Consistency of circuit lower bounds with bounded theories
aa r X i v : . [ c s . CC ] F e b Consistency of circuit lower bounds with bounded theories
Jan Bydˇzovsk´y ∗ Jan Kraj´ıˇcek † Igor C. Oliveira ‡ February 25, 2020
Abstract
Proving that there are problems in P NP that require boolean circuits of super-linear sizeis a major frontier in complexity theory. While such lower bounds are known for largercomplexity classes, existing results only show that the corresponding problems are hardon infinitely many input lengths . For instance, proving almost-everywhere circuit lowerbounds is open even for problems in MAEXP . Giving the notorious difficulty of provinglower bounds that hold for all large input lengths, we ask the following question:
Can we show that a large set of techniques cannot prove that NP is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations , we investigate circuit complexity from the perspective of logic.Among other results, we prove that for any parameter k ≥ T that computational class C 6⊆ i.o.
SIZE( n k ), where ( T, C ) is one of the pairs: T = T and C = P NP , T = S and C = NP , T = PV and C = P .In other words, these theories cannot establish infinitely often circuit upper bounds forthe corresponding problems. This is of interest because the weaker theory PV alreadyformalizes sophisticated arguments, such as a proof of the PCP Theorem [Pic15b]. Theseconsistency statements are unconditional and improve on earlier theorems of [KO17] and[BM18] on the consistency of lower bounds with PV . ∗ Institute of Discrete Mathematics and Geometry, Technische Universit¨at Wien – [email protected] † Faculty of Mathematics and Physics, Charles University in Prague – [email protected] ‡ Department of Computer Science, University of Warwick – [email protected] Introduction
Understanding the computational power of polynomial size boolean circuits is one of themost mysterious questions in computer science. Despite major efforts to address this prob-lem and significant progress in several restricted settings (e.g. [Mul99, Ros10, MW18]), it isconsistent with current knowledge that every problem in NP can be computed by circuits con-taining no more than 4 n gates [FGHK16]. This bound is much weaker than the lower boundresults conjectured by most (but not all) researchers in the field. For instance, it is reason-able to expect that computing k -clique on n -vertex graphs requires circuits of size n Ω( k ) , butwe appear to be very far from establishing a result of this form for unrestricted boolean circuits. Fixed-polynomial size circuit lower bounds.
Given the difficulty of proving strongerlower bounds for problems in NP , a natural research direction is to investigate super-linear andfixed-polynomial circuit size lower bounds for problems in larger complexity classes. This lineof work was started by Kannan [Kan82], who showed that for each k ≥ p ∩ Π p that cannot be computed by circuits of size n k . The result was subsequentlyimproved by K¨obler and Watanabe [KW98], who obtained the same lower bound for the class ZPP NP ⊆ Σ p ∩ Π p , and by Cai [Cai07], who showed it for S p ⊆ ZPP NP . Two incomparableresults were then obtained by Vinodchandran [Vin05] and Santhanam [San09], who provedthat PP * SIZE [ n k ] and MA / * SIZE [ n k ], respectively. Modulo the use of a single bit of advice on each input length, Santhanam’s lower boundis known to imply all aforementioned results. Unfortunately, there exist barriers to adaptinghis techniques to prove super-linear lower bounds for smaller classes such as NP , as explainedby Aaronson and Wigderson [AW09]. Establishing such lower bounds is also open for P NP ,and constitutes an important frontier in the area of fixed-polynomial size lower bounds. In-terestingly, it is known that proving that P NP * SIZE [ n k ] for all k is equivalent to showing astronger Karp-Lipton collapse under the assumption that NP ⊆ SIZE [ poly ] [CMMW19]. Werefer to [CC06, GZ11] for more information about uniform complexity classes around P NP .While existing circuit lower bounds might not be entirely satisfactory from the perspec-tive of the uniform complexity of the problems, there is another important issue with theseresults: they only establish hardness on infinitely many input lengths . Could it be the casethat some natural problems are easy on some input lengths and hard on others? Perhaps theexistence of exceptional mathematical structures of certain sizes might affect (non-uniform)complexity theory around some input lengths? This possibility seems unlikely, but we are farfrom understanding the situation. For instance, a basic question in complexity that remainsopen is whether the nondeterministic time-hierarchy theorem can be extended to an almost-everywhere result (see [BFS09]). On the algorithmic side, an intriguing example is that thenatural problem of generating canonical prime numbers admits a faster algorithm on infinitelymany input lengths [OS17], but showing that the algorithm succeeds on all input lengths isopen. More recent works such as [FS17] and [MW18] show that quite often some control can We use
SIZE [ s ] to denote the set of languages computable by circuits of size at most s ( n ) on every largeenough input length. We say that a language L is in i.o. SIZE [ s ] if there is a language L ′ ∈ SIZE [ s ] such that L and L ′ agree on infinitely many input lengths. Indeed, a proof that E NP * SIZE [ n . ] would be considered a breakthrough by some researchers in the field. Some of our consistency results can be interpreted from this perspective: it is possible to establish stronger“logical” Karp-Lipton collapses if NP ⊆ SIZE [ poly ] and this inclusion is provable in certain theories [CK07]. In the sense of https://en.wikipedia.org/wiki/Exceptional_object
2e obtained over the set of hard input lengths. Still, proving an almost-everywhere circuit sizelower bound beyond 4 n gates remains open even for problems in MATIME [2 n ] (see [BFT98] fora related lower bound).Addressing these questions without further assumptions (i.e. unconditionally) appears tobe extremely challenging. In this work, we attempt to provide formal evidence that some prob-lems in lower uniform complexity classes are hard on every large enough input length . This canbe done via the investigation of circuit complexity from the perspective of mathematical logic.More precisely, we are interested in unconditional results showing that lower bounds such as NP * i.o. SIZE [ n ] are consistent with certain logical theories. To obtain interesting results,we consider theories that can formalize a variety of techniques from algorithms, complexity,and related areas. We focus on first-order theories in the standard sense of mathematical logic,which offers a principled way of investigating consistency statements of the form above. Wedescribe next the theories relevant to our work.
