{"paper_meta":{"paper_id":"arxiv:0710.0360","title":"0710.0360","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0710.0360v1 [cs.CC] 1 Oct 2007\nInterpolation in Valiant’s theory\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nAbstract. We investigate the following question: if a polynomial can\nbe evaluated at rational points by a polynomial-time boolean algorithm,\ndoes it have a polynomial-size arithmetic circuit? We argue that this\nquestion is certainly difficult. Answering it negatively would indeed imply\nthat the constant-free versions of the algebraic complexity classes VP and\nVNP defined by Valiant are different. Answering this question positively\nwould imply a transfer theorem from boolean to algebraic complexity.\nOur proof method relies on Lagrange interpolation and on recent re-\nsults connecting the (boolean) counting hierarchy to algebraic complex-\nity classes. As a byproduct we obtain two additional results:\n(i) The constant-free, degree-unbounded version of Valiant’s hypothe-\nsis VP ̸= VNP implies the degree-bounded version. This result was\npreviously known to hold for fields of positive characteristic only.\n(ii) If exponential sums of easy to compute polynomials can be computed\nefficiently, then the same is true of exponential products. We point\nout an application of this result to the P=NP problem in the Blum-\nShub-Smale model of computation over the field of complex numbers.\n1\nIntroduction\nMotivation – The starting point of this paper is a question raised by Christos\nPapadimitriou in a personal communication to Erich Kaltofen1:\nQuestion\n(*)\nIf a multivariate polynomial P can be evaluated by a (boolean)\npolynomial-time algorithm on rational inputs, does that imply that\nP can be computed by a polynomial-size arithmetic circuit? In such\na circuit, the only allowed operations are additions, subtractions,\nand multiplications.\nThis question can be interpreted in several ways: one should at least state\nin which ring the coefficients of P lie, and which constants can be used by the\narithmetic circuits. Here we will focus on polynomials with integer coefficients,\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA.\n1 At the Oberwolfach complexity theory workshop where a part of this work was\npresented in June 2007, several participants told P.K. that they had independently\nthought of the same question.\n\n2\nPascal Koiran and Sylvain Perifel\nand most of the paper will deal with constant-free circuits. In Section 3.4 we\nstudy the case of circuits with rational constants computing polynomials with\ninteger coefficients (as we shall see, this is more natural than it might seem at\nfirst sight).\nAs pointed out by Papadimitriou, Strassen’s “Vermeidung von Divisionen”\n(see for instance [7], chapter 7) shows that for evaluating a low-degree polynomial\nP, divisions would not increase exponentially the power of arithmetic circuits. It\nis indeed a natural question whether, more generally, all boolean operations can\nbe replaced efficiently by additions, subtractions and multiplications. Questions\nof the same flavour (can “looking at bits” help for arithmetic computations?)\nhave been studied before. In particular, Kaltofen and Villard have shown that\nlooking at bits does help for computing the determinant [10].\nDiscussion – It is not clear what the correct answer to question (*) should\nbe. In this paper we will argue that answering it either way seems difficult.\nA natural strategy for obtaining a negative answer to question (*) would be\nto exhibit a family of polynomials that are easy to evaluate on rational inputs\nbut hard to evaluate by arithmetic circuits. Unfortunately, there seems to be\na lack of candidate polynomials. Another difficulty is that a negative answer\nwould imply the separation of the algebraic complexity classes VP0 and VNP0.\nThis observation is our main contribution to the study of question (*), and it\nis established in Theorem 8. The classes VP0 and VNP0 are constant-free ver-\nsions of the classes VP (of “easily computable polynomial families”) and VNP\n(of “easily definable polynomial families”) introduced by Valiant (precise defi-\nnitions are given in the next section). The separation VP0 ̸= VNP0 seems very\nplausible, but it also seems very difficult to establish. As explained at the be-\nginning of the introduction, we study in Section 3.4 the case of circuits with\nrational constants computing polynomials with integer coefficients. Allowing ra-\ntional constants makes the hypothesis that question (*) has a negative answer\nstronger than in the constant-free case. Accordingly, we obtain a stronger con-\nclusion: we can now show that the hypothesis would imply a superpolynomial\nlower bound on the size of arithmetic circuits computing the permanent.\nObtaining a positive answer to question (*) also seems difficult since it would\nimply the following transfer theorem: FP = ♯P ⇒VP = VNP (assuming that\nFP = ♯P, the permanent must be in FP; a positive answer to question (*) would\ntherefore imply that the permanent is in VP, and that VP = VNP by complete-\nness of the permanent). Unfortunately, in spite of all the work establishing close\nconnections between the boolean model of computation and the algebraic mod-\nels of Valiant and of Blum, Shub and Smale [2, 4, 5, 8, 9, 13–15] no such transfer\ntheorem is known. In fact, we do not know of any hypothesis from boolean com-\nplexity theory that would imply the equality VP = VNP (but transfer theorems\nin the opposite direction were established in [4]).\nSummary of results – Most of our results are derived from Theorem 3,\nour main theorem: if the evaluation of a family of polynomials (fn) at integer\npoints is a problem that lies in the (non uniform) counting hierarchy, the hy-\npothesis VP0 = VNP0 implies that (fn) can be evaluated by polynomial-size\n\nInterpolation in Valiant’s theory\n3\narithmetic circuits. Theorem 8, which contains our main contribution to the\nstudy of question (*), follows immediately since polynomial-time problems lie\nin the counting hierarchy. The proof of Theorem 3 relies on techniques from [1,\n6] and on Lagrange interpolation. Besides the application to question (*), we\nderive two additional results from Theorem 3:\n– The elements of the complexity classes VP, VNP and of their constant-free\nversions are families of polynomials of polynomially bounded degree. We\nshow in Theorem 5 that the collapse VP0 = VNP0 would imply the same\ncollapse for the unbounded versions of VP0 and VNP0. For fields of positive\ncharacteristic, the same result (and its converse) was obtained with different\ntechniques by Malod [16, 17].\n– Our third application of Theorem 3 is to the “P = NP?” problem in the\nBlum-Shub-Smale model of computation over C (or more generally, fields of\ncharacteristic 0). One natural strategy for separating PC from NPC would\nbe to exhibit a problem A in NPC \\ PC. Drawing on results from [13], we\nshow that this strategy is bound to fail for a fairly large class of “simple”\nproblems A, unless one can prove that VP0 ̸= VNP0. The class of “simple”\nproblems that we have in mind is NP(C,+,−,=). This is the class of NP prob-\nlems over the set of complex numbers endowed with addition, subtraction,\nand equality tests (there is therefore no multiplication in this structure).\nIt contains many natural problems, such as Subset Sum and Twenty Ques-\ntions [2, 19], that most likely belong to NPC \\ PC. As an intermediate result,\nwe show in Theorem 5 that if exponential sums of easy to compute poly-\nnomials can be computed efficiently, then the same is true of exponential\nproducts.\n2\nPreliminaries\n2.1\nValiant’s Classes\nIn Valiant’s model, one computes families of polynomials. A book-length treat-\nment of this topic can be found in [4]. We fix a field K of characteristic zero.\nAn arithmetic circuit is a circuit whose inputs are indeterminates x1, . . . , xu(n)\ntogether with arbitrary constants of K; there are +, −and ×-gates, and we\ntherefore compute multivariate polynomials. The polynomial computed by an\narithmetic circuit is defined in the usual way by the polynomial computed by its\noutput gate. The size of a circuit is the number of gates.\nThus a family (Cn) of arithmetic circuits computes a family (fn) of polynomi-\nals, fn ∈K[x1, . . . , xu(n)]. The class VPnb defined in [17] is the set of families (fn)\nof polynomials computed by a family (Cn) of polynomial-size arithmetic circuits,\ni.e., Cn computes fn and there exists a polynomial p(n) such that |Cn| ≤p(n)\nfor all n. We will assume without loss of generality that the number u(n) of\nvariables is bounded by a polynomial function of n. The subscript “nb” indi-\ncates that there is no bound on the degree of the polynomial, in contrast with\nthe original class VP of Valiant where a polynomial bound on the degree of the\n\n4\nPascal Koiran and Sylvain Perifel\npolynomial computed by the circuit is required. Note that these definitions are\nnonuniform.\nThe class VNP is the set of families of polynomials defined by an exponential\nsum of VP families. More precisely, (fn( ̄x)) ∈VNP if there exists (gn( ̄x, ̄y)) ∈VP\nand a polynomial p such that | ̄y| = p(n) and fn( ̄x) = P\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ). Sim-\nilarly, the class VΠP is the set of families of polynomials defined by an expo-\nnential product of VPnb families. More precisely, (fn( ̄x)) ∈VΠP if there ex-\nists (gn( ̄x, ̄y)) ∈VPnb and a polynomial p such that | ̄y| = p(n) and fn( ̄x) =\nQ\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ).\nWe can also define constant-free circuits: the only constant allowed is then\n1 (in order to allow the computation of constant polynomials). In this case, we\ncompute polynomials with integer coefficients. If f is a polynomial with integer\ncoefficients, we denote by τ(f) the size of a smallest constant-free circuit com-\nputing f. For classes of families of polynomials, we will use the superscript 0 to\nindicate the absence of constant: for instance, we will write VP0\nnb. For bounded-\ndegree classes, we are to be more careful because we also want to avoid the\ncomputation of constants of exponential bitsize: we first need the following def-\ninition.\nDefinition 1. Let C be an arithmetic circuit. The formal degree of a gate of C\nis defined by induction:\n– the formal degree of an input is 1;\n– the formal degree of a gate + or −is the maximum of the formal degrees of\nits inputs;\n– the formal degree of a gate × is the sum of the formal degrees of its inputs.\nNow, the formal degree of a circuit is the formal degree of the output gate.\nWe are now able to define constant-free degree-bounded Valiant’s classes. A\nfamily of polynomials (fn) belongs to VP0 if it is computable by a family of\ncircuits of size and formal degree bounded by a polynomial function of n. The\nclass VNP0 is then defined accordingly by a sum of VP0 families, in the same\nway as VNP is defined from VP.\nRemark 1. The hypothesis τ(PERn) = nO(1) used in [6] is implied by the hy-\npothesis VNP0 ⊂VP0\nnb and hence by VP0 = VNP0. As mentioned in [6], the\nconverse τ(PERn) = nO(1) ⇒VNP0 = VP0 is not known to hold, because the\nfamily (PERn) is not known to be VNP0-complete in a constant-free context\n(the proof of completeness of Valiant [20] indeed uses the constant 1/2). We\nwill mostly be concerned by the hypothesis VP0 = VNP0, but we will come to\nthe hypothesis τ(PERn) = nO(1) in Section 3.4 when dealing with circuits with\nconstants.\n2.2\nCounting Classes\nIn this paper we will encounter several counting classes, in particular the counting\nhierarchy defined below. Let us first see two classes of functions, ♯P and GapP.\n\nInterpolation in Valiant’s theory\n5\nDefinition 2.\n– The class ♯P is the set of functions f : {0, 1}∗→{0, 1}∗such\nthat there exist a language A ∈P and a polynomial p(n) satisfying\nf(x) = #{y ∈{0, 1}p(|x|) : (x, y) ∈A}.\n– A function f is in GapP if it is the difference of two functions in ♯P.\nReturning to classes of languages we recall the definition of the counting hierar-\nchy, introduced by Wagner [21]. It contains all the polynomial hierarchy PH and\nis contained in PSPACE. It is defined via the “majority” operator C as follows.\nDefinition 3.\n– If K is a complexity class, the class C.K is the set of lan-\nguages A such that there exist a language B ∈K and a polynomial p(n)\nsatisfying\nx ∈A ⇐⇒#{y ∈{0, 1}p(|x|) : (x, y) ∈B} ≥2p(|x|)−1.\n– The i-th level CiP of the counting hierarchy is defined recursively by C0P = P\nand Ci+1P = C.CiP. The counting hierarchy CH is the union of all these levels\nCiP.\nLevel 1 of CH, that is, C.P, is also called PP. Since Valiant’s classes are nonuni-\nform, we will rather work with nonuniform versions of these boolean classes, as\ndefined now following Karp and Lipton [11].\nDefinition 4. If K is a complexity class, the class K/poly is the set of languages\nA such that there exist a language B ∈K, a polynomial p(n) and a family of\nwords (called advices) (an)n≥0 satisfying\n– for all n ≥0, |an| ≤p(n);\n– for all word x, x ∈A ⇐⇒(x, a(|x|)) ∈B.