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Incomputability, Undecidability and Nondeterminism: from formal systems to human minds Filippo Cacace Università Campus Biomedico, Roma Ponte di Legno,

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December This presentation is based on the work by S. Barry Cooper and Piergiorgio Odifreddi See: Incomputability in Nature unpublished manuscript, 2001

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December The questions we want to ask 1. Does incomputability occur in Nature? 2. Is a theory of incomputability useful for our description of the Universe?

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Objections to incomputability in 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

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Objections to incomputability 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 mathematics may be theoretical, but they play no role in physics Conclusion: the origins of incomputabilty in mathematics may be theoretical, but they play no role in physics

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Birth of incomputability theory: Turing Theorem "There is no effective procedure (algorithm) for deciding whether or not a generic program ever 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: NumberProgramOutput 1p 1 r 1 2p 2 r 2 ……… r i is computed by p i. If the program does not halt, r i =0

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Turing Theorem (2) r ij is the j-th digit of r i. Now consider the real r*: r*=0.(r 11 +1)(r 22 +1)(r 33 +1)… (i.e. if r 1 =0.4923…,r 2 =0.1135…, r 3 = , then r*=0.528…) r* cannot be in the list (since is different from any other r i ), 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…

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Turing (1936) Gödel (1931) "There is no formal axiomatic system from which all 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.

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Incomputable sets The incomputability of real numbers is equivalent to the incomputability of sets of integers. Such sets are called recursively enumerable sets (or computably enumerable sets) The incomputability of real numbers is equivalent to the incomputability of sets of integers. Such sets are called recursively enumerable sets (or computably enumerable sets) Most "incomputable" sets come with descriptions which can be tied in closely to diagonalising techniques Most "incomputable" sets come with descriptions which can be tied in closely to diagonalising techniques For example, all incomputable sets of natural numbers are solutions to diophantine equations, but in order to construct such a set, you must use diagonalising techniques For example, all incomputable sets of natural numbers are solutions to diophantine equations, but in order to construct such a set, you must use diagonalising techniques The lack of "natural" examples of incomputable sets has been used as an argument against being interested in such things The lack of "natural" examples of incomputable sets has been used as an argument against being interested in such things

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December The search for "overt" incomputability in phisycs The most straightforward approach would be to take mathematical equations known to accurately describe some natural phenomenon, and to extract solutions exhibiting incomputability in some generally convincing form The most straightforward approach would be to take mathematical equations known to accurately describe some natural phenomenon, and to extract solutions exhibiting incomputability in some generally convincing form There are a few proposals that illustrate incomputability which emerges in "natural mathematics" There are a few proposals that illustrate incomputability which emerges in "natural mathematics" see for example: M.B. Pour-El and J.J. Richards, Computability in Analysis and Physics, Springer-Verlag, 1989

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December The search for "overt" incomputability in phisycs Despite these proposals, sceptics find plenty of scope for arguing that the mathematics involved is not typical, of for throwing doubt on the role of incomputability in it. Despite these proposals, sceptics find plenty of scope for arguing that the mathematics involved is not typical, of for throwing doubt on the role of incomputability in it. Moreover, there can e quite plausible obstacles to overtness. For instance, there may be mathematical constraints on what incomputabilities can be described Moreover, there can e quite plausible obstacles to overtness. For instance, there may be mathematical constraints on what incomputabilities can be described

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December What is missing in our description of the Universe? Key element in many controversies: interaction between the local and the global, and among different levels of descriptions. Key element in many controversies: interaction between the local and the global, and among different levels of descriptions. There are breakdowns in the reductive structures commonly relied on in science There are breakdowns in the reductive structures commonly relied on in science Examples: – quantum wave reduction and associated non- locality – reductive nature of evolution – origins and exact nature of consciousness

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Traditional reductions are no longer adequate Traditional reductions are no longer adequate Example: rushing stream Example: 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

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December A mathematical analogy A relation is definable from some other relations/functions on a given domain if it can be described in terms of those relations/functions A relation is definable from some other relations/functions on a given domain if it can be described 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.

