Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds

0.42 MB·

by  Michael A. Forbes

Succinct Hitting Sets and Barriers to Proving Algebraic
Circuits Lower Bounds
Michael A.Forbes AmirShpilka† BenLee Volk†
∗
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2 Abstract

n We formalizea frameworkof algebraicallynaturallower boundsforalgebraiccircuits. Just
a
as with the natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower
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bounds, our notion of algebraically natural lower bounds captures nearly all lower bound
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techniquesknown. However,unlikethebooleansetting, therehasbeennoconcreteevidence
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demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general

] algebraic circuits, as there is little understanding whether algebraic circuits are expressive
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enoughtosupport“cryptography”secureagainstalgebraiccircuits.
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Following a similar result of Williams [Wil16] in the boolean setting, we show that the
s. existence of analgebraicnaturalproofsbarrierisequivalent tothe existence ofsuccinct deran-
c domization of the polynomial identity testing problem. That is, whether the coefficient vec-
[
torsofpolylog(N)-degreepolylog(N)-sizecircuitsisahittingsetfortheclassofpoly(N)-degree

1 poly(N)-sizecircuits. Further,wegiveanexplicituniversalconstruction showingthatif such
v asuccincthittingsetexists,thenouruniversalconstructionsuffices.
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Further, we assess the existing literature constructing hitting sets for restricted classes of
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algebraic circuits and observe that none of them are succinct as given. Yet, we show how to
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5 modify some of these constructions to obtain succinct hitting sets. This constitutes the first
0 evidencesupportingtheexistenceofanalgebraicnaturalproofsbarrier.
1. OurframeworkissimilartotheGeometricComplexityTheory(GCT)programofMulmu-
0 leyandSohoni[MS01],exceptthathereweemphasizeconstructivenessoftheproofswhilethe
7 GCTprogramemphasizessymmetry. Nevertheless,oursuccincthittingsetshaverelevanceto
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theGCTprogramastheyimplylowerboundsforthecomplexityofthedefiningequationsof
:
v polynomialscomputedbysmallcircuits.
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X
r
a
∗Simons Institute for the Theory of Computing, at the University of California, Berkeley. E-mail:
[email protected]. Thisworkwas performedwhen the author was atStanford University, while supported
bytheNSF,includingNSFCCF-1617580,andtheDARPASafewareprogram.
†Department of Computer Science, Tel Aviv University, Tel Aviv, Israel, E-mails: [email protected],
[email protected]. TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanCommunity’s
SeventhFrameworkProgramme(FP7/2007-2013) undergrantagreementnumber257575andfromtheIsraelScience
Foundation(grantnumber552/16).
1 Introduction
Computational complexity theory studies the limits of efficient computation, and a particular
goal is to quantify the power of different computational resources such as time, space, non-
determinism,andrandomness. Suchquestionscanbeinstantiatedasaskingtoproveequalitiesor
separations betweencomplexity classes, such as resolving P versus NP. Indeed,there have been
various successes: the (deterministic) time-hierarchy theorem showing that P = EXP ([HS65]),
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circuit lowerboundsshowingthatAC0 = P([Ajt83,FSS84,Yao85,Ha˚s89]),andinteractiveproofs
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showing IP = PSPACE ([LFKN92, Sha90]). However, for each of these seminal works we have
now established barriers for why their underlying techniques cannot resolve questions such as P
versus NP. Respectively, the above results are covered by the barriers of relativization of Baker,
Gill and Solovay [BGS75], natural proofs of Razborov and Rudich [RR97], and algebraization of
Aaronsonand Wigderson[AW09]. Inthis workwerevisit thenaturalproofsbarrier ofRazborov
and Rudich [RR97] and seek to understand how it extends to a barrier to algebraic circuit lower
bounds. While previous works have considered versions of an algebraic natural proofs barrier,
wegivethefirstevidenceofsuchabarrieragainstrestrictedalgebraicreasoning.
Natural Proofs: The setting of Razborov and Rudich [RR97] is that of non-uniform complexity,
whereinsteadofconsideringaTuringmachinesolvingaproblemonallinputsizes,oneconsiders
a modelsuch as boolean circuits wherethe computationaldevice can change with thesize ofthe
input. While circuits are at least as powerful as Turing machines, and can even (trivially) com-
pute undecidable languages, their ability to solve computational problems of interest can seem
closertouniformcomputation. Forexample,ifcircuitscansolveNP-hardproblemsthenthereare
unexpected implications for uniform computation similar to P = NP (the polynomial hierarchy
collapses ([KL82])). As such, obtaining lower bounds for boolean circuits was seen as a viable
methodtoindirectlytackleTuringmachinelowerbounds,withthebenefitofbeingabletoappeal
to more combinatorial methods and thus bypassing the relativization barrier of Baker, Gill and
Solovay[BGS75]whichseemstoobstructmostmethodsthatcanexploituniformity.
