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The Strength of Multilinear Proofs

Ran Raz, Iddo Tzameret
2008 Computational Complexity  
We show that this 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.  ... 
doi:10.1007/s00037-008-0246-0 fatcat:qgyc6hxogbc5rdejpnqdjbo57m

Simple Hard Instances for Low-Depth Algebraic Proofs [article]

Nashlen Govindasamy, Tuomas Hakoniemi, Iddo Tzameret
2022 arXiv   pre-print
We prove super-polynomial lower bounds on 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.  ... 
arXiv:2205.07175v1 fatcat:5rpfawkfqrco7bkbbkwcao4mgu

Non-deterministic exponential time has two-prover interactive protocols

L�szl� Babai, Lance Fortnow, Carsten Lund
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.  ... 
doi:10.1007/bf01200056 fatcat:7fo5g7yc5jdr3mpfeyh5bsb6ey

Non-deterministic exponential time has two-prover interactive protocols

L. Babai, L. Fortnow, C. Lund
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.  ... 
doi:10.1007/bf01200430 fatcat:alxouhut3fcuth7uhfqr7zuyjm

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

Noga Alon, Mrinal Kumar, Ben Lee Volk, Marc Herbstritt
2018 Computational Complexity Conference  
We prove a lower bound 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  ... 
doi:10.4230/lipics.ccc.2018.11 dblp:conf/coco/AlonKV18 fatcat:75gqziq4ejfjxmrc3p7k4zbohi

Resolution over linear equations and multilinear proofs

Ran Raz, Iddo Tzameret
2008 Annals of Pure and Applied Logic  
We develop and study 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.  ... 
doi:10.1016/j.apal.2008.04.001 fatcat:n6f6awapx5e4vljh3k7yicciqm

Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits [article]

Noga Alon, Mrinal Kumar, Ben Lee Volk
2017 arXiv   pre-print
We prove a lower bound 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.  ... 
arXiv:1708.02037v3 fatcat:c6dnrpzmkvfitdytesarela4di

Improved Low-Depth Set-Multilinear Circuit Lower Bounds [article]

Deepanshu Kush, Shubhangi Saraf
2022 arXiv   pre-print
Our 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.  ... 
arXiv:2205.00611v1 fatcat:76itrp6dj5b3zcfhvwcy5dotje

Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials

Daniel Pellegrino, Pilar Rueda, Enrique A. Sánchez-Pérez
2015 RACSAM  
Surveying 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.  ... 
doi:10.1007/s13398-015-0224-8 fatcat:43u2hzwlvvfk5a24otuuguwr64

Secure obfuscation in a weak multilinear map model: A simple construction secure against all known attacks [article]

Eric Miles, Amit Sahai, Mark Zhandry
2016 IACR Cryptology ePrint Archive  
Until recently, 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]  ... 
dblp:journals/iacr/MilesSZ16a fatcat:bjixdan3zvfdxnzokftzafvdae

Pseudo-commutative monads and pseudo-closed 2-categories

Martin Hyland, John Power
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.  ... 
doi:10.1016/s0022-4049(02)00133-0 fatcat:4k3d6pbidfcxzavhpwq57ad4ea

Weak compactness and strongly summing multilinear operators [article]

Daniel Pellegrino, Pilar Rueda, Enrique A. Sanchez-Perez
2013 arXiv   pre-print
However, for strongly summing 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.  ... 
arXiv:1311.4685v1 fatcat:5533fxgfljeljf5dgy56yn4iae

Algebraic Proofs over Noncommutative Formulas [article]

Iddo Tzameret
2010 arXiv   pre-print
We study possible formulations 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.  ... 
arXiv:1004.2159v2 fatcat:y4bunmfvcfevthvl64zhge3hue

Generalized polynomial identities

Louis Halle Rowen
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.  ... 
doi:10.1016/0021-8693(75)90170-2 fatcat:r3rclw67kvc7hda7zflzc6odyi

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

Tsuyoshi Ito, Thomas Vidick
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science  
At 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.  ... 
doi:10.1109/focs.2012.11 dblp:conf/focs/ItoV12 fatcat:njeobvhslfaopkver2ul3u2dk4
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