Bounded Arithmetic.
Bounded arithmetic theories are fragments of Peano Arithmeticwith close connections to computational complexity and proof complexity. Such theories havebeen widely investigated by logicians and complexity theorists since the 1970’s. Among themost influential theories we have Cook’s equational theory PV [Coo75] and its correspondingfirst-order formalization [KPT91] (see also [Jeˇr06]), Buss’s theories S and T [Bus86], andextensions of these theories by variants of the pigeonhole principle developed primarily byJeˇr´abek [Jeˇr04, Jeˇr05, Jeˇr07, Jeˇr09] (such as theory APC extending PV ). The objects ofstudy in these theories are natural numbers (representing finite binary strings), and the basicfunctions and relations are given by polynomial-time ( p -time) algorithms in some programmingscheme. For instance, Cook [Coo75] relied on Cobham’s theorem [Cob65] that all p -timefunctions can be generated from few initial ones by composition and bounded recursion onnotation. For convenience, the language L ( PV ) we adopt here is the language of PV , havinga function symbol for each p -time algorithm. There are relation symbols = and ≤ with theirusual meaning, and all other relations we want to include are represented by their characteristicfunctions. The specific axiomatization of PV is not important here: everything will also workfor the theory of all true universal L ( PV )-sentences (to be denoted by True ), and PV ⊆ True .We only note that PV proves induction for all p -time predicates by formalizing binary search,cf. [KPT91, Kra95].The original language of theories S and T as defined in [Bus86] is a finite subset of L ( PV ), but we consider theories S ( PV ) and T ( PV ) in the richer language L ( PV ). (We willadd to these theories even more axioms, which makes any consistency statement stronger.)The principal axioms of the two theories are length-induction (LIND) and induction (IND),respectively, accepted for Σ b ( PV )-formulas. Theory S ( PV ) is close to PV (it is ∀ Σ b ( PV )-conservative over it), but T ( PV ) appears to be significantly stronger (cf. [Kra95]). Theories Note also that establishing a consistency statement is a necessary step before the corresponding circuitlower bound can be unconditionally established, since a true statement is always consistent with a sound theory. In this paper we use PV to refer to its first-order formulation (cf. [Kra95, Section 5.3]). This does not necessarily imply that PV can prove the relevant properties of its function symbols. Forinstance, the AKS algorithm [AKS02] for testing primality appears as some symbol f AKS ∈ L ( PV ), but PV mightnot be able to prove that x is prime if and only if f AKS ( x ) = 1. We review later in the text some definitions necessary in this work. For more information about standardconcepts in bounded arithmetic, we refer to a reference such as [Kra95]. V , S ( PV ) and T ( PV ) and their extensions by a form of the pigeonhole principle (oftenreferred to as dWPHP or sWPHP ) are actually quite strong for the purposes of complexitytheory. They are now known to formalize many key theorems in algorithms, combinatorics,complexity, and related fields (cf. [WP87, Bus86, Kra95, Raz95, Jeˇr05, Jeˇr04, Jeˇr07, Jeˇr09,CN10, Pic14, Pic15a, Pic15b, BKZ15, LC11, Oja04, Le14, MP17] and references therein).Recall that the class of Σ b ( PV )-formulas consists of formulas of the form ∃ y ≤ t ( x ) . . . ∃ y k ≤ t k ( x ) A ( x, y ) , where the t i are L ( PV )-terms not involving y i , and A is quantifier free. The definition ofthis class in the original language of S is a bit more complicated (distinguishing two kindsof bounded quantifiers), but in our language L ( PV ) it is equivalent to this simpler definition.The class of Σ b ( PV )-formulas is defined similarly, but the formula A can also be the negationof a Σ b ( PV )-formula (these negations are Π b ( PV )-formulas). The predicates definable over thenatural numbers by Σ b ( PV )-formulas and by Σ b ( PV )-formulas are exactly the predicates fromΣ p = NP and from Σ p , respectively. We shall denote the theory of all true ∀ Σ b ( PV )-sentencesby True . Our results.