There have been many important lower bounds obtained for restricted classes of circuits:
constant-depth circuits ([Ajt83, FSS84, Yao85, Ha˚s89]), constant-depth circuits with prime mod-
ulargates([Raz87,Smo87]),aswellaslowerboundsformonotonecircuits([Raz85,AB87,Tar88]).
Razborov and Rudich [RR97] observed that many of these lower bounds prove more than just a
lower bound for a single explicit function. Indeed, they observed that such lower bounds often
distinguish functions computable by small circuits from random functions, and in fact they do so
efficiently. Specifically,anaturalproperty PisasubsetofbooleanfunctionsP f : 0,1 n
n 1
0,1 withthefollowingproperties.1 ⊆ ∪ ≥ { { } →
{ }}
1. Usefulness: If f : 0,1 n 0,1 iscomputable bypoly(n)-sizecircuits than f hasproperty
{ } → { }
P.
2. Largeness: Randomfunctions f : 0,1 n 0,1 donothavethepropertyPwithnoticeable
{ } → { }
probability,thatis,withprobabilityatleast1/poly(N) = 2 O(n).
−
3. Constructivity: Givenatruth-tableofafunction f : 0,1 n 0,1 ,ofsize N = 2n,deciding
{ } → { }
whether f hasthepropertyPcanbecheckedinpoly(N) = 2O(n) time.
1TheRazborovandRudich[RR97]definitionofanaturalpropertyactuallyappliestothecomplementoftheprop-
ertyPweusehere.Thisisatrivialdifferenceforbooleancomplexity,butisimportantforalgebraiccomplexityasthere
naturalpropertiesareone-sided,seeSection1.2.
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To obtain a circuit lower bound, a priori one only needs to obtain a (non-trivial) property P that
is useful in the above sense. However, Razborov and Rudich [RR97] showed that (possibly after
asmallmodification)mostcircuitlowerbounds(suchasthoseforconstant-depthcircuits([Ajt83,
FSS84, Yao85, Ha˚s89, Raz87, Smo87])) yield large and constructive properties, and called such
lowerboundsnaturalproofs.
Further, Razborov and Rudich [RR97] argued that standard cryptographic assumptions im-
plythatnaturalproofscannotyieldsuper-polynomiallowerboundsagainstanyrestrictedclassof
circuits that is sufficiently rich to implement cryptography. That is, a pseudorandom function is an
efficiently computable function f : 0,1 n 0,1 λ 0,1 such that when sampling the key
{ } ×{ } → { }
k 0,1 λ at random the resulting distribution of functions f( ,k) is computationally indistin-
∈ { } ·
guishablefromatrulyrandomfunctionf : 0,1 n 0,1 . Theexistenceofpseudorandomfunc-
{ } → { }
tions follows from the existence of one-way functions ([HILL99, GGM86]) which is essentially the
weakest interesting cryptographic assumption. There are even candidate constructions of pseu-
dorandom functions computable by polynomial-size constant-depth threshold circuits (TC0) as
givenbyNaorandReingold[NR97],whosesecurityrestsontheintractability ofdiscrete-logand
factoring-type assumptions (see also Krause and Lucks [KL01]). As such, it is widely-believed
that there are pseudorandom functions, even ones computationally indistinguishable from ran-
domexcepttoadversariesrunninginexp(λΩ(1))-time.
In contrast, Razborov and Rudich [RR97] showedthat a natural proof usefulagainst poly(n)-
size circuits can distinguish a pseudorandomfunction from a truly random function in poly(2n)-
time, which would contradict the believed exp(λΩ(1))-indistinguishability when taking λ to be a
largeenoughpolynomialinn. Thatis,supposePisanaturalproperty. Thenforapseudorandom
function f( , ) and each value k 0,1 λ of the key, the resulting function f( ,k) : 0,1 n
· · ∈ { } · { } →
0,1 hasapoly(n)-sizecircuit,andhaspropertyP(byusefulness). Incontrast,randomfunctions
{ }
will not have property P with noticeable probability (by largeness). As the property is construc-
tive,thisgivesapoly(2n)-timealgorithmdistinguishing f( ,k)fromarandomfunction,asdesired.
·
While the natural proofs barrier has proved difficult to overcome, there are results that seem
to circumvent it. For example, the barrier does not seem to apply to the lower bounds obtained
formonotonecircuits([Raz85]),astherethenotionofa“randommonotonefunction”isnotwell-
defined. Further, there are results (such as Williams’ [Wil14] result of ACC0 = NEXP) that cir-
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cumvent the natural proofs barrier by incorporating techniques from uniform complexity. Other
work has demonstrated that relaxing the notion of natural proof can avoid the implications to
breaking cryptography. Chow [Cho11] has shown that almost natural proofs (which relax large-
nessslightly)canprovesuper-polynomialcircuitlowerbounds(underplausiblecryptographicor
complexity-theoretic assumptions). Williams [Wil16] has shown, among other results, that some
circuitlowerbounds(suchasforEXPorNEXP)areequivalenttoconstructive(non-trivial)proper-
tiesusefulagainstsmallcircuits, whichyethavenoneedforanysortoflargeness. Chapman and
Williams [CW15] have shown that obtaining circuit lower bounds for a self-checkable problem
(such as SAT) is essentially equivalent to obtaining a natural property against circuits that “check
theirwork”. Theseworkssuggestthattheexactimplicationsofthenaturalproofsbarrierremains
notfullyunderstood.
