1 Introduction
A propositional proof system (to be abbreviated pps) in the sense of Cook and Reckhow [Reference Cook and Reckhow5] is a polynomial time (ptime, shortly) binary relation $P(x,y)$ such that $\exists x P(x,y)$ defines exactly TAUT, the set of propositional tautologies (in the DeMorgan language for definiteness). The efficiency of a pps P is measured by the lengthsofproofs function $s_P$ : for $\tau \in \mbox {TAUT}$ put
A pps P for which $s_P(\tau )$ is bounded above by $\tau ^c$ for some independent $c \geq 1$ is called pbounded. As pointed out by Cook and Reckhow [Reference Cook and Reckhow5], the $\mathcal {N}\mathcal {P}$ vs. $co\mathcal {N}\mathcal {P}$ problem (asking whether the computational complexity class $\mathcal {N}\mathcal {P}$ is closed under complementation) can be equivalently restated as a question whether a pbounded pps exists. The existence of a pbounded pps is thus a fundamental problem of proof complexity.
A pps P is not pbounded iff there exists an infinite subset $H \subseteq \mbox {TAUT}$ such that for any $c \geq 1$ , for only finitely many $\tau \in H$ it holds that $s_P(\tau ) \le \tau ^c$ . Any such set H will be said to be hard for P.
There are essentially only two classes of formulas known that make plausible candidates for being hard for strong pps: reflection principles and $\tau $ formulas coming from proof complexity generators. The former class is a classic topic of proof complexity and its exposition can be found in [Reference Krajíček22, Section 19.2].
The latter formulas are constructed as follows. Take a function $g : {\{0,1\}^*} \rightarrow {\{0,1\}^*}$ that stretches all size n inputs to size $m = m(n)> n$ (and hence the complement of its range $rng(g)$ is infinite) and such that its restriction $g_n$ to ${\{0,1\}^n}$ is computed by a size $m^{O(1)}$ circuit $C_n$ . For each $b \in {\{0,1\}^m} \setminus rng(g_n)$ encode naturally (as in the proof of the $\mathcal {N}\mathcal {P}$ completeness of SAT) the statement
by a size $m^{O(1)}$ tautology $\tau (g)_b$ . A function g is said to be hard for P iff the set $\bigcup _{n \geq 1}\{\tau (g)_b\ \ b \in \{0,1\}^{m(n)} \setminus rng(g_n)\}$ is hard for P, and we speak of the function g as of a proof complexity generator in this context.
We shall actually restrict ourselves hereFootnote ^{1} to the rudimentary case of generators g computed in time polynomial in n (except the example of function ${{\mathbf {tt}}}_{s,k}$ defined below that is computed in time polynomial in m) and, in fact, Lemma 4.2 shows that the question whether g could be nonuniform (i.e., computed by a family of circuits that need not to come from a common algorithm) is to some extent irrelevant.
The $\tau (g)_b$ formulas were defined in [Reference Krajíček14] motivated by problems in bounded arithmetic and independently (and with an apparently different motivation) in Alekhnovich et al. [Reference Alekhnovich, BenSasson, Razborov and Wigderson1]. Unfortunately the authors of [Reference Alekhnovich, BenSasson, Razborov and Wigderson1] did not pursue the topicFootnote ^{2} and developing the theory was a rather lonely affair until recently. The theory of proof complexity generators has now a number of facets and it is linked not only to bounded arithmetic and proof complexity but also to various topics in computational complexity theory. To give the reader an idea let us mention (just by key words and phrases) some topics that have a nontrivial contact with the theory:

• lengthsofproofs lower bounds, feasible interpolation, implicit proof systems, proof search,

• circuit complexity, the minimum circuit size problem, natural proofs, nondeterministic circuits,

• bounded arithmetic, Gödel’s incompleteness theorem, provability of upper and lower bounds, forcing with random variables,

• Nisan–Wigderson generators, structural complexity, $\mathcal {N}\mathcal {P}$ search problems, Kolmogorov complexity, learning theory,