\nRemark that the advice only depends on the size of x: it must therefore be the\nsame for all words of same length.\n2.3\nSequences of Integers\nOur aim now is to introduce a notion of complexity of a sequence of integers.\nIn order to avoid dealing with the sign of integers separately, we assume that\nwe can retrieve it from the boolean encoding of the integers. For example, the\nsign could be given by the first bit of the encoding and the absolute value by\nthe remaining bits.\nDefinition 5. A sequence of exponential bitsize is a sequence of integers (a(n, k))\nsuch that there exists a polynomial p(n) satisfying:\n1. a(n, k) is defined for n, k ∈N and 0 ≤k < 2p(n);\n2. for all n > 1, for all k < 2p(n), the bitsize of a(n, k) is ≤2p(n).\n\n6\nPascal Koiran and Sylvain Perifel\nFrom a(n, k), the following language is then defined:\nBit(a) = {(1n, k, j, b)| the j-th bit of a(n, k) is b},\nThe reader should be aware that the above definition and the next one are not\nquite the same as in [6]: we use a unary encoding for n instead of a binary\nencoding.\nDefinition 6. A sequence a(n, k) of exponential bitsize is definable in CH/poly\nif the language Bit(a) is in CH/poly.\nRemark 2. We shall also meet sequences with more than two parameters (n, k),\nfor example a(n, α(1), . . . , α(n)) for some integers α(i). In order to see it as a\nsequence with two parameters, (α(1), . . . , α(n)) will be considered as the encoding\nof a single integer. The parameter n might also be given as a subscript, as in\nfn(k), which should better be written f(n, k).\nLet us now propose a similar definition for families of polynomials.\nDefinition 7. Let (fn(x1, . . . , xu(n))) be a family of polynomials with integer\ncoefficients. We say that (fn) can be evaluated in CH/poly at integer points if\nthe following conditions are satisfied:\n1. the number u(n) of variables is polynomially bounded;\n2. the degree of fn as well as the bitsize of the coefficients of fn are bounded by\n2p(n) for some polynomial p(n);\n3. the language {(1n, i1, . . . , iu(n), j, b)| the j-th bit of fn(i1, . . . , iu(n)) is b} is\nin CH/poly.\nRemark 3. The same definition can be made for other complexity classes than\nCH/poly. For instance, if we replace CH/poly by P we obtain the notion of “poly-\nnomial time evaluation at integer points”. This notion will be useful for the study\nof question (*).\nThe following lemma is obvious from these definitions.\nLemma 1. The family (fn(x1, . . . , xu(n))) can be evaluated in CH/poly at inte-\nger points if and only if the sequence of integers a(n, i1, . . . , iu(n)) = fn(i1, . . . , iu(n))\nis definable in CH/poly.\nThe following theorem of [1, Theorem 4.1] will also be useful due to its corollary\nbelow.\nTheorem 1. Let BitSLP be the following problem: given a constant-free arith-\nmetic circuit computing an integer N, and given i ∈N in binary, decide whether\nthe i-th bit of the binary representation of N is 1. Then BitSLP is in CH.\nCorollary 1. If (fn) ∈VP0\nnb then it can be evaluated in CH/poly at integer\npoints.\n\nInterpolation in Valiant’s theory\n7\nThe results of this paper rely on the following link between Valiant’s classes\nand the counting hierarchy, [6, Lemmas 2.5 and 2.12].\nLemma 2. If VP0 = VNP0 then CH/poly = P/poly.\nIn particular, Lemma 2 was used to show that big sums and products are com-\nputable in the counting hierarchy, [6, Theorem 3.7]. As already mentioned, the\ncontext is not exactly the same as in [6] because we use a unary encoding. We\nnow give a version of this result which is just an easy “scaling up” of [6, Theo-\nrem 3.7] (it is enough to define a′(2p(n), k) = a(n, k) and to apply the result of\nB ̈urgisser).\nTheorem 2. Let p(n) be a polynomial and suppose a = (a(n, k))n∈N,k≤2p(n) is\ndefinable in CH/poly. Consider the sequences\nb(n) =\n2p(n)\nX\nk=0\na(n, k) and d(n) =\n2p(n)\nY\nk=0\na(n, k).\nThen (b(n))n∈N and (d(n))n∈N are definable in CH/poly.\nSuppose now that (s(n))n∈N and (t(n))n∈N are definable in CH/poly. Then the\nsequence of products (s(n)t(n))n∈N, and, if t(n) > 0, the sequence of quotients\n(⌊s(n)/t(n)⌋)n∈N, are definable in CH/poly.\n3\nInterpolation\nWe now begin the main technical developments.\n3.1\nCoefficients\nThe following lemma is Valiant’s criterion [20], see also [4, Prop. 2.20] and [12,\nTh. 2.3].\nLemma 3. Let a : (1n, i) 7→a(1n, i) be a function of GapP/poly, where n is\ngiven in unary and i in binary. Let p(n) be a polynomial and define the following\nsequence of polynomials:\nfn(x1, . . . , xp(n)) =\n2p(n)−1\nX\ni=0\na(1n, i)xi1\n1 · · · x\nip(n)\np(n) ,\nwhere ij is the j-th bit in the binary expression of i.\nThen (fn) ∈VNP0.\nHere is a “scaled up” generalization of [6, Th. 4.1(2)] to multivariate poly-\nnomials.\n\n8\nPascal Koiran and Sylvain Perifel\nLemma 4. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf VP0 = VNP0 then (fn) ∈VP0\nnb.\nProof. Expand a in binary: a(n, α(1), . . . , α(n)) = P2n\ni=0 ai(n, ̄α)2i. Let hn be the\nfollowing polynomial:\nhn(x1,1, x1,2, . . . , x1,n, x2,1, . . . , xn,n, z1, . . . , zn) =\n2n\nX\ni=0\nX\n ̄α\nai(n, ̄α)zi1\n1 · · · zin\nn xα(1)\n1\n1,1 xα(1)\n2\n1,2 · · · xα(1)\nn\n1,n xα(2)\n1\n2,1 · · · xα(n)\nn\nn,n .\nThen we have:\nhn(x20\n1 , x21\n1 , . . . , x2n\n1 , x20\n2 , . . . , x2n\nn , 220, 221, . . . , 22n) = fn(x1, . . . , xn).\nSince VP0 = VNP0, by Lemma 2 the nonuniform counting hierarchy collapses,\ntherefore computing the i-th bit ai(n, ̄α) of a(n, ̄α) on input (1n, ̄α, i) is in\nGapP/poly (and even in P/poly). By Lemma 3, (hn) ∈VNP0. By the hypothesis\nVP0 = VNP0, (hn) ∈VP0 and thus using repeated squaring for computing big\npowers yields (fn) ∈VNP0\nnb.\n⊓⊔\n3.2\nInterpolation\nLet us now state two lemmas on interpolation polynomials.\nLemma 5 (multivariate Lagrange interpolation). Let p(x1, . . . , xn) be a\npolynomial of degree ≤d. Then\np(x1, . . . , xn) =\nX\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\n Y\njk̸=ik\nxk −jk\nik −jk\n \n,\nwhere the integers jk range from 0 to d.\nProof. The proof goes by induction on the number n of variables. For n = 1,\nthis is the usual Lagrange interpolation formula: we have\np(x) =\nd\nX\ni=0\np(i)\nY\nj̸=i\nx −j\ni −j\nbecause both polynomials are of degree ≤d and coincide on at least d+1 distinct\npoints.\n\nInterpolation in Valiant’s theory\n9\nFor n + 1, the induction case n = 1 yields\np(x1, . . . , xn+1) =\nd\nX\nin+1=0\np(x1, . . . , xn, in+1)\nY\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\n.\nBy induction hypothesis, this is equal to\nd\nX\nin+1=0\n \n \nX\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\nY\njk̸=ik\nxk −jk\nik −jk\n \n \nY\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\nwhich is the desired result.\n⊓⊔\nLemma 6. Let a(n) = Q2n−1\ni=0\nQ\nj̸=i(i −j), where j ranges from 0 to 2n −1. Let\npi1,...,in( ̄x) be the following family of polynomials:\npi1,...,in(x1, . . . , xn) =\nn\nY\nk=1\n \na(n)\nY\njk̸=ik\nxk −jk\nik −jk\n \n,\nwhere the integers jk range from 0 to 2n −1 and the integers ik are given in\nbinary and range from 0 to 2n −1. Then the coefficients of pi1,...,in are integers\ndefinable in the counting hierarchy, as is a(n).\nProof. As a first step, note that the coefficient of the monomial xα1\n1 · · · xαn\nn\nin pn\nis equal to the product of the coefficients of the monomials xαk\nk\nin the univariate\npolynomials a(n) Q\njk̸=ik\nxk−jk\nik−jk . Hence we just have to check that these different\ncoefficients of univariate polynomials are themselves definable in the counting\nhierarchy. Let us first focus on the univariate polynomial Q\njk̸=ik(xk −jk), that\nis, let us forget the multiplicative term b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) for the\nmoment.\nWe use the same argument as [6, Cor. 3.9]. Namely, we remark that the\ncoefficients of this polynomial are bounded in absolute value by 22n2\n. Therefore\nin the univariate polynomial Q\njk̸=ik(xk −jk) we can replace the variable xk\nby 22n2\nand there will be no overlap of the coefficients of the different powers\nof xk, thus we can recover the coefficients of the monomial from the value of\nthis product. By the first part of Theorem 2, we can evaluate in the counting\nhierarchy the polynomial at the point 22n2\n, because it is a product of exponential\nsize. So the coefficients are definable in the counting hierarchy.\nIt is now enough to note that the first part of Theorem 2 implies that a(n)\nas well as b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) are also definable in the counting\nhierarchy.\n⊓⊔\nRemark that the sequence a(n) of Lemma 6 is introduced only so as to obtain\ninteger coefficients. We will then divide by a(n) in the next proofs.\n\n10\nPascal Koiran and Sylvain Perifel\n3.3\nMain Results\nLet us now state the main theorem.\nTheorem 3. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials.\nSuppose (fn) can be evaluated in CH/poly at integer points. If VP0 = VNP0 then\n(fn) ∈VP0\nnb.\nProof. The goal is to use the interpolation formula of Lemma 5:\nfn(x1, . . . , xu(n)) =\nX\n0≤i1,...,iu(n)≤d\nbi1,...,iu(n)( ̄x),\n(1)\nwhere bi1,...,iu(n)( ̄x) = fn(i1, . . . , iu(n)) Qu(n)\nk=1\nQ\njk̸=ik\nxk−jk\nik−jk . We will show that\nthe coefficients of bi1,...,iu(n) and fn are definable in CH/poly. The conclusion of\nthe theorem will then follow from Lemma 4.\nIn order to show that the coefficients of bi1,...,iu(n) are definable in CH/poly,\nwe note that the polynomial pi1,...,in and the sequence a(n) of Lemma 6 satisfy\nthe relation\nbi1,...,iu(n)( ̄x) = a(u(n))−u(n)fn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x).\nBy Lemma 6, the coefficients of pi1,...,iu(n)( ̄x) are definable in CH. By hypothesis,\n(fn) can be evaluated in CH/poly at integer points. This implies by Lemma 1\nthat fn(i1, . . . , iu(n)) is definable in CH/poly. This is also the case of the product\nfn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x) by Theorem 2. Now, the same theorem enables us\nto divide by a(u(n))u(n), thereby showing that the coefficients of bi1,...,iu(n)( ̄x)\nare definable in CH/poly. It then follows from (1) and another application of\nTheorem 2 that the coefficients of fn are definable in CH/poly. Therefore by\nLemma 4, (fn) ∈VP0\nnb under the hypothesis VP0 = VNP0.\n⊓⊔\nWe now derive some consequences of Theorem 3.\nTheorem 4. Let (fn( ̄x, ̄ǫ)) ∈VP0\nnb. Let\ngn( ̄x) =\nX\n ̄ǫ\nfn( ̄x, ̄ǫ) and hn( ̄x) =\nY\n ̄ǫ\nfn( ̄x, ̄ǫ).\nIf VNP0 = VP0 then (gn) and (hn) are in VP0\nnb.\nProof. By Corollary 1 (fn) can be evaluated in CH/poly at integer points. Now,\nusing Lemma 1 before and after the first part of Theorem 2 shows that (gn) and\n(hn) can also be evaluated in CH/poly at integer points. The result then follows\nby Theorem 3.\n⊓⊔\nThe following is now immediate.\nTheorem 5. The hypothesis VP0 = VNP0 implies that VP0\nnb = VNP0\nnb and\nVP0\nnb = VΠP0.\n\nInterpolation in Valiant’s theory\n11\nRemark 4. It is not clear whether the converse of the first implication in The-\norem 5 (VP0 = VNP0 =⇒VP0\nnb = VNP0\nnb) holds true. This is related to the\nissue of large constants in arithmetic circuits: it seems difficult to rule out the\npossibility that some polynomial family in VNP0 (for instance, the permanent or\nthe hamiltonian) does not lie in VP0 but is still computable by polynomial-size\narithmetic circuits using integer constants of exponential bit size.\nThe converse does hold if arbitrary constants are allowed: we indeed have\nVPnb = VNPnb =⇒VP = VNP. But in this non-constant-free context, it is not\nclear whether VP = VNP unconditionally implies VPnb = VNPnb: indeed, in this\ncontext the generalized Riemann hypothesis would be needed to make the proof\nof Lemma 2 work (see [6] for details).\nAs mentioned in the introduction, another corollary concerns a transfer the-\norem with classes of algebraic complexity in the BSS model. Blum, Shub and\nSmale [2, 3] have defined the classes P and NP over the real and complex fields.\nIt was extended to arbitrary structures by Poizat [18]. Here we use nonuniform\nversions of these classes, hence the notations P and NP.\nTheorem 7 below proves that, over a field of characteristic zero, if we separate\n(the nonuniform versions of) P and NP thanks to a “simple” NP problem, then\nwe separate (the constant-free versions of) VP and VNP. The class of “simple”\nproblems here is NP where the multiplication is not allowed, i.e., the only oper-\nations are +, −and =. It contains in particular Twenty Questions and Subset\nSum. We will need a result from [13]:\nTheorem 6. Let K be a field of characteristic zero. If VP0\nnb = VΠP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nBy Theorem 5, the following is immediate.\nTheorem 7. Let K be a field of characteristic zero. If VP0 = VNP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nAt last, as a corollary of Theorem 3 again, we obtain the following result\nconcerning question (*), suggesting that it will be hard to refute. As pointed out\nin the introduction, this result does not give any evidence concerning the answer\nto question (*) since the separation VP0 ̸= VNP0 is very likely to be true.\nTheorem 8. If question (*) has a negative answer then VP0 ̸= VNP0. More\nprecisely, let (fn) be a family of multivariate polynomials which can be evaluated\nin polynomial time at integer points (in the sense of Remark 3). If VP0 = VNP0\nthen (fn) ∈VP0\nnb.