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Uncomputably Noisy Ergodic Limits

Jeremy Avigad
2012 Notre Dame Journal of Formal Logic  
function needs to be to approximate the limit to within a given epsilon.  ...  V'yugin has shown that there are a computable shift-invariant measure on Cantor space and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages A_n  ...  Intuitively, h k is a "noisy" function of complexity k.  ... 
doi:10.1215/00294527-1716757 fatcat:td7nm6j5njhrpnv5n7zkqnf32u

Uncomputably Noisy Ergodic Limits

Jeremy Avigad
2018
function needs to be to approximate the limit to within a given ε.  ...  V'yugin has shown that there are a computable shift-invariant measure on 2N and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages Anf.  ...  Intuitively, h k is a "noisy" function of complexity k.  ... 
doi:10.1184/r1/6493031 fatcat:kx2io5g4mvfjzlnp2sqb2tvbcq

Cold dynamics in cellular automata: a tutorial

Guillaume Theyssier
2022 Natural Computing  
A rich zoo of properties is presented and discussed: nilpotency and asymptotic, generic or mu-variants, unique ergodicity, convergence, bounded-changeness, freezingness.  ...  Besides dynamical considerations, we also focus on computational aspects: we show how such 'cold cellular automata' can still compute under their dynamical constraint, and what are their computational limitation  ...  It appears that in many cases the resulting noisy CA is ergodic [52] as a probabilistic CA. In fact, proving that a particular noisy CA is not ergodic turns out to be difficult.  ... 
doi:10.1007/s11047-022-09886-2 fatcat:e4ppv36pgbfgpdn46df54ivb7u

Signatures of infinity: Nonergodicity and resource scaling in prediction, complexity, and learning

James P. Crutchfield, Sarah Marzen
2015 Physical Review E  
We introduce a simple analysis of the structural complexity of infinite-memory processes built from random samples of stationary, ergodic finite-memory component processes.  ...  We draw out consequences for the resource divergences that delineate the structural hierarchy of ergodic processes and for processes that are themselves hierarchical.  ...  All these are uncomputable, though, even if one is given a generative model.  ... 
doi:10.1103/physreve.91.050106 pmid:26066103 fatcat:zbfuofbzxjd5vcjsuxhpr6ayvy

Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning [article]

James P. Crutchfield, Sarah Marzen
2015 arXiv   pre-print
We introduce a simple analysis of the structural complexity of infinite-memory processes built from random samples of stationary, ergodic finite-memory component processes.  ...  We draw out consequences for the resource divergences that delineate the structural hierarchy of ergodic processes and for processes that are themselves hierarchical.  ...  All these are uncomputable, though, even if one is given a generative model.  ... 
arXiv:1504.00386v1 fatcat:3cicfswlijdyjjwf4txyk76u6a

Ensemble Inference Methods for Models With Noisy and Expensive Likelihoods [article]

Oliver R. A. Dunbar, Andrew B. Duncan, Andrew M. Stuart, Marie-Therese Wolfram
2022 arXiv   pre-print
However, model complexity often leads to parameter-to-data maps which are expensive to evaluate and are only available through noisy approximations.  ...  The objective evaluations (blue circles) are noisy leading to rapid fluctuations around a visible convex objective function (defined by the, in practice uncomputable, infinite time-average limit.)  ...  For ergodic, mixing dynamical systems a central limit theorem may sometimes be proven to hold [2] , or empirically observed, for data drawn at random from the invariant measure.  ... 
arXiv:2104.03384v2 fatcat:xskk7dxnm5dttb4x42ab45edee

On the Theoretical Properties of the Exchange Algorithm [article]

Guanyang Wang
2021 arXiv   pre-print
or 'Does the exchange algorithm admit a Central Limit Theorem?' have not been answered yet.  ...  We compare the exchange algorithm with the original Metropolis--Hastings algorithm and provide both necessary and sufficient conditions for the geometric ergodicity of the exchange algorithm.  ...  of a Markov chain Central Limit Theorem (CLT).  ... 
arXiv:2005.09235v4 fatcat:bzbcwknmwrg2ncviqt7ta6duq4

Multiuser MIMO Achievable Rates With Downlink Training and Channel State Feedback

