2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
Why are Proof Complexity Lower Bounds Hard?
Jan´ Pich Rahul Santhanam Department of Computer Science Department of Computer Science University of Oxford University of Oxford Oxford, United Kingdom Oxford, United Kingdom [email protected] [email protected]
Abstract—We formalize and study the question of whether its limits? Or are these problems fundamentally intractable there are inherent difficulties to showing lower bounds on in some sense? propositional proof complexity. Since complexity theory is the theory of intractability, We establish the following unconditional result: Proposi- tional proof systems cannot efficiently show that truth tables it is natural to apply it to the seeming intractability of of random Boolean functions lack polynomial size non-uniform complexity-theoretic questions themselves. Since the early proofs of hardness. Assuming a conjecture of Rudich, proposi- days of complexity theory, progress on complexity lower tional proof systems also cannot efficiently show that random bounds has gone hand-in-hand with the formalization of k-CNFs of linear density lack polynomial size non-uniform various sorts of barriers to progress. In the 70s, analogies proofs of unsatisfiability. Since the statements in question assert the average-case hardness of standard NP problems (MCSP between complexity theory and recursion theory were devel- and 3-SAT respectively) against co-nondeterministic circuits oped, with various concepts and techniques from recursion for natural distributions, one interpretation of our result is theory being adapted to the resource-bounded setting. How- that propositional proof systems are inherently incapable of ever, Baker, Gill & Solovay [7] observed in the late 70s that efficiently proving strong complexity lower bounds in our popular machine simulation and diagonalization techniques formalization. Another interpretation is that an analogue of the Razborov-Rudich ‘natural proofs’ barrier holds in proof from recursion theory relativized, i.e., continued to work complexity: under reasonable hardness assumptions, there even when machines were given access to an arbitrary are natural distributions on hard tautologies for which it is oracle. By giving an oracle relative to which P = NP and infeasible to show proof complexity lower bounds for strong another oracle relative to which P = NP [7], they proved that enough proof systems. no relativizing techniques could solve the NP vs P question. For the specific case of the Extended Frege (EF) proposi- It seemed that techniques of a fundamentally different sort tional proof system, we show that at least one of the following cases holds: (1) EF has no efficient proofs of superpolynomial were required. circuit lower bound tautologies for any Boolean function or (2) After this barrier result, attention shifted to a more finitis- There is an explicit family of tautologies of each length such tic setting. Rather than considering uniform machines that that under reasonable hardness assumptions, most tautologies work for all inputs, Boolean circuits corresponding to finite are hard but no propositional proof system can efficiently functions became the object of study. It is well-known that establish hardness for most tautologies in the family. Thus, P under reasonable hardness assumptions, either the Circuit can be simulated by polynomial-size Boolean circuits, and Lower Bounds program toward complexity separations cannot therefore super-polynomial lower bounds on Boolean circuit be implemented in EF, or there are inherent obstacles to size imply lower bounds against P. implementing the Cook-Reckhow program for EF. Perhaps the hope was that circuits are simpler and more Keywords-Computational complexity, proof complexity, cir- ‘combinatorial’ objects, and therefore more amenable to cuit lower bounds, barriers lower bounds via combinatorial and algebraic techniques. Indeed, initial results were promising. In a series of I. INTRODUCTION influential works [2], [21], [24], [48] applying the technique of random restrictions, super-polynomial lower bounds A. Motivation were shown for the Parity function against constant-depth Complexity theory is full of questions that are easy to circuits. Razborov [40] and Smolensky [47] developed the state but hard to answer. The most famous of these is the P polynomial approximation technique to give lower bounds vs NP problem [14], but there are numerous others such as against constant-depth circuits with prime modular gates. the NP vs coNP problem, the PSPACE vs P problem, and Razborov [39] used the method of approximations to show the BPP vs P problem. In all of these cases, despite decades that the Clique problem required super-polynomial size of effort, very little progress has been made. Is this because monotone circuits. This sequence of works introduced complexity theory is a young field, and we have not yet had several new lower bound techniques, and it seemed that the time to develop a deep understanding of computation and steady progress was being made toward the goal of
2575-8454/19/$31.00 ©2019 IEEE 1305 DOI 10.1109/FOCS.2019.00080 separating NP and P via a Circuit Lower Bounds Program. P . Equivalently, NP = coNP iff there is a hard sequence of tautologies for every propositional proof system P . Circuit Lower Bounds Program: Separate NP and Just as the Circuit Lower Bounds Program aims to make P by proving super-polynomial lower bounds for functions progress on the NP vs P question, the work of Cook and in NP against increasingly expressive classes of Boolean Reckhow suggests an approach to making progress on the circuits. NP vs coNP question.
Unfortunately, the Circuit Lower Bounds Program stalled Cook-Reckhow Program: Separate NP and coNP by in the early 90s. Even now, almost 30 years later, we proving super-polynomial lower bounds on the proof size still don’t know if there are explicit functions that require of tautologies in increasingly powerful proof systems P . super-polynomial depth-two threshold circuits, or constant- depth circuits with Mod 6 gates. However, even if no We note a couple of differences between the circuit breakthrough lower bounds were proved for a while, our complexity lower bound setting and the proof complexity understanding of barriers to lower bounds did evolve, due lower bound setting. In the circuit complexity setting, it mainly to work of Razborov [41], [42] and Razborov is easy to see by a counting argument that most Boolean & Rudich [45]. Razborov & Rudich [45] developed the functions do not have small circuits. The challenge is to concept of natural proofs, which captures known circuit show a circuit lower bound for an explicit Boolean function, lower bounds proved via combinatorial and algebraic i.e., a function in NP. In the proof complexity setting, we techniques, and showed that under standard cryptographic do not care about explicitness - we are satisfied with lower assumptions, natural proofs cannot prove super-polynomial bounds for any sequence of tautologies. In this setting, lower bounds for general Boolean circuits. counting arguments do not work to get non-constructive The natural proofs barrier has not proven to be as lower bounds, since the space of small proofs is comparable damaging to the finitistic approach to complexity as the in size to the space of tautologies. relativization barrier was to the machine-based approach. Despite these differences, progress on the Cook-Reckhow Circuits are still a popular model, and combinatorial and Program has been partly inspired by analogies with the algebraic techniques are still heavily used. Various methods Circuit Lower Bounds Program. Haken [23] showed super- to evade the natural proofs barrier have been proposed, polynomial lower bounds on the size of proofs of the Pi- though none have yet resulted in any breakthrough lower geonhole Principle in the Resolution proof system. Ajtai [3] bounds for explicit functions. obtained a significant extension of this result by showing that The meta-mathematics of circuit lower bounds is one of the Pigeonhole Principle is also super-polynomially hard for our motivations for the results in this paper, but the main Bounded-Depth Frege - a restricted version of the standard motivation comes from a different approach to complex- Frege system where the lines are bounded-depth circuits. ity theory, via proof complexity. The NP vs P problem Ajtai’s super-polynomial lower bound was strengthened to essentially asks if short proofs for mathematical theorems an exponential lower bound by [9]. Both [3] and [9] use are easy to find. Thus logic and proof are inherent to the strong versions of the random restriction technique used to problem. These aspects are emphasized not just in Steve prove lower bounds on bounded-depth circuits. Cook’s seminal paper [14] stating the problem, but already There has been significant effort devoted to proving in Godel’s famous letter to von Neumann in the early 1950s. lower bounds for constant-depth Frege proofs with modular Cook and Reckhow [18] defined the notion of a proposi- counting gates - the proof-theoretic analogue of constant- tional proof system: a polynomial-time computable surjec- depth circuits with modular counting gates - but without any tive function mapping strings to tautologies. The requirement success. Thus the Cook-Reckhow program is also stalled. that the range of the function is the set of tautologies What is even more unsatisfactory, though, is that unlike models the soundness of the proof system; the surjectiveness the Circuit Lower Bounds program, we do not have a good of the function models completeness; the polynomial-time explanation for why progress is stalled. There is no known computability of the function models efficient verifiability analogue of the natural proofs barrier for proof complexity, of proofs. A proof of a tautology φ with respect to a and indeed this is one of the main open questions asked in propositional proof system P is simply a string x such that the survey of Beame and Pitassi [10] from the late 90s. P (x)=φ. A natural complexity question then is how the Our main motivation in this paper is to formalize and minimum proof size grows as a function of the length of the study the question of whether there are inherent obstacles to tautology, with respect to a given propositional proof system proving proof complexity lower bounds for strong proposi- P . tional proof systems. Cook and Reckhow [18] show that NP = coNP iff There are several reasons why this question is interesting. there is a propositional proof system P that is polynomially First, as described above, it is motivated by the difficulty of bounded, i.e., all tautologies have polynomial-size proofs in making progress on the Cook-Reckhow program.