For an L ( PV )-formula ϕ ( x ) and an integer k ≥
1, the L ( PV )-sentence UB i.o.k ( ϕ )is defined as follows: ∀ ( n ) ∃ ( m ) ( m ≥ n ) ∃ C m ( | C m | ≤ m k ) ∀ x ( | x | = m ) , ϕ ( x ) ≡ ( C m ( x ) = 1) . (1)The sentence UB i.o.k ( ϕ ) formalizes that the m -bit boolean functions defined by ϕ (over differentinput lengths) are computed infinitely often ( i.o. ) by circuits of size m k . We unconditionally establish that almost-everywhere circuit lower bounds for complexityclasses contained in P NP are consistent with bounded arithmetic theories. Theorem 1 (Consistency of almost-everywhere circuit lower bounds with bounded theories) . Let k ≥ be any positive integer. For any of the following pairs of an L ( PV ) -theory T and auniform complexity class C : ( a ) T = T ( PV ) ∪ True and C = P NP , ( b ) T = S ( PV ) ∪ True and C = NP , ( c ) T = PV ∪ True and C = P ,there is an L ( PV ) -formula ϕ ( x ) defining a language L ∈ C such that T does not prove thesentence UB i.o.k ( ϕ ) . Our arguments are somewhat non-constructive and do not provide a single explicit formula ϕ ( x ) in each case of the result. Informally, Theorem 1 shows (in particular) the followingconsistency statements: T ( PV ) P NP ⊆ i.o. SIZE [ n k ], S ( PV ) NP ⊆ i.o. SIZE [ n k ], PV P ⊆ i.o. SIZE [ n k ]. The notation 1 ( n ) means that n is the length of another variable. We abuse notation and use | C m | todenote the number of gates in C m . We refer to [Pic15a] for a detailed discussion of the formalization of circuitcomplexity in bounded arithmetic.
4n other words, there are models of these theories (satisfying a large fraction of modern com-plexity theory) that contain explicit problems that require circuits of size n k on every largeenough input length. Another interpretation is that one can develop theories of compu-tational complexity that postulate the existence of hard problems (as new axioms) withoutever proving a contradictory statement. As alluded to above, given the expressive power ofthese theories, we view the consistency results as evidence that such lower bounds hold in thestandard mathematical universe. Nevertheless, if one strongly believes in an inclusion such as NP ⊆ SIZE [ n k ] for a large enough k , then Theorem 1 shows that even to prove this inclusionon infinitely many input lengths it will be necessary to use mathematical arguments that arebeyond the reasoning capabilities of the corresponding theories.We stress that True and True contain several statements of interest about algorithms,boolean circuits, extremal combinatorial objects, etc. Theorem 1 shows that even assumingsuch statements as axioms the corresponding theories cannot prove fixed-polynomial size circuitupper bounds. We note that the particular syntactic form of ϕ defining the hard language in Theorem 1items (a) and (c) is irrelevant as long as p -time functions and predicates are defined by openformulas of the language of PV and the SAT predicate used in the argument is defined by aΣ b -formula. Indeed, if two open L ( PV )-formulas define the same predicate then this universalstatement is included in theory True and hence (c) holds identically for all open formulasdefining the same language. An analogous observation applies to (a): languages in P NP aredefinable by ∆ b -formulas w.r.t. the theory and the universal statement stating their equiva-lence is thus in True . Related work and techniques.
Some works have investigated the unprovability of circuitlower bounds, or equivalently, the consistency of upper bounds . We refer to the introductionof [MP17] for more information about this line of work, and to Appendix A for some relatedremarks that might be of independent interest. Theorem 1 and our techniques are more directlyconnected to [CK07], [KO17], and [BM18]. We review the relevant results next.Cook and Kraj´ıˇcek [CK07] (see also [Kra98]) were the first to systematically investigatethe consistency of circuit lower bounds. They established several results showing that NP * SIZE [ poly ] is consistent with PV , S , and T under appropriate assumptions regarding thecollapse of PH . For instance, it was shown (in particular) that T NP ⊆ SIZE [ poly ] if PH * P NP . While their results are conditional , [CK07] considered consistency statements fora fixed language in NP with respect to all polynomial bounds. In [KO17], two of the authorsestablished an unconditional result showing that PV P ⊆ SIZE [ n k ], where k is any fixedinteger. This consistency statement was subsequently improved by [BM18], who considereda more natural formalization of the statement that a language has circuits of size O ( n k ) and A bit more precisely, the lower bound holds for every input length n ≥ n , where n is an element of themodel. Note that n might be a nonstandard element of this model. One can even contemplate the possibility that more advanced consistency results might allow the develop-ment of “logic-based” cryptography: protocols that are unconditionally secure against all efficient algorithmsthat can be proved correct in a given theory. For instance, T ( PV ) ∪ True proves the correctness of the AKS primality testing algorithm (see Footnote7), i.e., it shows that ∀ x ( ∃ y (1 < y < x ∧ y | x ) ↔ f AKS ( x ) = 0) since this sentence is in True . This implies thatthis theory proves that primality testing can be done by circuits of size n c for a fixed c on every large enoughinput length n . In other words, the languages have both Σ b and Π b definitions that are provably equivalent in the theory. All previous results referto the consistency of lower bounds on infinitely many input lengths, and Theorem 1 part (c)strictly improves upon [KO17] and [BM18]. In terms of techniques, the proof of Theorem 1 explores methods from complexity the-ory and mathematical logic to establish the unprovability of infinitely often upper bounds.We combine ideas from the conditional results of [CK07] with the unconditional approach of[KO17]. The general theme is to obtain computational information from proofs in the cor-responding bounded theories. For instance, under the assumption that there is a PV -proof π that a problem in P admits non-uniform circuits of size n k , we attempt to extract from π a more “uniform” construction of such circuits. The ideal plan is to contradict existinglower bounds against uniform circuits, such as those investigated in [SW14] and other works.However, as explained in [KO17], implementing this plan is not straightforward, since the“uniformity” one obtains from π does not match existing results in the area of uniform circuitlower bounds. Moreover, the proof of Theorem 1 creates additional difficulties because theuniform circuit lower bounds, already insufficient, only hold on infinitely many input lengths.In order to overcome this difficulty, we make use of further insights on the logical side of theargument. In turn, this requires appropriate extensions of the complexity-theoretic arguments. Extensions and open problems.
