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Compression Complexity
[article]

2017
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arXiv
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pre-print

The Kolmogorov complexity of x, denoted C(x), is the length of the shortest program that generates x. For such a simple definition, Kolmogorov complexity has a rich and deep theory, as well as applications to a wide variety of topics including learning theory, complexity lower bounds and SAT algorithms. Kolmogorov complexity typically focuses on decompression, going from the compressed program to the original string. This paper develops a dual notion of compression, the mapping from a string to

arXiv:1702.04779v1
fatcat:azs256nspbgndlcctui3lzmdvm
## more »

... ng from a string to its compressed version. Typical lossless compression algorithms such as Lempel-Ziv or Huffman Encoding always produce a string that will decompress to the original. We define a general compression concept based on this observation. For every m, we exhibit a single compression algorithm q of length about m which for n and strings x of length n >= m, the output of q will have length within n-m+O(1) bits of C(x). We also show this bound is tight in a strong way, for every n >= m there is an x of length n with C(x) about m such that no compression program of size slightly less than m can compress x at all. We also consider a polynomial time-bounded version of compression complexity and show that similar results for this version would rule out cryptographic one-way functions.##
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Bounding Rationality by Discounting Time
[article]

2009
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arXiv
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pre-print

The idea of discounting based on computation time was developed by

arXiv:0911.3162v1
fatcat:2kywohnrpre4fkq3o5yn7ew5fq
*Fortnow*[11] , where he used it for a variaton on the "program equilibria" framework devloped by Tennenholtz [12] ; moreover, a single ... As mentioned earlier,*Fortnow*[11] considers discounted computation time in this context to obtain a broader range of program equilibria rather than to model bounded rationality, and he allows only for ...##
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Robust Simulations and Significant Separations
[article]

2010
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arXiv
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pre-print

Our definition of robust simulations extends the notion of uniform hardness of Downey and

arXiv:1012.2034v1
fatcat:vfwmhda34ncfjgxzica7wcxd3q
*Fortnow*[DF03] . A set A is uniformly hard in the sense of Downey and*Fortnow*if A ∈ r.o.P. ... We use Proposition 27 to give an analogue of the result of Burhman,*Fortnow*and Thierauf [BFT98] that MAEXP ⊆ SIZE(poly) in the robustly often setting. ...##
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One Complexity Theorist's View of Quantum Computing
[article]

2000
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arXiv
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pre-print

For a background on relativization see the survey paper by

arXiv:quant-ph/0003035v1
fatcat:5f2kmepgsbdetcqiipkmuhif54
*Fortnow*[For94] .*Fortnow*and Rogers [FR99] observed that BQP ⊆ AWPP basically falls out of the characterization given in Section 2. ... Li [Li93] and Fenner,*Fortnow*, Kurtz and Li [FFKL93] defined and studied the class AWPP (stands for Almost-Wide Probabilistic Polynomial-time) extensively. ...##
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Complexity limitations on quantum computation
[article]

1998
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arXiv
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pre-print

The classes LWPP and AWPP were first defined by Fenner,

arXiv:cs/9811023v1
fatcat:mxiruxhcyneehe3qa6pey6wcoq
*Fortnow*and Kurtz [FFK94] and Fenner,*Fortnow*, Kurtz and Li [FFKL93] . ... Fenner,*Fortnow*, Kurtz and Li [FFKL93] give an interesting collapse for AWPP relative to generic oracles. ...##
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Multi-outcome and Multidimensional Market Scoring Rules
[article]

2012
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arXiv
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pre-print

Hanson's market scoring rules allow us to design a prediction market that still gives useful information even if we have an illiquid market with a limited number of budget-constrained agents. Each agent can "move" the current price of a market towards their prediction. While this movement still occurs in multi-outcome or multidimensional markets we show that no market-scoring rule, under reasonable conditions, always moves the price directly towards beliefs of the agents. We present a modified

arXiv:1202.1712v1
fatcat:5i4xsqwxrrbcnedjkl75a7kyca
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... present a modified version of a market scoring rule for budget-limited traders, and show that it does have the property that, from any starting position, optimal trade by a budget-limited trader will result in the market being moved towards the trader's true belief. This mechanism also retains several attractive strategic properties of the market scoring rule.##
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Sophistication Revisited
[chapter]