1306 Second, propositional proof systems correspond naturally system satisfying the ‘feasible interpolation’ property can to certain classes of algorithms that are used in practice for efficiently prove circuit lower bound formulas corresponding solving the Satisfiability problem. For example, the Tree to s = nω(1) for any Boolean function. The advantage of this Resolution proof system corresponds to branching algo- result is that it applies to any Boolean function; the weakness rithms. Showing lower bounds for a proof system P trans- is that it only holds for propositional proof systems with lates to constructing hard instances for the corresponding feasible interpolation, and systems such as Extended Frege class A(P ) of algorithms, which is an interesting question and Frege are known not to have feasible interpolation under in its own right. The question of whether proof complexity cryptographic assumptions [12], [33]. lower bounds are hard thus connects to the question of We can define proof lower bound formulas for a given whether provably hard instances exist for various classes of propositional proof system R in an analogous way. Given algorithms. a formula φ and a parameter s, the corresponding R-proof Third, propositional proof systems can be interpreted lower bound formula lbR(φ, s) states that there is no R-proof not only algorithmically, as in the previous para, but also of φ of size s. Just as with circuit lower bound formulas, this meta-mathematically, as a setting within which barriers is a universal statement: there no proof for φ of size at most to progress in complexity theory can be formalized and s in R. This can be encoded by a DNF of size poly(|φ|,s), understood. This duality is also important in this paper, where the propositional variables correspond to the bits of and enables us to use our ideas to show the unprovability the proof. of certain strong complexity hypotheses within a finitistic Now we can ask about the proof complexity of R-proof context. In the terminology of Aaronson [1], these are new lower bound formulas. In this paper, we typically adopt ‘second-generation’ independence results. First generation a high standard for hardness - we require that the proof independence results are about unprovability in logical the- complexity of R-proof lower bound formulas is large for ories such as ZFC, PA and restrictions thereof. In this paper, every propositional proof system S arguing about these on the other hand, we focus on finitistic results in the context lower bound formulas. of propositional proof complexity, where proofs always exist We remark that upper bounds on the proof complexity for any tautology, but the question is whether they are of a of R-proof lower bound formulas for certain proof sys- reasonable size. tems R are implicit in previous work. A result of Cook We now proceed to discuss our model, and the hypotheses and Pitassi [17] implies that Resolution-proof lower bound we consider. formulas corresponding to the Pigeonhole principle can be efficiently proved within Extended Frege. Bellantoni, Pitassi B. The Setting and Urquhart [11] give constructive proofs of constant- We would like to study the question of whether proof depth Frege lower bounds that appear to be efficiently complexity lower bounds are inherently ‘hard’. The most implementable within Extended Frege. Thus some of the natural formalization of ‘hardness’ we can imagine in this strongest proof complexity lower bounds we have at present context is that tautologies corresponding to proof complexity seem also to be provable within standard propositional proof lower bounds are themselves hard to prove, even in strong systems. propositional proof systems. In contrast to these upper bound results, our emphasis This formalization is inspired by the analogy with circuit is on lower bound and impossibility results, which give evi- complexity. In his work on complexity barriers, Razborov dence that certain propositional proof systems are inherently [41], [42] considered the proof complexity of the circuit hard to analyze. lower bound formulas, which are propositional formulas Let us say a collection of R-proof complexity lower bounds is feasibly provable if the corresponding R-proof tt(fn,s) expressing that a Boolean function fn on n bits given by its truth table does not have Boolean circuits of lower bound formulas all have short S-proofs in some size s, for some parameter s. Intuitively the reason why propositional proof system S. Armed with this notion, we this statement can be expressed by propositional formulas can define a feasible version of the Cook-Reckhow program. (which are tautologies when f is indeed hard) is that it is a universal statement, saying that no circuit of size at Feasible Cook-Reckhow Program: For every propositional most s can compute the function corresponding to the given proof system R, show that for each k>0 there is a sequence { } truth table. This can be encoded by a DNF of size O(2ns3) of tautologies φn and a propositional proof system S, lb | |k where the propositional variables correspond to the bits of R(φn, φn ) has polynomial-size S-proofs. the circuit1. The main result of [41], [42] is that under standard cryptographic assumptions, no propositional proof Note that our requirement here is fairly mild: we allow the propositional proof system S to depend on the propositional 1Note that the encoding is exponentially large. More succinct encodings proof system R for which lower bounds are being shown. have been considered in recent work of Muller¨ and Pich [36] We even allow S to depend on the constant k corresponding
1307 to the lower bound being shown. Also note that we do Rudich originally stated his conjecture in the terminology not require that S has efficient proofs that the formulas of natural proofs; we find the reformulation below in terms φn are tautologies. All we ask is that these formulas are of proof systems more convenient. indeed tautologies, and that S can show that they don’t have efficient R-proofs. Rudich’s Conjecture: For any proof system R verifiable Since we are interested in impossibility results, the mild- in polynomial size, most Boolean functions on n bits do ness of our proof-theoretic requirements is an advantage not have short (i.e., poly(2n)) size R-proofs of hardness. - it makes our results stronger. However, we do insist on feasibility. Intuitively, in a finite resource-bounded world, it Suppose we wish to generate hard tautologies for seems unreasonable to allow the prover unlimited resources. some propositional proof system R. Simply pick the truth The results of [17] and [11] mentioned above consti- table of a random Boolean function fn, and consider the k tute partial progress toward the Feasible Cook-Reckhow circuit lower bound formula tt(fn,n ) for some fixed k. program. If R is Resolution or Constant-Depth Frege, the These circuit lower bound formulas are tautologies with feasibility condition is satisfied when S is Extended Frege high probability because most Boolean functions are hard. and the formulas {φn} are the pigeon-hole principles. Rudich’s Conjecture implies that these tautologies are The Feasible Cook-Reckhow Program can be interpreted also hard for R (in fact, even for R that is verifiable in as a feasible approach to proving NP = coNP. We wish polynomial size) with high probability. to get evidence for the difficulty or impossibility of im- Another useful perspective on Rudich’s Conjecture is to plementing this program. However, the mildness of our think of it as an average-case hardness hypothesis for the proof-theoretic requirements is an issue here. Consider any Minimum Circuit Size Problem. The Minimum Circuit Size efficiently computable sequence {φn} of tautologies of Problem MCSP is a fundamental problem in NP that is increasing length, such that the sequence {φn} requires believed to be intractable, but for which NP-completeness R-proofs of super-polynomial size. One way to define a has not yet been established. MCSP asks, given the truth proof system S in which these lower bounds are easily table of a Boolean function fn and a parameter s, whether provable is to simply add the corresponding proof lower fn has circuits of size at most s. Suppose we fix the k bound tautologies lbR(φn, |φn| ) as axioms to a standard parameter s as a function of n. Let us call this parameterized propositional proof system such as Extended Frege. Since version of the problem MCSP[s]. Then Rudich’s Conjecture these are tautologies, the resulting system S is still sound, says that for a large enough constant k, MCSP[nk] does and since the {φn} are efficiently computable, S can be not have any dense subsets computable by nondeterministic shown to be a propositional proof system in the Cook- polynomial-size circuits, where L denotes the complement Reckhow sense. Since the R-proof lower bound formulas of L. The conjecture that there are no dense subsets of the k corresponding to {φn} are axioms of S, they are also easily NO instances of MCSP[n ] computable in some circuit class provable. C can be seen as an assertion of zero-error average-case We are therefore led to consider situations where the hardness assumption against C with respect to the uniform hard tautologies are not efficiently computable. One natural distribution, since almost all instances of a given length N situation in which this happens is if the hard tautologies are are NO instances [25]. random in some sense. The Cook-Reckhow program aims Our notion of feasible proofs extends in a natural way to show a worst-case separation between NP and coNP.It to Rudich’s Conjecture, modulo a technicality about non- is plausible that stronger average-case separations hold, and uniform proofs. The Cook-Reckhow Program aims to sepa- the hypotheses we consider in this paper correspond to such rate the uniform classes NP and coNP. Rudich’s Conjecture, average-case separations for standard NP problems against on the other hand, asserts hardness for an NP problem (non-uniform) coNP over natural distributions. even against nondeterministic circuits, or equivalently proof The main hypothesis we consider is Rudich’s Conjecture systems verifiable in polynomial size. This non-uniformity [46]. Rudich’s main motivation in making this conjecture can be modelled easily within our propositional setting using was to strengthen the ‘natural proofs’ barrier of Razborov the notion of proof systems with advice due to Cook and and Rudich [45]. The ‘natural proofs’ barrier shows that Krajicek [16]. under the standard cryptographic assumption that one-way In analogy to the Feasible Cook-Reckhow Program, we functions exist, there are no dense subsets of the hard say that Rudich’s Conjecture admits feasible propositional Boolean functions computable by polynomial-size circuits, proofs if for every propositional proof system R with poly- where Boolean functions are represented explicitly by their omial advice and every constant k, there is a propositional truth tables. Rudich conjectured that something stronger proof system S such that for a 1 − o(1) fraction of Boolean was true: most hard functions do not even have short proofs functions fn on n bits, S proves efficiently that there are k k k of hardness that are verifiable in polynomial size, where no m size R-proofs of tt(fn,n ), where m = |tt(fn,n )|. ‘short’ means polynomial in the length of the truth table. Namely, for a significant fraction of Boolean functions f,
