Journal of Computer and System Sciences SS1494 journal of computer and system sciences 55, 2435 (1997) article no. SS971494 Natural Proofs Alexander A. Razborov* School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; and Steklov Mathematical Institute, Vavilova 42, 117966, GSP-1, Moscow, Russia and Steven Rudich- Computer Science Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15212 Received December 1, 1994; revised December 2, 1996 Baker, Gill, and Solovay [7], who used oracle separation We introduce the notion of natural proof. We argue that the known results for many major complexity classes to argue that proofs of lower bounds on the complexity of explicit Boolean functions relativizing proof techniques could not solve these in nonmonotone models fall within our definition of natural. We show, problems. Since relativizing proof techniques involving based on a hardness assumption, that natural proofs can not prove superpolynomial lower bounds for general circuits. Without the hard- diagonalization and simulation were the only available ness assumption, we are able to show that they can not prove exponen- tools at the time of their work, progress along known lines tial lower bounds (for general circuits) for the discrete logarithm was ruled out. 0 problem. We show that the weaker class of AC -natural proofs which Because of this, people began to study these problems is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser, from the vantage of Boolean circuit complexity, rather than Yao, and Ha# stad is inherently incapable of proving the bounds of Razborov and Smolensky. We give some formal evidence that natural machines. The new goal is to prove a stronger, nonuniform proofs are indeed natural by showing that every formal complexity version of P{NP, namely that SAT (or some other measure, which can prove superpolynomial lower bounds for a single problem in NP) does not have polynomial-size circuits. function, can do so for almost all functions, which is one of the two Many new proof techniques have been discovered and suc- ] requirements of a natural proof in our sense. 1997 Academic Press cesfully applied to prove lower bounds in circuit complexity, as exemplified by [11, 1, 40, 14, 27, 28, 3, 2, 37, 4, 29, 36, 8, 5, 23, 24, 15, 13, 17, 26, 6] among others, although the 1. INTRODUCTION lower bounds have not come up near the level of P or even It is natural to ask what makes lower bound questions NC. These techniques are highly combinatorial, and in prin- such as P =? PSPACE, P =? NP, and P =? NC so difficult to ciple they are not subject to relativization. They exist in a solve. A nontechnical reason for thinking they are difficult much larger variety than their recursion-theoretic prede- cessors. Even so, in this paper we give evidence of a general might be that some very bright people have tried and ? failedbut this is hardly satisfactory. A technical reason limitation on their ability to resolve P = NP and other hard along the same lines would be provided by a reduction to problems. these questions from another problem known to be really Section 2 introduces and formalizes the notion of a hard such as the Riemann hypothesis. Perhaps the ultimate natural proof. We argue that all lower bound proofs known to demonstration that P =? NP is a hard problem would be to date against nonmonotone Boolean circuits are natural, or show it to be independent of set theory (ZFC). can be represented as natural. In Section 3 we present diverse Another way to answer this question is to demonstrate examples of circuit lower bound proofs and show why they that known methods are inherently too weak to solve are natural in our sense. While Section 5 gives some general problems such as P =? NP. This approach was taken in theoretical reasons why proofs against circuits tend to be natural, Section 4 gives evidence that ``naturalizable'' proof techniques cannot prove strong lower bounds on circuit size. In particular, we show modulo a widely believed crypto- * Supported by Grant 93-6-6 of the Alfred P. Sloan Foundation, by Grant 93-011-16015 of the Russian Foundation for Fundamental graphic assumption that no natural proof can prove super- Research, and by an AMS-FSU grant. polynomial lower bounds for general circuits, and we show - Partially supported by NSF Grant CCR-9119319. unconditionally that no natural proof can prove exponential 0022-0000Â97 25.00 24 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. File: 571J 149401 . By:CV . Date:28:07:01 . Time:05:43 LOP8M. V8.0. Page 01:01 Codes: 6463 Signs: 4540 . Length: 60 pic 11 pts, 257 mm NATURAL PROOFS 25 lower bounds on the circuit size of the discrete logarithm Usefulness. The circuit size of any sequence of functions problem. f1 , f2 , ..., fn , ..., where fn # Cn , is