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Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes

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Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes Sam Buss1;2 Dima Grigoriev Department of Mathematics Computer Science and Engineering Univ. of Calif., San Diego Pennsylvania State University La Jolla, CA 92093-0112 University Park, PA 16802-6106 [email protected] [email protected] Russell Impagliazzo1;3 Toniann Pitassi1;4 Computer Science and Engineering Computer Science Univ. of Calif., San Diego University of Arizona La Jolla, CA 92093-0114 Tucson, AZ 85721-0077 [email protected] [email protected] Abstract e±cient search algorithms and in part by the desire to extend lower bounds on proposition proof complexity This paper gives nearly optimal lower bounds on the to stronger proof systems. minimum degree of polynomial calculus refutations of The Nullstellensatz proof system is a propositional Tseitin's graph tautologies and the mod p counting proof system based on Hilbert's Nullstellensatz and principles, p 2. The lower bounds apply to the was introduced in [1]. The polynomial calculus (PC) ¸ polynomial calculus over ¯elds or rings. These are is a stronger propositional proof system introduced the ¯rst linear lower bounds for polynomial calculus; ¯rst by [4]. (See [8] and [3] for subsequent, more moreover, they distinguish linearly between proofs over general treatments of algebraic proof systems.) In the ¯elds of characteristic q and r, q = r, and more polynomial calculus, one begins with an initial set of 6 generally distinguish linearly the rings Zq and Zr where polynomials and the goal is to prove that they cannot q and r do not have the identical prime factors. be simultaneously equal to zero over a ¯eld F .A polynomial calculus (PC) derivation of Pl from a set of 1 Introduction polynomials Q is a sequence of polynomials P1;:::;Pl such that each polynomial is either an initial polynomial The problem of recognizing when a proposition formula from Q, or follows from one of the following two rules: is a tautology is dual to the satis¯ability problem and (i) If Pi and Pj are previous polynomials, then cPi +dPj is therefore central to computer science. A principal can be derived, where c; d F ; (ii) if P is a previous 2 i method of establishing that a formula is a tautology polynomial, then xPi can be derived. The degree of a is to ¯nd a proof of it in a formal system such as PC derivation is the maximum degree of the Pi's. We resolution or (extended) Frege systems. In fact, many identify polynomials Pi with the equations Pi = 0 and algorithms for establishing propositional validity are a PC refutation of Q (a proof that the equations Q =0 essentially a search for a proof in a particular formal are not solvable over F ) is simply a PC derivation of 1 system. In recent years, several algebraic proof systems, (i.e., of 1 = 0). including the Nullstellensatz system and the polynomial The de¯nition of the polynomial calculus depends calculus (also called the `GrÄobner' system) have been implicitly on the choice of a ¯eld F such that all proposed: these systems are motivated in part by the polynomials are over the ¯eld F . A number of authors desire to identify powerful proof systems which support also consider the polynomial calculus over rings ([3, 2]). 1Supported in part by international grant INT-9600919/ME-103 The only di®erence in the de¯nition of the PC system from the NSF (USA) and the MSMT· (Czech republic) is that a PC refutation over a ring is a derivation of r 2 Supported in part by NSF grant DMS-9803515 (i.e., of r = 0) for some non-zero r in the ring. Our 3Supported in part by NSF grant CCR-9734911, Sloan Research Fellowship BR-3311, and US-Israel BSF grant 97-00188. main results apply to both ¯elds and rings. 