\n3.4\nArithmetic Circuits with Constants\nIn this section we investigate another interpretation of question (*): we still\nconsider polynomials with integer coefficients, but we allow rational constants\nin our circuits (it turns out that the constant 1/2 plays a special role due to its\nappearance in the completeness proof for the permanent). The hypothesis that\n\n12\nPascal Koiran and Sylvain Perifel\nquestion (*) has a negative answer is then stronger, and we obtain a stronger\nconclusion than in Theorem 8. Namely, we can conclude that τ(PERn) ̸= nO(1)\ninstead of VNP0 ̸= VP0 (see Remark 1). We recall that τ, the constant-free\narithmetic circuit complexity of a polynomial, is defined in Section 2.1.\nTheorem 9. As explained above, we consider here polynomials with integer co-\nefficients but circuits with rational constants. If question (*) has a negative an-\nswer, then τ(PERn) is not polynomially bounded.\nMore precisely, let (fn) be a family of multivariate polynomials which can\nbe evaluated in polynomial time at integer points (in the sense of Remark 3). If\nτ(PERn) is polynomially bounded, (fn) can be evaluated by a family of polynomial-\nsize arithmetic circuits that use only the constant 1/2.\nIt is easy to see that Theorem 9 follows from a slight modification of the\ndifferent lemmas above. Lemma 2 is replaced by the following stronger lemma,\nfrom [6].\nLemma 7. If τ(PERn) = nO(1) then CH/poly = P/poly.\nThen Lemma 4 is replaced by the following result, whose proof relies on an\ninspection of Valiant’s proof [20] of VNP-completeness of the permanent, see [6].\nLemma 8. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nFinally, Theorem 3 becomes the following.\nLemma 9. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials (with\ninteger coefficients). Suppose (fn) can be evaluated in CH/poly at integer points.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nTheorem 9 follows since the coefficient 2p(n) can be cancelled by multiplying by\nthe constant 2−p(n), which can be computed from scratch from the constant 1/2.\nAcknowledgments\nWe would like to thank Erich Kaltofen and Christos Papadimitriou for sharing their\nthoughts on question (*).\n\nInterpolation in Valiant’s theory\n13\nReferences\n1. E. Allender, P. B ̈urgisser, J. Kjeldgaard-Pedersen, and P. Bro Miltersen. On the\ncomplexity of numerical analysis. In IEEE Conference on Computational Com-\nplexity, pages 331–339, 2006.\n2. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n3. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n4. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory. Num-\nber 7 in Algorithms and Computation in Mathematics. Springer, 2000.\n5. P. B ̈urgisser. The complexity of factors of multivariate polynomials. Foundations\nof Computational Mathematics, 4(4):369–396, 2004.\n6. P. B ̈urgisser.\nOn defining integers in the counting hierarchy and proving lower\nbounds in algebraic complexity. In Proc. STACS 2007, pages 133–144, 2007. Full\nversion: ECCC Report No. 113, August 2006.\n7. P. B ̈urgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory.\nSpringer, 1997.\n8. H. Fournier and P. Koiran. Are lower bounds easier over the reals? In Proc. 30th\nACM Symposium on Theory of Computing, pages 507–513, 1998.\n9. H. Fournier and P. Koiran. Lower bounds are not easier over the reals: Inside PH. In\nProc. 27th International Colloquium on Automata, Languages and Programming,\nvolume 1853 of Lecture Notes in Computer Science, pages 832–843. Springer, 2000.\n10. E. Kaltofen and G. Villard. On the complexity of computing determinants. Com-\nputational Complexity, 13(3-4):91-130, 2004.\n11. R. Karp and R. Lipton. Turing machines that take advice. L’Enseignement Ma-\nth ́ematique, 28:191–209, 1982.\n12. P. Koiran. Valiant’s model and the cost of computing integers. Computational\nComplexity, 13(3-4):131–146, 2004.\n13. P. Koiran and S. Perifel. Valiant’s model: from exponential sums to exponential\nproducts. In Proc. MFCS 2006, volume 4162 of Lecture Notes in Computer Science,\npages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the complex field.\nIn Proc. MFCS 2007, volume 4708 of Lecture Notes in Computer Science, pages\n359-370, 2007.\n15. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the reals. In Proc.\nSTACS 2007, volume 4393 of Lecture Notes in Computer Science, pages 417–428.\nSpringer-Verlag, 2007. long version: http://prunel.ccsd.cnrs.fr/ensl-00103018.\n16. G. Malod. The complexity of polynomials and their coefficient functions. In Proc.\n22nd IEEE Conference on Computational Cmplexity, pages 193–204, 2007.\n17. Guillaume Malod.\nPolynˆomes et coefficients.\nPhD thesis, Universit ́e Claude\nBernard Lyon 1, July 2003.\nAvailable from http://tel.archives-ouvertes.fr/tel-\n00087399.\n18. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n19. M. Shub and S. Smale. On the intractability of Hilbert’s Nullstellensatz and an\nalgebraic version of “P=NP”. Duke Mathematical Journal, 81(1):47–54, 1995.\n20. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979.\n21. K. W. Wagner. The complexity of combinatorial problems with succinct input\nrepresentation. Acta Inform., 23(3):325–356, 1986.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0710.0360v1 [cs.CC] 1 Oct 2007\nInterpolation in Valiant’s theory\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nAbstract. We investigate the following question: if a polynomial can\nbe evaluated at rational points by a polynomial-time boolean algorithm,\ndoes it have a polynomial-size arithmetic circuit? We argue that this\nquestion is certainly difficult. Answering it negatively would indeed imply\nthat the constant-free versions of the algebraic complexity classes VP and\nVNP defined by Valiant are different. Answering this question positively\nwould imply a transfer theorem from boolean to algebraic complexity.\nOur proof method relies on Lagrange interpolation and on recent re-\nsults connecting the (boolean) counting hierarchy to algebraic complex-\nity classes. As a byproduct we obtain two additional results:\n(i) The constant-free, degree-unbounded version of Valiant’s hypothe-\nsis VP ̸= VNP implies the degree-bounded version. This result was\npreviously known to hold for fields of positive characteristic only.\n(ii) If exponential sums of easy to compute polynomials can be computed\nefficiently, then the same is true of exponential products. We point\nout an application of this result to the P=NP problem in the Blum-\nShub-Smale model of computation over the field of complex numbers.\n1\nIntroduction\nMotivation – The starting point of this paper is a question raised by Christos\nPapadimitriou in a personal communication to Erich Kaltofen1:\nQuestion\n(*)\nIf a multivariate polynomial P can be evaluated by a (boolean)\npolynomial-time algorithm on rational inputs, does that imply that\nP can be computed by a polynomial-size arithmetic circuit? In such\na circuit, the only allowed operations are additions, subtractions,\nand multiplications.\nThis question can be interpreted in several ways: one should at least state\nin which ring the coefficients of P lie, and which constants can be used by the\narithmetic circuits. Here we will focus on polynomials with integer coefficients,\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA.\n1 At the Oberwolfach complexity theory workshop where a part of this work was\npresented in June 2007, several participants told P.K. that they had independently\nthought of the same question."},{"paragraph_id":"p2","order":2,"text":"2\nPascal Koiran and Sylvain Perifel\nand most of the paper will deal with constant-free circuits. In Section 3.4 we\nstudy the case of circuits with rational constants computing polynomials with\ninteger coefficients (as we shall see, this is more natural than it might seem at\nfirst sight).\nAs pointed out by Papadimitriou, Strassen’s “Vermeidung von Divisionen”\n(see for instance [7], chapter 7) shows that for evaluating a low-degree polynomial\nP, divisions would not increase exponentially the power of arithmetic circuits. It\nis indeed a natural question whether, more generally, all boolean operations can\nbe replaced efficiently by additions, subtractions and multiplications. Questions\nof the same flavour (can “looking at bits” help for arithmetic computations?)\nhave been studied before. In particular, Kaltofen and Villard have shown that\nlooking at bits does help for computing the determinant [10].\nDiscussion – It is not clear what the correct answer to question (*) should\nbe. In this paper we will argue that answering it either way seems difficult.\nA natural strategy for obtaining a negative answer to question (*) would be\nto exhibit a family of polynomials that are easy to evaluate on rational inputs\nbut hard to evaluate by arithmetic circuits. Unfortunately, there seems to be\na lack of candidate polynomials. Another difficulty is that a negative answer\nwould imply the separation of the algebraic complexity classes VP0 and VNP0.\nThis observation is our main contribution to the study of question (*), and it\nis established in Theorem 8. The classes VP0 and VNP0 are constant-free ver-\nsions of the classes VP (of “easily computable polynomial families”) and VNP\n(of “easily definable polynomial families”) introduced by Valiant (precise defi-\nnitions are given in the next section). The separation VP0 ̸= VNP0 seems very\nplausible, but it also seems very difficult to establish. As explained at the be-\nginning of the introduction, we study in Section 3.4 the case of circuits with\nrational constants computing polynomials with integer coefficients. Allowing ra-\ntional constants makes the hypothesis that question (*) has a negative answer\nstronger than in the constant-free case. Accordingly, we obtain a stronger con-\nclusion: we can now show that the hypothesis would imply a superpolynomial\nlower bound on the size of arithmetic circuits computing the permanent.\nObtaining a positive answer to question (*) also seems difficult since it would\nimply the following transfer theorem: FP = ♯P ⇒VP = VNP (assuming that\nFP = ♯P, the permanent must be in FP; a positive answer to question (*) would\ntherefore imply that the permanent is in VP, and that VP = VNP by complete-\nness of the permanent). Unfortunately, in spite of all the work establishing close\nconnections between the boolean model of computation and the algebraic mod-\nels of Valiant and of Blum, Shub and Smale [2, 4, 5, 8, 9, 13–15] no such transfer\ntheorem is known. In fact, we do not know of any hypothesis from boolean com-\nplexity theory that would imply the equality VP = VNP (but transfer theorems\nin the opposite direction were established in [4]).\nSummary of results – Most of our results are derived from Theorem 3,\nour main theorem: if the evaluation of a family of polynomials (fn) at integer\npoints is a problem that lies in the (non uniform) counting hierarchy, the hy-\npothesis VP0 = VNP0 implies that (fn) can be evaluated by polynomial-size"},{"paragraph_id":"p3","order":3,"text":"Interpolation in Valiant’s theory\n3\narithmetic circuits. Theorem 8, which contains our main contribution to the\nstudy of question (*), follows immediately since polynomial-time problems lie\nin the counting hierarchy. The proof of Theorem 3 relies on techniques from [1,\n6] and on Lagrange interpolation. Besides the application to question (*), we\nderive two additional results from Theorem 3:\n– The elements of the complexity classes VP, VNP and of their constant-free\nversions are families of polynomials of polynomially bounded degree. We\nshow in Theorem 5 that the collapse VP0 = VNP0 would imply the same\ncollapse for the unbounded versions of VP0 and VNP0. For fields of positive\ncharacteristic, the same result (and its converse) was obtained with different\ntechniques by Malod [16, 17].\n– Our third application of Theorem 3 is to the “P = NP?” problem in the\nBlum-Shub-Smale model of computation over C (or more generally, fields of\ncharacteristic 0). One natural strategy for separating PC from NPC would\nbe to exhibit a problem A in NPC \\ PC. Drawing on results from [13], we\nshow that this strategy is bound to fail for a fairly large class of “simple”\nproblems A, unless one can prove that VP0 ̸= VNP0. The class of “simple”\nproblems that we have in mind is NP(C,+,−,=). This is the class of NP prob-\nlems over the set of complex numbers endowed with addition, subtraction,\nand equality tests (there is therefore no multiplication in this structure).\nIt contains many natural problems, such as Subset Sum and Twenty Ques-\ntions [2, 19], that most likely belong to NPC \\ PC. As an intermediate result,\nwe show in Theorem 5 that if exponential sums of easy to compute poly-\nnomials can be computed efficiently, then the same is true of exponential\nproducts.\n2\nPreliminaries\n2.1\nValiant’s Classes\nIn Valiant’s model, one computes families of polynomials. A book-length treat-\nment of this topic can be found in [4]. We fix a field K of characteristic zero.