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Optimality of size-degree tradeoffs for polynomial calculus

2010
*
ACM Transactions on Computational Logic
*

There are methods to turn short refutations

doi:10.1145/1838552.1838556
fatcat:yhp2rsc4u5d5fgmgc2hhuc7j3i
*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. ...##
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Semi-Algebraic Proofs, IPS Lower Bounds and the τ-Conjecture: Can a Natural Number be Negative?
[article]

2019
*
arXiv
*
pre-print

We introduce the binary value principle which is a simple subset-

arXiv:1911.06738v1
fatcat:ptrubfzcavaela42q7fkztc5ye
*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. ...##
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Automated Precision Analysis: A Polynomial Algebraic Approach

2010
*
2010 18th IEEE Annual International Symposium on Field-Programmable Custom Computing Machines
*

However, the flexibility

doi:10.1109/fccm.2010.32
dblp:conf/fccm/BolandC10
fatcat:2tjsaioncjhzvoirizzt6fkstu
*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. ...##
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Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields

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] ...

##
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Bounding Variable Values and Round-Off Effects Using Handelman Representations

2011
*
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
*

The precision used

doi:10.1109/tcad.2011.2161307
fatcat:4aqsnca3lbfutn35g44r4mbqtm
*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. ...##
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Proof Complexity (Dagstuhl Seminar 18051)

2018
*
Dagstuhl Reports
*

The study

doi:10.4230/dagrep.8.1.124
dblp:journals/dagstuhl-reports/AtseriasNPS18
fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu
*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. ...##
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The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs
[chapter]

2013
*
Lecture Notes in Computer Science
*

I will continue with R(k) systems, whose proof-search problem turned out to hide the

doi:10.1007/978-3-642-39071-5_1
fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m
*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. ...##
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Narrow Proofs May Be Maximally Long

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

Moreover, our lower bound generalizes to

doi:10.1109/ccc.2014.36
dblp:conf/coco/AtseriasLN14
fatcat:uek3kmbdobaehdcplxysvsmoaa
*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 ...##
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Narrow Proofs May Be Maximally Long
[article]

2014
*
arXiv
*
pre-print

Moreover, our lower bound generalizes to

arXiv:1409.2731v1
fatcat:luvihb4my5dnnmegzob5lomsne
*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 ...##
###
Separations in Proof Complexity and TFNP
[article]

2022
*
arXiv
*
pre-print

These results can be interpreted

arXiv:2205.02168v1
fatcat:e7cuqyewirhgnettqtdxkqehrm
*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. ...##
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Subtraction-free complexity, cluster transformations, and spanning trees
[article]

2014
*
arXiv
*
pre-print

Subtraction-free computational

arXiv:1307.8425v4
fatcat:f5muaguat5ffrhrcz2xmeaykp4
*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 ...##
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Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees

2014
*
Foundations of Computational Mathematics
*

Subtraction-free computational

doi:10.1007/s10208-014-9231-y
fatcat:2ypmfkpnqzfqdif7m6ks6npc4e
*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 ...##
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Sum of squares lower bounds for refuting any CSP
[article]

2017
*
arXiv
*
pre-print

We show that whenever the predicate P supports a t-wise uniform probability distribution on its satisfying assignments, the

arXiv:1701.04521v1
fatcat:x3vvztyaejbollddojoubeo6rq
*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. ...##
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Unifying Known Lower Bounds via Geometric Complexity Theory

2015
*
Computational Complexity
*

the partial derivatives technique (Nisan-Wigderson), the results

doi:10.1007/s00037-015-0103-x
fatcat:ifoaveduubhmxlokr4pygqzjqq
*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 ...##
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Unifying Known Lower Bounds via Geometric Complexity Theory

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

partial derivatives technique (Nisan-Wigderson), the results

doi:10.1109/ccc.2014.35
dblp:conf/coco/Grochow14
fatcat:z5eytgp64jh2pclri4tlmky53e
*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 ...
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