• pseudorandomness, oneway functions, indistinguishability obfuscation.
A more detailed presentation of key points of the theory and of the necessary background requires a text of a book length but the interested reader may look at [Reference Krajíček22, Section 19.46] (or at older [Reference Krajíček19, Chapters 29 and 30]) for an overview and further references. The introduction to Razborov’s [Reference Razborov28] is an interesting presentation of his ideas about the topic (including a formulation of a conjecture that stimulated some of my own work).
Be it as it may, the theory as it is now grew out of the motivation for the formulas in [Reference Krajíček14]: a logic question about the provability of the dual weak PHP principle (dWPHP) for ptime functions in a weak bounded arithmetic theory $S^1_2$ (cf. [Reference Krajíček14, Problem 7.7]). The $\mbox {dWPHP}(f)$ says that function f does not map any interval $[0,a]$ onto $[0,2a]$ (the term $2a$ can be altered to various other values, e.g., to $a^2$ , without changing the logical strength of the principle over $S^1_2$ ). Denote the theory resulting from adding to $S^1_2$ all instances of $\mbox {dWPHP}(f)$ for all (suitably defined) ptime functions f by $S^1_2 + \mbox {dWPHP}(\Delta ^b_1)$ . The problem (cf. [Reference Krajíček14, Problem 7.7]) is:

• Is $S^1_2 + \mbox {dWPHP}(\Delta ^b_1)$ equal to $S^1_2$ ? If not, is it at least $\Sigma ^b_1$ conservative over it?
This problem has a rather rich background and let me try to outline it in one paragraph. A task inherently difficult for bounded arithmetic (and for feasible algorithms) is to count a number of elements of a finite set. It was discovered by Woods [Reference Woods31] that explicit counting may be replaced in many arguments in combinatorics or number theory by the pigeonhole principle PHP for bounded formulas, a statement that no bounded formula defines the graph of a function mapping $[0,a+1]$ injectively into $[0,a]$ . It is still unknown whether this principle (denoted $\Delta _0$ PHP) is provable in bounded arithmetic (the problem is due to MacIntyre). Then Paris, Wilkie and Woods [Reference Paris, Wilkie and Woods27] found out that the weak PHP (no bounded formula defines the graph of a function mapping $[0,2a]$ injectively into $[0,a]$ ), denoted $\Delta _0$ WPHP, often suffices and that this principle is provable in bounded arithmetic (they used theory $I\Delta _0 + \Omega _1$ , nowadays it is replaced by a more convenient Buss’s theory $S_2$ ). In a parallel development Buss [Reference Buss3] defined a subtheory $S^1_2$ of $S_2$ and proved that functions with $\mathcal {N}\mathcal {P}$ graphs provably total in this theory are exactly those that are ptime computable. A final twist before the formulation of our problem was a theorem by A. Wilkie (unpublished but presented in [Reference Krajíček13, Theorem 7.3.7]) that functions with $\mathcal {N}\mathcal {P}$ graphs provably total in theory $S^1_2 + \mbox {dWPHP}(\Delta ^b_1)$ are computable in randomized ptime. It occurred to me that one may turn the table around and take the theory $S^1_2 + \mbox {dWPHP}(\Delta ^b_1)$ as a basis for formalizing randomized algorithms and to establish its link with randomized ptime analogous to the link between $S^1_2$ and deterministic ptime. Because randomized algorithms, and probabilistic constructions and arguments more generally, are ubiquitous in combinatorics and complexity theory I denoted in [Reference Krajíček14] the theory BT for “basic theory.” The link was eventually established by Jeřábek in his PhD Thesis and in a subsequent series of papers [Reference Jeřábek9–Reference Jeřábek12]. In order not to interfere with his work I decided to focus on the provability/conservativity problem above and on the related propositional logic side of things, and this led me to proof complexity generators.
Right from the beginning there were two working conjectures:

1. There are generators pseudosurjective for Extended Frege systems EF (cf. [Reference Krajíček14, Conjecture 7.9 and Corollary 7.10] and [Reference Krajíček15, Conjecture 4.1 and Corollary 4.2]).
This conjecture is related to the provability problem mentioned above and the notion of pseudosurjectivity implies the hardness as defined earlier. We shall touch upon it in Section 2; the reader can find details in [Reference Krajíček15, Reference Krajíček16].