One can adapt the methods used in the proof of Theorem1 to show that
APC and indeed theory S ( PV ) ∪ sWPHP ( PV ) UB i.o.k ( ϕ ) , (2)for some L ( PV )-formula ϕ ( x ) defining a language in ZPP NP [ O (log n )] . In contrast, existing(infinitely often) lower bounds for
ZPP NP seem to hold only when the NP oracle is adaptivelyqueried polynomially many times [KW98, Cai07], or with respect to non-adaptive queriesbut for a promise version of this class (see the discussion in [San09, Section 3.2]). There isstrong evidence that asking more queries increases computational power (see [CC06, CP08]and references therein), and it is known that polynomially many non-adaptive queries to an NP oracle are equivalent in power to logarithmic many adaptive queries [Hem89, BH91]. Theproblem of proving super-linear circuit lower bounds for ZPP
NPtt (i.e.
ZPP NP with non-adaptivequeries) was investigated recently by [DPV18], and in a sense the consistency statement in (2)addresses this question with respect to APC .On the one hand, this consistency statement feels less appealing than the results in Theorem1 due to its proximity to existing lower bounds in complexity theory. But on the other hand,it highlights the importance of APC in connection to frontier questions in complexity theory The formalizations in [KO17] and [BM18] differ on how the O ( · ) notation is handled, and we refer to thecorresponding papers for details. Here the sentences UB i.o.k ( ϕ ) refer to infinitely often upper bounds, and thisissue is not relevant. In model-theoretic terms, [KO17] and [BM18] provide models where the circuit lower bound holds on somelarge enough input length. A slight modification of the proof in [BM18] gives a fixed model with arbitrarilylarge hard input lengths (Moritz M¨uller, private communication). On the other hand, our results provide amodel where the lower bound holds on every large enough input length. This is obtained as in Theorem 1 parts (a) and (b) by proving the following “logical” Karp-Lipton collapse:If S ( PV ) ∪ sWPHP ( PV ) ⊢ NP ⊆ SIZE [ poly ] then PH collapses to ZPP NP [ O (log n )] . The proof of the latter adaptsthe argument in [CK07, Theorem 5.1 ( ii )], using randomization to obtain witnesses for the required dWPHP axioms and an NP oracle to check that they are correct. (A bit more formally, the idea is to first Skolemize thetheory, reducing the argument to the case of S ( PV ), then to handle the newly introduced function symbols bywitnessing them in the standard model through a probabilistic computation with an NP oracle.) APC . For instance, can one show that APC MA ⊆ SIZE [ n k ], partially addressing the use of non-uniform advice in [San09]? In connection to thisand related problems, it might be fruitful to investigate a potential extension of the equivalencein [CMMW19] to a result that relates consistency statements, witnessing theorems, and logicalKarp-Lipton theorems.It would also be interesting to improve our consistency results for S and T with respectto the uniformity of the hard problems, and to establish a non-trivial statement about theconsistency of circuit lower bounds with T (Theorem 1 part ( a ) extends to S using a similarargument and appropriate results from [CK07]).We include in Appendix A a discussion on the consistency of P = NP and its connectionto the unprovability of circuit lower bounds. In order to emphasize the main ideas, we assume some familiarity with logic, boundedarithmetic, and complexity theory. Everything needed can be found in [Kra95]. The interestedreader can consult [CN10] for a more recent reference in bounded arithmetic, [Bus97] for aconcise introduction, and [Pud13] for an accessible exposition. For more background in circuitcomplexity, we refer to [Juk12]. For a discussion of the formalization of complexity theory andcircuit complexity in bounded arithmetic, see [MP17] and references therein.Our proofs will rely on some results and arguments from [CK07] and [KO17], and we referto the detailed presentation in these papers instead of repeating the proofs here. In moredetail, what is needed from [CK07] is that some or their theorems can be modified to include
True or True . On the other hand, the proof of Theorem 1 (c) can only be followed if thereader is familiar with the simpler argument from [KO17].We use P NP [ ℓ ( n )] to denote the set of languages decided by a deterministic polynomial timemachine that makes at most ℓ ( n ) queries to an NP oracle. We will assume without loss ofgenerality that the oracle is some fixed NP -complete language such as formula satisfiability. This section proves Theorem 1. Let k be a positive integer. We argue in each item as follows.