1308 S can efficiently prove that there are no short R-proofs of rules out polynomial-size S-proofs in every propositional the circuit lower bound formula corresponding to f and size proof system S 2, even though S is allowed to depend on parameter nk. R. The second hypothesis we consider is a nondeterministic At first glance, this might seem strange. If no proposi- version of Feige’s Hypothesis [20]. Consider the following tional proof system S can efficiently prove a sequence of very natural distribution UΔ,N over 3-CNFs on N variables tautologies, then NP = coNP [18]. So why doesn’t our result with ΔN clauses, where Δ > 0 is any constant: we imply a significant complexity lower bound? pick each clause by selecting 3 literals uniformly and The reason is that the R-proof lower bound formu- independently at random from the 2N possible literals. las aren’t necessarily tautologies with high probability. If Feige’s Hypothesis is that there is no polynomial-time Rudich’s Conjecture is true, then they are tautologies. If algorithm that outputs ‘unsatisfiable’ with significant Rudich’s Conjecture is false, on the other hand, this isn’t probability over the distribution UΔ,N and never outputs clear. Indeed, our proof of Theorem 1 splits into two cases: ‘unsatisfiable’ on satisfiable formulas. We consider a the first in which Rudich’s Conjecture is true, and the nondeterministic version of Feige’s Hypothesis conjectured second in which Rudich’s Conjecture is false. If Rudich’s by Ryan O’Donnell [25], [38]. Conjecture is false, we show using standard techniques (together with an amplification argument) that there is a Nondeterministic Feige’s Hypothesis: For any propositional proof system R with polynomial advice in propositional proof system R verifiable in polynomial which most circuit lower bound tautologies for functions on size, with high probability over φ picked from UΔ,N , there n inputs have short proofs, for infinitely many n. In this case, are no polynomial size proofs of unsatisfiability for φ in R. the R-proof lower bound formulas aren’t tautologies, with high probability, and hence they cannot have polynomial- Just as Rudich’s conjecture implies that circuit lower size S-proofs (or indeed proofs of any size) for any sound bound formulas corresponding to random functions are hard propositional proof system S. tautologies for any propositional proof system with high The crux of our proof is the argument that when Rudich’s probability, Nondeterministic Feige’s Hypothesis implies Conjecture is true, then Rudich’s Conjecture does not admit that random 3-CNFs with ΔN clauses are hard tautologies feasible proofs. Thus Rudich’s Conjecture is self-defeating for any propositional proof system with high probability, in the propositional setting: its truth implies its unprovability. for any Δ > 0. Thus both hypotheses state that natural This is reminiscent of a comment by Scott Aaronson in his distributions on tautologies are hard. survey on independence results [1] for P vs NP about the Also, just as Rudich’s conjecture is an average-case ‘bizarre self-referential nature of P = NP - a conjecture that hardness hypothesis for MCSP, Nondeterministic Feige’s all but asserts the titanic difficulty of finding its own proof’. Hypothesis is an average-case hardness hypothesis for 3- We show that when considering the stronger statement that SAT against nondeterministic algorithms. Barak [8] has is Rudich’s Conjecture, this intuition can be made formal in advocated studying Feige’s hypothesis and its ilk, as offering the feasible setting. some of our best hopes of more insight into average-case We briefly explain the ideas of the proof. The general complexity. plan is to exploit the connections between proof complexity We say that Nondeterministic Feige’s Hypothesis admits and pseudorandomness discovered by [42], [45]. If Rudich’s feasible propositional proofs if for every propositional proof conjecture is true, then given any propositional proof system system R with polyomial advice and every constant k, there R with polynomial advice, most truth table tautologies for is a propositional proof system S such that with probability functions on n bits are indeed hard for R. We would like − 1 o(1) over φ chosen from UΔ,N , S proves efficiently that to use Rudich’s conjecture again to show that for any proof k there are no m size R-proofs of φ, where m = |φ|. system S,eventheR-proof lower bound formulas for the circuit lower bound tautologies based on fn do not have C. Our Results short proofs in S, for most fn. The main result of this paper is that we unconditionally The problem is that if S does not have short proofs of a rule out feasible proofs of Rudich’s Conjecture. certain formula, this can be for two different reasons. First, Theorem 1. Rudich’s Conjecture does not admit feasible the formula might be a tautology but without short proofs in propositional proofs. S. This is the good case for us - if all short proofs in S for R-lower bound formulas were for tautologies, we could use In other words, there is a propositional proof system R this to get a dense set of hard functions computable by small with polynomial advice such that no propositional proof nondeterministic circuits, and thereby get a contradiction to system S can prove lower bounds on the size of R-proofs for most circuit lower bound tautologies. We emphasize two 2In fact we rule out polynomial-size proofs even for systems S that take aspects of this result. First, it is unconditional. Second, it polynomial advice
1309 Rudich’s Conjecture. However, there is a second case: the is non-constructive: we show unconditionally that a certain formula is not a tautology at all, and so of course it does propositional proof system R with polynomial advice exists not have short proofs in S. This case is problematic, because for which it is hard to show lower bounds, but we are now some functions fn for which there are short S-proofs unable to say what R is. Such non-constructive techniques of R-lower bound formulas corresponding to fn might be have been used before as ingredients in diagonalization-style easy, and thus we don’t get a dense subset of hard functions. arguments, but we are unaware of any previous examples in To overcome this problem, we use Rudich’s Conjecture propositional proof complexity. again, to argue that the truth table of MCSP itself does not The second interpretation is more relevant to our original have small circuits. Thus, the circuit lower bound formula motivation of finding proof complexity barriers. As we men- with fn = MCSP is indeed a tautology. Moreover, the tioned, the crux of the proof of Theorem 1 is to show that sequence of such circuit lower bound tautologies has short if Rudich’s Conjecture is true, there is a propositional proof proofs in an appropriately defined propositional proof system system R (without advice) such that R-proof complexity R, where these circuit lower bound tautologies are simply lower bounds are hard to show in any propositional proof added as axioms to a standard proof system. system S. This is an analogue of the ‘natural proofs’ barrier It is still not clear how this helps us. An important step of Razborov and Rudich [45] in proof complexity. Just conceptually is to introduce the notion of pseudorandom as the ‘natural proofs’ barrier says that ‘feasible’ circuit tautologies, which are analogues of pseudorandom functions lower bounds for random functions are unlikely if one-way in the proof complexity setting. Intuitively, pseudorandom functions exist, our result says that if Rudich’s Conjecture is tautologies are a collection of tautologies which are derived true, there is a proof system R such that R-proof complexity in a specified way from pseudorandom sets, but unlike lower bounds for appopriately defined ‘random tautologies’ random tautologies, have short proofs in some predefined are hard to prove. proof system R. Thus they can be distinguished from Corollary 1. If Rudich’s Conjecture holds, then there is random tautologies by having short proofs in R, just as { } pseudorandom functions can be distinguished from random a pps R and a samplable sequence of distributions DN functions in the cryptographic setting by being computable on formulas of length poly(N), such that with probability − ω(1) from short seeds by a polynomial-time function. For the 1 1/N over φN sampled from DN , φN is a tautology actual proof of Theorem 1, we work with hitting tautologies, that does not have poly(N) size R-proofs, but there is a constant k such that no pps S has polynomial-size proofs of which correspond to hitting sets just as pseudorandom lb | |k tautologies correspond to pseudorandom sets. R(φN , φN ). Inspired by an idea of Razborov [42], we show how More succinctly, if random circuit lower bound tautologies to use Rudich’s Conjecture (for a third time!) to get a are hard for propositional proof systems, it is also hard collection W of hitting tautologies which are easy for the within a propositional framework to explain why they are proof system R we define. Since all these hitting tautologies hard. are easy for R, for any propositional proof system S, the It is natural to ask if Theorem 1 is a special phenomenon soundness of S implies that there are no short S-proofs of to do with circuit lower bounds, or is part of a more general R-lower bound formulas corresponding to these tautologies. phenomenon. We give evidence for the latter by showing Suppose that there were short S-proofs of R-lower bound a similar result for Nondeterministic Feige’s Hypothesis, formulas for random circuit lower bound formulas. We show which on the surface seems to have nothing to do with circuit how this together with the hitting property of W implies a lower bounds. However, unlike Theorem 1, our result for contradiction. Nondeterministic Feige’s Hypothesis is conditional. We now discuss two possible interpretations of Theorem 3 1. The first is a metamathematical interpretation, where Theorem 2. Assuming Rudich’s Conjecture , Nondetermin- we understand the result as saying something about the istic Feige’s Hypothesis does not admit feasible proposi- difficulty of proving strong complexity lower bounds. If tional proofs. strong lower bounds such as Rudich’s Conjecture hold, Nondeterministic Feige’s Hypothesis guarantees that ran- then Theorem 1 indicates that standard ‘slice-and-measure’ dom 3-CNFs of linear density are hard for any propositional techniques such as those used in existing circuit complexity proof system R. Now the question is how to leverage this and proof complexity lower bounds are likely to fail in to show that it is also hard to prove that a randomly chosen establishing such lower bounds. So either we must believe in the failure of Rudich’s Conjecture, which would itself be 3In fact, we don’t need the full strength of Rudich’s Conjecture to get very interesting and somewhat counterintuitive, or we should the desired consequence for Nondeterministic Feige’s Hypothesis. It would be prepared to step outside a finitistic mindset if we want be enough to solve Open Problem 3 in [46], which is about ‘stretching demi-bits’, and seems within reach of current techniques. In the interest of to make progress on very strong complexity conjectures. minimizing hypotheses used, we state the result as above, but it is useful Indeed, this is illustrated by our proof of Theorem 1, which to keep in mind that much weaker hypotheses suffice.