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December dynamic flow of water molecules dynamic flow of water molecules dynamics of molecules are known and computable dynamics of molecules are known and computable global form of the stream global form of the stream global form is connected to laws of dynamics but it is hard to deduce its properties from the laws global form is connected to laws of dynamics but it is hard to deduce its properties from the laws dynamic flow into the structure: applications of the form m+n to integers m, n addition can be computed appereance of two sets: even ad odd numbers properties of these sets require 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 analogy

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December … this is only a basic metaphor for other ways in which more or less unexpected gloabal characteristics of strucutres emerge quite deterministically from local infrastructure

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Does the Universe has a discrete model? The physical world is representable in a discretized way (in terms of its quantum structure), but it does not follow that the mathematics needed is correspondingly discrete The physical world is representable in a discretized way (in terms of its quantum structure), but it does not follow that the mathematics needed is correspondingly discrete Much of applied mathematics seems dependent on limiting processes and descriptions in terms of real numbers Much of applied mathematics seems dependent on limiting processes and descriptions in terms of real numbers Non linear phenomena imply that one cannot describe complex environments within a fixed level of approximation Non linear phenomena imply that one cannot describe complex environments within a fixed level of approximation

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December It seems more likely that the observed discreteness is something imposed by natural laws on an underlying indiscrete mathematical model It seems more likely that the observed discreteness is something imposed by natural laws on an underlying indiscrete mathematical model Even with a finite model, the algorithmic content associated with natural processes is very different from a finite, discrete model in that it entails uncompleted infinities Even with a finite model, the algorithmic content associated with natural processes is very different from a finite, discrete model in that it entails uncompleted infinities Real numbers in the form of uncompleted infinites feed into physical reality, and determine a mathematical model which is not discrete Real numbers in the form of uncompleted infinites feed into physical reality, and determine a mathematical model which is not discrete

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December The Universe is a Turing machine with oracles! To overcome the shortcomings of discrete models Turing introduced machines with oracles To overcome the shortcomings of discrete models Turing introduced machines with oracles An oracle is a (possibly infinite) set of natural numbers An oracle is a (possibly infinite) set of natural numbers In Turing machines with oracles (TMO), the program can ask wether or not a specific number is in the set, by calling a special function In Turing machines with oracles (TMO), the program can ask wether or not a specific number is in the set, by calling a special function Essentially, TMO are capable of working with real numbers, and generate a structure describable in terms of computale relations over real numbers Essentially, TMO are capable of working with real numbers, and generate a structure describable in terms of computale relations over real numbers

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Why real numbers are not TM computable? In the proof of Turing Theorem we have built an incomputable real by means of diagonalisation, a technique that does not have an obvious physical counterpart In the proof of Turing Theorem we have built an incomputable real by means of diagonalisation, a technique that does not have an obvious physical counterpart The incomputability of reals, however, is not generally due to "diagonalisable" situations: The incomputability of reals, however, is not generally due to "diagonalisable" situations: 1. All real numbers are derivable as the limit of a sequence of computable numbers (rational numbers)

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

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Incomputable reals = Chaos The impredictability of association between input and output is the essence of sensitive dependence on initial conditions of physical systems The impredictability of association between input and output is the essence of sensitive dependence on initial conditions of physical systems A matemathical model for chaotic situations cannot be discrete, and have the sort of ingredients needed to generate incomputable elements A matemathical model for chaotic situations cannot be discrete, and have the sort of ingredients needed to generate incomputable elements

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

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December A model for the Universe We are considering a scientific presentation of the Universe via some informative mathematical structure We are considering a scientific presentation of the Universe via some informative mathematical structure We make a comparison among this model and the Turing Universe We make a comparison among this model and the Turing Universe The basic premise of the model is that existence takes the most general form allowed by considerations of internal consistence The basic premise of the model is that existence takes the most general form allowed by considerations of internal consistence

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Mathematical indications are that a large amount if information content is needed for the emergence of anything like a classical Universe of well-defined individual objects Mathematical indications are that a large amount if information content is needed for the emergence of anything like a classical Universe of well-defined individual objects Such a Universe is rigid, in the sense that it has no non-trivial automorphism leaving its properties unchanged Such a Universe is rigid, in the sense that it has no non-trivial automorphism leaving its properties unchanged Recent mathematical results suggest that the Turing Universe (the structure of computable relations on reals) is not completely rigid: it has a certain degree of flexibility, even if there are relatively few (at most countably many) Turing automorphisms Recent mathematical results suggest that the Turing Universe (the structure of computable relations on reals) is not completely rigid: it has a certain degree of flexibility, even if there are relatively few (at most countably many) Turing automorphisms

