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Compression Complexity [article]

Stephen Fenner, Lance Fortnow
2017 arXiv   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
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.
arXiv:1702.04779v1 fatcat:azs256nspbgndlcctui3lzmdvm

Bounding Rationality by Discounting Time [article]

Lance Fortnow, Rahul Santhanam
2009 arXiv   pre-print
The idea of discounting based on computation time was developed by 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  ... 
arXiv:0911.3162v1 fatcat:2kywohnrpre4fkq3o5yn7ew5fq

Robust Simulations and Significant Separations [article]

Lance Fortnow, Rahul Santhanam
2010 arXiv   pre-print
Our definition of robust simulations extends the notion of uniform hardness of Downey and 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.  ... 
arXiv:1012.2034v1 fatcat:vfwmhda34ncfjgxzica7wcxd3q

One Complexity Theorist's View of Quantum Computing [article]

Lance Fortnow
2000 arXiv   pre-print
For a background on relativization see the survey paper by 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.  ... 
arXiv:quant-ph/0003035v1 fatcat:5f2kmepgsbdetcqiipkmuhif54

Complexity limitations on quantum computation [article]

Lance Fortnow, John D. Rogers
1998 arXiv   pre-print
The classes LWPP and AWPP were first defined by Fenner, 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.  ... 
arXiv:cs/9811023v1 fatcat:mxiruxhcyneehe3qa6pey6wcoq

Multi-outcome and Multidimensional Market Scoring Rules [article]

Lance Fortnow, Rahul Sami
2012 arXiv   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
more » ... 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.
arXiv:1202.1712v1 fatcat:5i4xsqwxrrbcnedjkl75a7kyca

Sophistication Revisited [chapter]

Luís Antunes, Lance Fortnow
2003 Lecture Notes in Computer Science  
Antunes, 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  ... 
doi:10.1007/3-540-45061-0_23 fatcat:iuclnlpp7ba4jig2pee6ogaova

Derandomizing from Random Strings [article]

Harry Buhrman and Lance Fortnow and Michal Koucký and Bruno Loff
2009 arXiv   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
more » ... 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.
arXiv:0912.3162v1 fatcat:u3omgcizevfdrige55uyvq2fh4

Complexity of Combinatorial Market Makers [article]

Yiling Chen, Lance Fortnow, Nicolas Lambert, David M. Pennock, Jennifer Wortman
2008 arXiv   pre-print
RELATED WORK 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.  ... 
arXiv:0802.1362v1 fatcat:taxeaucm2jdwnenfsjdsjyckxy

Computational Complexity [chapter]

Lance Fortnow, Steven Homer
2014 Handbook of the History of Logic  
doi:10.1016/b978-0-444-51624-4.50011-3 fatcat:2i32xxd6z5gw7gfyq7figio6my

Sophistication Revisited

Luís Antunes, Lance Fortnow
2007 Theory of Computing Systems  
Antunes, 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  ... 
doi:10.1007/s00224-007-9095-5 fatcat:txbx7utfjvh6xgktdjvntbq2zm

Prediction and Dimension [chapter]

Lance Fortnow, Jack H. Lutz
2002 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
more » ... 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.
doi:10.1007/3-540-45435-7_26 fatcat:6hjf2pi3qnfrleaeix2tq4wm6i

Uniformly hard languages

Rod Downey, Lance Fortnow
2003 Theoretical Computer Science  
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
more » ... 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.
doi:10.1016/s0304-3975(02)00810-1 fatcat:ntrsqttmkzclfdt2u2p7sfr2am

Matrix Multiplication and Binary Space Partitioning Trees : An Exploration [article]

CNP Slagle, Lance Fortnow
2020 arXiv   pre-print
Slagle 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  ... 
arXiv:2012.05365v1 fatcat:txccbeygmbabfaaqy3o3f7wghm

Repeated Matching Pennies with Limited Randomness [article]

Michele Budinich, Lance Fortnow
2011 arXiv   pre-print
It is in fact easy to come up with settings, as in Fortnow and Santhanam [4] , in which simply computing a best response strategy involves solving a computationally hard problem.  ... 
arXiv:1102.1096v2 fatcat:y7e4vr7icjbi3cajasnnc43jzy
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