AlgebraicNaturalProofs: Algebraiccircuitsarethemostnaturalmodelforcomputingpolyno-
mialsbyusingadditionandmultiplication. Whilemorerestrictedthangeneral(boolean)compu-
tation, proving lower bounds for algebraic circuits has proved challenging. Yet, we do not have
formal barrier results for understanding the difficulty of such lower bounds. While such lower
bounds are not a priori subject to the natural proofs barrier due to the formal differences in the
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computationalmodel,therelevanceoftheideasofnaturalproofstoalgebraiccircuitshasbeenre-
peatedlyasked. Aaronson-Drucker[AD08] as well as Grochow [Gro15]noticedthat many ofthe
prominentalgebraic circuit lowerbounds(suchas [Nis91a,NW97, Raz06,RY09])are algebraically
natural,inthattheyobeyaalgebraicformofusefulness,largeness,andconstructivity.
While this would seemingly then imply a Razborov and Rudich [RR97]-type barrier for ex-
isting techniques, there is a key piece missing: we have very little evidence for the existence of
algebraic pseudorandom functions. That is, the pseudorandom functions used by Razborov and
Rudich[RR97]arebooleanfunctions,andnaiveattemptstoalgebrizethemseeminglydonotyield
pseudorandompolynomials. Indeed,asalgebraiccircuitsareacomputationalmodelweakerthan
generalcomputation,it is conceivable thattheyare tooweaktoimplementcryptography,sothat
naturalproofsbarrierwouldnotapply. Incontrast,itisalsoconceivablethatalgebraiccircuitsare
soweakthattheycan compute“enough”cryptographytobesecureagainst algebraiccircuits, so
thatanaturalproofsbarrierwouldapply.
Our Work: In this work we formalize the study of pseudorandom polynomials by exhibiting
thefirstconstructionsprovablysecureagainstrestrictedclassesofalgebraiccircuits. Inparticular,
we follow Williams [Wil16] in treating the existence of a natural proofs barrier as the problem
of succinct derandomization: replacing randomness with pseudorandomness that further has a
succinctdescription. Werevisitexistingderandomizationofrestrictedclassesofalgebraiccircuits
andshow(vianon-trivialmodification)thattheycanbemadesuccinctinmanycases.
Recently, and independently of our work, Grochow, Kumar, Saks, and Saraf [GKSS17] ob-
servedasimilarconnectionbetweenanaturalproofsbarrierforalgebraiccircuitsandsuccinctde-
randomization. TheirworkalsopresentsconnectionswithGeometricComplexityTheory(which
wediscussbelowin Section1.7) andalgebraic proofcomplexity. However,unlike ourworkthey
donotpresentanyconstructionsofsuccinctderandomization.
1.1 AlgebraicComplexity
Wenowdiscussthealgebraic settingforwhich wewishtodiscussthenaturalproofsbarrier. Al-
gebraic complexity theorystudiesthe complexity ofsyntactic computation of polynomials using
algebraic operations. Themostnatural modelofcomputationis that ofan algebraic circuit, which
isadirectedacyclicgraphwhoseleavesarelabeledbyeithervariables x ,...,x orelementsfrom
1 n
the field F, and whose internal nodes are labeled by the algebraic operations of addition (+) or
multiplication ( ). Each node in the circuit computes a polynomial in the natural way, and the
×
circuit has oneor moreoutput nodes, which are nodesofout-degreezero. The size ofthecircuit is
definedtobethenumberofwires,andthedepthisdefinedtobethelengthofalongestpathfrom
an inputnodetotheoutputnode. Asusual,acircuit whoseunderlyinggraphisatreeiscalled a
formula. Onecanassociatevariouscomplexityclasseswithalgebraiccircuits,andthemostimpor-
tantonefor usis VP, whichtheclassesofn-variate polynomials withpoly(n)-degreecomputable
by poly(n)-size algebraic circuits. Thereis also VNP, which we will informally defineas the class
of“explicit”polynomials.