super-polynomial; i.e., for Natural proofs form a hierarchy according to the com- any constant k, for sufficiently large n, the circuit size of fn plexity of the combinatorial property involved in the proof. is greater than nk. We show without using any cryptographic assumption that A proof that some function does not have polynomial-sized AC 0-natural proofs, which are sufficient to prove the parity circuits is natural against PÂpoly if the proof contains, more lower bounds of [11, 40, 14], are inherently incapable of or less explicitly, the definition of a natural combinatorial proving the bounds for AC 0[q]-circuits of [29, 36, 8]. property C which is useful against PÂpoly. One application of natural proofs was given in [33]. It n Note that the definition of a natural proof, unlike that of was shown there that in certain fragments of bounded a natural combinatorial property, is not precise. This is arithmetic any proof of superpolynomial lower bounds for because while the notion of a property being explicitly general circuits would naturalize, i.e., could be recast as a defined in a journal paper is perfectly clear to the working natural proof. Combined with the material contained in mathematician, it is a bit slippery to formalize. This lack Section 4 of this paper, this leads to the independence of of precision will not affect the precision of our general such lower bounds from these theories (assuming our cryp- statements about natural proofs (see Section 4) because tographic hardness assumption). See also [19, 34] for inter- they will appear only in the form ``there exists (no) natural pretations of this approach in terms of the propositional proof...'' and should be understood as equivalent to ``there calculus, [10, 25] for further results in this direction, and exists (no) natural combinatorial property C ....'' [35] for an informal survey. n The definitions of natural property and natural proof can 1.1. Notation and definitions. We denote by Fn the set of be explained much less formally. First, a proof that some all Boolean functions in n variables. Most of the time, it will explicit function [gn] does not have polynomial-sized cir- be convenient to think of fn # Fn as a binary string of length cuits must plainly identify some combinatorial property Cn n 2 , called the truth-table of fn . fn is a randomly chosen func- of gn that is used in the proof. That is, the proof will show tion from Fn , and in general, we reserve the bold face in our that all functions fn that have this property, including gn formulae for random objects. itself, are hard to compute. In other words, Cn is useful.If k k The notation AC , NC is used in the standard sense to [gn]#NP; then the proof concludes P{NP. Our main denote nonuniform classes. AC 0[m], TC 0, and PÂpoly are contention, backed by evidence in the next section, is that the classes of functions computable by polynomial-size current proof techniques would strongly tend to make this bounded-depth circuits allowing MOD-m gates, bounded- Cn large and constructive as defined above. (Or at least these depth circuits allowing threshold gates, and unbounded- two conditions would hold for some subproperty C*ofn Cn.) depth circuits over a complete basis, respectively. In order to understand the definition of large more n intuitively, let N=2 . Largeness requires that |Cn*|Â|Fn| k 2. NATURAL PROOFS 1ÂN for some fixed k>0; i.e., fn has a nonnegligible chance of having property Cn . 2.1. Natural Combinatorial Properties Constructively is a more subtle notion to understand and justify. We take as our basic benchmark of ``constructive'' We start by defining what we mean by a ``natural com- O(n) that fn # Cn be decidable in time 2 , i.e., polynomial as a binatorial property''; natural proofs will be those that use a n natural combinatorial property. function of 2 . Now, this is exponential in the number n of Formally, by a combinatorial property of Boolean variables in fn , and this makes our concept somewhat functions we will mean a set of Boolean functions mysterious, especially since we are going to employ it for [C F | n # |]. Thus, a Boolean function f will possess studying computations which are polynomial in n! The best n n n justification we have is empirical: the vast majority of property Cn if and only if fn # Cn . (Alternatively, we will sometimes find it convenient to use function notation: properties of Boolean functions or n-vertex graphs (etc.) C ( f )=1 if f # C ; and C ( f )=0 if f  C .) The com- that one encounters in combinatorics are at worst exponen- n n n n n n n n tial-time decidable, and, as a matter of fact, known lower binatorial property Cn is natural if it contains a subset Cn* with the following two conditions: bounds proofs operate only with such properties.