\nAn arithmetic circuit is a circuit whose inputs are indeterminates x1, . . . , xu(n)\ntogether with arbitrary constants of K; there are +, −and ×-gates, and we\ntherefore compute multivariate polynomials. The polynomial computed by an\narithmetic circuit is defined in the usual way by the polynomial computed by its\noutput gate. The size of a circuit is the number of gates.\nThus a family (Cn) of arithmetic circuits computes a family (fn) of polynomi-\nals, fn ∈K[x1, . . . , xu(n)]. The class VPnb defined in [17] is the set of families (fn)\nof polynomials computed by a family (Cn) of polynomial-size arithmetic circuits,\ni.e., Cn computes fn and there exists a polynomial p(n) such that |Cn| ≤p(n)\nfor all n. We will assume without loss of generality that the number u(n) of\nvariables is bounded by a polynomial function of n. The subscript “nb” indi-\ncates that there is no bound on the degree of the polynomial, in contrast with\nthe original class VP of Valiant where a polynomial bound on the degree of the"},{"paragraph_id":"p4","order":4,"text":"4\nPascal Koiran and Sylvain Perifel\npolynomial computed by the circuit is required. Note that these definitions are\nnonuniform.\nThe class VNP is the set of families of polynomials defined by an exponential\nsum of VP families. More precisely, (fn( ̄x)) ∈VNP if there exists (gn( ̄x, ̄y)) ∈VP\nand a polynomial p such that | ̄y| = p(n) and fn( ̄x) = P\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ). Sim-\nilarly, the class VΠP is the set of families of polynomials defined by an expo-\nnential product of VPnb families. More precisely, (fn( ̄x)) ∈VΠP if there ex-\nists (gn( ̄x, ̄y)) ∈VPnb and a polynomial p such that | ̄y| = p(n) and fn( ̄x) =\nQ\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ).\nWe can also define constant-free circuits: the only constant allowed is then\n1 (in order to allow the computation of constant polynomials). In this case, we\ncompute polynomials with integer coefficients. If f is a polynomial with integer\ncoefficients, we denote by τ(f) the size of a smallest constant-free circuit com-\nputing f. For classes of families of polynomials, we will use the superscript 0 to\nindicate the absence of constant: for instance, we will write VP0\nnb. For bounded-\ndegree classes, we are to be more careful because we also want to avoid the\ncomputation of constants of exponential bitsize: we first need the following def-\ninition.\nDefinition 1. Let C be an arithmetic circuit. The formal degree of a gate of C\nis defined by induction:\n– the formal degree of an input is 1;\n– the formal degree of a gate + or −is the maximum of the formal degrees of\nits inputs;\n– the formal degree of a gate × is the sum of the formal degrees of its inputs.\nNow, the formal degree of a circuit is the formal degree of the output gate.\nWe are now able to define constant-free degree-bounded Valiant’s classes. A\nfamily of polynomials (fn) belongs to VP0 if it is computable by a family of\ncircuits of size and formal degree bounded by a polynomial function of n. The\nclass VNP0 is then defined accordingly by a sum of VP0 families, in the same\nway as VNP is defined from VP.\nRemark 1. The hypothesis τ(PERn) = nO(1) used in [6] is implied by the hy-\npothesis VNP0 ⊂VP0\nnb and hence by VP0 = VNP0. As mentioned in [6], the\nconverse τ(PERn) = nO(1) ⇒VNP0 = VP0 is not known to hold, because the\nfamily (PERn) is not known to be VNP0-complete in a constant-free context\n(the proof of completeness of Valiant [20] indeed uses the constant 1/2). We\nwill mostly be concerned by the hypothesis VP0 = VNP0, but we will come to\nthe hypothesis τ(PERn) = nO(1) in Section 3.4 when dealing with circuits with\nconstants.\n2.2\nCounting Classes\nIn this paper we will encounter several counting classes, in particular the counting\nhierarchy defined below. Let us first see two classes of functions, ♯P and GapP."},{"paragraph_id":"p5","order":5,"text":"Interpolation in Valiant’s theory\n5\nDefinition 2.\n– The class ♯P is the set of functions f : {0, 1}∗→{0, 1}∗such\nthat there exist a language A ∈P and a polynomial p(n) satisfying\nf(x) = #{y ∈{0, 1}p(|x|) : (x, y) ∈A}.\n– A function f is in GapP if it is the difference of two functions in ♯P.\nReturning to classes of languages we recall the definition of the counting hierar-\nchy, introduced by Wagner [21]. It contains all the polynomial hierarchy PH and\nis contained in PSPACE. It is defined via the “majority” operator C as follows.\nDefinition 3.\n– If K is a complexity class, the class C.K is the set of lan-\nguages A such that there exist a language B ∈K and a polynomial p(n)\nsatisfying\nx ∈A ⇐⇒#{y ∈{0, 1}p(|x|) : (x, y) ∈B} ≥2p(|x|)−1.\n– The i-th level CiP of the counting hierarchy is defined recursively by C0P = P\nand Ci+1P = C.CiP. The counting hierarchy CH is the union of all these levels\nCiP.\nLevel 1 of CH, that is, C.P, is also called PP. Since Valiant’s classes are nonuni-\nform, we will rather work with nonuniform versions of these boolean classes, as\ndefined now following Karp and Lipton [11].\nDefinition 4. If K is a complexity class, the class K/poly is the set of languages\nA such that there exist a language B ∈K, a polynomial p(n) and a family of\nwords (called advices) (an)n≥0 satisfying\n– for all n ≥0, |an| ≤p(n);\n– for all word x, x ∈A ⇐⇒(x, a(|x|)) ∈B.\nRemark that the advice only depends on the size of x: it must therefore be the\nsame for all words of same length.\n2.3\nSequences of Integers\nOur aim now is to introduce a notion of complexity of a sequence of integers.\nIn order to avoid dealing with the sign of integers separately, we assume that\nwe can retrieve it from the boolean encoding of the integers. For example, the\nsign could be given by the first bit of the encoding and the absolute value by\nthe remaining bits.\nDefinition 5. A sequence of exponential bitsize is a sequence of integers (a(n, k))\nsuch that there exists a polynomial p(n) satisfying:\n1. a(n, k) is defined for n, k ∈N and 0 ≤k < 2p(n);\n2. for all n > 1, for all k < 2p(n), the bitsize of a(n, k) is ≤2p(n)."},{"paragraph_id":"p6","order":6,"text":"6\nPascal Koiran and Sylvain Perifel\nFrom a(n, k), the following language is then defined:\nBit(a) = {(1n, k, j, b)| the j-th bit of a(n, k) is b},\nThe reader should be aware that the above definition and the next one are not\nquite the same as in [6]: we use a unary encoding for n instead of a binary\nencoding.\nDefinition 6. A sequence a(n, k) of exponential bitsize is definable in CH/poly\nif the language Bit(a) is in CH/poly.\nRemark 2. We shall also meet sequences with more than two parameters (n, k),\nfor example a(n, α(1), . . . , α(n)) for some integers α(i). In order to see it as a\nsequence with two parameters, (α(1), . . . , α(n)) will be considered as the encoding\nof a single integer. The parameter n might also be given as a subscript, as in\nfn(k), which should better be written f(n, k).\nLet us now propose a similar definition for families of polynomials.\nDefinition 7. Let (fn(x1, . . . , xu(n))) be a family of polynomials with integer\ncoefficients. We say that (fn) can be evaluated in CH/poly at integer points if\nthe following conditions are satisfied:\n1. the number u(n) of variables is polynomially bounded;\n2. the degree of fn as well as the bitsize of the coefficients of fn are bounded by\n2p(n) for some polynomial p(n);\n3. the language {(1n, i1, . . . , iu(n), j, b)| the j-th bit of fn(i1, . . . , iu(n)) is b} is\nin CH/poly.\nRemark 3. The same definition can be made for other complexity classes than\nCH/poly. For instance, if we replace CH/poly by P we obtain the notion of “poly-\nnomial time evaluation at integer points”. This notion will be useful for the study\nof question (*).\nThe following lemma is obvious from these definitions.\nLemma 1. The family (fn(x1, . . . , xu(n))) can be evaluated in CH/poly at inte-\nger points if and only if the sequence of integers a(n, i1, . . . , iu(n)) = fn(i1, . . . , iu(n))\nis definable in CH/poly.\nThe following theorem of [1, Theorem 4.1] will also be useful due to its corollary\nbelow.\nTheorem 1. Let BitSLP be the following problem: given a constant-free arith-\nmetic circuit computing an integer N, and given i ∈N in binary, decide whether\nthe i-th bit of the binary representation of N is 1. Then BitSLP is in CH.\nCorollary 1. If (fn) ∈VP0\nnb then it can be evaluated in CH/poly at integer\npoints."},{"paragraph_id":"p7","order":7,"text":"Interpolation in Valiant’s theory\n7\nThe results of this paper rely on the following link between Valiant’s classes\nand the counting hierarchy, [6, Lemmas 2.5 and 2.12].\nLemma 2. If VP0 = VNP0 then CH/poly = P/poly.\nIn particular, Lemma 2 was used to show that big sums and products are com-\nputable in the counting hierarchy, [6, Theorem 3.7]. As already mentioned, the\ncontext is not exactly the same as in [6] because we use a unary encoding. We\nnow give a version of this result which is just an easy “scaling up” of [6, Theo-\nrem 3.7] (it is enough to define a′(2p(n), k) = a(n, k) and to apply the result of\nB ̈urgisser).\nTheorem 2. Let p(n) be a polynomial and suppose a = (a(n, k))n∈N,k≤2p(n) is\ndefinable in CH/poly. Consider the sequences\nb(n) =\n2p(n)\nX\nk=0\na(n, k) and d(n) =\n2p(n)\nY\nk=0\na(n, k).\nThen (b(n))n∈N and (d(n))n∈N are definable in CH/poly.\nSuppose now that (s(n))n∈N and (t(n))n∈N are definable in CH/poly. Then the\nsequence of products (s(n)t(n))n∈N, and, if t(n) > 0, the sequence of quotients\n(⌊s(n)/t(n)⌋)n∈N, are definable in CH/poly.\n3\nInterpolation\nWe now begin the main technical developments.\n3.1\nCoefficients\nThe following lemma is Valiant’s criterion [20], see also [4, Prop. 2.20] and [12,\nTh. 2.3].\nLemma 3. Let a : (1n, i) 7→a(1n, i) be a function of GapP/poly, where n is\ngiven in unary and i in binary. Let p(n) be a polynomial and define the following\nsequence of polynomials:\nfn(x1, . . . , xp(n)) =\n2p(n)−1\nX\ni=0\na(1n, i)xi1\n1 · · · x\nip(n)\np(n) ,\nwhere ij is the j-th bit in the binary expression of i.\nThen (fn) ∈VNP0.\nHere is a “scaled up” generalization of [6, Th. 4.1(2)] to multivariate poly-\nnomials."},{"paragraph_id":"p8","order":8,"text":"8\nPascal Koiran and Sylvain Perifel\nLemma 4. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf VP0 = VNP0 then (fn) ∈VP0\nnb.\nProof. Expand a in binary: a(n, α(1), . . . , α(n)) = P2n\ni=0 ai(n, ̄α)2i. Let hn be the\nfollowing polynomial:\nhn(x1,1, x1,2, . . . , x1,n, x2,1, . . . , xn,n, z1, . . . , zn) =\n2n\nX\ni=0\nX\n ̄α\nai(n, ̄α)zi1\n1 · · · zin\nn xα(1)\n1\n1,1 xα(1)\n2\n1,2 · · · xα(1)\nn\n1,n xα(2)\n1\n2,1 · · · xα(n)\nn\nn,n .\nThen we have:\nhn(x20\n1 , x21\n1 , . . . , x2n\n1 , x20\n2 , . . . , x2n\nn , 220, 221, . . . , 22n) = fn(x1, . . . , xn).\nSince VP0 = VNP0, by Lemma 2 the nonuniform counting hierarchy collapses,\ntherefore computing the i-th bit ai(n, ̄α) of a(n, ̄α) on input (1n, ̄α, i) is in\nGapP/poly (and even in P/poly). By Lemma 3, (hn) ∈VNP0. By the hypothesis\nVP0 = VNP0, (hn) ∈VP0 and thus using repeated squaring for computing big\npowers yields (fn) ∈VNP0\nnb.\n⊓⊔\n3.2\nInterpolation\nLet us now state two lemmas on interpolation polynomials.\nLemma 5 (multivariate Lagrange interpolation). Let p(x1, . . . , xn) be a\npolynomial of degree ≤d. Then\np(x1, . . . , xn) =\nX\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\n Y\njk̸=ik\nxk −jk\nik −jk"},{"paragraph_id":"p9","order":9,"text":",\nwhere the integers jk range from 0 to d.\nProof. The proof goes by induction on the number n of variables. For n = 1,\nthis is the usual Lagrange interpolation formula: we have\np(x) =\nd\nX\ni=0\np(i)\nY\nj̸=i\nx −j\ni −j\nbecause both polynomials are of degree ≤d and coincide on at least d+1 distinct\npoints."},{"paragraph_id":"p10","order":10,"text":"Interpolation in Valiant’s theory\n9\nFor n + 1, the induction case n = 1 yields\np(x1, . . . , xn+1) =\nd\nX\nin+1=0\np(x1, . . . , xn, in+1)\nY\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\n.\nBy induction hypothesis, this is equal to\nd\nX\nin+1=0"},{"paragraph_id":"p11","order":11,"text":"X\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\nY\njk̸=ik\nxk −jk\nik −jk"},{"paragraph_id":"p12","order":12,"text":"Y\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\nwhich is the desired result.\n⊓⊔\nLemma 6. Let a(n) = Q2n−1\ni=0\nQ\nj̸=i(i −j), where j ranges from 0 to 2n −1. Let\npi1,...,in( ̄x) be the following family of polynomials:\npi1,...,in(x1, . . . , xn) =\nn\nY\nk=1"},{"paragraph_id":"p13","order":13,"text":"a(n)\nY\njk̸=ik\nxk −jk\nik −jk"},{"paragraph_id":"p14","order":14,"text":",\nwhere the integers jk range from 0 to 2n −1 and the integers ik are given in\nbinary and range from 0 to 2n −1. Then the coefficients of pi1,...,in are integers\ndefinable in the counting hierarchy, as is a(n).\nProof. As a first step, note that the coefficient of the monomial xα1\n1 · · · xαn\nn\nin pn\nis equal to the product of the coefficients of the monomials xαk\nk\nin the univariate\npolynomials a(n) Q\njk̸=ik\nxk−jk\nik−jk . Hence we just have to check that these different\ncoefficients of univariate polynomials are themselves definable in the counting\nhierarchy. Let us first focus on the univariate polynomial Q\njk̸=ik(xk −jk), that\nis, let us forget the multiplicative term b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) for the\nmoment.