A central open problem in algebraic complexity theory is proving super-polynomial lower
bound for the algebraic circuit size of any explicit polynomial, that is, proving VP = VNP. Sub-
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stantial attention has been given to this problem, using various techniques that leverage non-
trivial algebraic tools to study the syntactic nature of these circuits. Indeed, our knowledge of
algebraic lower bounds seem to surpass that of boolean circuits, as we have super-linear lower
boundsforgeneralcircuits([Str73,BS83])—agoalasyetunachievedinthebooleansetting. Sim-
ilarly, there are a wide array of super-polynomial or even exponential lower bounds known for
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various weaker models of computation such as non-commutative formulas ([Nis91a]), multilin-
ear formulas ([Raz09, RY08]), and homogeneousdepth-3 and depth-4 circuits ([NW97, GKKS16,
KSS14, FLMS14, KLSS14, KS14]). We refer the reader to Saptharishi [Sap16] for a continuously-
updatingcomprehensivecompendiumoftheselowerbounds.
However, this landscape might still feel reminiscent of the boolean setting, in that there are
various restricted models where lower bounds techniques are known, and yet lower bounds for
general circuits or formulas remain relatively poorly understood. Yet, there has been some sig-
nificant recent cause for optimism for obtaining general circuit lower bounds, as various depth-
reduction results ([VSBR83, AJMV98, AV08, Koi12, Tav15, GKKS16, CKSV16]) have shown that
n-variable degree-d polynomials computable by size-s algebraic circuits have sO(√d)-size depth-3
or homogeneous depth-4 formulas. Further, recent methods ([Kay12, GKKS14, KSS14, FLMS14,
KLSS14, KS14]) have proven (nd)Ω(√d) lower bounds computing explicit polynomials by homo-
geneousdepth-4formulas. Ifonecouldsimplypushthesemethodstoobtaina“(nd)ω(√d)”lower
boundthenthiswouldobtainsuper-polynomiallowerboundsforgeneralcircuits! Unfortunately,
allofthelowerboundsmethodsknownseemtoapplynotjusttocandidatehardpolynomials,but
alsocertaineasypolynomials,sothatnosuch“(nd)ω(√d)”lowerboundcanbebeobtainedasthis
wouldcontradictthedepth-reductiontheorems.
Giventhisstateofaffairs,itisunclearwhethertobeoptimisticorpessimisticregardingfuture
prospects for obtaining superpolynomial lower bounds for general algebraic circuits. To resolve
this uncertainty it is clearly important to formalize the barriers constraining our lower bound
techniques. Indeed, as mentionedabove all known lower-bound methodsapply not just to hard
polynomialsbutalsotoeasypolynomials—isthisintrinsictocurrentmethods? Thisisessentially
thequestionofwhetherthereisanalgebraicnaturalproofsbarrier,aswenowdescribe.
1.2 AlgebraicNaturalProofs
We now define the notion of an algebraically natural proof used in this paper. Intuitively, we
wanttoknowwhetherlowerboundsmethodscandistinguishbetweenlow-complexityandhigh-
complexity polynomials, so that they are useful in the sense of Razborov and Rudich [RR97]. In
particular, we want to knowif such distinguishers2 can be efficient, so that they are also construc-
tive. Severalworks,suchasAaronsonandDrucker[AD08]Grochow[Gro15](seealsoShpilkaand
Yehudayoff[SY10,Section3.9],andAaronson[Aar16,Section6.5.3])havenoticedthatmostallof
the lower bounds methods in algebraic complexity theory are themselves algebraic in a certain
sensewhichwenowdescribe.
The simplest example is to consider matrix rank, where the complexity ofan n n matrix M
×
<
is exactly captured by its determinant, which is a polynomial. That is, if M is of rank n then
detM = 0, andifrank = n thendetM = 0. Thekeyfeaturehereis thatdetM is apolynomialin
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thecoefficientsoftheunderlyingalgebraicobject,whichinthiscaseisthematrixM. Mostofthecentral
lowerboundstechniques,suchaspartialderivatives([NW97]),evaluation/coefficientdimension
([Nis91a,Raz06,RY09,FS13]),orshiftedpartialderivatives([Kay12,GKKS14])aregeneralizations
of this idea, specifically leveraging notionsof linear algebra and rank. Abstractly, thesemethods
takeann-variatepolynomial f,inspectsitscoefficients,andthenformsanexponentially-large(in
n) matrix M whose entries are polynomials in the coefficients of f. One then shows that if f is
f
<
simple thenrankM r, while foran explicit polynomial f onecan showthatrankM r. In
f 0 f0 ≥
2Grochow[Gro15]referredtodistinguishersastestpolynomials,astheytestwhetheraninputpolynomialisoflow-
orhigh-complexity.
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particular,bybasiclinearalgebrathisshowsthatthereissomer rsubmatrix M ofM suchthat
× ′f f
detM = 0forsimple f,yetdetM = 0,provingthat f isahardpolynomial.
′f ′f0 6 0
Wenowobservethattheabove outlinegivesanatural property P := f : detM = 0 in the
{ ′f }
senseofRazborovandRudich[RR97].