2003
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Lecture Notes in Computer Science
*

Antunes,

doi:10.1007/3-540-45061-0_23
fatcat:iuclnlpp7ba4jig2pee6ogaova
*Fortnow*, van Melkebeek and Vinodchandran [AFvMV06] consider logical depth as one instantiation of this more general theme and the authors propose several other variants and show many applications ...##
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Derandomizing from Random Strings
[article]

2009
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arXiv
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pre-print

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible

arXiv:0912.3162v1
fatcat:u3omgcizevfdrige55uyvq2fh4
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... not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.##
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Complexity of Combinatorial Market Makers
[article]

2008
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arXiv
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pre-print

RELATED WORK

arXiv:0802.1362v1
fatcat:taxeaucm2jdwnenfsjdsjyckxy
*Fortnow*et al. [11] study the computational complexity of finding acceptable trades among a set of bids in a Boolean combinatorial market. ... Following the notational conventions of*Fortnow*et al. [11] , we use ω ∈ φ to mean that the outcome ω satisfies the Boolean formula φ. Similarly, ω ∈ φ implies that the outcome ω does not satisfy φ. ...##
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Computational Complexity
[chapter]

2014
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Handbook of the History of Logic
*

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Sophistication Revisited

2007
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Theory of Computing Systems
*

Antunes,

doi:10.1007/s00224-007-9095-5
fatcat:txbx7utfjvh6xgktdjvntbq2zm
*Fortnow*, van Melkebeek and Vinodchandran [AFvMV06] consider logical depth as one instantiation of this more general theme and the authors propose several other variants and show many applications ...##
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Prediction and Dimension
[chapter]

2002
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Lecture Notes in Computer Science
*

Given a set X of sequences over a finite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X. (ii) The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequences in X. (iii) The feasible dimension of X is the polynomial-time effectivization of the classical

doi:10.1007/3-540-45435-7_26
fatcat:6hjf2pi3qnfrleaeix2tq4wm6i
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... f the classical Hausdorff dimension ("fractal dimension") of X. Predictability is known to be stable in the sense that the feasible predictability of X ∪ Y is always the minimum of the feasible predictabilities of X and Y. We show that deterministic predictability also has this property if X and Y are computably presentable. We show that deterministic predictability coincides with predictability on singleton sets. Our main theorem states that the feasible dimension of X is bounded above by the maximum entropy of the predictability of X and bounded below by the segmented self-information of the predictability of X, and that these bounds are tight.##
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Uniformly hard languages

2003
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Theoretical Computer Science
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Ladner (J. Assoc. Comput. Mach. 22 (1975) 155) showed that there are no minimal recursive sets under polynomial-time reductions. Given any recursive set A, Ladner constructs a set B such that B strictly reduces to A but B does not lie in P. The set B does have very long sequences of input lengths of easily computable instances. We examine whether Ladner's results hold if we restrict ourselves to "uniformly hard languages" which have no long sequences of easily computable instances. Under a hard

doi:10.1016/s0304-3975(02)00810-1
fatcat:ntrsqttmkzclfdt2u2p7sfr2am
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... ances. Under a hard to disprove assumption, we show that there exists a minimal recursive uniformly hard set under honest many-one polynomial-time reductions.##
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Matrix Multiplication and Binary Space Partitioning Trees : An Exploration
[article]

2020
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arXiv
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pre-print

Slagle

arXiv:2012.05365v1
fatcat:txccbeygmbabfaaqy3o3f7wghm
*Fortnow*University of Arizona December 11, 2020 We can define a more conservative pruning rule of |β + γ| ≤ / √ 2 ≤ | sin α| + | cos α| (2.3) For future analyses, we apply the more conservative ...##
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Repeated Matching Pennies with Limited Randomness
[article]

2011
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arXiv
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pre-print

It is in fact easy to come up with settings, as in

arXiv:1102.1096v2
fatcat:y7e4vr7icjbi3cajasnnc43jzy
*Fortnow*and Santhanam [4] , in which simply computing a best response strategy involves solving a computationally hard problem. ...
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