the ω-sequence β codingfor the halting problem: for all i ∈ N, β(i) = 1 if and only if the Turingmachine code i halts on all inputs. Once the scientist finds e ≤ p, thevalues of β(i), for i > p, are not needed, since the scientist converges on e.

We conclude that R ∈ PREDβα, for all ordinals α ∈ ωω, since to identify

each function ψ ∈ R only a fixed number of bits of β are needed.



Proof:

Example (sketch)

Just consider the scientist M that, on input

ψ(0)# . . .#ψ(t− 1)# ,

tests for all of the codes of Turing machines between 0 and t− 1 which halton all inputs and outputs the least code e such that φe[t] = ψ[t].

For sufficient large t ≥ p the scientist finds the code e of ψ to be e ≤ p ≤ t.

For the task described above it is enough p bits of the ω-sequence β codingfor the halting problem: for all i ∈ N, β(i) = 1 if and only if the Turingmachine code i halts on all inputs. Once the scientist finds e ≤ p, thevalues of β(i), for i > p, are not needed, since the scientist converges on e.

We conclude that R ∈ PREDβα, for all ordinals α ∈ ωω, since to identify

each function ψ ∈ R only a fixed number of bits of β are needed.



Conclusion

“The implications for our current understanding of science are profound.If the laws of physics turn out to be merely ‘local by-laws’, it might be thatwhilst our observable part of the universe favours the existence of life andhuman beings, other far more distant regions may exist where different lawspreclude the formation of life, at least as we know it.”

“If our results are correct, clearly we shall need new physical theories tosatisfactorily describe them.”

[...]“It varies by only a tiny amount — about one part in 100,000 — over

most of the observable universe, but it’s possible that much larger variationscould occur beyond our observable horizon,” Mr King said.”

[...]“It’s one of the biggest questions of modern science — are the laws of

physics the same everywhere in the universe and throughout its entirehistory? We’re determined to answer this burning question one way or theother.”

The team — from the University of New South Wales, SwinburneUniversity of Technology and the University of Cambridge — has submitteda report of the discovery for publication in the journal Physical ReviewLetters.



Conclusion










Conclusion










Conclusion










Conclusion










Conclusion










Bibliography I

[AB91] Leonard M. Adleman and Mario Blum. Inductive inference and unsolvability. Journal of Symbolic Logic,56(3):891–900, 1991.

[B74] J. Barzdins. Two theorems on the limiting synthesis of functions. In Theory of Algorithms and Programs,volume I, pages 82–88. Latvian State University, 1974.

[BB75] Lenore Blum and M. Blum. Toward a mathematical theory of inductive inference. Information andControl, 28:125–155, 1975.

[Cas11] John Case. Algorithmic scientific inference, within our computable expected reality, 2011. TechnicalReport, SUNY/Buffalo.

[Coo12] Barry S. Cooper. Mathematics, metaphysics and the multiverse. In M. J. Dinneen, B. Khoussainov, andA. Nies, editors, Computation, Physics and Beyond 2012, Lecture Notes in Computer Science, pages252–267. Springer, 2012. International Workshop on Theoretical Computer Science, WTCS 2012,Dedicated to Cristian S. Calude on the Occasion of His 60th Birthday, Auckland, New Zealand, February21-24, 2012.

[CS78] John Case and Carl Smith. Anomaly hierarchies of mechanized inductive inference. In Richard J. Lipton,Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, pages 314–319. ACM, San Diego,California, USA, 1978.

[CS83] John Case and Carl Smith. Comparison of identification criteria for machine inductive inference.Theoretical Computer Science, 25(2):193–220, 1983.

[Fel72] J. Feldman. Some decidability results on grammatical inference and complexity. Information and Control,20:244–262, 1972.

[FGJ+94] Lance Fortnow, W. Gasarch, Sanjay Jain, E. Kinber, M. Kummer, S. Kurtz, M. Plesrkoch, T. Slaman,R. Solovay, and F. Stephan. Extremes in the degrees of inferability. Annals of Pure and Applied Logic,66:231–276, 1994.



Bibliography II

[GH86] Robert Geroch and James B. Hartle. Computability and Physical Theories. Foundations of Physics,16(6):533–550, 1986.

[Gol67] E. M. Gold. Language identification in the limit. Information and Control, 10:447–474, 1967.

[JORS99] Sanjay Jain, Daniel N. Osherson, James S. Royer, and Arun Sharma. Systems That Learn. AnIntroduction to Learning Theory. The MIT Press, second edition, 1999.

[Kob81] K. Kobayashi. On compressibility of infinite sequences. Technical Report C–34, Research Reports onInformation Sciences, 1981.

[Lov69] D. W. Loveland. A variant of the Kolmogorov concept of complexity. Information and Control,15:115–133, 1969.

[Pei09] Charles Sanders Peirce. The Logic of Interdisciplinarity. The Monist Series, Elize Bisanz (Ed.). AkademieVerlag, 2009.

[Pop35] Karl R. Popper. The Logic of Scientific Discovery. Routledge, 1935. First English edition published in1959 by Hutchinson & Co. First published by Routledge in 1992.


Documents

Ger & HarSciPop LawsHierarchy The Incomputable · 1Departamento de Matem atica, Instituto Superior T ecnico [email protected] 2CMAF { Centro de Matem atica e Aplica˘c~oes Fundamentais