2. There is a generator hard for all proof systems (cf. [Reference Krajíček16, Section 2]).
We shall concentrate here on the second conjecture and we shall restrict our formulation to uniform generators (i.e., computed by algorithms not just by sequences of circuits) having the minimal required stretch $m(n) = n+1$ . It is easy to see that truncating any ptime generator to outputsize $n+1$ preserves the hardness over any pps simulating resolution (e.g., such a truncation can be applied to generators ${{\mathbf {tt}}}_{s,k}$ and $U^t$ defined later). It also allows for a particularly simple formulation of Conjecture 1.1: by [Reference Krajíček16, Section 1] (or [Reference Krajíček22, Lemma 19.4.1]) the second conjecture can be then restated without any reference to proof complexity notions as follows.
Conjecture 1.1. [Reference Krajíček16, Section 2]
There exists a ptime function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets. That is, the complement of $rng(g)$ is $\mathcal {N}\mathcal {P}$ immune.
A fundamental question of proof complexity is, in my view, whether the hardness of proving a tautology can be traced back to the hardness of computing some computational task associated with the formula. A paradigm of such a reduction is the method of feasible interpolation that applies to a wide range of proof systems albeit not to strong ones (cf. [Reference Krajíček22, Chapters 17 and 18]). One can interpret Conjecture 1.1 as stating a reduction of provability hardness to computational hardness for all proof systems in the following sense:

• short proofs, here witnesses to the membership in an infinite $\mathcal {N}\mathcal {P}$ set A,