1. Usefulness: For low-complexity f we have that f P as argued above. Further, P is a non-
∈
trivialpropertyas f / P.
0
∈
2. Constructivity: Foragiven f,decidingwhether“f P?” istantamounttocomputingdetM .
∈ ′f
Even though M might be exponentially-large, it is often polynomially-large in thesize of f
′f
(which isexponentialinthenumbern ofvariables in f). Astypically M isasimplematrix
′f
intermsof f,computingdetM isessentiallythecomplexityofcomputingthedeterminant,
′f
which is computable by small algebraic circuits ([Ber84, MV97]). Thus, the property P is
efficientlydecidableinthesizeofitsinput.
3. Largeness: Thelargenessconditionisintrinsichere,asthepropertyisgovernedbythevanish-
ingofanon-zeropolynomial;detM isnon-zeroasapolynomialasinparticulardetM =
′f ′f0 6
0. As non-zero polynomials evaluate to non-zero at random points with high probability
([Sch80,Zip79,DL78]),thismeansthatsuchdistinguisherscertifythatrandompolynomials
areofhigh-complexity.
Thus,weseethat theabove meta-methodformsaverynaturalsubclass ofnaturalproofs. As
algebraiccomputationisasubclassofgeneralcomputation,thisisweakerframeworkthanthefull
naturalproofsparadigm. While it mightthenseemthattheRazborovand Rudich[RR97]barrier
should apply to this framework also, we only need to prove lower bounds for algebraic circuits,
sothatwehaveaweakergoalthanobtainingbooleanlowerbounds. Hencethenecessitytostudy
thisrestrictedframeworkanditsbarriersinitsownright.
We now give a formalization ofthe above notion, which is implicit in prior work and known
to experts. To begin, we must first note that in comparing low-complexity to high-complexity
polynomials,wemustdetailthespaceinwhichthepolynomialsreside. Therearethreespacesof
primaryinterest.
1. F[x ,...,x ]d: Thespaceofn-variatepolynomialsoftotaldegreeatmostd. ThereareN :=
1 n n,d
(n+d)manymonomialsxa := xa1 xan inthisspace.
d 1 ··· n
2. F[x ,...,x ]d : Thespaceofhomogeneousn-variatepolynomialsoftotaldegreeexactlyd.
1 n hom
Thereare Nhom := (n+d−1)manymonomials xa inthisspace.
n,d d
3. F[x ,...,x ]d : The space of n-variate polynomials of individual degree at most d. There
1 n ideg
are Nideg := (d+1)n manymonomials xa inthisspace.
n,d
While this may seem pedantic, it is important to distinguish these spaces. That is, while homo-
geneousdegree-d polynomials capture nearly all of the interesting complexity of polynomials of
degreeatmostd,itistrivialtodistinguishthetwo. Thatis,considerthedistinguisherpolynomial
c that simply returns the constant coefficient (the coefficient of 1) of a polynomial f = ∑ c xa.
0 a a
ThispolynomialvanishesonF[x ,...,x ]d ford > 0,butdoesnotvanishontheconstantpoly-
1 n hom
nomial 1 F[x ,...,x ]d. However, it would be absurd to say that “1 is a hard polynomial for
1 n
∈
F[x ,...,x ]d ”. Thus,indiscussinghowpropertiescandistinguishpolynomialswemustspec-
1 n hom
ify the domain of interest. Indeed, to discuss lower bounds for homogeneous computation one
5
mustrestrictattentiontothespaceF[x]d , and likewisetodiscusslower boundsfor multilinear
hom
computationonemustrestrictattentiontothespaceF[x]1 .
ideg
Wenowpresentourdefinition,withenoughgeneralitytohandletheabovespacesofpolyno-
mialssimultaneously. Thatis,forafixedsetofmonomials (suchasallmonomialsofdegreeat
M
most d) we considerthe space span( ), which is definedas all linear combinations over mono-
M
mials in . We then identify a polynomial f span( ) defined by f = ∑ c xa with its
list of sucMh coefficients, which is a vector coeff∈(f) FM defined coeff (f)xa:∈=M(ca) . We
M a xa
M ∈ M ∈M
thenaskfordistinguisherD whichtakeasinputthese manycoefficients,whichcanseparate
|M|
low-complexitypolynomialsfromhigh-complexitypolynomials.
Definition 1.1 (Algebraically Natural Proof).Let F[x ,...,x ] be a set of monomials =
1 n
M ⊆ M
xa ,andletthesetspan( ) := ∑ c xa : c F bealllinearcombinationsofthesemonomials.
L{et}a span( ) and M F[ c{ xa∈M] bae classae∈s of}polynomials, where the latter is in many
a xa
C ⊆ M D ⊆ { } ∈M |M|
variables.
