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The Complexity of Infinite Computations In Models of Set Theory

2009
*
Logical Methods in Computer Science
*

second

doi:10.2168/lmcs-5(4:4)2009
fatcat:p5b7skes2bd3nf7qvbnmiygusm
*in*one*model**of*ZFC are \pi_2^0-*sets*, while*in*a different*model**of*ZFC both are analytic but non Borel*sets*. ... We show that a similar result holds for*the*class*of*languages*of**infinite*pictures which are recognized by B\"uchi tiling systems. ... Acknowledgement I wish to thank*the*referees for their very useful comments on a preliminary version*of*this paper. ...##
###
The complexity of the index sets of _0-categorical theories and of Ehrenfeucht theories
[article]

2006
*
arXiv
*
pre-print

We classify

arXiv:math/0610776v1
fatcat:l4ud2baig5d75j62hq324ggnpy
*the**computability*-theoretic*complexity**of*two index*sets**of*classes*of*first-order*theories*: We show that*the*property*of*being an _0-categorical*theory*is Π^0_3-complete; and*the*property ... Finally, as a corollary, we note that*the*properties*of*having only decidable*models*, and*of*having only*computable**models*, are both Π^1_1-complete. ... Both authors also wish to thank*the*University*of*Heidelberg and Klaus Ambos-Spies for their hospitality during*the*time this research was carried out. ...##
###
Cellular automata as models of complexity

1984
*
Nature
*

and 2 •

doi:10.1038/311419a0
fatcat:kbs6fi7g5vb6dhyuuzf6bdz6xq
*Infinite*distance at fixed speed*in*class 3 • Irregular propagation over an*infinite*range*in*class 4 • Recognized by idealized*computers*with a CPU containing a finite number*of*internal states ... initial configurations = Trivial formal language •*Set**of*configurations after any finite number*of*time steps = Regular language Formal Language*Theory*•*Set**of*configurations generated*in**the*first ...*Computation**theory*• For class 3 and 4: •*the*occurrence*of*a particular length n site value sequence*in**infinite*time is undecidable. • Decidable*in*finite time considering initial sequence length n0= ...##
###
Preface

2008
*
Archive for Mathematical Logic
*

*The*workshop brought researchers

*in*classical

*model*

*theory*and its applications together with researchers

*in*

*computable*

*model*

*theory*. ... Several topics

*in*

*model*

*theory*were presented, including three areas represented

*in*this volume: abstract

*model*

*theory*with links to

*computable*

*model*

*theory*(Laskowski),

*model*

*theory*

*of*fields (Martin-Pizarro ...

*The*editors would like to thank

*the*National Science Foundation for support under Special Year grant DMS 0532644. ...

##
###
Page 463 of Mathematical Reviews Vol. , Issue 86b
[page]

1986
*
Mathematical Reviews
*

For amusement, it is also proved that every nonstandard

*model**of*finite*set**theory*ZF~-© has a submodel satisfying*infinite**set**theory*ZF.” ... For an*infinite**set*A*of*natural numbers, let [A]” denote*the*fam- ily*of*all*infinite*subsets*of*A. ...##
###
Black Holes and Complexity via Constructible Universe

2020
*
Universe
*

*The*

*model*

*of*a general black hole is proposed based on formal tools from Zermelo–Fraenkel

*set*

*theory*like random forcing or minimal countable constructible

*model*Lα. ...

*The*Bekenstein–Hawking proportionality rule is shown to hold up to a multiplicative constant. Higher degrees

*of*randomness and algorithmic

*computational*

*complexity*are derived

*in*

*the*

*model*. ...

*The*requirements by anonymous referees improved

*the*readability

*of*

*the*paper very much. ...

##
###
CID - Computing with Infinite Data

2017
*
Bulletin of the European Association for Theoretical Computer Science
*

We will enlist expertise

dblp:journals/eatcs/Spreen17
fatcat:5wake7kvvbhihellbla7hgzvuq
*in*topology, logic, and*set**theory*, to address*the*question*of*which objects are representable and how they can be represented most efficiently. ... These ideas are greatly generalised*in*Weihrauch's type-two*theory**of*effectivity which aims to represent*infinite*data*of*any kind as streams*of*finite d ata. ...*infinite*objects, Markov*computability*and type-two*theory**of*effectivity, and to obtain a better understanding*of**the*first one*in*structural terms; and (II) to extend (effective) descriptive*theory*...##
###
Complexities Approach to Two Problems In Number Theory
[article]

