Incomputability, Undecidability and Nondeterminism: from formal systems to human minds

Incomputability, Undecidability Incomputability, Undecidability and Nondeterminism: from and Nondeterminism: from formal systems to human formal systems to human

mindsminds

Filippo CacaceFilippo Cacace

Università Campus Biomedico, RomaUniversità Campus Biomedico, Roma

[email protected]@unicampus.it

Ponte di Legno, 30-12-2003

F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December 2003

2

This presentation is based on This presentation is based on the work by S. Barry Cooperthe work by S. Barry Cooper

and Piergiorgio Odifreddiand Piergiorgio OdifreddiSee:See:

Incomputability in NatureIncomputability in Nature

unpublished manuscript, 2001unpublished manuscript, 2001


3

The questions we want to askThe questions we want to ask

1.1. Does incomputability occur in Nature?Does incomputability occur in Nature?

2.2. Is a theory of incomputability useful for our Is a theory of incomputability useful for our description of the Universe?description of the Universe?


4

Objections to incomputability Objections to incomputability in Physicsin Physics

– Despite the development of the mathematical theory of computability in the 30s (Godel, Turing, Post, Kleene) one is still left with uncertainty as to how it applies to the real world

– How would uncomputability be distinguishable from theoretically computable but very complex phenomena?

– If the Universe has a discrete and finite model incomputability does not have any relevance in its description


5

Objections to incomputability Objections to incomputability in Physics (2)in Physics (2)

– At a purely intuitive level, one has a seemingly unproblematic model of a deterministic, even mechanical, Universe

– Incomputability can be incorporated in the particular form of randomness (for example quantum indeterminacy) without any need for any theory of incomputability

Conclusion: the origins of incomputabilty in Conclusion: the origins of incomputabilty in mathematics may be theoretical, but they play no mathematics may be theoretical, but they play no role in physicsrole in physics


6

Birth of incomputability theory: Birth of incomputability theory: Turing TheoremTuring Theorem

"There is no effective procedure (algorithm) for "There is no effective procedure (algorithm) for deciding whether or not a generic program ever deciding whether or not a generic program ever halts"halts"

Proof. A computer program is a binary string, that is, a natural number (we include the input for the program in the program itself). We can list programs in ascending order. For each program we write the output:

Number Program Output

1 p1 r1

2 p2 r2

… … …

ri is computed by pi. If the program does not halt, ri =0


7

Turing Theorem (2)Turing Theorem (2)rij is the j-th digit of ri.

Now consider the real r*:

r*=0.(r11+1)(r22+1)(r33+1)…

(i.e. if r1=0.4923…,r2=0.1135…, r3=0.98712, then r*=0.528…)

r* cannot be in the list (since is different from any other ri), thus it is not computable. �

Why we cannot compute r*? One could take the i-th program, run it, take the i-th digit of its output and then add 1, thus obtaining the i-th digit of r*.

The problem is that you cannot know whether or not the i-th program ever halts or outputs a n-th digit: you can just sit there and wait…


8

Turing (1936) Turing (1936) G Göödel (1931)del (1931)

"There is no formal axiomatic system from which all "There is no formal axiomatic system from which all mathematical truth can be proved"mathematical truth can be proved"

Proof. If we had a complete axiomatic system for mathematics (that is, a system where each true proposition is a provable theorem) we could run through all possible proofs until we find a proof that a given program halts, or find a proof that it never halts. We could therefore decide if any computer program halts. Since we can't do this (Turing theorem) no formal axiomatic sysem can be complete.


9

Incomputable setsIncomputable sets The incomputability of real numbers is equivalent to the The incomputability of real numbers is equivalent to the

incomputability of sets of integers. Such sets are called incomputability of sets of integers. Such sets are called recursively enumerable sets recursively enumerable sets (or(or computably enumerable computably enumerable setssets))

Most "incomputable" sets come with descriptions which Most "incomputable" sets come with descriptions which can be tied in closely to diagonalising techniquescan be tied in closely to diagonalising techniques

For example, all incomputable sets of natural numbers are For example, all incomputable sets of natural numbers are solutions to diophantine equations, but in order to solutions to diophantine equations, but in order to construct such a set, you must use diagonalising construct such a set, you must use diagonalising techniques techniques

The lack of "natural" examples of incomputable sets has The lack of "natural" examples of incomputable sets has been used as an argument against being interested in such been used as an argument against being interested in such thingsthings


10

The search for "overt" The search for "overt" incomputability in phisycsincomputability in phisycs

The most straightforward approach would be to The most straightforward approach would be to take mathematical equations known to accurately take mathematical equations known to accurately describe some natural phenomenon, and to describe some natural phenomenon, and to extract solutions exhibiting incomputability in extract solutions exhibiting incomputability in some generally convincing formsome generally convincing form

There are a few proposals that illustrate There are a few proposals that illustrate incomputability which emerges in "natural incomputability which emerges in "natural mathematics"mathematics"

see for example:

M.B. Pour-El and J.J. Richards, Computability in Analysis and Physics, Springer-Verlag, 1989


11

The search for "overt" The search for "overt" incomputability in phisycsincomputability in phisycs

Despite these proposals, sceptics find plenty of Despite these proposals, sceptics find plenty of scope for arguing that the mathematics involved scope for arguing that the mathematics involved is not typical, of for throwing doubt on the role of is not typical, of for throwing doubt on the role of incomputability in it.incomputability in it.

Moreover, there can e quite plausible obstacles to Moreover, there can e quite plausible obstacles to overtness. For instance, there may be overtness. For instance, there may be mathematicalmathematical constraints on what constraints on what incomputabilities can be describedincomputabilities can be described


12

What is missing in our What is missing in our description of the Universe?description of the Universe?

Key element in many controversies: interaction Key element in many controversies: interaction between the local and the global, and among between the local and the global, and among different levels of descriptions.different levels of descriptions.

There are breakdowns in the reductive structures There are breakdowns in the reductive structures commonly relied on in sciencecommonly relied on in science

Examples:– quantum wave reduction and associated non-

locality– reductive nature of evolution– origins and exact nature of consciousness


13

Traditional reductions are no longer adequateTraditional reductions are no longer adequate

Example: rushing streamExample: rushing stream– Local dynamics based on a lower level description

are well understood– The continually evolving forms on the stream

surface seem to depend on globally emerging relationships not derivable from local analysis

– The form of the streams constrains the movement of the molecules of water, while at the same time being traceable back to those same movements


14

A mathematical analogyA mathematical analogy

A relation is A relation is definabledefinable from some other from some other relations/functions on a given domain if it can be relations/functions on a given domain if it can be described in terms of those relations/functionsdescribed in terms of those relations/functions

Example: the set of even integers is definable from + within the set of integers via the formula:

x (y) ( y + y = x ).– + is observable and can be algorithmically captured– But then, standing back from the structure (,+), we

observe something global: seems to fall into two distinct parts, with specific properties.


15

dynamic flow of water dynamic flow of water moleculesmolecules

dynamics of molecules are dynamics of molecules are known and computableknown and computable

global form of the streamglobal form of the stream

global form is connected global form is connected to laws of dynamics but it to laws of dynamics but it is hard to deduce its is hard to deduce its properties from the lawsproperties from the laws

dynamic flow into the dynamic flow into the structure: applications of structure: applications of the form the form m+n to integers to integers m, , n

addition can be computedaddition can be computed

appereance of two sets: appereance of two sets: even ad odd numberseven ad odd numbers

properties of these sets properties of these sets require further analysisrequire further analysis

For example, odd numbers flow into E, but numbers within E never get out of it. E is maximal subset having this property

A mathematical analogyA mathematical analogy


16

… … this is only a basic metaphor for other this is only a basic metaphor for other ways in which more or less unexpected ways in which more or less unexpected gloabal characteristics of strucutres gloabal characteristics of strucutres emerge quite deterministically from local emerge quite deterministically from local infrastructureinfrastructure


17

Does the Universe has a Does the Universe has a discrete model?discrete model?

The physical world is representable in a The physical world is representable in a discretized way (in terms of its quantum discretized way (in terms of its quantum structure), but it does not follow that the structure), but it does not follow that the mathematics needed is correspondingly discretemathematics needed is correspondingly discrete

Much of applied mathematics seems dependent Much of applied mathematics seems dependent on limiting processes and descriptions in terms on limiting processes and descriptions in terms of real numbersof real numbers

Non linear phenomena imply that one cannot Non linear phenomena imply that one cannot describe complex environments within a fixed describe complex environments within a fixed level of approximationlevel of approximation


18

It seems more likely that the observed It seems more likely that the observed discreteness is something discreteness is something imposedimposed by natural by natural laws on an underlying indiscrete mathematical laws on an underlying indiscrete mathematical modelmodel

Even with a finite model, the algorithmic content Even with a finite model, the algorithmic content associated with natural processes is very associated with natural processes is very different from a finite, discrete model in that it different from a finite, discrete model in that it entails entails uncompleted infinitiesuncompleted infinities

Real numbers in the form of uncompleted Real numbers in the form of uncompleted infinites feed into physical reality, and determine infinites feed into physical reality, and determine a mathematical model which is not discretea mathematical model which is not discrete


19

The Universe is a Turing The Universe is a Turing machine with oracles!machine with oracles!

To overcome the shortcomings of discrete To overcome the shortcomings of discrete models Turing introduced models Turing introduced machines with oraclesmachines with oracles

An oracle is a (possibly infinite) set of natural An oracle is a (possibly infinite) set of natural numbersnumbers

In Turing machines with oracles (TMO), the In Turing machines with oracles (TMO), the program can ask wether or not a specific number program can ask wether or not a specific number is in the set, by calling a special functionis in the set, by calling a special function

Essentially, TMO are capable of working with real Essentially, TMO are capable of working with real numbers, and generate a structure describable in numbers, and generate a structure describable in terms of terms of computale relations over real numberscomputale relations over real numbers


20

Why real numbers are not TM Why real numbers are not TM computable?computable?

In the proof of Turing Theorem we have built an In the proof of Turing Theorem we have built an incomputable real by means of diagonalisation, a incomputable real by means of diagonalisation, a technique that does not have an obvious technique that does not have an obvious physical counterpartphysical counterpart

The incomputability of reals, however, is not The incomputability of reals, however, is not generally due to "generally due to "diagonalisablediagonalisable" situations:" situations:

1. All real numbers are derivable as the limit of a sequence of computable numbers (rational numbers)


21

2. Even if one limits oneself to computable sequences of computable numbers, one still gets limiting reals which are not computable

3. These occurr every time that there is an absence of a computable modulus of convergence of the sequence

4. In other words, the link between input and output is broken: one does not know at any point in the sequence how close an approximation to the limiting real is currently being provided


22

Incomputable reals = ChaosIncomputable reals = Chaos

The impredictability of association between input The impredictability of association between input and output is the essence of sensitive and output is the essence of sensitive dependence on initial conditions of physical dependence on initial conditions of physical systemssystems

A matemathical model for chaotic situations A matemathical model for chaotic situations cannot be discrete, and have the sort of cannot be discrete, and have the sort of ingredients needed to generate incomputable ingredients needed to generate incomputable elementselements


23

Summary (so far)Summary (so far)– We have provided a preliminary basis for the belief

the the universe is not TM computable– Its algorithmic content is equivalent to the

structure of computable relations on reals– The computational counterpart of natural

processes could be Turing machines with oracles; however certain basic natural processes are not known to give rise to computable relations

– We now want to show that such phenomena are modelled via an analysis of invariance and definability of their basic Turing model


24

A model for the UniverseA model for the Universe

We are considering a scientific presentation of We are considering a scientific presentation of the Universe via some informative mathematical the Universe via some informative mathematical structurestructure

We make a comparison among this model and We make a comparison among this model and the Turing Universethe Turing Universe

The basic premise of the model is that existence The basic premise of the model is that existence takes the most general form allowed by takes the most general form allowed by considerations of internal consistenceconsiderations of internal consistence


25

Mathematical indications are that a large amount if Mathematical indications are that a large amount if information content is needed for the emergence of information content is needed for the emergence of anything like a classical Universe of well-defined anything like a classical Universe of well-defined individual objectsindividual objects

Such a Universe is Such a Universe is rigidrigid, in the sense that it has no , in the sense that it has no non-trivial automorphism leaving its properties non-trivial automorphism leaving its properties unchangedunchanged

Recent mathematical results suggest that the Recent mathematical results suggest that the Turing UniverseTuring Universe (the structure of computable (the structure of computable relations on reals) is not completely rigid: it has a relations on reals) is not completely rigid: it has a certain degree of flexibility, even if there are certain degree of flexibility, even if there are relatively few (at most relatively few (at most countably manycountably many) Turing ) Turing automorphismsautomorphisms


26

Basic structure of matterBasic structure of matter

Mathematical indications are that a low level of Mathematical indications are that a low level of information content goes with nonrigidity and information content goes with nonrigidity and lack of Turing-invariant individualslack of Turing-invariant individuals

The correspondent predition for the Universe The correspondent predition for the Universe would be that its most basic components may would be that its most basic components may materialise ambigously (lack of materialise ambigously (lack of definabilitydefinability))

The prediction is confirmed by classic The prediction is confirmed by classic experiments on subatomic particles, quantum experiments on subatomic particles, quantum vacuum, etc.vacuum, etc.