Apolynomial D isanalgebraic -naturalproofagainst ,alsocalledadistinguisher,if
∈ D D C
1. D isanon-zeropolynomial.
2. Forall f , D vanishesonthecoefficientvectorof f,thatis, D(coeff (f)) = 0. ♦
∈ C M
WewillprimarilyinterestedintakingthesetofmonomialsMtocorrespondtooneoftheabove
three setsof polynomials, F[x]d, F[x]d and F[x]d , to which we define the relevant coefficient
hom ideg
vectors as as coeff , coeffhom and coeffideg. We will use “coeff” if the space of polynomials is
n,d n,d n,d
clearfromcontext.
Thus,torevisit thecomparisonwith Razborovand Rudich[RR97], condition(2) saysthatthe
distinguisherDisusefulagainsttheclass . Condition(1)indicatesthatthepropertyisnon-trivial,
C
and in particular is large, as a non-zero polynomial will evaluate to non-zero at a random point
with high probability ([Sch80, Zip79, DL78]). Finally, the fact that distinguisher D comes from
the restricted class is the constructivity requirement, and the main question is how simple the
D
distinguisherDcanbe.
Further, note how the above distinguishers naturally have a one-sided nature to them as in
algebraic complexity one typically seeks lower bounds against any field of coefficients. In using
theabovetodefinetheRazborovandRudich[RR97]stylepropertyP := f : D(coeff (f)) = 0 ,
{ M }
wenotethatthecomplementproperty P = span( ) P = f : D(coeff (f)) = 0 cannotbe
¬ M \ { M 6 }
expressedintheaboveframework. Thatis,fornon-zeropolynomials pandq,itcannotbethatthe
product pq vanisheseverywhere(over large enoughfields),sothat in particular it cannot bethat
p(α) = 0iffq(α) = 0.
6
We argued above that most of the main lower bound techniques fall into the above algebraic
natural proof paradigm where the distinguisher has polynomial-size algebraic circuits, so that
theproofis VP-natural. This motivatesthefollowing questionabout algebraic VP-natural proofs
againstVP.
Question 1.2. For the space of total degree polynomials F[x ,...,x ]d, is there an algebraic poly(N )-
1 n n,d
sizenaturalproofforlowerboundsagainstpoly(n,d)-sizecircuits?
While one could make a detailed study of existing lower bounds to prove the intuitive fact
that VP-natural properties suffice for them, our attention will be to studying the limits of this
framework. Thatsaid,itisworthmentioningthatthereareknowntechniquesforalgebraiccircuit
lowerboundsthatfalloutsidethisframework.
6
First,theshiftedpartialderivativetechniqueofGupta,Kamath,KayalandSaptharishi[Kay12,
GKKS14]is notcurrentlyknownto be VP-natural. That is, while it doesfall intothe above rank-
based meta-method (and thus the algebraic natural proof paradigm), the matrices involved are
actually quasi-polynomially largeintheirinput,sothemethodisonlyquasiVP-natural. However,
astheshiftedpartialtechniqueprovesexponentiallowerboundstherequiredquasiVP-naturalness
stillseemsrathermodest.
In contrast, there are actually methods which completely fall of the algebraic framework (con-
structiveornot). Thatis,asdiscussedbelowinSection1.7,thisalgebraicdistinguisherframework
is limited to proving border complexity lower bounds, where border complexity is always upper
bounded by usual complexity notions. For the tensor rank model, distinguishers actually prove
border rank lower bounds. In contrast, the substitutionmethod ([BCS97, Chapter 6],[Bla¨14]) can
prove tensor rank lower bounds which are higher than known border rank upper bounds (for
explicit tensors), giving a separation between these two complexities and thus showing the sub-
stitution method is not captured by the algebraic natural proof framework. However, all such
knownseparationsarebyatmostamultiplicative constantfactor, sotheinability ofthesubstitu-
tion method to be algebraically natural does not currently seem to be a serious deficiency in the
frameworkdevelopedhere.
1.3 PseudorandomPolynomials
Having given our formal definition of algebraic natural proofs, we now explain our notion of
thealgebraic naturalproofbarrier. Inparticular, as algebraically natural proofsconcernthe zeros
of (non-zero) polynomials computable by small circuits, this naturally leads us to the polynomial
identitytesting(PIT)problem.
PolynomialIdentityTesting: Polynomialidentitytestingisthefollowingalgorithmicproblem:
given an algebraic circuit D computing an N-variate polynomial, decide whether D computes
theidentically zeropolynomial. Theproblemadmitsasimpleefficient randomizedalgorithmby
theSchwartz-Zippel-DeMillo-Lipton Lemma[Sch80, Zip79, DL78]. That is, evaluations ofa low-
degree non-zero polynomial at random points will be non-zero with high probability. Thus, to
checknon-zeronessitisenoughtoevaluateDonarandominputαandobservewhetherD(α) = 0,
which is clearly efficient. However, the best known deterministic algorithms run in exponential
time. Designingan efficient deterministicalgorithmforPITis anothermajor openproblemin al-
gebraiccomplexity,withintricateandbidirectionalconnectionstoprovingalgebraicandboolean
circuitlowerbounds[HS80,Agr05,KI04].