1310 3-CNF of linear density is hard for a given proof system R. implemented within the proof system, or there are inherent If we had a way of constructing pseudorandom or hitting obstacles to implementing the Cook-Reckhow program for tautologies based on Nondeterministic Feige’s Hypothesis, the proof system. We state the result for Extended Frege, but an approach similar to the one for Rudich’s Conjecture might essentially the same proof also gives the analogous result for work. However, we have no idea how to do this, and are Frege. forced to adopt a different strategy. Theorem 3. Assume Rudich’s Conjecture and that E does We use a connection to a hardness hypothesis we call not have sub-exponential size nondeterministic circuits. Ei- the MKTP Hardness Hypothesis. We will not describe this ther EF does not efficiently prove circuit lower bound hypothesis formally here, but it is analogous to Rudich’s tautologies tt(fn,s) for any sequence {fn} of functions and Conjecture: while Rudich’s Conjecture is about most func- ω s(n)=n (1), or there are sets SN of formulas of size N tions lacking short proofs of hardness, the MKTP Hardness that can be generated in time poly(N), such that for each Hypothesis is about most strings lacking short proofs that c>0 and all large enough N, most φ ∈ SN are tautologies they are hard in the sense of KT-complexity. KT-complexity and require EF-proofs of size at least N ω(1), and yet no is a notion of time-bounded Kolmogorov complexity defined 2 propositional proof system S can prove in size N c that for by Allender [4] that is closely related to circuit complexity. c most φ ∈ SN , φ requires EF proofs of size at least N . We leave the details to the main body of the paper and continue with a sketch of the proof. The proof of Theorem 3 uses ideas mentioned earlier It is shown in [25] that there is an average-case reduction together with a complexity-theoretic derandomization tech- from Feige’s Hypothesis to the question of whether a string nique [26], [37] to generate explicit sets of tautologies. has high KT-complexity. We observe that this reduction D. Related Work translates to our setting, and with some technical work, show how to leverage it to get an implication from feasible Perhaps the most interesting recent work on the difficulty proofs of Nondeterministic Feige’s Hypothesis to feasible of showing proof complexity lower bounds is by Grochow proofs of the MKTP Hardness Hypothesis (with parameters and Pitassi [22]. They define a proof system called the chosen appropriately). We then show that feasible proofs of Ideal Proof System (IPS). IPS is verifiable in randomised the MKTP Hardness Hypothesis do not exist under strong polynomial time, but is not known to be a propositional enough complexity assumptions, and in particular assuming proof system 4. It is shown in [22] that IPS lower bounds Rudich’s Conjecture. This second part of the argument is for DNF tautologies imply that VNP = VP, where VNP analogous to the proof of Theorem 2, but there are some is the algebraic complexity analogue of NP and VP is the additional difficulties that need to be overcome. algebraic complexity analogue of P. Since separating VNP A common theme to Theorems 1 and 2 is that the hardness and VP is believed to be a hard problem, this gives evidence of proving that strings are “random-like” has implications for that showing IPS lower bounds is hard. The work of [22] barriers in proof complexity. differs from ours in considering a proof system that is not Theorem 2 has implications for both meta-mathematics of known to be propositional, and giving evidence for the complexity lower bounds and for proof complexity barriers, hardness of proof complexity lower bounds based on the just as with Theorem 1. The implication for proof complex- presumed hardness of circuit complexity separations. In our ity barriers is especially interesting in this case since random work, we formalize the question of proof complexity lower CNFs are a very natural class of formulas to analyze in bounds within propositional proof complexity, and study the terms of proof complexity. Similar to Corollary 1, we get inherent limitations of propositional proof systems, rather from Theorem 2 that under certain hardness assumptions, than reducing to circuit complexity questions. namely Rudich’s Conjecture and Nondeterministic Feige’s In terms of the finitistic meta-mathematics of complexity Hypothesis, there is a propositional proof system R with separations, the work that is perhaps closest to ours in spirit advice such that proving lower bounds on R is hard for any is by Razborov [42]. Razborov shows that under certain propositional proof system S. complexity assumptions, no proof system P with feasible The final result we highlight in this introduction concerns interpolation can prove any super-polynomial circuit com- the standard Extended Frege proof system. Our previous plexity lower bound. Our work differs from that of Razborov results give ‘proofs of principle’: under natural hardness in showing an unconditional result, and in considering assumptions, there are propositional proof systems that are arbitrary propositional proof systems, rather than ones with hard to analyze. While these results show fundamental feasible interpolation. The ‘feasible interpolation’ condition obstacles to proving strong complexity lower bounds in a is a fairly restrictive one, and there is evidence that Frege and feasible way, it is unclear how relevant they are to analysis EF do not have feasible interpolation [12], [32]. Rudich [46] of standard proof systems such as Frege and Extended Frege. 4However, as pointed out by an anonymous referee, by using Adleman’s We show that for these proof systems, a ‘lose-lose theorem’ trick, IPS is a propositional proof system with polynomial advice, when holds: Either the Circuit Lower Bounds program cannot be defined over large enough fields
1311 uses his Conjecture to extend Razborov’s result to propo- for each x ∈{0, 1}n, x ∈ TAUT iff there exists y ∈{0, 1}∗ sitional proof systems that have the ‘feasible disjunction’ such that R(x, y, wn) holds. We call an advice string wn property. However, the feasible disjunction property is not good for R if it satisfies the preceding condition, and we well understood, and it is not clear if Frege or EF have this call an advice sequence {wi} good for R if for each n, wn property. There is also work on proof complexity generators is good for R. Given x ∈ TAUT and advice string w|x|, any [6], [28], [29], [44] where fairly general conjectures are y for which R(x, y, w|x|) holds is called an R-proof of x made about unprovability of circuit lower bounds in EF, with advice w|x|. A pps R with advice is p-bounded if there but it is not known how to connect these conjectures to exists an advice sequence {wi} good for R and a polynomial complexity assumptions. It is also important to note that p such that for each x ∈ TAUT, there is an R-proof of x our formalization is different from the ones considered in with advice w|x| of size at most p(|x|). these papers - our notion of ‘feasible proofs’ aims to prove Proposition 2. NP ⊆ coNP/poly iff there exists a p- complexity separations implicitly rather than explicitly as bounded pps with polynomial advice. done using the circuit lower bound tautologies. This implicit formulation allows us to prove negative results even for Let P, Q be propositional proof systems. P simulates Q arbitrary propositional proof systems. if there is a polynomial p such that whenever there is an We focus in this paper on the setting of propositional proof s-size Q-proof of φ, there is a p(s)-size P -proof of φ. P is complexity. There is a lot of work on independence results optimal if it simulates every propositional proof system. in the setting of bounded arithmetic, which can be thought of P admits instantiation property if whenever there is an as a uniform version of propositional proof complexity. For s-size P -proof of φ(x1,...,xn), P proves each instance of PV n example, it is known unconditionally that Cook’s theory 1 φ, i.e. φ(a1,...,an) where ai ∈{0, 1} , by a proof of size cannot prove super-polynomial lower bounds on EF (which s. is the propositional proof system analogue of PV1) [13], [19], [32]. Some of our results in this paper have analogues B. Formalizing Lower Bounds in the Propositional Setting in the bounded arithmetic setting, which we plan to explore Circuit lower bounds. An s(n)-size circuit lower bound in a separate work. O n for a function f ∈Fn can be expressed by a 2 ( )-size II. PRELIMINARIES propositional formula tt(f,s), Fn denotes the set of all Boolean functions on n inputs. f(y) = C(y) A. Proof complexity y∈{0,1}n Propositional proof systems were defined by Cook and where the formula f(y) = C(y) says that a circuit C Reckhow as surjective polynomial-time computable func- represented by poly(s) variables does not output f(y) on { }∗ tions from 0, 1 to the set of tautologies (represented as input y. strings). We use an equivalent notion that is better suited to defining proof systems with advice. Let TAUT denote Definition 3. Given a Boolean function f ∈Fn and size the set of strings over the binary alphabet that encode parameter s, tt(f,s) is a propositional DNF formula of size n DNF tautologies, under some standard efficient encoding O˜(2 s3) over O˜(s) variables expressing that f does not of formulas as strings. Given this standard encoding of have Boolean circuits of size s. | | formulas as strings, for any formula φ, φ denotes the length More explicitly, tt(f,s) is defined by taking an OR over of the encoding of φ, in bits. all y ∈{0, 1}n of the predicate f(y) = C(y), and expressing Definition 1. A propositional proof system (pps) is a f(y) = C(y) as a DNF formula of size O˜(s3) over O˜(s) polynomial-time computable relation R(·, ·) such that for propositional variables using standard techniques. each ∈{ }∗, ∈ TAUT iff there exists ∈{ }∗ x 0, 1 x y 0, 1 Definition 4. Given pps R, propositional formula φ and such that R(x, y) holds. Given x ∈ TAUT, any y for size function s : N → N, lbR(φ, s) is a propositional DNF which holds is called an -proof of . A pps R(x, y) R x formula of size poly(|φ| + s) over poly(|φ| + s) variables is polynomially bounded (p-bounded) if there exists a R expressing that there is no R-proof of φ having size s. polynomial p such that for each x ∈ TAUT, there is an R-proof of x of size at most p(|x|). More explicitly, the formula lbR(φ, s) contains s variables y ,...,ys encoding R-proofs of length s and poly(|φ| + s) Proposition 1. NP = coNP iff there exists a p-bounded pps. 1 auxiliary variables encoding the computation of the relation Definition 2. Given a : N → N, a propositional proof system R, to verify that y1,...,ys does not constitute an R-proof with a bits of advice is a relation R(x, y, z) computable in of φ. polynomial time such that for each n ∈ N, there is wn ∈ We extend the above definition in a natural way to ppses {0, 1}∗ of length a(n) satisfying the following condition: with advice. In the process, we overload the notation lb: the
