Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata

Modanese, Augusto

doi:10.1007/978-3-030-79416-3_18

Cited by 2 publications

(4 citation statements)

References 22 publications

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“…Our main motivation is to analyze to what extent-if at all-the addition of randomness to the model is able to make up for inherent limitations of it. For instance, models of sublinear-time cellular automata are usually restricted to a local view of their input [16] and are also unable to cope with long unary subwords [15]. As a result of our analysis, we are able to demonstrate yet another connection between randomness and counting-as has been observed in the past in multiple areas of complexity theory (for various examples thereof, see [1,8]).…”

Section: Introductionmentioning

confidence: 57%

“…Theorem 2 indicates that unconditional time-efficient derandomization results for PACA are beyond reach of current techniques, so perhaps one should consider space-efficient derandomization instead. Indeed, it is straightforward to show that PACA can be simulated by space-efficient machines (e.g., using an adaptation of the algorithm from [15]); hence, it is possible to recast the many constructions of PRGs that fool space-bounded machines (e.g., [9,17]) as PRGs that fool PACA. Nevertheless, we expect better constructions can be obtained from exploiting the locality of PACA (which space-bounded machines do not suffer from).…”

Section: Pseudorandom Generatorsmentioning

confidence: 99%

“…Nevertheless, these efforts seem to have been almost exclusively devoted to the linear-or real-time case-to the detriment of the sublinear-time one [16]. This is particularly unfortunate since, as it was recently shown in [15], the study of sublineartime variants of cellular automata might help better direct efforts in resolving outstanding problems in computational complexity theory.…”

Section: Introductionmentioning

confidence: 99%

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Sublinear-Time Probabilistic Cellular Automata

Modanese¹

2022

Preprint

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We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA), accordingly. We consider both one-and two-sided error versions of the model (in the same spirit as the classical Turing machine classes RP and BPP) and establish a separation between the classes of languages they can recognize all the way up to o( √ n) time. We also prove that the derandomization of T (n)-time PACA (to polynomial-time deterministic cellular automata) for various regimes of T (n) = ω(log n) implies non-trivial derandomization results for the class RP (e.g., P = RP). Last but not least, as our main contribution we give a full characterization of the constant-time PACA classes: For one-sided error, the class is equal to that of the deterministic model; that is, we prove that constant-time one-sided error PACA can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we characterize the respective class in terms of a linear threshold condition and prove that it lies in-between the class of strictly locally testable languages (SLT) and that of locally threshold testable languages (LTT) while being incomparable to the locally testable languages (LT). ACM Subject Classification Theory of computation → Formal languages and automata theory Keywords and phrases Cellular automata, local computation, probabilistic modelsAcknowledgements I would like to thank Thomas Worsch for the many helpful discussions and feedback.

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Section: Introductionmentioning

confidence: 57%

Section: Pseudorandom Generatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sublinear-Time Probabilistic Cellular Automata

Modanese¹

2022

Preprint

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“…Theorem 6 and the original speedup of Oliveira and Santhanam can be interpreted as hardness magnification theorems. Hardness magnification is an approach to strong complexity lower bounds by reducing them to seemingly much weaker lower bounds developed in a series of recent papers [33,27,31,25,9,10,7,6,8,26,11], see [6] for a more comprehensive survey. For example, it turns out that in order to prove that functions computable in nondeterministic quasipolynomialtime are hard for NC 1 it suffices to show that a parameterized version of the minimum circuit size problem MCSP is hard for AC 0 [2].…”

Section: Learning Speedupmentioning

confidence: 99%

Learning algorithms from circuit lower bounds

Pich¹

2020

Preprint

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We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small circuits attempting to compute hard functions. As our main result we prove that if it is possible to find efficiently, in a particular interactive way, errors of many p-size circuits attempting to solve hard problems, then p-size circuits can be PAC learned over the uniform distribution with membership queries by circuits of subexponential size. The opposite implication holds as well. This provides a new characterisation of learning algorithms and extends the natural proofs barrier of Razborov and Rudich. The proof is based on a method of exploiting Nisan-Wigderson generators introduced by Krajíček (2010) and used to analyze complexity of circuit lower bounds in bounded arithmetic.An interesting consequence of known constructions of learning algorithms from circuit lower bounds is a learning speedup of Oliveira and Santhanam (2016). We present an alternative proof of this phenomenon and discuss its potential to advance the program of hardness magnification.

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Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata

Cited by 2 publications

References 22 publications

Sublinear-Time Probabilistic Cellular Automata

Sublinear-Time Probabilistic Cellular Automata

Learning algorithms from circuit lower bounds

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