Thetwoflavorsinwhichtheproblemappearsarethewhite-boxmodel,inwhichthealgorithm
is allowed to inspect the structure of the circuit, and the black-box model, in which the algorithm
isonlyallowedtoaccessevaluationsofthecircuitoninputsofitschoice,suchastherandomized
algorithm describedabove. Itcan be easily seenthat efficient deterministicblack-box algorithms
areequivalenttoconstructingsmallhitting sets: ahittingsetforclass F[c ,...,c ]ofcircuits
1 N
D ⊆
isaset FN suchthatforanynon-zerocircuit D ,thereexistsα suchthat D(α) = 0.
H ⊆ ∈ D ∈ H 6
While small hitting setsexist for VP, little progresshas been made for explicitly constructingany
non-trivialhittingsetsforgeneralalgebraiccircuits(orevensolvingPITinthewhite-boxmodel).
In contrast, there has been substantial work developing efficient deterministic white- and black-
boxPITalgorithmsfornon-trivialrestrictedclassesofalgebraiccomputation,seeforexamplethe
surveysofSaxena[Sax09,Sax14]andShpilka-Yehudayoff[SY10].
7
Succinct Derandomization: We now define our notions of an pseudorandom polynomials by
connecting the algebraic natural proof framework with hitting sets. Consider a class of poly-
C
nomials, say within the space of polynomials of bounded total degree F[x ,...,x ]d. If D is an
1 n
algebraicnaturalproofagainst thenwehave:
C
1. D isanon-zeropolynomial.
2. D vanishesontheset := coeff (f) : f ofcoefficientvectorsofpolynomialsin .
n,d
H { ∈ C} C
Puttogether,theseconditionsareequivalenttosayingthatthat isnotahittingsetfor D. Thus,
H
we see that there are algebraically natural proofs if and only if coefficient-vectors of simple poly-
nomials are not hitting sets. Thus, we see that the existence of an algebraic natural proofs bar-
riercan berephrasedas whetherPITcan be derandomizedusingsuccinctpseudorandomness. A
completely analogous statement was proven by Williams [Wil16] in boolean setting, where the
existenceoftheRazborovandRudich[RR97]naturalproofsbarrierwasshownequivalenttosuc-
cinct derandomization of ZPE, those problems solvable in zero-error 2O(n)-time. However, that
equivalencethereisslightlymoreinvolved,whileitisimmediatehere.
We now give the formal definition mirroring the above discussion, in the same generality of
Definition1.1.
Definition 1.3 (Succinct Hitting Set).Let F[x ,...,x ] be a set of monomials = xa ,
1 n a
M ⊆ M { }
and let the set span( ) := ∑ c xa : c F be all linear combinations of these monomials.
Let span( ) aMnd {F[xac∈M a ] be cala∈sses}of polynomials, where the latter is in many
a xa
C ⊆ M D ⊆ { } ∈M |M|
variables.
is a -succinct hitting set for if := coeff (f) : f is a hitting set for . That is,
D C isnCon-zeroiff D isnon-zero,DthatHis,ther{eissomMe f su∈chCth}at D(coeff (f)) =D0. ♦
∈ D |H ∈ C M 6
The above argument showing the tension between algebraic natural proofs and pseudoran-
dompolynomialscanbesummarizedinthefollowingtheorem,whichfollowsimmediatelyfrom
thedefinitions.
Theorem 1.4.Let F[x ,...,x ] be a set of monomials = xa , and let the set span( ) :=
1 n a
M ⊆ M { } M
∑ c xa : c F be all linear combinations of these monomials. Let span( ) and
F{[ xca∈M a ]becalas∈seso}fpolynomials, wherethelatterisin manyvariablCes.⊆ M D ⊆
a xa
{ } ∈M |M|
Thenthereisaalgebraic -naturalproofagainst iff isnota -succincthittingsetfor .
D C C C D
Instantiatingthis claim with being thespaceofdegree-dmonomials, wegetthefollowing
M
quantitativeversionoftheabove.
Corollary 1.5.Let F[x ,...,x ]d be the class of poly(n,d)-size circuits of total degree at most d.
1 n
C ⊆
Then there is an algebraic poly(N )-natural proof against iff is not a poly(n,d)-succinct hitting set
n,d
C C
forpoly(N )-sizecircuitsin N variables.
n,d n,d
In the common regime when d = poly(n), we have that poly(n) = polylog(N ). That is, this
n,d
existence of an algebraic natural proofs barrier is equivalent to saying that coefficient-vectors of
polylogarithmic-sizecircuitsformahittingsetofpolynomial-size.
Withthisequivalencein hand,wecannowphrasethequestionofanalgebraic naturalproofs
barrier.