1312 number of parameters tells us whether we are dealing with nondeterministic variant has been conjectured by O’Donnell standard ppses, or ppses with advice. [25], [38]. We state each of these hardness assumptions in turn. Definition 5. Given pps R with advice function a : N → N, propositional formula φ, size function s : N → N and advice Rudich’s Conjecture: There is a constant c>0 string w of length a(|φ|), lbR(φ, s, w) is a propositional such that for every language L,ifL ⊆ MCSP[nc] and DNF formula of size poly(|φ| + s) over poly(|φ| + s) ω L ∈ NSIZE(poly), then L is 1/N (1)-sparse. variables expressing that there is no R-proof with advice w of φ having size s. In other words, any set in NSIZE(poly) consisting C. Rudich’s Conjecture and Nondeterministic Feige’s Hy- only of truth tables of hard Boolean functions (where pothesis ’hard’ means that the circuit complexity is greater than nk) is sub-polynomially sparse. Rudich’s conjecture was ⊆{ }∗ Given a language L 0, 1 , the n-slice Ln of L is originally stated in terms of the notion of natural proofs, ∩{ }n N → L 0, 1 . Given a function : [0, 1], we say that a but the formulation above is equivalent. ⊆{ }∗ language L 0, 1 is -sparse if for each large enough Next, we state the nondeterministic variant of Feige’s | | n ≤ n, Ln /2 (n). L is -dense if for each large enough hypothesis. Given a positive constant Δ, U ,N is the | | n Δ n, Ln /2 >(n). Note that -density is not simply the distribution over 3-CNFs on N variables obtained by negation of -sparsity. picking Δn 3-clauses independently at random, where We say that a function f : N → N is poly-constructible if n each 3-clause is picked by choosing 3 literals uniformly there is a Turing machine transducer that, on input 1 , halts and independently at random from the set of 2N literals with f(n) on its output tape within poly(n) steps. over N variables. Given a language L, L denotes the complement of L. We will be working with two fundamental NP problems Nondeterministic Feige’s Hypothesis: For any constant - MCSP (Minimum Circuit Size Problem) and 3-SAT (Sat- Δ > 0, for every language L,ifL ⊆ 3 − SAT and isfiability Problem for 3-CNFs). ∈ NSIZE L (poly), then Prφ∼UΔ,N [L(φ) = 1] = o(1). Given a string y of length N, fn(y) is the Boolean N function on log(N) bits whose truth table is the 2 log( ) - As stated, this hypothesis is about the hardness of bit initial prefix of y. Conversely, given a Boolean function proving unsatisfiability. Since we are concerned with proof f on n bits, tt(f) is the truth table of f. Given a Boolean systems for tautologies in this paper, we will simply circuit C, fC denotes the Boolean function computed by C, interprets a proof that φ is a tautology as a proof that φ and
1313 ω 1 − 1/N (1) fraction of Boolean functions fn over n = Theorem 4. [26] If there is a constant γ>0 such that E ⊆ γn log(N) bits, when N is large enough. This is equivalent io − NSIZE(2 ), then there is a complexity-theoretic 1/N - to saying that for every large enough constant d>0, there PRG with seed length O(log(N)) against nondeterministic d d are no non-uniform proofs of size m of tt(fn,n ) for most size N. d fn, since a lower bound of size n for d ≥ c implies a lower c We next define the crucial concept of R-easy hitting and bound of size n . We say that the conjecture admits feasible pseudorandom tautologies. proofs if given a fixed pps R with polynomial advice, there is a pps S that efficiently proves lower bounds on the size Definition 11. Let R be a pps, : N → R be non-negative, of the R-proofs for random truth table tautologies, thereby and s : N → N be a size function. Let C be a circuit class. We feasibly giving evidence that R does not witness a refutation say W is a set of -pseudorandom (resp. -hitting) tautolo- of Rudich’s conjecture. Note that we allow S to depend gies against C that is s-easy for R if there is a polynomial- ∗ ∗ on R, and moreover we only require S to provide efficient time computable function f : {0, 1} →{0, 1} and a ∗ proofs of lower bounds on R-proof size for 1−o(1) fraction sequence {HN }, where for large enough n, HN ⊆{0, 1} of truth-table tautologies, even though Rudich’s conjecture is an (N)-discrepancy (resp. (N)-hitting) set against C, implies more strongly that all but a negligible fraction of and moreover: truth table tautologies require large R-proofs. 1) W = N f(HN ) 2) For all but finitely many φ ∈ W , we have that φ ∈ Definition 8. We say that Nondeterministic Feige’s Hy- TAUT pothesis admits feasible propositional proofs if for every 3) Each tautology φ ∈ W has R-proofs of size s(|φ|). Δ > 0, for every pps R with polynomial advice and for ∈ TAUT − ω(1) 4) Pry∼UN [f(y) ]=1 1/N every integer d>0, there is a pps S, such that if {wm} 5) Pry∼U [f(y) has R-proofs of size s(|f(y)|)] = o(1) is a sequence of poly-size advice strings good for R, then N for all large enough N, with probability 1 − o(1) over φ On occasion, we will use the above definition for an sampled from UΔ,N , there are polynomial-sized S-proofs of R that is a pps with advice. In the default case that the d lbR(φ, m ,wm), where m = |φ|. parameter s is polynomially bounded, we will simply call the corresponding pseudorandom or hitting tautologies R- D. Pseudorandomness easy. Intuitively, the above definition captures the notion of tau- Definition 9. For fixed integers N and t and a parameter tologies that can be computed efficiently (using the function ≥ ⊆{ }N 0, we say that HN 0, 1 is an -hitting set against f) from hitting or pseudorandom sets and have short proofs ∩ ∅ size t (resp. nondeterministic size t)ifHN S = for every in R, while if f is applied to random strings, the resulting ⊆{ }N -dense S 0, 1 computable by circuits (resp. nondeter- formulas are tautologies with very high probability but lack ⊆{ }N ministic circuits) of size t. We say that HN 0, 1 is an short R-proofs with significant probability. -discrepancy set against size t if for every circuit C of size | − |≤ E. Kolmogorov Complexity t on N bits, Prx∼UN [C(x)=1] Pry∈HN [C(y)=1] . Given an integer s, we say that an -hitting set (resp. an KT-complexity was proposed in [4], as a variant of -discrepancy set) HN is s-succinct if for each y ∈ HN , Levin’s notion of time-bounded Kolmogorov complexity fn(y) has circuits of size at most s. that is closely connected to circuit complexity. Indeed, it is known that KT(tt(f)) and the minimum circuit size of f are N → N N → N Proposition 3. Let s : and t : be size polynomially-related to each other. Fix a universal random- N → R functions, and let : be non-negative. For any access Turing machine U that simulates all Turing machines ∈ N N , there are s(n)-succinct (N)-hitting sets over N- efficiently. Informally, the KT-complexity of a string y is MCSP bit strings against non-deterministic size t(N) iff [s] the minimum of |d| + t, where d is a string for describing has no (N)-dense subsets in non-deterministic size t(N). y implicitly and t is the time it takes to output y. More d Definition 10. Let : N → N satisfy (N) ≤ N for each formally, we have the definition below, where U denotes n, and let t : N → N be a size function. Let : N → R be the Turing machine U with random access to the string d: non-negative. We say that a sequence of functions { N }, N G Definition 12. Let y = y1 ···yN ∈{0, 1} . The KT- { }(N) →{ }N where each GN : 0, 1 0, 1 is a complexity- complexity of y is defined as follows. theoretic -PRG with seed length (N) against size t (resp. d nondeterministic size s) if there is a Turing machine which KT(y):=min{|d|+t | U (i)=yi in t steps for any 1 ≤ i ≤ N +1}. N ∈{ }(N) given 1 and z 0, 1 computes GN (z) in time Here, yN+1 is defined as ⊥ (a stop symbol). 2O((N)), and moreover for each large enough N, the range We have the following simple facts about KT- complexity, of GN is an (N)-discrepancy set against size t(N) (resp. non-deterministic size t(N)). in analogy to corresponding facts about the standard notion of Kolmogorov complexity.
1314 Proposition 4. 1) For any string y, KT(y) ≤|y| + We next define certain formulas expressing that a string O(log(|y|)) is random in the sense of KT-complexity. 2) Given any non-negative integer r, for each n, for Definition 14. Given a string of length and size − r y N at least 1 1/2 fraction of strings y of length n, parameter , random is a propositional DNF formula KT ≥| |− s (y, s) (y) y r. of size O˜(Ns3) over O˜(s) variables expressing that y does The first item follows by using y as its own description, not have KT-complexity at most s. and the second item follows from a straightforward counting More explicitly, random(y, s) is defined by taking an OR argument. d over all i ∈ [N +1]of the predicate yi = Us−|d|(i), where The following computational problem is naturally associ- d KT Ut (i) denotes the simulation of U on i for t steps, and ated with -complexity. d 3 expressing yi = Us−|d|(i) as a DNF formula of size O˜(s ) Definition 13. Given a size function s : N → N, MKTP[s] over O˜(s) propositional variables using standard techniques. is the set of strings y such that KT(y) ≤ s(|y|). Definition 15. Let s : N → N be such that s(N) ≤ N for all Note that for any poly-constructible s, MKTP[s] is in N. We say that the MKTP[s] Hardness Hypothesis admits NP, simply by guessing a string d and a number t such that feasible propositional proofs if for every large enough inte- d |d| + t ≤ s(N), and checking that U (i)=yi in t steps for ger d>0 and every pps R with polynomial advice, there is ≤ ≤ each 1 i N +1. Note also that unlike in the case of a pps S, such that if {wm} is a sequence of poly-sized advice MCSP, we measure s as a function of the length N of the strings good for R, then for all large enough N, for a 1 − input string. o(1) fraction of strings y of length N, there are polynomial- KT d The following lemma bounds the complexity of sized S-proofs of lbR(random(y, s(N)),m ,wm), where satisfiable 3-CNFs. m = |random(y, s(N))|.