( a ) We consider two cases. If the polynomial hierarchy PH collapses to P NP , then we candefine a language L ∈ P NP such that L / ∈ i.o. SIZE [ n k ]. More precisely, L computes on inputlength n as the lexicographic first truth-table corresponding to a function h : { , } n → { , } that cannot be computed by circuits of size n k . This language can be easily specified using aconstant number of quantifiers over strings of length poly ( n ) (cf. [Kan82]). By the equivalencebetween languages in Σ pi and predicates definable by Σ bi ( PV ) formulas (see e.g. [Kra95, Theorem3.2.12]), there is an L ( PV )-formula ϕ L ( x ) that defines L (using the correspondence between { , } ∗ and N ). Since T ( PV ) ∪ True is sound and L is hard on every large enough inputlength, this theory cannot prove the sentence UB i.o.k ( ϕ L ).Assume now that PH does not collapse to P NP . Let ϕ SAT ( x ) be a Σ b ( PV )-formula that7efines the formula satisfiability problem (SAT). We take a particular formulation of ϕ SAT ( x )for which the input encoding is paddable, meaning that inputs of the satisfiability problem oflength ℓ < m can be easily converted into equivalent inputs of length m . If T ( PV ) ∪ True doesnot prove UB i.o.k ( ϕ SAT ), we are done, given that this formula defines a language in NP ⊆ P NP .Suppose T ( PV ) ∪ True ⊢ UB i.o.k ( ϕ SAT ). This formula has unbounded existential quantifiers,but since T ( PV ) ∪ True is axiomatized by bounded formulas, Parikh’s theorem (cf. [Kra95,Section 5.1]) implies that there is an L ( PV )-term t ( x ) such that UB i.o.k ( ϕ SAT ) is provable in thetheory even if the existential quantifiers are bounded by t (1 ( n ) ). In particular, m and | C m | inEquation (1) can be bounded as m, | C m | ≤ n O (1) . (3)By our assumption on paddability, a circuit C m deciding satisfiability on formulas encodedusing m bits also works for all formulas of length n ≤ m . But by (3), | C m | ≤ n O (1) and hence C m can serve as a polynomial size circuit solving SAT on formulas of size n . Consequently,if T ( PV ) ∪ True proves that SAT is infinitely often in SIZE [ n k ] it also proves that SAT isin SIZE [ poly ( n )]. We now invoke the argument of [CK07, Theorem 5.1 ( iii )] who showed (inparticular) that if T proves that SAT ∈ SIZE [ poly ] then PH collapses to P NP . Their proof canbe adapted to T ( PV ) ∪ True , since all sentences in True are witnessed by FP NP functions andadding these sentences as new axioms does not affect the required witnessing theorem. Thiscollapse of PH is in contradiction to our assumption in this case of the proof, which completesthe argument.( b ) Consider the formula ϕ SAT ( x ) defined in item ( a ) above. If S ∪ True UB i.o.k ( ϕ SAT ) thereis nothing else to prove. Otherwise, by the same argument via Parikh’s Theorem it followsthat S ∪ True proves that SAT admits polynomial size circuits on every input length. Nowthe argument in [CK07, Theorem 5.1 ( ii )] (easily modifiable to handle True because axiomsin it are universal sentences) implies that every language L ∈ PH is also in P NP [ c · log n ] for some c ∈ N . In particular, every such language is in P NP [ n ] . Consequently, by Kannan’s construction[Kan82] there is a language L hard ∈ P NP [ n ] such that L hard / ∈ i.o. SIZE [ n k +2 ]. We will need thefollowing lemma. Lemma 1. If NP ⊆ i.o. SIZE [ n k ] then P NP [ n ] ⊆ i.o. SIZE [ n k +2 ] .Proof. Let L be a language in P NP [ n ] decided by a deterministic polynomial-time oracle machine M running in time at most q ( n ). For convenience, we assume without loss of generality that M makes exactly n queries before accepting or rejecting an input string, regardless of the answersprovided by its NP oracle O .We consider the language L aux containing all tuples ( a, j, b , . . . , b n , ( t ) , c ), where | a | = n ,1 ≤ j ≤ n , each b i ∈ { , } , t ∈ N is a padding parameter, and c ∈ { , } is a control bit, whichsatisfy the following conditions: In more detail, adding a function symbol for these witnessing functions turns sentences from
True intouniversal sentences, and universal sentences do not influence witnessing theorems. For example, if ∀ x ∃ y ( y ≤ s ( x ) ∧ A ( x, y )) is in True and f is the symbol for the associated witnessing function, the universal sentence willbe ∀ x ( f ( x ) ≤ s ( x ) ∧ A ( x, f ( x ))). The proof of the lemma uses ideas from the proof of [Kra93, Proposition 1.3] showing that S can defineall FP NP [ wit,O (log n )] functions, extending a proof from [Bus86] that T can define all FP NP functions. A similarargument was employed in the proof of [FSW09, Theorem 10] (without the infinitely often condition). c = 0, then when M computes on a and its first j queries are answered according to b , . . . , b j , for each i ≤ j if y i ∈ { , } ⋆ is the i -th query and b i = 1 we have y i ∈ O .– If c = 1, the machine M accepts a within q ( n ) steps under oracle answers b i for 1 ≤ i ≤ n .Since O ∈ NP and M is a deterministic polynomial time machine, L aux ∈ NP . Using thehypothesis of the lemma, for infinitely many values of n there exists t ≤ n and a circuit D n for L aux of size at most C ( n + log n + n + t + 1) k ≤ n k +1 (for large enough n ) that decides L aux with respect to our parameter n (the input length for an instance of L ). Note that theparameter t allows us to hit the “good” input lengths without technical considerations aboutthe input encoding employed in the definition of L aux .We will use D n (with the correct value t non-uniformly hardcoded in the input) as a sub-routine in order to solve L on inputs of length n , as described next. First, we recover thecorrect oracle answers d , . . . , d n for a given input string a . This is done in n steps, where the i -th step recovers d i . To recover d , we use D n to compute D n ( a, j , ~b z }| { , ⋆, . . . , ⋆, ( t ) , c , (4)where each ⋆ can be replaced by an arbitrary bit. If the output is 1, the first query made by M on a has a positive answer with respect to O (since positive queries must be strings in O by the definition of tuples in L aux when c = 0). Otherwise, we must have d = 0. Next, weinvoke D n ( a, j , ~b z }| { d , , ⋆, . . . , ⋆, ( t ) , c , (5)knowing that the answer to the first query is correct. By the same argument, we are able torecover d , and proceeding similarly, we can recover all correct answers d , . . . , d n . Finally, byinvoking D n ( a, n, d , . . . , d n , ( t ) ,