Giuseppe Caire, Nihar Jindal, Mari Kobayashi, Niranjay Ravindran
2010 IEEE Transactions on Information Theory  
We consider a MIMO fading broadcast channel and compute achievable ergodic rates when channel state information is acquired at the receivers via downlink training and it is provided to the transmitter  ...  These results, as ours, rely on the behavior of the noisy prediction error for small . C.  ...  Section III develops bounds on the ergodic rates achievable by the baseline scheme.  ... 
doi:10.1109/tit.2010.2046225 fatcat:aks2ndmqzbbcxhwibswhtvpcni

Symbolic Dynamics for Discrete Adaptive Games [article]

Cosma Rohilla Shalizi (Center for the Study of Complex Systems, University of Michigan; Santa Fe Institute) and David J. Albers (Physics Department, University of Wisconsin-Madison; the Santa Fe Institute)
2003 arXiv   pre-print
The transients are, however, bad news for methods based on either ergodic or thermodynamic limits.  ...  The proofs of these assertions are somewhat complicated results from ergodic theory, well-summarized in Ref. [32] . D.  ... 
arXiv:cond-mat/0207407v2 fatcat:g7ldqi2ncvh2fmdtzsgaasf6je

Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization [article]

Yuval R. Sanders, Dominic W. Berry, Pedro C. S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, Ryan Babbush
2020 arXiv   pre-print
However, today, our ability to evaluate quantum heuristics through experimentation is limited since the only available quantum computers are small and noisy [1] .  ...  The idea is motivated by the search of configuration space in the classically non-ergodic phase associated with hard optimization problems.  ... 
arXiv:2007.07391v2 fatcat:vd4yv7w2mzapbnatgkrf3r23ie

Separation Theorems for Phase-Incoherent Multiple-User Channels [article]

Hamidreza Ebrahimzadeh Saffar, Ehsan Haj Mirza Alian, Patrick Mitran
2011 arXiv   pre-print
In all cases, the input signals are assumed to undergo non-ergodic phase shifts due to the channel.  ...  We now further upper bound C n (θ) and then take the limit and intersection.  ...  In many systems, it is difficult to know phase shifts at the transmitter side due to the delay and resource limits in feedback transmission.  ... 
arXiv:1110.3062v1 fatcat:nhoftq7jj5gdzmd6ndz3yvhjdy

Lempel-Ziv complexity analysis of one dimensional cellular automata

E. Estevez-Rams, R. Lora-Serrano, C. A. J. Nunes, B. Aragón-Fernández
2015 Chaos  
Lempel-Ziv complexity carries a number of practical advantages while avoiding the uncomputable nature of Kolmogorov randomness.  ...  As Kolmogorov randomness is uncomputable, a practical alternative to equation (12) is needed.  ...  The approach avoids the limitations in system size, inherent in using compression software and allows to analyze large systems.  ... 
doi:10.1063/1.4936876 pmid:26723145 fatcat:q6q3vqy2anbzzbazon54tl7dg4

An Algorithmic Approach to Emergence [article]

Charles Alexandre Bédard, Geoffroy Bergeron
2022 arXiv   pre-print
Because algorithmic complexity is uncomputable, so is the structure function.  ...  They are expressed in terms of models that distillate the structures from the apparently noisy boundary conditions.  ... 
arXiv:2205.12997v1 fatcat:d5t3hpjvurbtbk62vhz7wg2h5i

Exact Complexity: The Spectral Decomposition of Intrinsic Computation [article]

James P. Crutchfield, Christopher J. Ellison, Paul M. Riechers
2013 arXiv   pre-print
Note that W 1 = |1 π W |, which is always the case for an ergodic process.  ...  Note that W 1 = |1 π W |, again, since the timelike subprocess is ergodic.  ... 
arXiv:1309.3792v1 fatcat:ev24dje3gbfr5huf6zbltpaggi

Optimal high-level descriptions of dynamical systems [article]

David H. Wolpert, Joshua A. Grochow, Eric Libby, Simon DeDeo
2015 arXiv   pre-print
Furthermore, AIC is formally uncomputable, so one has to settle for results concerning asymptotic behavior.  ...  Indeed, even if we allowed an infinitely expandable RAM, such a cost would be uncomputable, in general.  ... 
arXiv:1409.7403v2 fatcat:tsilyydsgbeifevfscsfxxikry
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