\nWe use the same argument as [6, Cor. 3.9]. Namely, we remark that the\ncoefficients of this polynomial are bounded in absolute value by 22n2\n. Therefore\nin the univariate polynomial Q\njk̸=ik(xk −jk) we can replace the variable xk\nby 22n2\nand there will be no overlap of the coefficients of the different powers\nof xk, thus we can recover the coefficients of the monomial from the value of\nthis product. By the first part of Theorem 2, we can evaluate in the counting\nhierarchy the polynomial at the point 22n2\n, because it is a product of exponential\nsize. So the coefficients are definable in the counting hierarchy.\nIt is now enough to note that the first part of Theorem 2 implies that a(n)\nas well as b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) are also definable in the counting\nhierarchy.\n⊓⊔\nRemark that the sequence a(n) of Lemma 6 is introduced only so as to obtain\ninteger coefficients. We will then divide by a(n) in the next proofs."},{"paragraph_id":"p15","order":15,"text":"10\nPascal Koiran and Sylvain Perifel\n3.3\nMain Results\nLet us now state the main theorem.\nTheorem 3. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials.\nSuppose (fn) can be evaluated in CH/poly at integer points. If VP0 = VNP0 then\n(fn) ∈VP0\nnb.\nProof. The goal is to use the interpolation formula of Lemma 5:\nfn(x1, . . . , xu(n)) =\nX\n0≤i1,...,iu(n)≤d\nbi1,...,iu(n)( ̄x),\n(1)\nwhere bi1,...,iu(n)( ̄x) = fn(i1, . . . , iu(n)) Qu(n)\nk=1\nQ\njk̸=ik\nxk−jk\nik−jk . We will show that\nthe coefficients of bi1,...,iu(n) and fn are definable in CH/poly. The conclusion of\nthe theorem will then follow from Lemma 4.\nIn order to show that the coefficients of bi1,...,iu(n) are definable in CH/poly,\nwe note that the polynomial pi1,...,in and the sequence a(n) of Lemma 6 satisfy\nthe relation\nbi1,...,iu(n)( ̄x) = a(u(n))−u(n)fn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x).\nBy Lemma 6, the coefficients of pi1,...,iu(n)( ̄x) are definable in CH. By hypothesis,\n(fn) can be evaluated in CH/poly at integer points. This implies by Lemma 1\nthat fn(i1, . . . , iu(n)) is definable in CH/poly. This is also the case of the product\nfn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x) by Theorem 2. Now, the same theorem enables us\nto divide by a(u(n))u(n), thereby showing that the coefficients of bi1,...,iu(n)( ̄x)\nare definable in CH/poly. It then follows from (1) and another application of\nTheorem 2 that the coefficients of fn are definable in CH/poly. Therefore by\nLemma 4, (fn) ∈VP0\nnb under the hypothesis VP0 = VNP0.\n⊓⊔\nWe now derive some consequences of Theorem 3.\nTheorem 4. Let (fn( ̄x, ̄ǫ)) ∈VP0\nnb. Let\ngn( ̄x) =\nX\n ̄ǫ\nfn( ̄x, ̄ǫ) and hn( ̄x) =\nY\n ̄ǫ\nfn( ̄x, ̄ǫ).\nIf VNP0 = VP0 then (gn) and (hn) are in VP0\nnb.\nProof. By Corollary 1 (fn) can be evaluated in CH/poly at integer points. Now,\nusing Lemma 1 before and after the first part of Theorem 2 shows that (gn) and\n(hn) can also be evaluated in CH/poly at integer points. The result then follows\nby Theorem 3.\n⊓⊔\nThe following is now immediate.\nTheorem 5. The hypothesis VP0 = VNP0 implies that VP0\nnb = VNP0\nnb and\nVP0\nnb = VΠP0."},{"paragraph_id":"p16","order":16,"text":"Interpolation in Valiant’s theory\n11\nRemark 4. It is not clear whether the converse of the first implication in The-\norem 5 (VP0 = VNP0 =⇒VP0\nnb = VNP0\nnb) holds true. This is related to the\nissue of large constants in arithmetic circuits: it seems difficult to rule out the\npossibility that some polynomial family in VNP0 (for instance, the permanent or\nthe hamiltonian) does not lie in VP0 but is still computable by polynomial-size\narithmetic circuits using integer constants of exponential bit size.\nThe converse does hold if arbitrary constants are allowed: we indeed have\nVPnb = VNPnb =⇒VP = VNP. But in this non-constant-free context, it is not\nclear whether VP = VNP unconditionally implies VPnb = VNPnb: indeed, in this\ncontext the generalized Riemann hypothesis would be needed to make the proof\nof Lemma 2 work (see [6] for details).\nAs mentioned in the introduction, another corollary concerns a transfer the-\norem with classes of algebraic complexity in the BSS model. Blum, Shub and\nSmale [2, 3] have defined the classes P and NP over the real and complex fields.\nIt was extended to arbitrary structures by Poizat [18]. Here we use nonuniform\nversions of these classes, hence the notations P and NP.\nTheorem 7 below proves that, over a field of characteristic zero, if we separate\n(the nonuniform versions of) P and NP thanks to a “simple” NP problem, then\nwe separate (the constant-free versions of) VP and VNP. The class of “simple”\nproblems here is NP where the multiplication is not allowed, i.e., the only oper-\nations are +, −and =. It contains in particular Twenty Questions and Subset\nSum. We will need a result from [13]:\nTheorem 6. Let K be a field of characteristic zero. If VP0\nnb = VΠP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nBy Theorem 5, the following is immediate.\nTheorem 7. Let K be a field of characteristic zero. If VP0 = VNP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nAt last, as a corollary of Theorem 3 again, we obtain the following result\nconcerning question (*), suggesting that it will be hard to refute. As pointed out\nin the introduction, this result does not give any evidence concerning the answer\nto question (*) since the separation VP0 ̸= VNP0 is very likely to be true.\nTheorem 8. If question (*) has a negative answer then VP0 ̸= VNP0. More\nprecisely, let (fn) be a family of multivariate polynomials which can be evaluated\nin polynomial time at integer points (in the sense of Remark 3). If VP0 = VNP0\nthen (fn) ∈VP0\nnb.\n3.4\nArithmetic Circuits with Constants\nIn this section we investigate another interpretation of question (*): we still\nconsider polynomials with integer coefficients, but we allow rational constants\nin our circuits (it turns out that the constant 1/2 plays a special role due to its\nappearance in the completeness proof for the permanent). The hypothesis that"},{"paragraph_id":"p17","order":17,"text":"12\nPascal Koiran and Sylvain Perifel\nquestion (*) has a negative answer is then stronger, and we obtain a stronger\nconclusion than in Theorem 8. Namely, we can conclude that τ(PERn) ̸= nO(1)\ninstead of VNP0 ̸= VP0 (see Remark 1). We recall that τ, the constant-free\narithmetic circuit complexity of a polynomial, is defined in Section 2.1.\nTheorem 9. As explained above, we consider here polynomials with integer co-\nefficients but circuits with rational constants. If question (*) has a negative an-\nswer, then τ(PERn) is not polynomially bounded.\nMore precisely, let (fn) be a family of multivariate polynomials which can\nbe evaluated in polynomial time at integer points (in the sense of Remark 3). If\nτ(PERn) is polynomially bounded, (fn) can be evaluated by a family of polynomial-\nsize arithmetic circuits that use only the constant 1/2.\nIt is easy to see that Theorem 9 follows from a slight modification of the\ndifferent lemmas above. Lemma 2 is replaced by the following stronger lemma,\nfrom [6].\nLemma 7. If τ(PERn) = nO(1) then CH/poly = P/poly.\nThen Lemma 4 is replaced by the following result, whose proof relies on an\ninspection of Valiant’s proof [20] of VNP-completeness of the permanent, see [6].\nLemma 8. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nFinally, Theorem 3 becomes the following.\nLemma 9. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials (with\ninteger coefficients). Suppose (fn) can be evaluated in CH/poly at integer points.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nTheorem 9 follows since the coefficient 2p(n) can be cancelled by multiplying by\nthe constant 2−p(n), which can be computed from scratch from the constant 1/2.\nAcknowledgments\nWe would like to thank Erich Kaltofen and Christos Papadimitriou for sharing their\nthoughts on question (*)."},{"paragraph_id":"p18","order":18,"text":"Interpolation in Valiant’s theory\n13\nReferences\n1. E. Allender, P. B ̈urgisser, J. Kjeldgaard-Pedersen, and P. Bro Miltersen. On the\ncomplexity of numerical analysis. In IEEE Conference on Computational Com-\nplexity, pages 331–339, 2006.\n2. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n3. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n4. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory. Num-\nber 7 in Algorithms and Computation in Mathematics. Springer, 2000.\n5. P. B ̈urgisser. The complexity of factors of multivariate polynomials. Foundations\nof Computational Mathematics, 4(4):369–396, 2004.\n6. P. B ̈urgisser.\nOn defining integers in the counting hierarchy and proving lower\nbounds in algebraic complexity. In Proc. STACS 2007, pages 133–144, 2007. Full\nversion: ECCC Report No. 113, August 2006.\n7. P. B ̈urgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory.\nSpringer, 1997.\n8. H. Fournier and P. Koiran. Are lower bounds easier over the reals? In Proc. 30th\nACM Symposium on Theory of Computing, pages 507–513, 1998.\n9. H. Fournier and P. Koiran. Lower bounds are not easier over the reals: Inside PH. In\nProc. 27th International Colloquium on Automata, Languages and Programming,\nvolume 1853 of Lecture Notes in Computer Science, pages 832–843. Springer, 2000.\n10. E. Kaltofen and G. Villard. On the complexity of computing determinants. Com-\nputational Complexity, 13(3-4):91-130, 2004.\n11. R. Karp and R. Lipton. Turing machines that take advice. L’Enseignement Ma-\nth ́ematique, 28:191–209, 1982.\n12. P. Koiran. Valiant’s model and the cost of computing integers. Computational\nComplexity, 13(3-4):131–146, 2004.\n13. P. Koiran and S. Perifel. Valiant’s model: from exponential sums to exponential\nproducts. In Proc. MFCS 2006, volume 4162 of Lecture Notes in Computer Science,\npages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the complex field.\nIn Proc. MFCS 2007, volume 4708 of Lecture Notes in Computer Science, pages\n359-370, 2007.\n15. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the reals. In Proc.\nSTACS 2007, volume 4393 of Lecture Notes in Computer Science, pages 417–428.\nSpringer-Verlag, 2007. long version: http://prunel.ccsd.cnrs.fr/ensl-00103018.\n16. G. Malod. The complexity of polynomials and their coefficient functions. In Proc.\n22nd IEEE Conference on Computational Cmplexity, pages 193–204, 2007.\n17. Guillaume Malod.\nPolynˆomes et coefficients.\nPhD thesis, Universit ́e Claude\nBernard Lyon 1, July 2003.\nAvailable from http://tel.archives-ouvertes.fr/tel-\n00087399.\n18. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n19. M. Shub and S. Smale. On the intractability of Hilbert’s Nullstellensatz and an\nalgebraic version of “P=NP”. Duke Mathematical Journal, 81(1):47–54, 1995.\n20. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979.\n21. K. W. Wagner. The complexity of combinatorial problems with succinct input\nrepresentation. Acta Inform., 23(3):325–356, 1986."}],"pages":[{"page":1,"text":"arXiv:0710.0360v1 [cs.CC] 1 Oct 2007\nInterpolation in Valiant’s theory\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nAbstract. We investigate the following question: if a polynomial can\nbe evaluated at rational points by a polynomial-time boolean algorithm,\ndoes it have a polynomial-size arithmetic circuit? We argue that this\nquestion is certainly difficult. Answering it negatively would indeed imply\nthat the constant-free versions of the algebraic complexity classes VP and\nVNP defined by Valiant are different. Answering this question positively\nwould imply a transfer theorem from boolean to algebraic complexity.\nOur proof method relies on Lagrange interpolation and on recent re-\nsults connecting the (boolean) counting hierarchy to algebraic complex-\nity classes. As a byproduct we obtain two additional results:\n(i) The constant-free, degree-unbounded version of Valiant’s hypothe-\nsis VP ̸= VNP implies the degree-bounded version. This result was\npreviously known to hold for fields of positive characteristic only.\n(ii) If exponential sums of easy to compute polynomials can be computed\nefficiently, then the same is true of exponential products. We point\nout an application of this result to the P=NP problem in the Blum-\nShub-Smale model of computation over the field of complex numbers.\n1\nIntroduction\nMotivation – The starting point of this paper is a question raised by Christos\nPapadimitriou in a personal communication to Erich Kaltofen1:\nQuestion\n(*)\nIf a multivariate polynomial P can be evaluated by a (boolean)\npolynomial-time algorithm on rational inputs, does that imply that\nP can be computed by a polynomial-size arithmetic circuit? In such\na circuit, the only allowed operations are additions, subtractions,\nand multiplications.\nThis question can be interpreted in several ways: one should at least state\nin which ring the coefficients of P lie, and which constants can be used by the\narithmetic circuits. Here we will focus on polynomials with integer coefficients,\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA.\n1 At the Oberwolfach complexity theory workshop where a part of this work was\npresented in June 2007, several participants told P.K. that they had independently\nthought of the same question."},{"page":2,"text":"2\nPascal Koiran and Sylvain Perifel\nand most of the paper will deal with constant-free circuits. In Section 3.4 we\nstudy the case of circuits with rational constants computing polynomials with\ninteger coefficients (as we shall see, this is more natural than it might seem at\nfirst sight).\nAs pointed out by Papadimitriou, Strassen’s “Vermeidung von Divisionen”\n(see for instance [7], chapter 7) shows that for evaluating a low-degree polynomial\nP, divisions would not increase exponentially the power of arithmetic circuits. It\nis indeed a natural question whether, more generally, all boolean operations can\nbe replaced efficiently by additions, subtractions and multiplications. Questions\nof the same flavour (can “looking at bits” help for arithmetic computations?)\nhave been studied before. In particular, Kaltofen and Villard have shown that\nlooking at bits does help for computing the determinant [10].\nDiscussion – It is not clear what the correct answer to question (*) should\nbe. In this paper we will argue that answering it either way seems difficult.\nA natural strategy for obtaining a negative answer to question (*) would be\nto exhibit a family of polynomials that are easy to evaluate on rational inputs\nbut hard to evaluate by arithmetic circuits. Unfortunately, there seems to be\na lack of candidate polynomials. Another difficulty is that a negative answer\nwould imply the separation of the algebraic complexity classes VP0 and VNP0.\nThis observation is our main contribution to the study of question (*), and it\nis established in Theorem 8. The classes VP0 and VNP0 are constant-free ver-\nsions of the classes VP (of “easily computable polynomial families”) and VNP\n(of “easily definable polynomial families”) introduced by Valiant (precise defi-\nnitions are given in the next section). The separation VP0 ̸= VNP0 seems very\nplausible, but it also seems very difficult to establish. As explained at the be-\nginning of the introduction, we study in Section 3.4 the case of circuits with\nrational constants computing polynomials with integer coefficients. Allowing ra-\ntional constants makes the hypothesis that question (*) has a negative answer\nstronger than in the constant-free case. Accordingly, we obtain a stronger con-\nclusion: we can now show that the hypothesis would imply a superpolynomial\nlower bound on the size of arithmetic circuits computing the permanent.\nObtaining a positive answer to question (*) also seems difficult since it would\nimply the following transfer theorem: FP = ♯P ⇒VP = VNP (assuming that\nFP = ♯P, the permanent must be in FP; a positive answer to question (*) would\ntherefore imply that the permanent is in VP, and that VP = VNP by complete-\nness of the permanent). Unfortunately, in spite of all the work establishing close\nconnections between the boolean model of computation and the algebraic mod-\nels of Valiant and of Blum, Shub and Smale [2, 4, 5, 8, 9, 13–15] no such transfer\ntheorem is known. In fact, we do not know of any hypothesis from boolean com-\nplexity theory that would imply the equality VP = VNP (but transfer theorems\nin the opposite direction were established in [4]).\nSummary of results – Most of our results are derived from Theorem 3,\nour main theorem: if the evaluation of a family of polynomials (fn) at integer\npoints is a problem that lies in the (non uniform) counting hierarchy, the hy-\npothesis VP0 = VNP0 implies that (fn) can be evaluated by polynomial-size"},{"page":3,"text":"Interpolation in Valiant’s theory\n3\narithmetic circuits. Theorem 8, which contains our main contribution to the\nstudy of question (*), follows immediately since polynomial-time problems lie\nin the counting hierarchy. The proof of Theorem 3 relies on techniques from [1,\n6] and on Lagrange interpolation. Besides the application to question (*), we\nderive two additional results from Theorem 3:\n– The elements of the complexity classes VP, VNP and of their constant-free\nversions are families of polynomials of polynomially bounded degree. We\nshow in Theorem 5 that the collapse VP0 = VNP0 would imply the same\ncollapse for the unbounded versions of VP0 and VNP0. For fields of positive\ncharacteristic, the same result (and its converse) was obtained with different\ntechniques by Malod [16, 17].\n– Our third application of Theorem 3 is to the “P = NP?” problem in the\nBlum-Shub-Smale model of computation over C (or more generally, fields of\ncharacteristic 0). One natural strategy for separating PC from NPC would\nbe to exhibit a problem A in NPC \\ PC. Drawing on results from [13], we\nshow that this strategy is bound to fail for a fairly large class of “simple”\nproblems A, unless one can prove that VP0 ̸= VNP0. The class of “simple”\nproblems that we have in mind is NP(C,+,−,=). This is the class of NP prob-\nlems over the set of complex numbers endowed with addition, subtraction,\nand equality tests (there is therefore no multiplication in this structure).\nIt contains many natural problems, such as Subset Sum and Twenty Ques-\ntions [2, 19], that most likely belong to NPC \\ PC. As an intermediate result,\nwe show in Theorem 5 that if exponential sums of easy to compute poly-\nnomials can be computed efficiently, then the same is true of exponential\nproducts.\n2\nPreliminaries\n2.1\nValiant’s Classes\nIn Valiant’s model, one computes families of polynomials. A book-length treat-\nment of this topic can be found in [4]. We fix a field K of characteristic zero.\nAn arithmetic circuit is a circuit whose inputs are indeterminates x1, . . . , xu(n)\ntogether with arbitrary constants of K; there are +, −and ×-gates, and we\ntherefore compute multivariate polynomials. The polynomial computed by an\narithmetic circuit is defined in the usual way by the polynomial computed by its\noutput gate. The size of a circuit is the number of gates.\nThus a family (Cn) of arithmetic circuits computes a family (fn) of polynomi-\nals, fn ∈K[x1, . . . , xu(n)]. The class VPnb defined in [17] is the set of families (fn)\nof polynomials computed by a family (Cn) of polynomial-size arithmetic circuits,\ni.e., Cn computes fn and there exists a polynomial p(n) such that |Cn| ≤p(n)\nfor all n. We will assume without loss of generality that the number u(n) of\nvariables is bounded by a polynomial function of n. The subscript “nb” indi-\ncates that there is no bound on the degree of the polynomial, in contrast with\nthe original class VP of Valiant where a polynomial bound on the degree of the"},{"page":4,"text":"4\nPascal Koiran and Sylvain Perifel\npolynomial computed by the circuit is required. Note that these definitions are\nnonuniform.\nThe class VNP is the set of families of polynomials defined by an exponential\nsum of VP families. More precisely, (fn( ̄x)) ∈VNP if there exists (gn( ̄x, ̄y)) ∈VP\nand a polynomial p such that | ̄y| = p(n) and fn( ̄x) = P\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ). Sim-\nilarly, the class VΠP is the set of families of polynomials defined by an expo-\nnential product of VPnb families. More precisely, (fn( ̄x)) ∈VΠP if there ex-\nists (gn( ̄x, ̄y)) ∈VPnb and a polynomial p such that | ̄y| = p(n) and fn( ̄x) =\nQ\n ̄ǫ∈{0,1}p(n) gn( ̄x, ̄ǫ).\nWe can also define constant-free circuits: the only constant allowed is then\n1 (in order to allow the computation of constant polynomials). In this case, we\ncompute polynomials with integer coefficients. If f is a polynomial with integer\ncoefficients, we denote by τ(f) the size of a smallest constant-free circuit com-\nputing f. For classes of families of polynomials, we will use the superscript 0 to\nindicate the absence of constant: for instance, we will write VP0\nnb. For bounded-\ndegree classes, we are to be more careful because we also want to avoid the\ncomputation of constants of exponential bitsize: we first need the following def-\ninition.\nDefinition 1. Let C be an arithmetic circuit. The formal degree of a gate of C\nis defined by induction:\n– the formal degree of an input is 1;\n– the formal degree of a gate + or −is the maximum of the formal degrees of\nits inputs;\n– the formal degree of a gate × is the sum of the formal degrees of its inputs.\nNow, the formal degree of a circuit is the formal degree of the output gate.\nWe are now able to define constant-free degree-bounded Valiant’s classes. A\nfamily of polynomials (fn) belongs to VP0 if it is computable by a family of\ncircuits of size and formal degree bounded by a polynomial function of n. The\nclass VNP0 is then defined accordingly by a sum of VP0 families, in the same\nway as VNP is defined from VP.\nRemark 1. The hypothesis τ(PERn) = nO(1) used in [6] is implied by the hy-\npothesis VNP0 ⊂VP0\nnb and hence by VP0 = VNP0. As mentioned in [6], the\nconverse τ(PERn) = nO(1) ⇒VNP0 = VP0 is not known to hold, because the\nfamily (PERn) is not known to be VNP0-complete in a constant-free context\n(the proof of completeness of Valiant [20] indeed uses the constant 1/2). We\nwill mostly be concerned by the hypothesis VP0 = VNP0, but we will come to\nthe hypothesis τ(PERn) = nO(1) in Section 3.4 when dealing with circuits with\nconstants.\n2.2\nCounting Classes\nIn this paper we will encounter several counting classes, in particular the counting\nhierarchy defined below. Let us first see two classes of functions, ♯P and GapP."},{"page":5,"text":"Interpolation in Valiant’s theory\n5\nDefinition 2.\n– The class ♯P is the set of functions f : {0, 1}∗→{0, 1}∗such\nthat there exist a language A ∈P and a polynomial p(n) satisfying\nf(x) = #{y ∈{0, 1}p(|x|) : (x, y) ∈A}.\n– A function f is in GapP if it is the difference of two functions in ♯P.\nReturning to classes of languages we recall the definition of the counting hierar-\nchy, introduced by Wagner [21]. It contains all the polynomial hierarchy PH and\nis contained in PSPACE. It is defined via the “majority” operator C as follows.\nDefinition 3.\n– If K is a complexity class, the class C.K is the set of lan-\nguages A such that there exist a language B ∈K and a polynomial p(n)\nsatisfying\nx ∈A ⇐⇒#{y ∈{0, 1}p(|x|) : (x, y) ∈B} ≥2p(|x|)−1.\n– The i-th level CiP of the counting hierarchy is defined recursively by C0P = P\nand Ci+1P = C.CiP. The counting hierarchy CH is the union of all these levels\nCiP.\nLevel 1 of CH, that is, C.P, is also called PP. Since Valiant’s classes are nonuni-\nform, we will rather work with nonuniform versions of these boolean classes, as\ndefined now following Karp and Lipton [11].\nDefinition 4. If K is a complexity class, the class K/poly is the set of languages\nA such that there exist a language B ∈K, a polynomial p(n) and a family of\nwords (called advices) (an)n≥0 satisfying\n– for all n ≥0, |an| ≤p(n);\n– for all word x, x ∈A ⇐⇒(x, a(|x|)) ∈B.\nRemark that the advice only depends on the size of x: it must therefore be the\nsame for all words of same length.\n2.3\nSequences of Integers\nOur aim now is to introduce a notion of complexity of a sequence of integers.\nIn order to avoid dealing with the sign of integers separately, we assume that\nwe can retrieve it from the boolean encoding of the integers. For example, the\nsign could be given by the first bit of the encoding and the absolute value by\nthe remaining bits.\nDefinition 5. A sequence of exponential bitsize is a sequence of integers (a(n, k))\nsuch that there exists a polynomial p(n) satisfying:\n1. a(n, k) is defined for n, k ∈N and 0 ≤k < 2p(n);\n2. for all n > 1, for all k < 2p(n), the bitsize of a(n, k) is ≤2p(n)."},{"page":6,"text":"6\nPascal Koiran and Sylvain Perifel\nFrom a(n, k), the following language is then defined:\nBit(a) = {(1n, k, j, b)| the j-th bit of a(n, k) is b},\nThe reader should be aware that the above definition and the next one are not\nquite the same as in [6]: we use a unary encoding for n instead of a binary\nencoding.\nDefinition 6. A sequence a(n, k) of exponential bitsize is definable in CH/poly\nif the language Bit(a) is in CH/poly.\nRemark 2. We shall also meet sequences with more than two parameters (n, k),\nfor example a(n, α(1), . . . , α(n)) for some integers α(i). In order to see it as a\nsequence with two parameters, (α(1), . . . , α(n)) will be considered as the encoding\nof a single integer. The parameter n might also be given as a subscript, as in\nfn(k), which should better be written f(n, k).\nLet us now propose a similar definition for families of polynomials.\nDefinition 7. Let (fn(x1, . . . , xu(n))) be a family of polynomials with integer\ncoefficients. We say that (fn) can be evaluated in CH/poly at integer points if\nthe following conditions are satisfied:\n1. the number u(n) of variables is polynomially bounded;\n2. the degree of fn as well as the bitsize of the coefficients of fn are bounded by\n2p(n) for some polynomial p(n);\n3. the language {(1n, i1, . . . , iu(n), j, b)| the j-th bit of fn(i1, . . . , iu(n)) is b} is\nin CH/poly.