Question1.6(AlgebraicNaturalProofsBarrier). Isthereapolylog(N)-succincthittingsetforcircuits
ofpoly(N)-size?
Again,wenotethatQuestion1.6wasalsoraisedbyGrochow,Kumar,Saks,andSaraf[GKSS17],
who presented a definition similar to Definition 1.3 and also observed the implication in Theo-
rem1.4.
8
SuccinctGenerators: While theabove equivalence already sufficesfor studyingthebarrier, the
notion of a hitting set is sometimes fragile. A more robust way to obtain hitting sets for a class
F[c ,...,c ]istoobtainagenerator,whichisapolynomialmapG : Fℓ FN suchthatD
1 N
D ⊆ → ∈ D
isanon-zeroiff D G 0,thatis,thecompositionD(G(y)) = 0isnon-zeroasapolynomialiny.
◦ 6≡ 6
ℓ
Hereonemeasuresthequalityofthegeneratorbyaskingtominimize theseed-length . Bypoly-
nomial interpolation, it follows that constructing small hitting sets is equivalent to constructing
ℓ
generatorswith small,seeforexampleShpilka-Yehudayoff[SY10].
However, in our setting we want succinct generators so that the polynomial-map G is com-
putablebyasmallalgebraiccircuit. Inparticular,convertingasuccincthittingset toagenerator
H
usingthestandardinterpolationmethodswouldgiveageneratorwhichhascircuitsizepoly( ).
|H|
However,aswearetryingtohitpolynomialson N variables,thiswouldyieldapoly(N)-sizegen-
erator whereas we would want a generator of complexity polylog(N). As such, we now define
succinctgeneratorsandgiveatighterrelationshipwithsuccincthittingsets.
Definition1.7.Let F[x ,...,x ] beaset ofmonomials = xa ,andletthe setspan( ) :=
1 n a
M ⊆ M { } M
∑ c xa : c F be all linear combinations of these monomials. Let span( ) and
F{[ xca∈M a ] be cala∈sses}of polynomials, where the latter is in many varCiab⊆les. FurMther, let D ⊆
a xa ′
F[{x1,.}..∈,xMn,y1,...,yℓ]beanotherclassofpolynomials. |M| C ⊆
ThenapolynomialmapG : Fℓ F isa -succinctgeneratorfor computablein if
M ′
→ C D C
1. mThiaelpcoolmynpoumteidalbGy(txhe,yx)a-:c=oo∑rdxian∈aMteGofxaG(.y)·xa isapolynomialfrom C′,where Gxa(y)isthepolyno-
2. Foreveryvalueα Fℓ,thepolynomial G(x,α) .
∈ ∈ C
3. G is agenerator for for . Thatis, D isa non-zero polynomial in F[c] iff D G 0in F[y],
D ∈ D ◦ 6≡
meaning that D(coeff (G(x,y))) = 0 as a polynomial in F[y], where these coefficients are taken
intheringF[y][x] sothMatcoeff (G6 (x,y)) F[y] . ♦
M
M ∈
Conditions (2) and item 3 are equivalent, over large enough fields, to the property that the
outputofthegenerator G(x,Fℓ) = G(x,α) : α Fℓ is a -succinct hitting setfor . However,
{ ∈ } C D
thegeneratorresultis a prioristrongeras it saysthat thehittingsetcan besuccinctly indexedby
apolynomialin .
′
C
Also, note that the computability of the generator implies -succinctness, that is, that its
′ ′
image G(x,α) : α FℓC areallcircuitswhichare -circuits,atleaCstassumingthat isaclassof
′ ′
{ ∈ } C C
polynomialswhichisclosedundersubstitution. However,sometimestheactuallysuccinctness
C
canbemorestringentthan forrestrictedclassesofcomputation.
′
C
We now give our first result, which uses the construction of a universal circuit to show that
there is a explicit universal construction of a succinct generator, that is, this circuit is a succinct
generator if there are any succinct hitting sets. Further, this shows that any succinct hitting set
(even infinite) implies a quasipolynomial deterministic black-box PIT algorithm. To make this
theoremclear,letVP denotetheclassofsmalllow-degreecircuitsinmvariables.
m
Theorem (Informal summary of Section 3). There is an explicit polylog(N)-size circuit which is a
VP -succinctgeneratorforVP iffthereisanyVP -succincthittingsetforVP . Further,
polylog(N) N polylog(N) N
the existence of any VP -succinct hitting set for VP implies an explicit poly(N)polylog(N)-size
polylog(N) N
hittingsetforVP .
N
Note that Aaronson and Drucker [AD08] proposed a candidate universal algebraic pseudo-
randomfunctionbasedongenericprojectionsofdeterminants,butdidnotproveitsuniversality.
Theirconstructiondoesnotseemsufficientfortheaboveresult,asdiscussedinSection3.
9