2016
*
arXiv
*
pre-print

problems

arXiv:1610.04026v3
fatcat:5udl5eurnzh63c6ysh6b7kta6y
*in**the*non-logical discipline.Futhermore,we show that resource-bounded Kolmogorov*Complexity*and*computational**complexity*can at least provide tips or principles to mathematical problems*in**the*... not belong to P (period number)*in*their paper,another is*the*problem about existence*of*bounded coefficients*of*continued fraction expansion*of*transcendental number.Thus we show a new approach to mathematical ... Different*computation**Models**The*two*complexity*,that is Kolmogorov*complexity*and*computational**complexity*can solve problems by*the**model**in*[8] : we can refine*the*input*of**models*as*the*program p,*the*...##
###
Infinite Time Turing Machines: Supertask Computation
[article]

2002
*
arXiv
*
pre-print

By doing so, they provide a natural

arXiv:math/0212047v1
fatcat:sdwiu5e5bbfwbkux4c354ghxcu
*model**of*infinitary*computability*, a theoretical*setting*for*the*analysis*of**the*power and limitations*of*supertask algorithms. ...*Infinite*time Turing machines extend*the*operation*of*ordinary Turing machines into transfinite ordinal time. ...*the*known*complexity*hierarchies*of*descriptive*set**theory*(which provide a way*of*measuring*the**complexity**of*various*sets**of*natural numbers or reals*in*terms*of**the**complexity**of*their definitions) ...##
###
Infinite-Time TuringMachines
[chapter]

2008
*
Hypercomputation
*

By doing so, they provide a natural

doi:10.1007/978-0-387-49970-3_4
fatcat:b3xt6bt4vzcxnfzrgir6rtxcue
*model**of*infinitary*computability*, a theoretical*setting*for*the*analysis*of**the*power and limitations*of*supertask algorithms. ...*Infinite*time Turing machines extend*the*operation*of*ordinary Turing machines into transfinite ordinal time. ...*the*known*complexity*hierarchies*of*descriptive*set**theory*(which provide a way*of*measuring*the**complexity**of*various*sets**of*natural numbers or reals*in*terms*of**the**complexity**of*their definitions) ...##
###
Page 4750 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

98h:03056
stance

*complexity*conjecture”*of*K.-I. Ko et al. [*in*Structure*in**complexity**theory*(Berkeley, Calif., 1986), 197-217, Lecture Notes*in**Comput*. ... Here is a finite*model*and L[#] is a numerated version*of**the*L-*theory**of*#. ...##
###
The unity between quantum field computation, real computation, and quantum computation
[article]

2001
*
arXiv
*
pre-print

Quantum field

arXiv:quant-ph/0109015v1
fatcat:bt6x7xaibjecdb2xge5mlunbhq
*computation*(as enhanced by Wightman's*model**of*quantum field*theory*) involves*computation*over*the*continuum which is remarkably related to*the*real*computation**model**of*Smale. ...*The*unity*of*these*computing**models*, which seem to have grown largely independently, could shed new light into questions*of**computational**complexity*, into*the*central P (Polynomial time) versus NP (Non-deterministic ... These domains are*in**the*form*of*semialgebraic*sets**in**the*language used*in*real*computation*. ...##
###
Computational power of neural networks: a characterization in terms of Kolmogorov complexity

1997
*
IEEE Transactions on Information Theory
*

*The*

*computational*power

*of*recurrent neural networks is shown to depend ultimately on

*the*

*complexity*

*of*

*the*real constants (weights)

*of*

*the*network. ...

*In*particular, we reveal a full and proper hierarchy

*of*nonuniform

*complexity*classes associated with networks having weights

*of*increasing Kolmogorov

*complexity*. ... ACKNOWLEDGMENT

*The*authors wish to thank E. Sontag for his encouragement and support via many discussions and D. Harel for asking

*the*question that led them to Corollary 5.4. ...

##
###
Page 5883 of Mathematical Reviews Vol. , Issue 98I
[page]

1998
*
Mathematical Reviews
*

This answers an open question

*of*Papadimitriou. (6) New Kolmogorov-based constructions*of**the*following relativized worlds: (a) There exists an*infinite**set**in*P with no sparse*infinite*subsets*in*NP. ... size*of*a*set*using CD*complexity*avoiding*the*random string needed by Sipser. ...##
###
Page 2879 of Mathematical Reviews Vol. , Issue 2003d
[page]

2003
*
Mathematical Reviews
*

For example,

*the*statement (*in**the*Kolmogorov*complexity*parlance) K(x) <n corresponds,*in**the*combinatorial approach, to saying that x belongs to a*set**of*cardinality about 2”. ...*In*particular it is shown that there is a proper hierarchy*of**the*notions*of*learning from standard text,*in**the*basic switching*model*,*in**the*newtext switching*model*and*in**the*restart switch- ing*model*...
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