• imply an upper bound on compression for some strings in A, using g as the decompressing algorithm.
With a bit of imagination a direct parallel between the conjecture and feasible interpolation may be seen when we restrict the conjecture. The conjecture can be equivalently stated as asserting that all $\mathcal {N}\mathcal {P}$ sets disjoint with $rng(g)$ are finite. A restriction of the conjecture may state the finiteness just for a subclass of all $\mathcal {N}\mathcal {P}$ sets. A natural restriction of Conjecture 1.1 in this sense, given a specific proof system P, is the restriction to $\mathcal {N}\mathcal {P}$ sets A from the class of those sets for which P can prove in polynomial size (the tautologies expressing for all lengths $n \geq 1$ ) that $A \cap rng(g) = \emptyset $ . This class of $\mathcal {N}\mathcal {P}$ set is the resultant $Res^P_g$ of [Reference Krajíček16] and the reader can find details there. Conjecture 1.1 restricted to P then says that $Res^P_g$ contains only finite sets. This looks in form similar to feasible interpolation: there we deduce feasible separability of two $\mathcal {N}\mathcal {P}$ sets whose disjointness can be proved efficiently in P, here we deduce the finiteness of an $\mathcal {N}\mathcal {P}$ set if it can be proved efficiently in P that it is disjoint from a particular $\mathcal {N}\mathcal {P}$ set, namely $rng(g)$ . Note also that the conjecture restricted to P implies that P is not pbounded.
Let us give two examples of potential generators (a third one will be discussed in Section 4). An illuminating example of a possibly strong generator is the truthtable function ${{\mathbf {tt}}}_{s,k}$ sending a size s circuit in k inputs to its truthtable (a size $2^k$ string), cf. [Reference Krajíček16] or [Reference Krajíček22, Section 19.5]. Circuits of size s can be coded by $10 s \log s$ bits and so to make the function stretching we assume that $n:= 10 s \log s < m(n):=2^k$ (hence size s circuits are coded by $n < m$ bits). It is computed in (uniform) time $O(s m) = 2^{O(k)}$ , so it is ptime if $s = 2^{\Omega (k)}$ .
The $\tau $ formulas determined by this generator state circuit lower bounds for particular Boolean functions: $\tau ({{\mathbf {tt}}}_{s,k})_b \in \mbox {TAUT}$ iff the function with truthtable b has circuit complexity bigger than s. This makes the formulas attractive but also hard to approach as we know very little about the size of general circuits.
It is known that the first working conjecture above implies that the $\tau $ formulas determined by the truth table function ${{\mathbf {tt}}}_{s,k}$ (with $s = 2^{\epsilon k}$ for any $0 < \epsilon < 1$ ) are hard for EF (cf. [Reference Krajíček16] or [Reference Krajíček19, Section 30.1]). On the other hand, unless $\mathcal {N}\mathcal {E} \cap co\mathcal {N}\mathcal {E}$ has size $s(k)$ circuits, the generator ${{\mathbf {tt}}}_{s,k}$ cannot be hard for all proof systemsFootnote ^{3} and hence it is not a good candidate for Conjecture 1.1 (cf. [Reference Krajíček19, p. 198]).
Our second example follows [Reference Krajíček24, Remark 6.1] and concerns timebounded Kolmogorov complexity. Recall that the complexity measure $K^t(w)$ is the minimal size of a program that prints w in time at most $t(w)$ (cf. [Reference Allender and Watanabe2]). The point is that a proof complexity generator with stretch $m \geq n + \omega (1)$ produces strings w of $K^t$ complexity smaller than $m = w$ . For example, if g stretches n bits to $m = 2n$ bits and runs in ptime $t(m)$ (which is also polynomial in n) then for all size m strings $w \in rng(g_n)$ and $n>> 0$ ,
In fact, as discussed in [Reference Krajíček24, Section 6.1], for a fixed polynomial time $t(n)$ sufficient for the computation of g one can consider the universal Turing machine $U^t$ underlying the definition of $K^t$ as a generator itself.Footnote ^{4} Then for any pps P simulating EF, if some $\tau (U^t)$ formulas have short Pproofs (e.g., by proving tautologies expressing the lower bound $K^t(w) \geq 2m/3$ ), so do some $\tau (g)$ formulas. That is, if there is any g computable in time t and hard for P then $U^t$ must be hard as well.
The paper is organized as follows. In Section 2 we consider the possibility of disproving (or at least of limiting possible g in) Conjecture 1.1 by finding a feasible way to witness that the complement of $rng(g)$ is not empty.
In Section 3 we discuss a new definition of hardness, the $\bigvee $ hardness, that strengthens (presumably) the hardness as defined above (but is weaker, also presumably, than the notion of pseudosurjectivity mentioned earlier). The reason for introducing the new notion is that a particular generator from the class of gadget generators introduced in [Reference Krajíček, Glymour, Wang and Westerstahl18] and recalled here in Section 4 is the $\bigvee $ hardestFootnote ^{5} among all generators but (presumably) not the hardest under the definition of the hardness as given above: in [Reference Krajíček, Glymour, Wang and Westerstahl18] we used for this result the notion of iterability that is in strength between hardness and pseudosurjectivity mentioned in Section 2, as it was at hand but that is not good for Conjecture 1.1. Namely, it is known (cf. [Reference Krajíček16]) that if there is any iterable map for a given pps (containing resolution) then ${{\mathbf {tt}}}_{s,k}$ is iterable for it too and hence hard. But by the remark above ${{\mathbf {tt}}}_{s,k}$ is unlikely to be hard for all proof systems.
This new notion of $\bigvee $ hardness is equivalent to the hardness as defined above for a class of pps satisfying the strong feasible disjunction property (Section 3). This class has the property that all pps not in it are automatically not pbounded.