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Basic structure of matter Mathematical indications are that a low level of information content goes with nonrigidity and lack of Turing-invariant individuals Mathematical indications are that a low level of information content goes with nonrigidity and lack of Turing-invariant individuals The correspondent predition for the Universe would be that its most basic components may materialise ambigously (lack of definability) The correspondent predition for the Universe would be that its most basic components may materialise ambigously (lack of definability) The prediction is confirmed by classic experiments on subatomic particles, quantum vacuum, etc. The prediction is confirmed by classic experiments on subatomic particles, quantum vacuum, etc.

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Nonlocality Elaborating such low level of information at levels of Turing universe at which rigidity sets in will produce new information content corresponding to a Turing invariant element Elaborating such low level of information at levels of Turing universe at which rigidity sets in will produce new information content corresponding to a Turing invariant element The prediction is that there is a level of material existence which does not display the ambiguities of the quantum level, and whose interaction with the quantum level have the effect of removing such ambiguity (collapse of wave function) The prediction is that there is a level of material existence which does not display the ambiguities of the quantum level, and whose interaction with the quantum level have the effect of removing such ambiguity (collapse of wave function) Since there is no obvios mathematical reason why quantum ambiguity should remain locally constrained, non-locality may appear in the collapse Since there is no obvios mathematical reason why quantum ambiguity should remain locally constrained, non-locality may appear in the collapse

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Nonlocality (2) The way in which definability asserts itself in the Turing universe is not kwown to be computable, which would explain the difficulties in predicting exactly how such a collapse might materialise in practice, and the apparent randomness involved

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Elementary particles One can only speculate about the origins of subatomic structure One can only speculate about the origins of subatomic structure One guess is that when we look at one such particle we are observing an instantiation of a relation, defined on some lower level lacking any sort of observable form One guess is that when we look at one such particle we are observing an instantiation of a relation, defined on some lower level lacking any sort of observable form This could be envisaged as a kind of formless soup of information content out of which emerge peacks of definability in the form of subatomic particles This could be envisaged as a kind of formless soup of information content out of which emerge peacks of definability in the form of subatomic particles

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Origin of physical laws The conjecture is that there is a parallel between natural laws and relations definable in an appropriate fragment of the Turing universe The conjecture is that there is a parallel between natural laws and relations definable in an appropriate fragment of the Turing universe The prediction is that a Universe with sufficiently developed information content to replicate the Turing universe will manifest corresponding material relations The prediction is that a Universe with sufficiently developed information content to replicate the Turing universe will manifest corresponding material relations The early Universe (immediately after the Big Bang) could not replicate such a fragment, due to the homogenisation of its information content The early Universe (immediately after the Big Bang) could not replicate such a fragment, due to the homogenisation of its information content

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December Complex systems A Turing definable relation does not necessarily yield a computable relationship with the defining factors A Turing definable relation does not necessarily yield a computable relationship with the defining factors However, working with relations at a given level of abstraction, there may well be computable relationships emerging, which may become the basis for a new level of investigation However, working with relations at a given level of abstraction, there may well be computable relationships emerging, which may become the basis for a new level of investigation The prediction is that invariant relations can emerge at higher level, relating to new entities reducible to one at previous levels The prediction is that invariant relations can emerge at higher level, relating to new entities reducible to one at previous levels

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F. Cacace - Incomputability, Undecidability and Nondeterminism - Ponte di Legno, December … to human minds A sufficiently complex part of the Universe could parallel the same processes that define basic relations of the Universe A sufficiently complex part of the Universe could parallel the same processes that define basic relations of the Universe Such a system could simulate within it phenomena within the exterior world Such a system could simulate within it phenomena within the exterior world Mental processes, while being a microcosm of the greater universe, do appear to mirror part of its complexity and hence its capaciy for global definition Mental processes, while being a microcosm of the greater universe, do appear to mirror part of its complexity and hence its capaciy for global definition

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