\nRemark 3. The same definition can be made for other complexity classes than\nCH/poly. For instance, if we replace CH/poly by P we obtain the notion of “poly-\nnomial time evaluation at integer points”. This notion will be useful for the study\nof question (*).\nThe following lemma is obvious from these definitions.\nLemma 1. The family (fn(x1, . . . , xu(n))) can be evaluated in CH/poly at inte-\nger points if and only if the sequence of integers a(n, i1, . . . , iu(n)) = fn(i1, . . . , iu(n))\nis definable in CH/poly.\nThe following theorem of [1, Theorem 4.1] will also be useful due to its corollary\nbelow.\nTheorem 1. Let BitSLP be the following problem: given a constant-free arith-\nmetic circuit computing an integer N, and given i ∈N in binary, decide whether\nthe i-th bit of the binary representation of N is 1. Then BitSLP is in CH.\nCorollary 1. If (fn) ∈VP0\nnb then it can be evaluated in CH/poly at integer\npoints."},{"page":7,"text":"Interpolation in Valiant’s theory\n7\nThe results of this paper rely on the following link between Valiant’s classes\nand the counting hierarchy, [6, Lemmas 2.5 and 2.12].\nLemma 2. If VP0 = VNP0 then CH/poly = P/poly.\nIn particular, Lemma 2 was used to show that big sums and products are com-\nputable in the counting hierarchy, [6, Theorem 3.7]. As already mentioned, the\ncontext is not exactly the same as in [6] because we use a unary encoding. We\nnow give a version of this result which is just an easy “scaling up” of [6, Theo-\nrem 3.7] (it is enough to define a′(2p(n), k) = a(n, k) and to apply the result of\nB ̈urgisser).\nTheorem 2. Let p(n) be a polynomial and suppose a = (a(n, k))n∈N,k≤2p(n) is\ndefinable in CH/poly. Consider the sequences\nb(n) =\n2p(n)\nX\nk=0\na(n, k) and d(n) =\n2p(n)\nY\nk=0\na(n, k).\nThen (b(n))n∈N and (d(n))n∈N are definable in CH/poly.\nSuppose now that (s(n))n∈N and (t(n))n∈N are definable in CH/poly. Then the\nsequence of products (s(n)t(n))n∈N, and, if t(n) > 0, the sequence of quotients\n(⌊s(n)/t(n)⌋)n∈N, are definable in CH/poly.\n3\nInterpolation\nWe now begin the main technical developments.\n3.1\nCoefficients\nThe following lemma is Valiant’s criterion [20], see also [4, Prop. 2.20] and [12,\nTh. 2.3].\nLemma 3. Let a : (1n, i) 7→a(1n, i) be a function of GapP/poly, where n is\ngiven in unary and i in binary. Let p(n) be a polynomial and define the following\nsequence of polynomials:\nfn(x1, . . . , xp(n)) =\n2p(n)−1\nX\ni=0\na(1n, i)xi1\n1 · · · x\nip(n)\np(n) ,\nwhere ij is the j-th bit in the binary expression of i.\nThen (fn) ∈VNP0.\nHere is a “scaled up” generalization of [6, Th. 4.1(2)] to multivariate poly-\nnomials."},{"page":8,"text":"8\nPascal Koiran and Sylvain Perifel\nLemma 4. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf VP0 = VNP0 then (fn) ∈VP0\nnb.\nProof. Expand a in binary: a(n, α(1), . . . , α(n)) = P2n\ni=0 ai(n, ̄α)2i. Let hn be the\nfollowing polynomial:\nhn(x1,1, x1,2, . . . , x1,n, x2,1, . . . , xn,n, z1, . . . , zn) =\n2n\nX\ni=0\nX\n ̄α\nai(n, ̄α)zi1\n1 · · · zin\nn xα(1)\n1\n1,1 xα(1)\n2\n1,2 · · · xα(1)\nn\n1,n xα(2)\n1\n2,1 · · · xα(n)\nn\nn,n .\nThen we have:\nhn(x20\n1 , x21\n1 , . . . , x2n\n1 , x20\n2 , . . . , x2n\nn , 220, 221, . . . , 22n) = fn(x1, . . . , xn).\nSince VP0 = VNP0, by Lemma 2 the nonuniform counting hierarchy collapses,\ntherefore computing the i-th bit ai(n, ̄α) of a(n, ̄α) on input (1n, ̄α, i) is in\nGapP/poly (and even in P/poly). By Lemma 3, (hn) ∈VNP0. By the hypothesis\nVP0 = VNP0, (hn) ∈VP0 and thus using repeated squaring for computing big\npowers yields (fn) ∈VNP0\nnb.\n⊓⊔\n3.2\nInterpolation\nLet us now state two lemmas on interpolation polynomials.\nLemma 5 (multivariate Lagrange interpolation). Let p(x1, . . . , xn) be a\npolynomial of degree ≤d. Then\np(x1, . . . , xn) =\nX\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\n Y\njk̸=ik\nxk −jk\nik −jk\n \n,\nwhere the integers jk range from 0 to d.\nProof. The proof goes by induction on the number n of variables. For n = 1,\nthis is the usual Lagrange interpolation formula: we have\np(x) =\nd\nX\ni=0\np(i)\nY\nj̸=i\nx −j\ni −j\nbecause both polynomials are of degree ≤d and coincide on at least d+1 distinct\npoints."},{"page":9,"text":"Interpolation in Valiant’s theory\n9\nFor n + 1, the induction case n = 1 yields\np(x1, . . . , xn+1) =\nd\nX\nin+1=0\np(x1, . . . , xn, in+1)\nY\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\n.\nBy induction hypothesis, this is equal to\nd\nX\nin+1=0\n \n \nX\n0≤i1,...,in≤d\np(i1, . . . , in)\nn\nY\nk=1\nY\njk̸=ik\nxk −jk\nik −jk\n \n \nY\njn+1̸=in+1\nxn+1 −jn+1\nin+1 −jn+1\nwhich is the desired result.\n⊓⊔\nLemma 6. Let a(n) = Q2n−1\ni=0\nQ\nj̸=i(i −j), where j ranges from 0 to 2n −1. Let\npi1,...,in( ̄x) be the following family of polynomials:\npi1,...,in(x1, . . . , xn) =\nn\nY\nk=1\n \na(n)\nY\njk̸=ik\nxk −jk\nik −jk\n \n,\nwhere the integers jk range from 0 to 2n −1 and the integers ik are given in\nbinary and range from 0 to 2n −1. Then the coefficients of pi1,...,in are integers\ndefinable in the counting hierarchy, as is a(n).\nProof. As a first step, note that the coefficient of the monomial xα1\n1 · · · xαn\nn\nin pn\nis equal to the product of the coefficients of the monomials xαk\nk\nin the univariate\npolynomials a(n) Q\njk̸=ik\nxk−jk\nik−jk . Hence we just have to check that these different\ncoefficients of univariate polynomials are themselves definable in the counting\nhierarchy. Let us first focus on the univariate polynomial Q\njk̸=ik(xk −jk), that\nis, let us forget the multiplicative term b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) for the\nmoment.\nWe use the same argument as [6, Cor. 3.9]. Namely, we remark that the\ncoefficients of this polynomial are bounded in absolute value by 22n2\n. Therefore\nin the univariate polynomial Q\njk̸=ik(xk −jk) we can replace the variable xk\nby 22n2\nand there will be no overlap of the coefficients of the different powers\nof xk, thus we can recover the coefficients of the monomial from the value of\nthis product. By the first part of Theorem 2, we can evaluate in the counting\nhierarchy the polynomial at the point 22n2\n, because it is a product of exponential\nsize. So the coefficients are definable in the counting hierarchy.\nIt is now enough to note that the first part of Theorem 2 implies that a(n)\nas well as b(n, ik) = a(n)/ Q\njk̸=ik(ik −jk) are also definable in the counting\nhierarchy.\n⊓⊔\nRemark that the sequence a(n) of Lemma 6 is introduced only so as to obtain\ninteger coefficients. We will then divide by a(n) in the next proofs."},{"page":10,"text":"10\nPascal Koiran and Sylvain Perifel\n3.3\nMain Results\nLet us now state the main theorem.\nTheorem 3. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials.\nSuppose (fn) can be evaluated in CH/poly at integer points. If VP0 = VNP0 then\n(fn) ∈VP0\nnb.\nProof. The goal is to use the interpolation formula of Lemma 5:\nfn(x1, . . . , xu(n)) =\nX\n0≤i1,...,iu(n)≤d\nbi1,...,iu(n)( ̄x),\n(1)\nwhere bi1,...,iu(n)( ̄x) = fn(i1, . . . , iu(n)) Qu(n)\nk=1\nQ\njk̸=ik\nxk−jk\nik−jk . We will show that\nthe coefficients of bi1,...,iu(n) and fn are definable in CH/poly. The conclusion of\nthe theorem will then follow from Lemma 4.\nIn order to show that the coefficients of bi1,...,iu(n) are definable in CH/poly,\nwe note that the polynomial pi1,...,in and the sequence a(n) of Lemma 6 satisfy\nthe relation\nbi1,...,iu(n)( ̄x) = a(u(n))−u(n)fn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x).\nBy Lemma 6, the coefficients of pi1,...,iu(n)( ̄x) are definable in CH. By hypothesis,\n(fn) can be evaluated in CH/poly at integer points. This implies by Lemma 1\nthat fn(i1, . . . , iu(n)) is definable in CH/poly. This is also the case of the product\nfn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x) by Theorem 2. Now, the same theorem enables us\nto divide by a(u(n))u(n), thereby showing that the coefficients of bi1,...,iu(n)( ̄x)\nare definable in CH/poly. It then follows from (1) and another application of\nTheorem 2 that the coefficients of fn are definable in CH/poly. Therefore by\nLemma 4, (fn) ∈VP0\nnb under the hypothesis VP0 = VNP0.\n⊓⊔\nWe now derive some consequences of Theorem 3.\nTheorem 4. Let (fn( ̄x, ̄ǫ)) ∈VP0\nnb. Let\ngn( ̄x) =\nX\n ̄ǫ\nfn( ̄x, ̄ǫ) and hn( ̄x) =\nY\n ̄ǫ\nfn( ̄x, ̄ǫ).\nIf VNP0 = VP0 then (gn) and (hn) are in VP0\nnb.\nProof. By Corollary 1 (fn) can be evaluated in CH/poly at integer points. Now,\nusing Lemma 1 before and after the first part of Theorem 2 shows that (gn) and\n(hn) can also be evaluated in CH/poly at integer points. The result then follows\nby Theorem 3.\n⊓⊔\nThe following is now immediate.\nTheorem 5. The hypothesis VP0 = VNP0 implies that VP0\nnb = VNP0\nnb and\nVP0\nnb = VΠP0."},{"page":11,"text":"Interpolation in Valiant’s theory\n11\nRemark 4. It is not clear whether the converse of the first implication in The-\norem 5 (VP0 = VNP0 =⇒VP0\nnb = VNP0\nnb) holds true. This is related to the\nissue of large constants in arithmetic circuits: it seems difficult to rule out the\npossibility that some polynomial family in VNP0 (for instance, the permanent or\nthe hamiltonian) does not lie in VP0 but is still computable by polynomial-size\narithmetic circuits using integer constants of exponential bit size.\nThe converse does hold if arbitrary constants are allowed: we indeed have\nVPnb = VNPnb =⇒VP = VNP. But in this non-constant-free context, it is not\nclear whether VP = VNP unconditionally implies VPnb = VNPnb: indeed, in this\ncontext the generalized Riemann hypothesis would be needed to make the proof\nof Lemma 2 work (see [6] for details).\nAs mentioned in the introduction, another corollary concerns a transfer the-\norem with classes of algebraic complexity in the BSS model. Blum, Shub and\nSmale [2, 3] have defined the classes P and NP over the real and complex fields.\nIt was extended to arbitrary structures by Poizat [18]. Here we use nonuniform\nversions of these classes, hence the notations P and NP.\nTheorem 7 below proves that, over a field of characteristic zero, if we separate\n(the nonuniform versions of) P and NP thanks to a “simple” NP problem, then\nwe separate (the constant-free versions of) VP and VNP. The class of “simple”\nproblems here is NP where the multiplication is not allowed, i.e., the only oper-\nations are +, −and =. It contains in particular Twenty Questions and Subset\nSum. We will need a result from [13]:\nTheorem 6. Let K be a field of characteristic zero. If VP0\nnb = VΠP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nBy Theorem 5, the following is immediate.\nTheorem 7. Let K be a field of characteristic zero. If VP0 = VNP0 then\nNP(K,+,−,=) ⊆P(K,+,−,×,=).\nAt last, as a corollary of Theorem 3 again, we obtain the following result\nconcerning question (*), suggesting that it will be hard to refute. As pointed out\nin the introduction, this result does not give any evidence concerning the answer\nto question (*) since the separation VP0 ̸= VNP0 is very likely to be true.\nTheorem 8. If question (*) has a negative answer then VP0 ̸= VNP0. More\nprecisely, let (fn) be a family of multivariate polynomials which can be evaluated\nin polynomial time at integer points (in the sense of Remark 3). If VP0 = VNP0\nthen (fn) ∈VP0\nnb.\n3.4\nArithmetic Circuits with Constants\nIn this section we investigate another interpretation of question (*): we still\nconsider polynomials with integer coefficients, but we allow rational constants\nin our circuits (it turns out that the constant 1/2 plays a special role due to its\nappearance in the completeness proof for the permanent). The hypothesis that"},{"page":12,"text":"12\nPascal Koiran and Sylvain Perifel\nquestion (*) has a negative answer is then stronger, and we obtain a stronger\nconclusion than in Theorem 8. Namely, we can conclude that τ(PERn) ̸= nO(1)\ninstead of VNP0 ̸= VP0 (see Remark 1). We recall that τ, the constant-free\narithmetic circuit complexity of a polynomial, is defined in Section 2.1.\nTheorem 9. As explained above, we consider here polynomials with integer co-\nefficients but circuits with rational constants. If question (*) has a negative an-\nswer, then τ(PERn) is not polynomially bounded.\nMore precisely, let (fn) be a family of multivariate polynomials which can\nbe evaluated in polynomial time at integer points (in the sense of Remark 3). If\nτ(PERn) is polynomially bounded, (fn) can be evaluated by a family of polynomial-\nsize arithmetic circuits that use only the constant 1/2.\nIt is easy to see that Theorem 9 follows from a slight modification of the\ndifferent lemmas above. Lemma 2 is replaced by the following stronger lemma,\nfrom [6].\nLemma 7. If τ(PERn) = nO(1) then CH/poly = P/poly.\nThen Lemma 4 is replaced by the following result, whose proof relies on an\ninspection of Valiant’s proof [20] of VNP-completeness of the permanent, see [6].\nLemma 8. Let\nfn(x1, . . . , xn) =\nX\nα(1),...,α(n)\na(n, α(1), . . . , α(n))xα(1)\n1\n· · · xα(n)\nn\n,\nwhere the integers α(i) range from 0 to 2n −1 and a(n, α(1), . . . , α(n)) is a se-\nquence of integers of absolute value < 22n definable in CH/poly.