Section 2 is complemented in Section 5 where we link possible limitations to the stretch g can have to the task of proving lower bounds on timebounded Kolmogorov complexity. We argue that known results imply that these approaches are not likely to work without proving first superpolynomial lower bounds for (uniform and nonuniform) computations.
We also indicate in Section 6 how to modify the notion of a generator (and the conjecture and results in Sections 2 and 5) to address the hardness of proof search instead of lengthsofproofs.
In Section 7 we discuss a way how to restrict Conjecture 1.1 and we show, under a hypothesis, that the conjecture holds relative to all feasibly infinite $\mathcal {N}\mathcal {P}$ sets: sets for which there is a ptime function picking arbitrarily large elements of the set. The paper is concluded by some remarks in Section 8.
Basic proof complexity background can be found in [Reference Krajíček22, Chapter 1]; the topic of hard formulas (including a brief introduction to the theory of proof complexity generators) is in [Reference Krajíček22, Chapter 19]. When we use some proof complexity notions and facts in a formal statement we define them first (and give a reference). But we also use proof complexity background in various informal remarks and there we only refer to the original source and/or to a place in [Reference Krajíček22] where it can be found.
2 Witnessing the dWPHP
The dWPHP for a function g extending n bits to $m = m(n)$ bits is formalized by the formula
Notation $\forall 1^{(n)}$ means that the universal quantifier ranges over all strings $1\dots 1$ of any length n. To witness this formula means to find a witness y for the existential quantifier given $1^{(n)}$ as input. This task became known recently in complexity theory as the range avoidance problem.Footnote ^{6}
Witnessing is a classic notion of proof theoryFootnote ^{7} and, in particular, many fundamental results in bounded arithmetic are formulated as follows: if a theory T proves a formula of a certain syntactic complexity then it can be witnessed (i.e., its leading $\exists $ can be witnessed) by a function from a certain computational class C. Such statements are known for many basic bounded arithmetic theories, many natural syntactic classes of formulas and computational classes of functions.
Unprovability results are generally difficult and usually conditional, and we shall use one below. But in the relativized setup (in our situation this would mean that g is given by an oracle) many unconditional unprovability results are known and they are usually derived by showing that a principle at hand cannot be witnessed by a function in some particular class C (for dWPHP see the end of this section).
We now give an application of the conditional unprovability result of [Reference Krajíček23]. Consider theory ${T_{{\tiny{\mathrm{PV}}}}}$ whose language has a kary function symbol $f_M$ attached to every ptime clocked machine M with k inputs, all $k \geq 1$ . The symbol $f_M$ is naturally interpreted on $\mathbf N$ by the function M computes. The axioms of ${T_{{\tiny{\mathrm{PV}}}}}$ are all universal sentences in the language true in $\mathbf N$ under this interpretation.
The hypothesis used in the unprovability result is this.
Hypothesis (H). There exists a constant $d \geq 1$ such that every language in $\mathcal {P}$ can be decided by circuits of size $O(n^d)$ : $\mathcal {P} \subseteq \mbox {Size}(n^d)$ .
The possibility that (H) is true with $d=1$ is attributed to Kolmogorov, but it is not a hypothesis accepted by mainstream complexity theory. However, there are no technical results supporting the skepticism. In fact, (H) has a number of interesting consequences such as $\mathcal {P} \neq \mathcal {N}\mathcal {P}$ or $\mathcal {E} \subseteq Size(2^{o(n)})$ (the latter is bad for universal derandomization, but it is good for proof complexity; cf. [Reference Krajíček17, Reference Krajíček23]).
The following theorem uses $g := {{\mathbf {tt}}}_{s,k}$ with $s = 2^{\epsilon k}$ for a fixed $0 <\epsilon < 1$ for our ptime function. The dWPHP for this function can be expressed by the formula
where $n = n(m) := 10 s \log s$ . We chose m as the natural parameter: m and n are polynomially related and determine each other, so this indeed expresses dWPHP.
Theorem 2.1. [Reference Krajíček23]
Assume hypothesis (H). Then for every $0 < \epsilon < 1$ and $s = s(k) := 2^{\epsilon k}$ the theory ${T_{{\tiny{\mathrm{PV}}}}}$ does not prove the sentence (1).
The proof of this theorem in [Reference Krajíček23] goes by showing that (1) cannot be witnessed in a particular interactive way discussed below. However, we want to stress that the unprovability result itself, perhaps proved from other hypotheses (or unconditionally) not using witnessing methods (but using model theory instead, for example) implies the impossibility to witness (1) in a particular way.
To illustrate the idea simply we start by showing that (1) cannot be witnessed by a ptime function f, assuming (H). The property that f witnesses (1) is itself a universal statement
and hence, if true, an axiom of ${T_{{\tiny{\mathrm{PV}}}}}$ . As this axiom easily implies (1) we get a contradiction with Theorem 2.1.
In fact, it is easy to see (as pointed out by one of the referees) that for any specific $0 < \epsilon < 1$ the existence of a ptime witnessing function f for (1) with $s = 2^{\epsilon k}$ is equivalent to the existence of a language in $\mathcal {E} \setminus Size(2^{\epsilon n})$ : the set
for any potential ptime f consists of the collection of characteristic functions of a language in $\mathcal {E}$ for input lengths $\ell \geq 1$ , and vice versa.
Consider now an interactive model of witnessing via Student–Teacher computation. In this model of computation (cf. [Reference Krajíček, Pudlák, Sgall and Rovan25, Reference Krajíček, