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The Strength of Multilinear Proofs

2008
*
Computational Complexity
*

We show that this

doi:10.1007/s00037-008-0246-0
fatcat:qgyc6hxogbc5rdejpnqdjbo57m
*proof*system is fairly strong, even when restricted to*multilinear*arithmetic formulas*of*a very small depth. Specifically, we show*the*following: 1. ... Polynomial size*proofs*manipulating depth 3*multilinear*arithmetic formulas for*the*functional pigeonhole principle; 3. ...*The*work*of**the*second author was carried out in partial fulfillment*of**the*requirements for*the*Ph.D. degree at Tel Aviv University. ...##
###
Simple Hard Instances for Low-Depth Algebraic Proofs
[article]

2022
*
arXiv
*
pre-print

We prove super-polynomial lower bounds on

arXiv:2205.07175v1
fatcat:5rpfawkfqrco7bkbbkwcao4mgu
*the*size*of*propositional*proof*systems operating with constant-depth algebraic circuits over fields*of*zero characteristic. ... Specifically, we construct a polynomial f computable with small-size constant-depth circuits, such that*the**multilinear*polynomial computing 1/f over Boolean values and its appropriate set-*multilinear*... Algebraic*Proof*Systems One important strand*of**proof*complexity deals with algebraic*proof*systems*of*increasing*strength*. ...##
###
Non-deterministic exponential time has two-prover interactive protocols

1991
*
Computational Complexity
*

*The*first part

*of*

*the*

*proof*

*of*

*the*main result extends recent techniques

*of*polynomial extrapolation

*of*truth values used in

*the*single prover case. ...

*The*second part is a verification scheme for

*multilinearity*

*of*an nvariable function held by an oracle and can be viewed as an independent result on program verification. ... Remarkably, in all

*of*these cases,

*the*universal player can be replaced by a probabilistic polynomial time player without reducing

*the*

*strength*

*of*

*the*models. ...

##
###
Non-deterministic exponential time has two-prover interactive protocols

1992
*
Computational Complexity
*

*The*first part

*of*

*the*

*proof*

*of*

*the*main result extends recent techniques

*of*polynomial extrapolation

*of*truth values used in

*the*single prover case. ...

*The*second part is a verification scheme for

*multilinearity*

*of*an nvariable function held by an oracle and can be viewed as an independent result on program verification. ... Remarkably, in all

*of*these cases,

*the*universal player can be replaced by a probabilistic polynomial time player without reducing

*the*

*strength*

*of*

*the*models. ...

##
###
Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

2018
*
Computational Complexity Conference
*

We prove a lower bound

doi:10.4230/lipics.ccc.2018.11
dblp:conf/coco/AlonKV18
fatcat:75gqziq4ejfjxmrc3p7k4zbohi
*of*Ω(n 2 / log 2 n) on*the*size*of*any syntactically*multilinear*arithmetic circuit computing some explicit*multilinear*polynomial f (x 1 , . . . , x n ). ... Our approach expands and improves upon a result*of*Raz, Shpilka and Yehudayoff ([31]), who proved a lower bound*of*Ω(n 4/3 / log 2 n) for*the*same polynomial. ... In a follow up work [28] , Raz further*strengthed**the*results in [29] and showed that there is a family*of**multilinear*polynomials in n variables which can be computed by a poly(n) size syntactically ...##
###
Resolution over linear equations and multilinear proofs

2008
*
Annals of Pure and Applied Logic
*

We develop and study

doi:10.1016/j.apal.2008.04.001
fatcat:n6f6awapx5e4vljh3k7yicciqm
*the*complexity*of*propositional*proof*systems*of*varying*strength*extending resolution by allowing it to operate with disjunctions*of*linear equations instead*of*clauses. ...*The*latter are different than previous*proofs*, and apply to*multilinear**proofs**of*Tseitin mod p graph tautologies over any field*of*characteristic 0. ... This work was carried out in partial fulfillment*of**the*requirements for*the*Ph.D. degree*of**the*second author. ...##
###
Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits
[article]

2017
*
arXiv
*
pre-print

We prove a lower bound

arXiv:1708.02037v3
fatcat:c6dnrpzmkvfitdytesarela4di
*of*Ω(n^2/^2 n) on*the*size*of*any syntactically*multilinear*arithmetic circuit computing some explicit*multilinear*polynomial f(x_1, ..., x_n). ... Our approach expands and improves upon a result*of*Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound*of*Ω(n^4/3/^2 n) for*the*same polynomial. ... Acknowledgments Part*of*this work was done while*the*first author was visiting Tel Aviv University. ...##
###
Improved Low-Depth Set-Multilinear Circuit Lower Bounds
[article]

2022
*
arXiv
*
pre-print

Our

arXiv:2205.00611v1
fatcat:76itrp6dj5b3zcfhvwcy5dotje
*proof*techniques are another variation on those*of*LST, and enable us to show an improved lower bound (matching that*of*Raz)*of**the*form n^Ω(log n), albeit for*the*same polynomial f in VNP (*the*NW ...*The*techniques*of*LST give a different route to set-*multilinear*formula lower bounds, and allow them to obtain a lower bound*of**the*form (log n)^Ω(log n) for*the*size*of*general set-*multilinear*formulas ... [KS16] shows a quasi-polynomial separation between*the**strength**of*homogeneous ΣΠΣΠ and ΣΠΣΠΣ circuits, which could be considered as some evidence towards*the*validity*of*this possibility. ...##
###
Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials