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nFinally, Theorem 3 becomes the following.\nLemma 9. Let (fn(x1, . . . , xu(n))) be a family of multivariate polynomials (with\ninteger coefficients). Suppose (fn) can be evaluated in CH/poly at integer points.\nIf τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =\nnO(1).\nTheorem 9 follows since the coefficient 2p(n) can be cancelled by multiplying by\nthe constant 2−p(n), which can be computed from scratch from the constant 1/2.\nAcknowledgments\nWe would like to thank Erich Kaltofen and Christos Papadimitriou for sharing their\nthoughts on question (*)."},{"page":13,"text":"Interpolation in Valiant’s theory\n13\nReferences\n1. E. Allender, P. B ̈urgisser, J. Kjeldgaard-Pedersen, and P. Bro Miltersen. On the\ncomplexity of numerical analysis. In IEEE Conference on Computational Com-\nplexity, pages 331–339, 2006.\n2. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n3. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n4. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory. Num-\nber 7 in Algorithms and Computation in Mathematics. Springer, 2000.\n5. P. B ̈urgisser. The complexity of factors of multivariate polynomials. Foundations\nof Computational Mathematics, 4(4):369–396, 2004.\n6. P. B ̈urgisser.\nOn defining integers in the counting hierarchy and proving lower\nbounds in algebraic complexity. In Proc. STACS 2007, pages 133–144, 2007. Full\nversion: ECCC Report No. 113, August 2006.\n7. P. B ̈urgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory.\nSpringer, 1997.\n8. H. Fournier and P. Koiran. Are lower bounds easier over the reals? In Proc. 30th\nACM Symposium on Theory of Computing, pages 507–513, 1998.\n9. H. Fournier and P. Koiran. Lower bounds are not easier over the reals: Inside PH. In\nProc. 27th International Colloquium on Automata, Languages and Programming,\nvolume 1853 of Lecture Notes in Computer Science, pages 832–843. Springer, 2000.\n10. E. Kaltofen and G. Villard. On the complexity of computing determinants. Com-\nputational Complexity, 13(3-4):91-130, 2004.\n11. R. Karp and R. Lipton. Turing machines that take advice. L’Enseignement Ma-\nth ́ematique, 28:191–209, 1982.\n12. P. Koiran. Valiant’s model and the cost of computing integers. Computational\nComplexity, 13(3-4):131–146, 2004.\n13. P. Koiran and S. Perifel. Valiant’s model: from exponential sums to exponential\nproducts. In Proc. MFCS 2006, volume 4162 of Lecture Notes in Computer Science,\npages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the complex field.\nIn Proc. MFCS 2007, volume 4708 of Lecture Notes in Computer Science, pages\n359-370, 2007.\n15. P. Koiran and S. Perifel. VPSACE and a transfer theorem over the reals. In Proc.\nSTACS 2007, volume 4393 of Lecture Notes in Computer Science, pages 417–428.\nSpringer-Verlag, 2007. long version: http://prunel.ccsd.cnrs.fr/ensl-00103018.\n16. G. Malod. The complexity of polynomials and their coefficient functions. In Proc.\n22nd IEEE Conference on Computational Cmplexity, pages 193–204, 2007.\n17. Guillaume Malod.\nPolynˆomes et coefficients.\nPhD thesis, Universit ́e Claude\nBernard Lyon 1, July 2003.\nAvailable from http://tel.archives-ouvertes.fr/tel-\n00087399.\n18. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n19. M. Shub and S. Smale. On the intractability of Hilbert’s Nullstellensatz and an\nalgebraic version of “P=NP”. Duke Mathematical Journal, 81(1):47–54, 1995.\n20. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979.\n21. K. W. Wagner. The complexity of combinatorial problems with succinct input\nrepresentation. Acta Inform., 23(3):325–356, 1986."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"sis VP ̸= VNP implies the degree-bounded version. This result was","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"out an application of this result to the P=NP problem in the Blum-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"nitions are given in the next section). The separation VP0 ̸= VNP0 seems very","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"imply the following transfer theorem: FP = ♯P ⇒VP = VNP (assuming that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"FP = ♯P, the permanent must be in FP; a positive answer to question (*) would","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"therefore imply that the permanent is in VP, and that VP = VNP by complete-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"plexity theory that would imply the equality VP = VNP (but transfer theorems","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"pothesis VP0 = VNP0 implies that (fn) can be evaluated by polynomial-size","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"show in Theorem 5 that the collapse VP0 = VNP0 would imply the same","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"– Our third application of Theorem 3 is to the “P = NP?” problem in the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"problems A, unless one can prove that VP0 ̸= VNP0. The class of “simple”","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"problems that we have in mind is NP(C,+,−,=). This is the class of NP prob-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"and a polynomial p such that | ̄y| = p(n) and fn( ̄x) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"ists (gn( ̄x, ̄y)) ∈VPnb and a polynomial p such that | ̄y| = p(n) and fn( ̄x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"Remark 1. The hypothesis τ(PERn) = nO(1) used in [6] is implied by the hy-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"nb and hence by VP0 = VNP0. As mentioned in [6], the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"converse τ(PERn) = nO(1) ⇒VNP0 = VP0 is not known to hold, because the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"will mostly be concerned by the hypothesis VP0 = VNP0, but we will come to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"the hypothesis τ(PERn) = nO(1) in Section 3.4 when dealing with circuits with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"f(x) = #{y ∈{0, 1}p(|x|) : (x, y) ∈A}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"– The i-th level CiP of the counting hierarchy is defined recursively by C0P = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"and Ci+1P = C.CiP. The counting hierarchy CH is the union of all these levels","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"Bit(a) = {(1n, k, j, b)| the j-th bit of a(n, k) is b},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"ger points if and only if the sequence of integers a(n, i1, . . . , iu(n)) = fn(i1, . . . , iu(n))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"Lemma 2. If VP0 = VNP0 then CH/poly = P/poly.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"rem 3.7] (it is enough to define a′(2p(n), k) = a(n, k) and to apply the result of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"Theorem 2. Let p(n) be a polynomial and suppose a = (a(n, k))n∈N,k≤2p(n) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"b(n) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"a(n, k) and d(n) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"fn(x1, . . . , xp(n)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"fn(x1, . . . , xn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"If VP0 = VNP0 then (fn) ∈VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"Proof. Expand a in binary: a(n, α(1), . . . , α(n)) = P2n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"i=0 ai(n, ̄α)2i. Let hn be the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"hn(x1,1, x1,2, . . . , x1,n, x2,1, . . . , xn,n, z1, . . . , zn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"n , 220, 221, . . . , 22n) = fn(x1, . . . , xn).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"Since VP0 = VNP0, by Lemma 2 the nonuniform counting hierarchy collapses,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"VP0 = VNP0, (hn) ∈VP0 and thus using repeated squaring for computing big","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"p(x1, . . . , xn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"k=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"jk̸=ik","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"Proof. The proof goes by induction on the number n of variables. For n = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"p(x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"j̸=i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"For n + 1, the induction case n = 1 yields","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"p(x1, . . . , xn+1) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"in+1=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"jn+1̸=in+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"in+1=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"k=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"jk̸=ik","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"jn+1̸=in+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"Lemma 6. Let a(n) = Q2n−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"j̸=i(i −j), where j ranges from 0 to 2n −1. Let","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"pi1,...,in(x1, . . . , xn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"k=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"jk̸=ik","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"jk̸=ik","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"jk̸=ik(xk −jk), that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"is, let us forget the multiplicative term b(n, ik) = a(n)/ Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"jk̸=ik(ik −jk) for the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"jk̸=ik(xk −jk) we can replace the variable xk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"as well as b(n, ik) = a(n)/ Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"jk̸=ik(ik −jk) are also definable in the counting","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"Suppose (fn) can be evaluated in CH/poly at integer points. If VP0 = VNP0 then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"fn(x1, . . . , xu(n)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"where bi1,...,iu(n)( ̄x) = fn(i1, . . . , iu(n)) Qu(n)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"k=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"jk̸=ik","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"bi1,...,iu(n)( ̄x) = a(u(n))−u(n)fn(i1, . . . , iu(n))pi1,...,iu(n)( ̄x).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"nb under the hypothesis VP0 = VNP0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"gn( ̄x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"fn( ̄x, ̄ǫ) and hn( ̄x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"If VNP0 = VP0 then (gn) and (hn) are in VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"Theorem 5. The hypothesis VP0 = VNP0 implies that VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"nb = VNP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"nb = VΠP0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"orem 5 (VP0 = VNP0 =⇒VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"nb = VNP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"VPnb = VNPnb =⇒VP = VNP. But in this non-constant-free context, it is not","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"clear whether VP = VNP unconditionally implies VPnb = VNPnb: indeed, in this","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"ations are +, −and =. It contains in particular Twenty Questions and Subset","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"nb = VΠP0 then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"NP(K,+,−,=) ⊆P(K,+,−,×,=).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"Theorem 7. Let K be a field of characteristic zero. If VP0 = VNP0 then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"NP(K,+,−,=) ⊆P(K,+,−,×,=).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"to question (*) since the separation VP0 ̸= VNP0 is very likely to be true.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"Theorem 8. If question (*) has a negative answer then VP0 ̸= VNP0. More","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"in polynomial time at integer points (in the sense of Remark 3). If VP0 = VNP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"conclusion than in Theorem 8. Namely, we can conclude that τ(PERn) ̸= nO(1)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"instead of VNP0 ̸= VP0 (see Remark 1). We recall that τ, the constant-free","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"Lemma 7. If τ(PERn) = nO(1) then CH/poly = P/poly.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"fn(x1, . . . , xn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"If τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"If τ(PERn) = nO(1) then there exists a polynomial p(n) such that τ(2p(n)fn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"algebraic version of “P=NP”. Duke Mathematical Journal, 81(1):47–54, 1995.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":31720,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}