4Supported in part by NSF grant CCR-9457783 and US-Israel The mod p counting principle can be formulated as BSF grant 95-00238. n a set MODp of constant-degree polynomials expressing the negation of the counting principle, and the present paper gives linear lower bounds on the degree of n polynomial calculus refutations of MODp over ¯elds of characteristic q = p. A couple lower bounds on the 6 degree of Nullstellensatz proofs of the mod p counting F of characteristic di®erent from 2: There will be rn=2 principles have been given in prior work: [1] gave underlying variables, one for each edge of Gn. We will non-constant lower bounds and [3] gave lower bounds of denote the variable corresponding to the edge e = i; j ² f g the form n . For the polynomial calculus, the best lower from i to j by ye = y i;j . For each variable ye,we n 2 f g bound on the degree of PC refutations of MODp was have the equation ye 1 = 0; this forces the variables Kraj¶³·cek's ­(log log n) lower bound based on a general to take on values of¡ either 1 or 1, with y = 1 ¡ e ¡ lower bound for symmetrically speci¯ed polynomials [6]. corresponding to the presence of e in the subgraph E0. A couple polynomial calculus lower bounds have Secondly, corresponding to each vertex i in Gn, we will been obtained for other families of tautologies. have the equation 1 + y i;j1 y i;j2 y i;jr = 0, where f g f g ¢¢¢ f g Razborov [9] established pn lower bounds on the j1;:::;jrare the neighbors of i in Gn. This corresponds degree of polynomial calculus proofs of the pigeon-hole to saying that the degree of i in the subgraph E0 is odd. principle. Kraj¶³·cek [6] proves log log n lower bounds for This set of equations, representing the Tseitin mod 2 a wide variety of symmetric tautologies. graph formula, will be denoted by TSn(2). Recently, Grigoriev [5] succeeded in giving very sim- For any prime p, we can generalize the above ple linear lower bounds on the degree of Nullstellensatz principle to obtain a mod p version as follows. Again, refutations of the Tseitin mod 2 graph tautologies. we ¯x an underlying r-regular, undirected graph Gn, The present work is motivated by this paper, and in and then let Gn0 be the corresponding directed graph particular by the idea of working in the Fourier basis where each undirected edge is replaced by two directed which greatly simpli¯es the argument. edges. Each vertex in Gn0 will have an associated The present paper establishes linear lower bounds label, or charge in [0;p 1] such that the sum of the to the polynomial calculus by proving that over a ¯eld vertex charges is congruent¡ to 1 mod p. The mod p n of characteristic q - p, any PC refutation of the MODp principle states that it is impossible to assign values polynomials requires degree ± n, for a constant ± which in [0;p 1] to each of the directed edges so that: depends on p and q. In section¢ 8 we generalize this (i) for any¡ pair of complementary edges i; j and j; i , linear lower bound to the polynomial calculus over v( i; j )+v( j; i ) 0 (mod p), and (ii) forh everyi vertexh i h i h i ´ rings Zq provided p and q are relatively prime. i, the sum of the edge values coming out of vertex i is As it is well-known to be easy to give constant congruent to the charge of that vertex mod p. Again, degree polynomial calculus (and even Nullstellensatz) this is impossible since if we sum the edges in pairs, we n refutations of the MODp polynomials over Fp, our obtain 0 mod p, but summing them by vertices gives n results imply that the MODp polynomials have a the total charge of 1 mod p. linear gap between proof complexity for the polynomial Let F be a ¯nite ¯eld with characteristic q = p that 6 calculus over Fp and over Fq. contains a primitive p-th root of unity !. Assume all It follows from a result of Kraj¶³·cek [7] that our charges of vertices are 1, and that n 1 (mod p). We ´ linear lower bounds on the degree of PC refutations can express the mod p Tseitin principle for Gn0 as the imply exponential lower bounds of AC0[q]-Frege proofs unsatis¯ability of the following system of polynomials of the mod p principles when Mod-q gates are present over F : We have rn underlying variables ye, one only at the top (root) of formulas. for every directed edge e. For each variable ye we p have the equation ye 1 = 0; this forces variables to ¡ 2 p 1 2 Tseitin tautologies: polynomial version take on values in 1;!;! ;:::;! ¡ . (The power of ! corresponds to the value assigned to e.) Secondly, Tseitin's (mod 2) graph tautologies are based on the for each vertex i in Gn0 , we will have the equation following idea. Let Gn be a connected undirected y i;j1 y i;j2 ;:::;y i;jr ! = 0, where j1;:::;jr are the h i h i h i ¡ graph on n vertices, where each node in the graph has neighbors of i. Third, for each edge e = i; j we have h i an associated charge of either 0 or 1, and where the the equation y i;j y j;i 1 = 0. This set of equations, h i h i ¡ total sum of the charges is odd. Then it is impossible to representing the Tseitin mod p formula, will be denoted choose a subset of the edges E0 from E so that for every by TSn(p).
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