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Optimality of size-degree tradeoffs for polynomial calculus

Nicola Galesi, Massimo Lauria
2010 ACM Transactions on Computational Logic  
There are methods to turn short refutations in Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) into refutations of low degree.  ...  We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω( √ m).  ...  ACKNOWLEDGMENT We would like to thank the anonymous reviewers for the careful work and the several suggestions. Some of them were substantial and really improved this article.  ... 
doi:10.1145/1838552.1838556 fatcat:yhp2rsc4u5d5fgmgc2hhuc7j3i

Semi-Algebraic Proofs, IPS Lower Bounds and the τ-Conjecture: Can a Natural Number be Negative? [article]

Yaroslav Alekseev and Dima Grigoriev and Edward A. Hirsch and Iddo Tzameret
2019 arXiv   pre-print
We introduce the binary value principle which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic  ...  complexity.  ...  Acknowledgement We wish to thank Michael Forbes, Dima Itsykson, Toni Pitassi and Dima Sokolov for useful discussions at various stages of this work.  ... 
arXiv:1911.06738v1 fatcat:ptrubfzcavaela42q7fkztc5ye

Automated Precision Analysis: A Polynomial Algebraic Approach

David Boland, George A. Constantinides
2010 2010 18th IEEE Annual International Symposium on Field-Programmable Custom Computing Machines  
However, the flexibility in the number representation is one of the key factors that can only be exploited on FPGAs, unlike GPUs and general purpose processors, and hence ignoring this potential significantly  ...  We also show it achieves comparable bounds to recent literature in a small fraction of the execution time, with greater scalability.  ...  individual monomial from the left hand side of equations (11) and (12) , using a polynomial of the form (17) , and we then note that the sum of polynomials created in this fashion would be a GHR.  ... 
doi:10.1109/fccm.2010.32 dblp:conf/fccm/BolandC10 fatcat:2tjsaioncjhzvoirizzt6fkstu

Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields

Antoine Joux, Vanessa Vitse
2011 Journal of Cryptology  
In 2008 and 2009, Gaudry and Diem proposed an index calculus method for the resolution of the discrete logarithm on the group of points of an elliptic curve defined over a small degree extension field  ...  In this paper, we study a variation of this index calculus method, improving the overall asymptotic complexity when n = Ω( 3 log 2 q).  ...  Their complexity is exponential in the size of the largest prime factor of the group cardinality; more precisely, the running time is of the order of the square root of this largest prime factor [36]  ... 
doi:10.1007/s00145-011-9116-z fatcat:zemlvnib35b4rbrua4gnv3ytxy

Bounding Variable Values and Round-Off Effects Using Handelman Representations

D. Boland, G. A. Constantinides
2011 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
The precision used in an algorithm affects the error and performance of individual computations, the memory usage and the potential parallelism for a fixed hardware budget.  ...  We demonstrate the proposed procedure on an iteration of the conjugate gradient algorithm, achieving proofs of bounds that can translate to global wordlength savings ranging from a few bits to proving  ...  ACKNOWLEDGEMENTS Supported in part by EU FP7 Project REFLECT.  ... 
doi:10.1109/tcad.2011.2161307 fatcat:4aqsnca3lbfutn35g44r4mbqtm

Proof Complexity (Dagstuhl Seminar 18051)

Albert Atserias, Jakob Nordström, Pavel Pudlák, Rahul Santhanam, Michael Wagner
2018 Dagstuhl Reports  
The study of proof complexity was initiated in [Cook and Reckhow 1979] as a way to attack the P vs.  ...  Proof complexity also gives a way of studying subsystems of Peano Arithmetic where the power of mathematical reasoning is restricted, and to quantify how complex different mathematical theorems are measured  ...  -Proof Complexity Russell Impagliazzo, Pavel Pudlák, and Jiří Sgall. Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Computational Complexity, 8(2):127-144, 1999.  ... 
doi:10.4230/dagrep.8.1.124 dblp:journals/dagstuhl-reports/AtseriasNPS18 fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs [chapter]

Albert Atserias
2013 Lecture Notes in Computer Science  
I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification.  ...  For example, resolution admits polynomial-size proofs of the least-number principle (every finite linear order has a least element) [40] , which underlies many inductive proofs.  ...  Acknowledgments We thank the comments of Allen Van Gelder and an anonymous referee on the preliminary draft of this paper.  ... 
doi:10.1007/978-3-642-39071-5_1 fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordstrom
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well.  ...  Proof size and space in PCR is defined in analogy with resolution, and the measure corresponding to width of clauses is (total) degree of polynomials.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
doi:10.1109/ccc.2014.36 dblp:conf/coco/AtseriasLN14 fatcat:uek3kmbdobaehdcplxysvsmoaa

Narrow Proofs May Be Maximally Long [article]

Albert Atserias, Massimo Lauria, Jakob Nordström
2014 arXiv   pre-print
Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well.  ...  Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
arXiv:1409.2731v1 fatcat:luvihb4my5dnnmegzob5lomsne

Separations in Proof Complexity and TFNP [article]

Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Tao
2022 arXiv   pre-print
These results can be interpreted in the language of total search problems. We show that , , are captured by unary-SA, unary-NS, and Reversible Resolution, respectively.  ...  Consequently, relative to an oracle, ⊈ and ⊈.  ...  Acknowledgements We thank Albert Atserias, Ilario Bonacina, Pritish Kamath, and David Steurer for discussions.  ... 
arXiv:2205.02168v1 fatcat:e7cuqyewirhgnettqtdxkqehrm

Subtraction-free complexity, cluster transformations, and spanning trees [article]

Sergey Fomin, Dima Grigoriev, Gleb Koshevoy
2014 arXiv   pre-print
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division.  ...  We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M.  ...  For each of the N local moves, the number of macroscopic operations involved is O(1), so the bit complexity of each move is polynomial in the size of the numbers involved (which is going to be logarithmic  ... 
arXiv:1307.8425v4 fatcat:f5muaguat5ffrhrcz2xmeaykp4

Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees

Sergey Fomin, Dima Grigoriev, Gleb Koshevoy
2014 Foundations of Computational Mathematics  
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division.  ...  We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M.  ...  For each of the N local moves, the number of macroscopic operations involved is O(1), so the bit complexity of each move is polynomial in the size of the numbers involved (which is going to be logarithmic  ... 
doi:10.1007/s10208-014-9231-y fatcat:2ypmfkpnqzfqdif7m6ks6npc4e

Sum of squares lower bounds for refuting any CSP [article]

Pravesh K. Kothari and Ryuhei Mori and Ryan O'Donnell and David Witmer
2017 arXiv   pre-print
We show that whenever the predicate P supports a t-wise uniform probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree d = Θ(n/Δ^2/(t-1)Δ) (which runs in time  ...  Together with recent work of Lee et al. [LRS15], our result also implies that any polynomial-size semidefinite programming relaxation for refutation requires at least Ω(n^(t+1)/2) constraints.  ...  Acknowledgment We would like to thank the Institute for Mathematical Sciences, National University of Singapore in 2016; a visit there was where some of the initial research for this work began.  ... 
arXiv:1701.04521v1 fatcat:x3vvztyaejbollddojoubeo6rq

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2015 Computational Complexity  
the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC^0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen  ...  We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including:  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1007/s00037-015-0103-x fatcat:ifoaveduubhmxlokr4pygqzjqq

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC 0 [p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen  ...  We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representationtheoretic framework suggested by geometric complexity theory (GCT), including: the  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1109/ccc.2014.35 dblp:conf/coco/Grochow14 fatcat:z5eytgp64jh2pclri4tlmky53e
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