2015
*
RACSAM
*

Surveying

doi:10.1007/s13398-015-0224-8
fatcat:43u2hzwlvvfk5a24otuuguwr64
*the*spirit*of*absolute summability on*multilinear*operators and homogeneous polynomials. ...*The*lack*of*such results in*the*theory*of*non-linear summing operators justifies*the*introduction*of*a class*of*polynomials and*multilinear*operators that satisfies at once all related nonlinear results ... Acknowledgment:*The*authors are indebted to M. López Pellicer for his valuable suggestions that helped to improve*the*presentation*of**the*paper. ...##
###
Secure obfuscation in a weak multilinear map model: A simple construction secure against all known attacks
[article]

2016
*
IACR Cryptology ePrint Archive
*

Until recently,

dblp:journals/iacr/MilesSZ16a
fatcat:bjixdan3zvfdxnzokftzafvdae
*the*strongest*proofs**of*security available for iO candidates were in a generic model that only allows "honest" use*of**the**multilinear*map. ... In particular,*the*authors [Crypto'16] recently gave a polynomial-time attack on several iO candidates when instantiated with*the**multilinear*maps*of*Garg, Gentry, and Halevi [Eurocrypt'13], and also proposed ... This is important because a significant amount*of*work has already gone into understanding*the**strengths*and limitations*of**the*original GGH13*multilinear*map candidate. •*The*construction in [GMS16] ...##
###
Pseudo-commutative monads and pseudo-closed 2-categories

2002
*
Journal of Pure and Applied Algebra
*

*The*former give rise to

*the*latter: if T is pseudo-commutative, then

*the*2-category T -Alg,

*of*strict T -algebras and pseudo-maps

*of*algebras, is pseudo-closed. ... In particular,

*the*2-category

*of*symmetric monoidal categories, is pseudo-closed. ... Acknowledgements We are grateful to a referee for some perceptive remarks and suggestions concerning

*the*material

*of*this paper. ...

##
###
Weak compactness and strongly summing multilinear operators
[article]

2013
*
arXiv
*
pre-print

However, for strongly summing

arXiv:1311.4685v1
fatcat:5533fxgfljeljf5dgy56yn4iae
*multilinear*operators and polynomials - one*of**the*most natural extensions*of**the*linear case to*the*non linear framework - weak compactness does not hold in general. ... We show that a subclass*of**the*class*of*strongly summing*multilinear*operators/polynomials, sharing its main properties such as Grothendieck's Theorem, Pietsch Domination Theorem and Dvoretzky-Rogers Theorem ...*Proof*. It is a particular case*of*[10, Theorem 2.2] analogous to*the**proof**of*Theorem 3.2. We shall need*the*following result to prove*the*sufficiency*of**the*Factorization Theorem. ...##
###
Algebraic Proofs over Noncommutative Formulas
[article]

2010
*
arXiv
*
pre-print

We study possible formulations

arXiv:1004.2159v2
fatcat:y4bunmfvcfevthvl64zhge3hue
*of*algebraic propositional*proof*systems operating with noncommutative formulas. ... ., polynomially verifiable) algebraic analog*of*Frege*proofs*, different from that given in [BIKPRS96,GH03]. ... Simulations, short*proofs*and separations for OFPC In this section we are concerned with*the*relative*strength**of*OFPC. ...##
###
Generalized polynomial identities

1975
*
Journal of Algebra
*

*of*rings with involution to yield a straightforward

*proof*

*of*a major theorem

*of*Herstein-Martindale-Amitsur; a generalization

*of*this result is then given in

*the*GI theory. ... In Section 2, new, reasonably easy

*proofs*are given

*of*major results

*of*Amitsur, Martindale and Jain characterizing primitive and prime rings with generalized identities;

*the*assumptions made in

*the*hypotheses ... A$-"f

*the*

*multilinearization*off. COROLLARY. For any W-algie R, f is a W-nontrivial GI

*of*R ;f f is a W-nontrivial GI

*of*R.

*Proof*. Immediate. ...

##
###
A Multi-prover Interactive Proof for NEXP Sound against Entangled Provers

2012
*
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
*

At

doi:10.1109/focs.2012.11
dblp:conf/focs/ItoV12
fatcat:njeobvhslfaopkver2ul3u2dk4
*the*heart*of*our result is a*proof*that Babai, Fortnow, and Lund's*multilinearity*test is sound even in*the*presence*of*entanglement between*the*provers, and our analysis*of*this test could be*of*independent ... shared entanglement between*the*provers could weaken or strengthen*the*computational power*of*multi-prover interactive*proofs*. ... s*multilinearity*test to*the*entangled-prover setting.*The*authors also thank Scott Aaronson, Dmitry Gavinsky, Oded Regev, and an anonymous referee for helpful suggestions. ...
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