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The Complexity of Infinite Computations In Models of Set Theory

Olivier Finkel, Wolfgang Thomas
2009 Logical Methods in Computer Science  
second 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.  ... 
doi:10.2168/lmcs-5(4:4)2009 fatcat:p5b7skes2bd3nf7qvbnmiygusm

The complexity of the index sets of _0-categorical theories and of Ehrenfeucht theories [article]

Steffen Lempp, Theodore A. Slaman
2006 arXiv   pre-print
We classify 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.  ... 
arXiv:math/0610776v1 fatcat:l4ud2baig5d75j62hq324ggnpy

Cellular automata as models of complexity

Stephen Wolfram
1984 Nature  
and 2 • 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 TheorySet 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=  ... 
doi:10.1038/311419a0 fatcat:kbs6fi7g5vb6dhyuuzf6bdz6xq

Preface

Douglas Cenzer, Valentina Harizanov, David Marker, Carol Wood
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.  ... 
doi:10.1007/s00153-008-0111-5 fatcat:5fmakldyfvfnhdw53js6k3p74a

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

Jerzy Król, Paweł Klimasara
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.  ... 
doi:10.3390/universe6110198 fatcat:mil5ho2sgjekpbe5ejunh37vgm

CID - Computing with Infinite Data

Dieter Spreen
2017 Bulletin of the European Association for Theoretical Computer Science  
We will enlist expertise 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  ... 
dblp:journals/eatcs/Spreen17 fatcat:5wake7kvvbhihellbla7hgzvuq

Complexities Approach to Two Problems In Number Theory [article]

Yang Bai, Xiuli Wang
2016 arXiv   pre-print
problems 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  ... 
arXiv:1610.04026v3 fatcat:5udl5eurnzh63c6ysh6b7kta6y

Infinite Time Turing Machines: Supertask Computation [article]

Joel David Hamkins
2002 arXiv   pre-print
By doing so, they provide a natural 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)  ... 
arXiv:math/0212047v1 fatcat:sdwiu5e5bbfwbkux4c354ghxcu

Infinite-Time TuringMachines [chapter]

2008 Hypercomputation  
By doing so, they provide a natural 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)  ... 
doi:10.1007/978-0-387-49970-3_4 fatcat:b3xt6bt4vzcxnfzrgir6rtxcue

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]

A. C. Manoharan
2001 arXiv   pre-print
Quantum field 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.  ... 
arXiv:quant-ph/0109015v1 fatcat:bt6x7xaibjecdb2xge5mlunbhq

Computational power of neural networks: a characterization in terms of Kolmogorov complexity

J.L. Balcazar, R. Gavalda, H.T. Siegelmann
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.  ... 
doi:10.1109/18.605580 fatcat:bbpev5jiqrh4xhofiaygqtw55q

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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