Lemma 1. Let {φn} be any sequence of satisfiable 3-CNFs, III. IMPOSSIBILITY OF FEASIBLY PROVING RUDICH’S such that |φn| = n for each n. Then there is a constant δ>0 CONJECTURE KT ≤ − such that for large enough n, (φn) (1 δ)n. Lemma 2. If Rudich’s Conjecture holds, then there is a We consider a family of hypotheses parameterized by pps R such that for any k, there is a collection of R-easy k k a size function s : N → N such that s(N)=N −ω(log(N)). 1/N -hitting tautologies against nondeterministic size N . Proof: Suppose Rudich’s Conjecture holds. Then there MKTP ⊆ MKTP [s] Hardness Hypothesis: If L [s] is a constant c such that any NSIZE(poly) subset of ∈ NSIZE and L (poly), then L is o(1)-sparse. MCSP[nc] is 1/N ω(1)-sparse. We assume wlog that c>1. Since MCSP[nc] contains almost all strings of length N for ≤ MKTP For s s , it is obvious that the [s] Hardness any N>0, it follows that MCSP[nc] does not have circuits MKTP Hypothesis implies the [s ] Hardness Hypothesis, of polynomial size almost everywhere. thus the hypothesis is more believable for larger s. In fact, { } ω We define a sequence zN of N-bit strings as follows. If for any s = log(N) (1), Rudich’s Conjecture implies the n N =2 for some non-negative integer n, then zN is defined MKTP corresponding hardness hypothesis for . to be the truth table of MCSP[mc] on inputs of length n.If N ω(1) N−2 log( ) Proposition 5. Let s(N)=log(N) be any size func- not, then zN = z2 log(N) 0 . By the observation tion. Rudich’s Conjecture implies the MKTP[s] Hardness in the previous para, we have that for any polynomially Hypothesis. bounded function t, for all large enough N, fn(zN ) does not have circuits of size t(n). Proof: Follows immediately from the standard fact that Next we use the strings {zN } to define a sequence of for any string x, KT(x) is upper bounded by a fixed hitting sets. Since Rudich’s Conjecture holds, by Proposition polynomial in the circuit size of . fn(x) 3, we have that for any k and each large enough N, there c k n is a n -succinct 1/N -hitting set HN ⊆{0, 1} against The following proposition, giving a connection between k nondeterministic size N . the MKTP[s] Hardness Hypothesis for a given s and hit- Consider the sequence of sets {HN } defined by HN = ting sets composed of strings with low KT-complexity, is {y ⊕ zN |y ∈ HN } for each N. We claim that {HN } is also completely analogous to Proposition 3. a sequence of 1/N k-hitting sets against nondeterministic k Proposition 6. Let s : N → N and t : N → N be size size N . Indeed, if not, for infinitely many N, there must k N k functions, and let : N → R be non-negative. For any N ∈ be a 1/N -dense subset SN of {0, 1} with N size N, there is an (N)-hitting set HN ⊆ MKTP[s] over N-bit nondeterministic circuits such that SN ∩HN = ∅. Define the strings against non-deterministic size t(N) iff MKTP[s] has subsets SN = y ⊕ zN |y ∈ SN . It is clear that for each N k no (N)-dense subsets in non-deterministic size t(N). such that SN has N size nondeterministic circuits, so must
1315 SN , simply by negating the i’th input variable in the circuit have that u ∈ TAUT, and hence this case is sound. In case for SN iff zi =1. Also, SN has the same density as SN ,as (b), u is accepted only if it is a truth-table formula expressing c it is just a linear translate of SN . Now, by definition of HN , that fn(zN ⊕ y) does not have circuits of size n , the proof c we have that HN ∩SN = ∅ iff HN ∩SN = ∅. Thus we have encodes a circuit of n accepting fn(y), and moreover N ≥ k c that for infinitely many N, there is an 1/N -dense subset N0 satisfies that fn(zN ) does not have circuits of size 2n + k SN of N-bit strings with N size nondeterministic circuits 3. It is clear that in this case, u is accepted only if it is a such that SN ∩ HN = ∅, contradicting the assumption that tautology, since the upper bound on the circuit complexity k k HN is an 1/N -hitting set against nondeterministic size N of fn(y) witnessed by C and the lower bound on the circuit for all large enough N. complexity of fn(zN ) guaranteed by N being large enough k Now we define our candidate collection of 1/N -hitting together imply that fn(zN ⊕y) does not have circuits of size tautologies against nondeterministic size N k, and a pps nc. R such that these hitting tautologies all have short proofs To show that R is polynomial-time computable, define a in R. We argue that if Rudich’s Conjecture holds, all the poly-time Turing machine M that decides R by operating conditions in Definition 11 are satisfied, and therefore our as follows. On input (u, v), it first checks if v begins with |u| candidate collection is indeed hitting. a 0. If so, it first checks that v =012 . If not, it rejects. If Define the poly-time computable function f : {0, 1}∗ → ∗ c yes, it uses brute force search over all possible assignments {0, 1} by f(x)=tt(fn(x), log(|x|) ). Let W = k to u to check that u is a tautology. If this brute force search N f(HN ). W is our candidate set of 1/N -hitting tau- k succeeds, it accepts, otherwise it rejects. If v begins with 1, tologies against nondeterministic size N . M checks that v =1
1316 that L ∈ NP. By assumption, there are infinitely many N 2 to derive a contradiction. As in the proof of Lemma 2, for which the fraction of strings y of length N in L is at let {HN } be the sequence of hitting sets associated with least γ, hence L is not 1/N ω(1)-sparse. Contradiction. W , and let f(x)=tt(fn(x), log(|x|) c) be the poly- time computable function associated with W . Since W is k k The pps R given by the proof of Lemma 2 does not have a 1/N -hitting against nondeterministic size N , and LS is a natural form, but as we observe in Section V, we can take R 1 − o(1)-dense set computable by nondeterministic circuits to be Extended Frege with circuit lower bound axioms, and of size at most N k, there must be an infinite sequence of this has implications our ability to show proof complexity strings {yN }, where each yN ∈ HN , such that for large lower bounds for Extended Frege. enough N, yN ∈ LS. But since W is R-easy, we have that c for large enough N, lbR(f(yN ), |f(yN )| ) is false, and by Lemma 3. If Rudich’s Conjecture holds, then Rudich’s soundness of S, it does not have polynomial-size S-proofs, Conjecture does not admit feasible propositional proofs. or indeed S-proofs of any size. This contradicts the inference Proof: By Lemma 2, under the assumption that that yN is in LS for large enough S, since for each large Rudich’s Conjecture holds, there is a pps R such that for enough string z ∈ LS, there are polynomial-sized S-proofs k c any k, there is a collection W of R-easy 1/N -hitting of lbR(f(z), |f(z)| ). tautologies against nondeterministic size N k. Assume, for the sake of contradiction, that Rudich’s The following is a restatement of Theorem 1. Conjecture admits feasible propositional proofs. This means Theorem 5. Rudich’s Conjecture does not admit feasible that for each pps R and integer d>0, there is a pps S, propositional proofs. such that for all large enough N, for a 1 − o(1) fraction of strings y of length N, there are polynomial-sized S-proofs Proof: Suppose Rudich’s Conjecture holds. Then d d d of lbR(tt(fn(y), log(N) ),m ), where m = |tt(fn,n )|. Lemma 3 gives us the desired conclusion. Consider the pps S that corresponds to the pps R from Now suppose Rudich’s Conjecture fails, and that Rudich’s Lemma 2, with d chosen to be the constant c from Rudich’s Conjecture admits feasible propositional proofs. We show Conjecture, as in the proof of Lemma 2. We have that for that the failure of Rudich’s conjecture implies that there large enough N, tautologies from W of size N have R- exists a pps R with polynomially bounded advice such that proofs of size at most N, and hence of size at most N c, for each d>0 and infinitely many n, for an Ω(1) fraction assuming wlog (as in the proof of Lemma 2) that c>1.On of Boolean functions f ∈Fn, there are polynomial-size R- the other hand, even though f(y) is a tautology with all but proofs of tt(f,nd). We then show how to use the pps R with negligible probability for y a randomly chosen N-bit string, advice to contradict the assumption that Rudich’s Conjecture it follows from the fact that W is a collection of hitting admits feasible propositional proofs. tautologies that there are N c-size R-proofs for f(y) with If Rudich’s Conjecture fails, then for all constants c>0, probability o(1). there is a constant a>0 and a language Lc such that Lc ⊆ c a Now we use the assumption about S to contradict the MCSP[n ] and Lc ∈ NSIZE(poly),butLc is 1/N -dense hitting property of W . Let b be a constant such that for a for infinitely many input lengths N. We show that in this 1 − o(1) fraction of strings y of length N, there are rb- case, it is also true that for all constants c > 0, there is a c c c size S-proofs of lbR(tt(fn(y), log(N) ),m ), where m = constant γ>0 and a language L such that L ⊆ MCSP[n ] c c c |tt(fn,n )|, and r = |lbR(tt(fn(y), log(N) ),m )|. Let a and L ∈ NSIZE(poly),butL is γ-dense for infinitely many be a constant such that rb