1) with c = 1 and using the correct oracle answers, we candecide if a ∈ L . Clearly, this entire computation can be performed by a circuit of size at most O ( n · | D n | ) = O ( n k +2 ), which completes the proof.It follows from Lemma 1 and the properties of L hard that there is a language L ∈ NP suchthat L / ∈ i.o. SIZE [ n k ]. Consequently, if ϕ L ( x ) is a formula that defines L then S ∪ True UB i.o.k ( ϕ L ). This completes the proof of item ( b ).( c ) We follow the overall strategy of the proof of [KO17, Theorem 2.1] (which combines theproof of [SW14, Theorem 1.1] with other ideas), but the infinitely often statement consideredhere introduces certain difficulties. In particular, it is not clear how to adapt the proof in[SW14] to show that P is not contained infinitely often in P -uniform SIZE [ n k ]. In general,combining different computations that succeed infinitely often might not produce a compu-tation that succeeds infinitely often. We explain below how the argument from [KO17] canbe modified to establish the stronger statement in part ( c ). (For simplicity of notation, werestrict our discussion to PV , but the argument works for PV ∪ True as well.)Let g k ′ for k ′ = 3 k be the PV function symbol provided by [KO17, Lemma 3.1]. Recall that PV proves that any uniform algorithm h running in time at most n k ′ − will fail to compute g k ′ , even if h is given a certain amount of advice that can depend on the input length. If PV UB i.o.k ( g k ′ ) we are done. Otherwise, applying the KPT Theorem (see e.g. [KO17, Theorem4.1]) to sentence UB i.o.k ( g k ′ ) (note crucially that UB i.o.k ( g k ′ ) has the right quantifier complexity),9e obtain a fixed r ∈ N (independent of n ) and PV function symbols f , . . . , f r such that oninput 1 ( n ) each function f i outputs n ≤ n i ≤ n a i (represented as 1 ( n i ) ) and a circuit C in i of sizeat most n ki that is a candidate circuit for g k ′ on inputs of length n i (the upper bound n a i isprovable in PV ). As usual in applications of the KPT Theorem, each function f i in addition to1 ( n ) might also depend on potential counter-examples to the correctness of the pairs ( n j , C jn j )for j < i . In other words, from the provability of UB i.o.k ( g k ′ ) theory PV proves the universalclosure of the following disjunction: [ f (1 ( n ) ) = (1 ( n ) , C n ) ∧ | C n | ≤ n k ∧ ( | x | = n → C n ( x ) = g k ′ ( x ))] ∨ [ f (1 ( n ) , x ) = (1 ( n ) , C n ) ∧ | C n | ≤ n k ∧ ( | x | = n → C n ( x ) = g k ′ ( x ))] ∨ (6) . . . ∨ [ f r (1 ( n ) , x , . . . , x r − ) = (1 ( n r ) , C rn r ) ∧ | C rn r | ≤ n kr ∧ ( | x r | = n r → C rn r ( x r ) = g k ′ ( x r ))] . Modifying the strategy of [KO17], we argue that either PV UB i.o.k ( e f ) for a certain PV function symbol e f that depends on f (we are done in this case), or PV proves that the circuit C n output by f (1 ( n ) ) does not succeed in computing g k ′ on inputs of length n = n (1 ( n ) )for infinitely many values of n . It will be important that such values of n are polynomiallygapped, and that an infinite set S of strings of the form 1 ( n ) corresponding to them can beenumerated by a PV function symbol u (1 ( ℓ ) ). This allows us to eliminate one disjunct inthe sentence obtained from the KPT Theorem if we quantify not over 1 ( n ) for all n but justover strings 1 ( n ) in the image of u (1 ( ℓ ) ), since on these specific 1 ( n ) the function f neversucceeds in generating a circuit that correctly computes g k ′ on input length n = n (1 ( n ) ),and in addition (as we explain below) there is a PV function symbol that provably producescounter-examples. The proof of our result can be completed by iterating the argument r timeswhile focusing on the relevant input lengths. The idea is similar in spirit to [KO17], but theargument is more involved because intuitively we need to consider a chain S r ⊆ . . . ⊆ S ofinfinite sets of input parameters: If j ≤ i then PV proves that function f j from the KPTdisjunction (with appropriate counter-examples) does not succeed on 1 ( n ) ∈ S i (assuming theprovability of certain auxiliary sentences UB i.o.k ( e f j )). We provide the details next.Recall that in the terminology of [KO17] the function symbol e f decides L succ , a paddedversion of the language L dc encoding the direct connection language of the circuits generatedby f . Our definition of e f is analogous to the construction in [KO17], but we need to