27

NonlocalityNonlocality Elaborating such low level of information at Elaborating such low level of information at

levels of Turing universe at which rigidity sets in levels of Turing universe at which rigidity sets in will produce new information content will produce new information content corresponding to a corresponding to a Turing invariantTuring invariant element element

The prediction is that there is a level of material The prediction is that there is a level of material existence which does not display the ambiguities existence which does not display the ambiguities of the quantum level, and whose interaction with of the quantum level, and whose interaction with the quantum level have the effect of removing the quantum level have the effect of removing such ambiguity (collapse of wave function)such ambiguity (collapse of wave function)

Since there is no obvios mathematical reason Since there is no obvios mathematical reason why quantum ambiguity should remain locally why quantum ambiguity should remain locally constrained, non-locality may appear in the constrained, non-locality may appear in the collapsecollapse


28

Nonlocality (2)Nonlocality (2)

The way in which definability asserts itself in the The way in which definability asserts itself in the Turing universe is not kwown to be computable, Turing universe is not kwown to be computable, which would explain the difficulties in predicting which would explain the difficulties in predicting exactly how such a collapse might materialise in exactly how such a collapse might materialise in practice, and the apparent randomness involvedpractice, and the apparent randomness involved


29

Elementary particlesElementary particles

One can only speculate about the origins of One can only speculate about the origins of subatomic structuresubatomic structure

One guess is that when we look at one such One guess is that when we look at one such particle we are observing an instantiation of a particle we are observing an instantiation of a relationrelation, defined on some lower level lacking any , defined on some lower level lacking any sort of observable formsort of observable form

This could be envisaged as a kind of formless This could be envisaged as a kind of formless soup of information content out of which emerge soup of information content out of which emerge peacks of definability in the form of subatomic peacks of definability in the form of subatomic particlesparticles


30

Origin of physical lawsOrigin of physical laws

The conjecture is that there is a parallel between The conjecture is that there is a parallel between natural laws and relations definable in an natural laws and relations definable in an appropriate fragment of the Turing universeappropriate fragment of the Turing universe

The prediction is that a Universe with sufficiently The prediction is that a Universe with sufficiently developed information content to replicate the developed information content to replicate the Turing universe Turing universe will manifest corresponding material will manifest corresponding material relationsrelations

The early Universe (immediately after the Big The early Universe (immediately after the Big Bang) could not replicate such a fragment, due to Bang) could not replicate such a fragment, due to the homogenisation of its information contentthe homogenisation of its information content


31

Complex systemsComplex systems A Turing definable relation does not necessarily A Turing definable relation does not necessarily

yield a computable relationship with the defining yield a computable relationship with the defining factorsfactors

However, working with relations at a given level However, working with relations at a given level of abstraction, there may well be computable of abstraction, there may well be computable relationships emerging, which may become the relationships emerging, which may become the basis for a new level of investigationbasis for a new level of investigation

The prediction is that invariant relations can The prediction is that invariant relations can emerge at higher level, relating to new entities emerge at higher level, relating to new entities reducible to one at previous levelsreducible to one at previous levels


32

… … to human mindsto human minds

A sufficiently complex part of the Universe could A sufficiently complex part of the Universe could parallel the same processes that define basic parallel the same processes that define basic relations of the Universerelations of the Universe

Such a system could simulate within it Such a system could simulate within it phenomena within the exterior worldphenomena within the exterior world

Mental processes, while being a microcosm of Mental processes, while being a microcosm of the greater universe, do appear to mirror part of the greater universe, do appear to mirror part of its complexity and hence its capaciy for global its complexity and hence its capaciy for global definitiondefinition

Incomputability, Undecidability and Nondeterminism: from formal systems to human minds

Documents

Transcript of Incomputability, Undecidability and Nondeterminism: from formal systems to human minds