1317 Next we argue that L is γ-dense for infinitely many N , for R, then for all large enough n, for a 1 − o(1) fraction of for some constant γ>0. Indeed, for any large enough N Boolean functions fn ∈Fn, there are polynomial-sized S- a+1 a 2 2 d of the form N such that Lc +1 is 1/N -dense on inputs proofs of lbR(tt(fn,n ),m ,wm), where m = |tt(fn,n )|. of length N, we have that the fraction of strings y of length Let R be the pps defined above. Then by the soundness a N in L is at least 1 − (1 − 1/N a)N ≥ 1 − 1/e. This is of the corresponding pps S given by the assumption that because y ∈ L iff some block yi ∈ L, and each block yi Rudich’s Conjecture admits feasible propositional proofs, it independently has probability at least 1/N a of being in L. follows that for all large enough n, for a 1 − o(1) fraction Finally we argue that L ∈ NSIZE(poly). Indeed, this of Boolean functions fn ∈Fn, lbR(tt(fn,n2),m2,wm) is follows directly from the fact that L ∈ NSIZE(poly), and true, and hence for all large enough N, for a 1 − o(1) that deciding membership of y ∈ L reduces to breaking y fraction of strings u of length N, tt(fn(u),n2) does not have up into blocks yi and checking if at least one yi is in L. quadratic size R-proofs with good advice wm, contradicting Indeed, if L has nondeterministic circuits of size at most the statement at the end of the previous para. N q for some constant q, L has nondeterministic circuits of q/a size at most (N )1+ . In fact, the proof of Theorem 5 gives that Rudich’s Thus we have established our claim about the existence Conjecture does not even admit feasible non-uniform propo- of L corresponding to each constant c > 0. We fix such an sitional proofs, i.e., the impossibility result extends to ppses q L as follows: choose a c>3 for which Lc ∈ NSIZE(N ) S that use polynomial advice. a for some q, and assume wlog that Lc is 1/N -dense Lemma 3 implies that there is a proof complexity- for some a ≥ q. Let L be the language given by the theoretic analogue of the Razborov-Rudich ”natural proofs” above construction corresponding to c = c − 1. Then we barrier for some pps R under Rudich’s conjecture. Namely, c− have that L ⊆ MCSP[n 1] and L has nondeterministic there is a samplable sequence of distributions under which a circuits of size at most (N )2. Moreover L is 1 − 1/e- random formula is a tautology that requires large R-proofs, dense for infinitely many input lengths N . Since L has yet no pps S can prove this efficiently. nondeterministic circuits of size at most (N )2, there is The following is a restatement of Corollary 1. a nondeterministic machine M running in time at most (N )2 polylog(N ) and using at most (N )2 polylog(N) bits Corollary 2. If Rudich’s Conjecture holds, then there is { } of advice deciding L. a pps R and a samplable sequence of distributions DN Now we define a pps R with advice corresponding to L. on formulas of length poly(N), such that with probability − ω(1) R has at most N 2 bits of advice, and is defined as follows: 1 1/N over φN sampled from DN , φN is a tautology |x| R(x, y, z)=1iff (a) y =12 and x ∈ TAUT, or (b) if that does not have poly(N) size R-proofs, but there is a x is of the form tt(fn(u),n2) for some string u such that constant k such that no pps S has polynomial-size proofs of lb | |k n = log(|u|) and M accepts u with witness y and advice R(φN , φN ). z. Proof: This follows from Lemma 3, by using the same Let {wm} be a good sequence of advice strings cor- pps R as in the proof of the Lemma, and the samplable responding to the correct advice for the nondeterministic sequence of distributions {DN } defined by choosing fn ∈ { } k machine M. We claim that R is a pps such that wm is Fn at random and outputting the formula tt(fn,n ). good for R. Indeed, any tautology φ has a trivial truth-table proof in R, even without advice. To argue soundness, note IV. IMPLAUSIBILITY OF FEASIBLY PROVING that in case (b), if M accepts u with witness y and correct NONDETERMINISTIC FEIGE’S HYPOTHESIS ∈ advice z, then u L and therefore fn(u) does not have The following lemma giving an average-case reduction c−1 ≥ 2 tt 2 circuits of size n n . Hence (fn(u),n ) is indeed a from SAT to MCSP is based on Theorem 38 in [25], but tautology. the proof is slightly different, as we need to be careful about We claim that for infinitely many N, for at least a fraction the output length of the reduction. at least 1−1/e of strings u of length N, tt(fn(u),n2) has R- proofs of size at most 2, where |tt 2 . Indeed, Lemma 4. For any large enough integer Δ > 0, there is a m m = (fn(u),n ) ∗ ∗ L is 1−1/e-dense for infinitely many N, and for each such polynomial-time computable function g : {0, 1} →{0, 1} N, the nondeterministic machine M accepts u with some satisfying the following conditions whenever N is large witness y and good advice wm in quadratic time, giving enough and a power of 2: quadratic size R-proofs of tt(fn(u),n2), as per definition of 1) For any 3-CNF φ on N variables with ΔN clauses, the pps R. |g(φ)| =3ΔN(log(2N)) Now we use the assumption that Rudich’s Conjecture 2) g is poly-time invertible admits feasible propositional proofs. This implies that for 3) For at least a (1−oN (1)) fraction of strings y of length every pps R with polynomial advice, there is a pps S, such KN, y = g(φ) for some 3-CNF φ on N variables with that if {wm} is a sequence of poly-sized advice strings good ΔN clauses
1318 4) If φ is a satisfiable 3-CNF on N variables with ΔN string for R at length m. clauses, then KT(g(φ)) ≤|g(φ)|−ΔN/12 We show that for every pps R with polynomial advice and every d>0, there is a pps S such that for all Proof: When N is a power of 2, picking a formula large enough N, with probability 1 − o(1) over y ∼ UN , φ from U ,N corresponds to picking a string zφ of length Δ there are polynomial-size S-proofs of lbR(random(y, N − 3ΔN(log(2N)) at random - imagine z to be made up of d γN/log(N)),m ,wm), where m = |random(y, N − ΔN equal size blocks each of length 3log(2N), and each γN/log(N))| and wm is a good advice string for R at length block to represent a sequence of 3 literals, each described m. by log(2N) bits. We simply set g(φ)=zφ. The first con- Given a string y of length N, define trunc(y) as the dition is obviously satisfied. The second condition follows largest prefix of y of length 3ΔN log(2N ) for some N from the bijection between 3-CNFs over N variables with that is a power of 2. Note that |trunc(y)|≥|y|/4. ΔN clauses and strings of length 3ΔN(log(2N) described Fix a pps R with good advice {wm} satisfying the above. The third condition follows from the simple form of criterion in the previous para. We define a pps R with the bijection. polynomial advice as follows. R(u, v, w)=1iff either (a) For the fourth condition, we show that for any satisfiable random − v =
1319 random(y, N − γN/log(N) has md-size R-proofs for large As in the proof of Lemma 2, the hardness hypothesis is enough N, then φ has cd-size R-proofs for large enough used to construct a collection of hitting tautologies. For each = |φ| when trunc(y)=g(φ). >0 and each positive integer k, we will get a collection We want to show that for every pps R with advice W of R-easy -hitting tautologies against nondeterministic k and every d>0, there is a pps S such that for all size N for some pps R, as follows. MKTP large enough N, with probability 1 − o(1) over y ∼ UN , Using the [δN/log(N)] hardness hypothesis in there are polynomial-size S-proofs of lbR(random(y, N − Proposition 6, for each >0 and positive integer k>0, d { } γN/log(N)),m ,wm), where m = |random(y, N − we get a sequence HN of -hitting sets supported on N- KT γN/log(N))| and wm is a good advice string for R at bit strings of -complexity at most δN/log(N) against k length m. Let S be a pps such that for all large enough n, nondeterministic size N . { } with probability 1 − o(1) over φ sampled from UΔ,n, there We define a sequence zN of N-bit strings as follows: cd are polynomial-size S -proofs of lbR (φ, ,w), where zN is the lexicographically first N-bit string such that = |φ|. KT(zN ) ≥ N −1. The existence of zN for every N follows We define S based on S as follows. S(u, v)=1iff from Proposition 4. Now we define a new sequence {HN } of d k either (a) u = lbR(random(y, N − γN/log(N)),m ,wm) -hitting sets against nondeterministic size N by including for some y of length N and m = |random(y, N − y of length N in HN iff y ⊕ zN is in HN . The argument γN/log(N))|, and moreover v is an S -proof of that HN is a hitting set is exactly the same as in the proof − cd lbR (g 1(trunc(y)), ,w) for some w that contains wm of Lemma 2. ∗ as a substring in the appropriate location (note that a good Define the poly-time computable function f : {0, 1} → ∗ advice string w for R is the concatenation of all good advice {0, 1} by f(x)=random(x, N − γN/log(N). Let W = strings wm form=|random(y, N − γN/log(N))|, where N f(HN ). W is our candidate set of -hitting tautologies |u| k trunc(y)=g(φ)), or (b) v =12 and u is a tautology. against nondeterministic size N . Completeness of S follows immediately from item (b) of Condition (i) in Definition 11 is satisfied trivially. Con- the definition of S. Soundness follows the fact that if R does dition (ii) follows from the fact that for strings x and x , not have cd-size proofs that φ is a tautology, then R does not KT(x ⊕ x ) ≤ KT(x)+KT(x )+O(log(|x|)). Condition have md-size proofs that y = g(φ)y is KT-random for any (iv) follows from the second item of Proposition 4. y such that g(φ)=trunc(y). Polynomial-time verifiability We next define a pps R with polynomial advice such that of S is clear from the definition. W is R-easy. R is defined exactly as in the proof of Lemma We still need to argue that for all large enough N, with 2, except that linear advice is needed to encode zN . The probability 1 − o(1) over y ∼ UN , there are polynomial- proof that R is a pps with advice is along the same lines of d size S-proofs of lbR(random(y, N −γN/log(N)),m ,wm) before. where wm is good advice for R.Wehavebyas- To argue condition (iii) in Definition 11, note that any sumption that for all large enough n (and in partic- tautology in W of the form random(y⊕zN ,N−γN/log(N) ular for n a power of two), with probability 1 − for y ∈ HN has a short proof given by the program KT o(1) over φ chosen from Un,Δ, there are polynomial- witnessing that y has -complexity at most δN/log(N). cd MKTP − size S -proofs of lbR (φ, ,w). But this follows by Regarding condition (v), this follows from the [N item (a) of the definition of S since the polynomial- γN/log(N)] Hypothesis exactly as condition (v) in the − cd size S -proofs v of lbR (g 1(trunc(y)), ,w) also func- proof of Lemma 2 follows from Rudich’s Conjecture. Now MKTP tion as polynomial-size S-proofs of lbR(random(y, N − note that the [δN]/ log(N) Hypothesis implies the d MKTP − γN/log(N)),m ,wm), and by the third item of Lemma [N γN/log(N)] Hypothesis for any constants δ 4, this happens for at least a 1 − o(1) fraction of strings y and γ. of length N for N large enough. The way that the set W of -hitting tautologies is used to argue that there are no feasible proofs of MKTP[N − Theorem 6. For each 1/2 >γ>0, there is δ>0 such γN/log(N)] Hardness Hypothesis is exactly analogous to MKTP that if the [δN/log(N)] hardness hypothesis holds, the proof of Lemma 3. then the MKTP[N −γN/log(N)] hardness hypothesis does not admit feasible propositional proofs. The following is a restatement of Theorem 2. Proof: The proof is analogous to the proof of Lemma 3, Corollary 3. Under Rudich’s Conjecture, Nondeterministic and how it builds on Lemma 2. Hence, rather than giving all Feige’s Hypothesis does not have feasible propositional details, we explain the structure of the proof, and describe proofs. the aspects in which the proof differs from the proofs of Lemma 2 and Lemma 3. Proof: Suppose, for the sake of contradiction, that Non- Given γ>0, we choose δ>0 to be any constant less deterministic Feige’s Hypothesis has feasible propositional than γ. proofs. By Lemma 5, there is a constant γ>0 such