changethe amount of padding in order to accommodate the new setting. Here f (1 ( n ) ) might generatecandidate circuits for g k ′ on larger input lengths. Moreover, the circuits for e f obtained fromthe provability of UB i.o.k ( e f ) are only guaranteed to work infinitely often. Handling thesecomplications in the case of f (and in subsequent cases) will be possible because g k ′ is hardon every large enough input length and the relevant input lengths ( n = n (1 ( n ) ) ≤ n a in thecase of f ) are provably computable in polynomial time .In more detail, let L dc encode the direct connection language of the sequence of circuits C n on n ≤ n a input bits produced by f (1 ( n ) ). Similarly to [KO17], our language L succ willbe a succinct version of L dc . This time we compress the tuples encoding C n to: h Bin ( n ) , ( n / k ) , u, v, w, t i , (7) For simplicity of notation, we left out in each row of (6) the condition n i ≥ n , for i = 1 , , . . . , r . As opposed to [KO17], which focuses on larger input lengths after each iteration of the argument. t is arbitrary. (The parameter t is needed in connection to an infinitely oftencircuit upper bound for L succ , since it makes this language paddable. The use of t here isdifferent than in [KO17], where it appears only for convenience and as a function of other inputparameters.) Under our assumptions, a p -time algorithm e f deciding L succ can be defined in PV . Suppose that PV ⊢ UB i.o.k ( e f ). Recall that, assuming C n is a correct circuit for g k ′ , asmall circuit for e f allows one to obtain a short advice string representing a circuit that decidesthe tuples of C n , which in turn allows us to compute g k ′ in time ≪ n k ′ − . Arguing in PV andadapting the proof of [KO17, Lemma 3.2] in the natural way (i.e. by padding t appropriatelyand using the almost-everywhere hardness of g k ′ ), it follows that for infinitely many choices of1 ( n ) , C n does not compute g k ′ on inputs of length n = n (1 ( n ) ). Equivalently, PV ⊢ ∀ ( ℓ ) ∃ ( n ) ( n ≥ ℓ ) ∃ x ( | x | = n (1 ( n ) )) , g k ′ ( x ) = C | x | ( x ) . (8)Using Herbrand’s Theorem and in analogy to [KO17, Lemma 3.2], there are PV functionsymbols u and e witnessing these existential quantifiers. Furthermore, provably in PV wehave | u (1 ( ℓ ) ) | ≤ ℓ c for some constant c . Therefore, we can take S as the infinite set of strings1 ( n ) obtained from u (1 ( ℓ ) ) over all choices of ℓ , and e (1 ( ℓ ) ) witnesses that the correspondingcircuits C n are incorrect over the associated input lengths n = n (1 ( n ) ).The formula obtained from our initial application of the KPT Theorem to UB i.o.k ( g k ′ ) cannow be simplified in PV to a formula equivalent to: ∀ ( n ) ∈ S , “KPT disjunct for j ∈ [2 , r ] under the counter-example x = e (1 ( ℓ ) )” , (9)where the quantifier ∀ ( n ) ∈ S is expressed in PV by “ ∀ ( ℓ ) ∀ ( n ) such that 1 ( n ) = u (1 ( ℓ ) )”. Abit more precisely, the second and later disjuncts in the KPT expression (6) contain functions f i for i > ( n ) and on each x j for which j < i , where the x j are the variables forcounter-examples to the correctness of circuits C j | x j | . Now substitute everywhere 1 ( n ) = u (1 ( ℓ ) )and x = e (1 ( ℓ ) ). By the choice of u and e , this substitution provably falsifies the firstdisjunct and also n (1 ( n ) ) ≥ n ≥ ℓ . Hence (6) is turned into a KPT expression with r − f ( u (1 ( ℓ ) ) , e (1 ( ℓ ) )) = (1 ( n ) , C n ) ∧ | C n | ≤ n k ∧ ( | x | = n → C n ( x ) = g k ′ ( x ))] ∨ . . . (10) ∨ [ f r ( u (1 ( ℓ ) ) , e (1 ( ℓ ) ) , x , . . . , x r − ) = (1 ( n r ) , C rn r ) ∧| C rn r | ≤ n kr ∧ ( | x r | = n r → C rn r ( x r ) = g k ′ ( x r ))] , where for convenience of notation we have omitted the conditions n i ≥ ℓ , for i = 2 , . . . , r . Onecan also replace f ( u (1 ( ℓ ) ) , e (1 ( ℓ ) )) by an equivalent term in the language of PV , say, f ′ (1 ( ℓ ) ),and similarly for each f i appearing in the expression above. We have therefore eliminated onedisjunct from the formula appearing in Equation (6).The result is proved as in [KO17] by iterating this argument in the natural way until somederived sentence UB i.o.k ( e f i ) is unprovable or one eliminates all disjuncts. The latter case leadsto a contradiction. (Intuitively, the sets S i contain infinitely many elements and on everystring 1 ( n ) one of the functions obtained from the initial KPT disjunction must succeed whengiven appropriate counter-examples. Eliminating all disjuncts contradicts the formula obtainedfrom KPT witnessing, or more precisely, one of the subsequent formulas derived from it in theargument presented above.)For instance, in the case we get r = 2 after the application of the KPT Theorem, assumingthat PV also proves UB i.o.k ( e f ′ ), and arguing identically as before, we now get functions u ′ and11 ′ computing witnesses (lengths and inputs) that circuits provided by f ′ fail infinitely oftento compute g k ′ . This is contradictory, because this time the formula from Equation (10)claims that f ′ must succeed (recall that f ′ (1 ( ℓ ) ) = f ( u (1 ( ℓ ) ) , e (1 ( ℓ ) ))) as there are no moredisjuncts if r = 2.This completes the proof of Theorem 1 item ( c ). Remark.
We note that in Theorem 1 it is possible to syntactically enforce the language L to be in the class C from the formula ϕ ( x ) and theory T . In general, this follows fromthe definability of these languages in the corresponding theories by formulas of appropriatecomplexity (see e.g. [Bus97, Section 2.6]). In more detail, for part ( a ), as we observed inthe concluding remarks of Section 1, the consistency result extends to theory S ( PV ). In thiscase, a language in P NP is definable in the theory via two provably equivalent Σ b and Π b formulas. (The provability of the equivalence needs to be done in S ( PV ).) For part ( b ), thecorresponding language L is definable in S ( PV ) by a Σ b ( PV ) formula. Lastly, for part ( c ) theproof presented above already implies the claim, since the language is given by a PV functionsymbol. Note that in parts ( b ) and ( c ) provability in the theory is not necessary: the syntacticform of the formula (i.e. Σ b ( PV ) and atomic PV formula, respectively) imply that the languageis in the corresponding class. Acknowledgements
We would like to thank J´an Pich, Rahul Santhanam, and Moritz M¨uller for several relateddiscussions. We are also grateful to the reviewers for comments that improved our presentation.This work was supported in part by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 615075and by a Royal Society University Research Fellowship. Jan Bydˇzovsk´y is currently partiallysupported by the Austrian Science Fund (FWF) under Project P31955.
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A Consistency of P = NP from unprovability of lower bounds Imagine that against most expectations P is actually equal to NP and there is a polynomialtime algorithm f (i.e. a PV function symbol) that finds a satisfying assignment for all satisfiableformulas. In other words, if ψ SAT ( x, y ) denotes an L ( PV )-formula that checks if y satisfies theformula encoded by x , then the sentence ϕ P = NP ( f ) def = ∀ x ∀ y [ ψ SAT ( x, y ) → ψ SAT ( x, f ( x ))] (11)is true in the standard model. Now suppose that in order to prove the universal statement ϕ P = NP ( f ) in Equation (11) you have to use concepts (definitions, predicates, etc.) that cannotbe defined as polynomial-time algorithms. To be more specific, assume that (11) is provableusing induction for non-deterministic polynomial-time algorithms (corresponding to theory T ( PV )), but not using induction for polynomial-time algorithms only (corresponding to theory PV ). Could we still maintain that the mere existence of f implies that the satisfiability problemis “feasible”?This question is more philosophical than mathematical, and we are not going to offer ananswer. Instead, we suggest to consider a strictly mathematical question. Conjecture 1.
For no polynomial-time algorithm f theory PV proves the sentence ϕ P = NP ( f ) . Informally, Conjecture 1 states that PV and by standard conservation results S are bothconsistent with P = NP . That is, either P = NP as often assumed, and hence the conjectureis trivially true, or P = NP but you cannot prove it using only polynomial-time concepts andreasoning. For this reason, Conjecture 1 is a formal weakening of the conjecture that P = NP .We do not claim any originality for the conjecture; not only it follows from P = NP butthe statement is also known to follow from the conjectures that bounded arithmetic does notcollapse to PV or that the Extended Frege propositional proof system is not polynomiallybounded. The conjecture must have been also one of the ideas leading Stephen Cook to hisseminal paper [Coo75]. We think it is a weakening of the P vs. NP conjecture that has anintrinsic relevance to it, and that it ought to be studied more (cf. [CK07] for more discussion).In this appendix, we observe that Conjecture 1 is related to the unprovability of circuitlower bounds . For a PV function symbol h and a circuit size parameter k ∈ N , consider thesentence LB a.e. k ( h ) def = ¬ UB i.o.k ( h ) , (12)where UB i.o.k ( h ) is the sentence from Equation (1). Intuitively, LB a.e. k ( h ) states that the languagedefined by h is hard on input length m for circuits of size m k whenever m ≥ n , for a fixedvalue n . Theorem 2 (Consistency of lower bounds with PV from the unprovability of lower bounds) . If there exists k ∈ N such that for no function symbol h theory PV proves the sentence LB a.e. k ( h ) ,then Conjecture 1 holds. PV does notprove that NP * SIZE [ n k ] for some k . Roughly speaking, Theorem 2 shows that if PV doesnot prove circuit lower bounds then P = NP is consistent with PV . Sketch of the proof of Theorem . The argument proceeds in the contrapositive. We formalizein PV the result that if P = NP then for each parameter k , P * i.o. SIZE [ n k ] (see e.g. [Lip94,Theorem 3]). Recall that this is obtained by combining the collapse of PH to P together withKannan’s argument [Kan82] showing that PH can define languages that are almost-everywherehard against circuits of fixed-polynomial size. The usual proof of this claim shows via acounting argument the existence of a truth-table of size 2 n that is hard against circuit size n k .A potential issue is that this result might not be available in PV .We overcome this difficulty as follows. From the provability in PV that P = NP , it followsthat the hierarchy T ( PV ) of bounded arithmetic theories T i ( PV ) collapses to PV [KPT91].Recall that the surjective weak pigeonhole principle sWPHP for PV function symbols is provablein T ( PV ) (see e.g. [Kra95]). Define a PV function symbol g that takes as input a circuit C ofsize n k and outputs the first n k +1 bits of the truth-table computed by C . From sWPHP ( g ) wenow derive in PV that the prefix of some truth-table is not computable by circuits of size n k ,if n is sufficiently large. We can (implicitly) extend the lexicographic first truth-table prefixsatisfying this property with zeroes, and use the resulting truth-table to define a PV -formula ϕ ( x ) with a constant number of bounded quantifiers that defines a language L that is hardagainst circuits of size n k , where the hardness is provable in PV . Since the provability in PV that P = NP implies the provability in PV that PH collapses to P , it follows that ϕ ( x )is equivalent in PV to the language defined by some PV function symbol h . In other words, PV ⊢ LB a.e. k ( hh