1320 that the MKTP[N − γN/log(N)] Hardness Hypothesis has 2) There is no pps Q such that there are polynomial- k k feasible propositional proofs. By Proposition 5, Rudich’s size Q-proofs of lbEF(tt(fn,n ),m ) (where m = k O n Conjecture implies the MKTP[δN/log(N)] Hardness Hy- |tt(fn,n )|) for a 1/2 ( ) fraction of Boolean func- pothesis for every δ>0. Now by applying Lemma 5 for tions fn for all constants k and all large enough n. the δ>0 in the statement of the Lemma, we have that Proof: Suppose that the first item is false, and that there the MKTP[N −γN/log(N)] Hardness Hypothesis does not is a sequence of Boolean functions {fn},fn ∈Fn, such that have feasible propositional proofs. Contradiction. k tt(fn,n ) has polynomial-size EF proofs for every k>0. V. A L OSE-LOSE THEOREM FOR EXTENDED FREGE Assume Rudich’s Conjecture. Then, using the falsehood of k Lemma 6. Assume there is an s-size EF-proof of tt(fn, 3n ) the first item and Lemma 6, the pps R in the statement EF for some k and sufficiently large n. Further, let gn be a of Lemma 2 can be taken to be . It then follows that EF function computable by a circuit of size nk. Then there is a Corollary 2 holds with R = , and this implies the second n k 2s + K2 -size EF-proof of tt(fn ⊕ gn,n ) where K is an item. absolute constant. The following is a more rigorous formulation of Theorem Proof: We will use the Extended Resolution proof 3. system to argue the lemma instead, taking advantage of the standard fact that Extended Resolution and Extended Frege Theorem 8. Assume that Rudich’s Conjecture holds, and n simulate each other [18]. that there is an >0 such that E ⊆io − NSIZE(2 ). Then Suppose a set of clauses S0 expresses C0 = fn ⊕gn for a at least one of the following is true: k k circuit C0 of size n , i.e. S0 is a negation of tt(fn ⊕gn,n ), 1) There is no sequence of Boolean functions {fn},fn ∈ k and a set of clauses S1 expresses C1 = gn where C1 is a Fn, such that tt(fn,n ) has polynomial-size EF- k circuit of size n coded by variables not occurring in S0. proofs for every k>0 a Let ui be the variable encoding the output of Ci on input 2) For each k>0 and each pps Q, there is a polynomial- a a. Further, for each n-bit string a let y be a new variable, time algorithm that on input 1N (where N =2n), and let Y be a set of clauses expressing that outputs a set SN of truth tables of Boolean functions a a ⊕ a on n bits such that for each large enough N, there y = u0 u1. k is F ∈ SN such that tt(fn(F ),n ) is a tautology, The set Y can contain extra variables that help to encode there are no N k-size EF-proofs of tt(fn(F ),nk), and a ⊕ a lb tt k k u0 u1. moreover EF( (fn(F ),n ),N ) does not have Q- k2 From S0 ∪ S1 ∪ Y we can derive a set of clauses S⊕ proofs of size N . expressing C0 ⊕ C1 = fn where C0 ⊕ C1 is a circuit of size k a n Proof: The basic idea is to derandomize the hard 3n outputting y on input a. This derivation takes K2 tautologies in Theorem 7 by using the assumption that steps, for an absolute constant K. Specifically, for each a,it E ⊆io − NSIZE n a ⊕ a a a ⊕ (2 ) in Theorem 4. Assume the first item derives fn(a)=u0 u1 = y from u0 = fn(a) gn(a) and a k is false, and let d be a constant such that the pps Q(x, y) is u = gn(a). Since we assume that we can refute tt(fn, 3n ) 1 verifiable in time (|x + y|)d. in size s and substituting some variables for constants does By a standard counting argument, we have that for most not increase the refutation size, we can refute S⊕ in size s, k n F of size N, when N is large enough, tt(fn(F ),n ) is a and S0 ∪S1 ∪Y in size s+K2 . We can now substitute the k n tautology. Since Rudich’s Conjecture is true, we have that n -size circuit computing gn in S1 and refute in size s+K2 for most F of size N, when N is large enough, there are the set of clauses S0 ∪ Y where Y is obtained from Y by k k no N -size EF proofs of tt(fn(F ),n ). By the falsity of substituting the circuit for gn. Note that the clauses in Y a a a a the first item, and applying Theorem 7, we have that for say just that y = u or y = ¬u . We can thus get rid of 0 0 most F of size N, there are no polynomial-size Q-proofs them with a substitution of literals which at most doubles k k k n of lbEF(tt(fn,n ),m ) (where m = |tt(fn,n )|), when N the resulting refutation size. This yields a 2s +2K2 -size is large enough. refutation of S . 0 We use the PRG given by Theorem 4 to simultaneously In fact, Lemma 6 holds even with ‘Extended Frege’ derandomize all three of these conditions, using the fact replaced by ‘Resolution’, using essentially the same proof. that each of them can be tested, on input F ,byafixed polynomial-size co-nondeterministic circuit. Indeed, the first Theorem 7. Assuming Rudich’s Conjecture, at least one of condition can be tested by a co-nondeterministic circuit of the following is true: size O˜(Nnk) which simply guesses a circuit of size nk and 1) There is no sequence of Boolean functions {fn},fn ∈ verifies that it does not compute F correctly. The second k Fn, such that tt(fn,n ) has polynomial-size EF- condition can be tested by a co-nondeterministic circuit that k proofs for every k>0 uses the verification algorithm for EF and has size O˜(N ).
1321 The third condition can be tested by a co-nondeterministic so called reflection principle for Q, cf. [31, Theorem 8.4.3.]. circuit that uses the verification algorithm for Q and has size Specifically, RefQ(π, φ, y) is a tautology with free variables 2 O˜(N dk ). encoding π, φ, y which states that if π is a Q-proof of a Applying Theorem 4, there is a complexity-theoretic formula φ, then φ is satisfied by y. 2 1/N 2dk -PRG with seed length O(log(N)) against nonde- Therefore, it suffices to show that for every system Q, dk2 terministic size N 2 . Let this PRG be {GN } and let SN proof system P admits p-size proofs of RefQ(π, φm,y) for be the range of GN 2dk2 . Clearly SN has size poly(N), and infinitely many m. If this was not the case, then for every k by a union bound, most F in SN satisfy all three conditions and for all sufficiently big n denoting the length of RefQ, k above, which proves the theorem. formula lbP (RefQ,n ) would be a tautology. Moreover, there would be a p-time algorithm A which given n outputs VI. THE FEASIBLE COOK-RECKHOW PROGRAM AND the tautology. This yields a contradiction with the Failure OPTIMALITY of Feasible Cook-Reckhow Program because A could be We observe that if we reason inside a proof system P , used to define an extension of EF admitting p-size proofs of k it is impossible to prove a lower bound on a stronger lbP (RefQ,n ) for every sufficiently big n. proof system Q. This is a straightforward consequence of a seminal work of Kraj´ıcekˇ and Pudlak´ [34]. We also explore consequences of switching the order of quantifiers in the hypothesis above about failure of the Theorem 9 (es. Kraj´ıcekˇ and Pudlak´ [34]). For any propo- Feasible Cook-Reckhow Program. sitional proof systems P, Q simulating EF, P does not n admit p-size proofs of lbQ(φn,nlog ) for any sequence of Definition 17 (Optimal proof system). A proof system P is propositional formulas φn of length n unless every sequence optimal if every proof system Q is simulated by P , i.e. there O n of tautologies with p-size Q-proofs admits n (log )-size P - is a polynomial p such that for every sequence of tautologies proofs. φn with Q-proofs of length sn, each φns has P -proofs of length p(sn). Proof sketch: Let ψn be a sequence of tautologies n with p-size Q-proofs. If P proves efficiently lbQ(φn,nlog ) Theorem 11. If optimal proof systems do not exist, then for O n for some φn , P proves ψn in size n (log ). The P -proof every proof system Q, there exists a proof system P such that proceeds as follows. If ¬ψn(a) held for some a, it could be for every sequence of tautologies φn there exists a constant k combined with a p-size Q-proof of ψn to get a contradiction k such that Q does not admit p-size proofs of lbP (φn,n ) and a p-size Q-proof of φn. for every sufficiently big n. Proof: Let Q be an arbitrary proof system and Q be We consider consequences of the failure of the Feasible a proof system simulating both Q and EF. Since Q is not Cook-Reckhow Program in a strong sense. This can be optimal, there is a proof system P such that Q does not linked to the existence of a certain variant of optimal proof have p-size proofs of RefP . By an argument similar to the systems. proof of Theorem 9, this implies that for every sequence Definition 16 (i.o. optimal proof system). A proof system P of tautologies φn there is a constant k such that Q (and is i.o. optimal if every proof system Q is i.o. simulated by hence also Q) does not have p-size proofs of formulas k P , i.e. there is a polynomial p such that for every sequence lbP (φn,n ). of tautologies φn with Q-proofs of length sn, infinitely many φns admit P -proofs of length p(sn). ACKNOWLEDGMENTS Failure of Feasible Cook-Reckhow Program. There exists We thank Udi Hrushovski, Igor Oliveira and Iddo a proof system P such that for every proof system Q, every Tzameret for useful conversations. Anonymous reviewers sequence of tautologies φn there exists a constant k such lb k provided helpful feedback on an earlier version of this that Q does not admit p-size proofs of P (φn,n ) for every manuscript. sufficiently big n. Both authors were supported in part by the European Re- Theorem 10. Failure of the Feasible Cook-Reckhow Pro- search Council under the European Union’s Seventh Frame- gram implies the existence of an i.o. optimal proof system. work Programme (FP7/ 2007-2014)/ERC Grant Agreement No. 615075. The second author was supported in part by Proof: Assume there is a proof system P that witnesses Grant 19-05497S of GA CR.ˇ the failure of the Feasible Cook-Reckhow Program. We will show that P i.o. simulates every proof system. W.l.o.g. P REFERENCES simulates EF since otherwise we could consider an extension of P by EF. It is known that every proof system Q can be [1] Aaronson S.; Is P Versus NP Formally Independent, Bulletin of the European Association of Theoretical Computer Science simulated by EF extended with axioms RefQ encoding the (EATCS), 81, pp. 109-136, 2003.
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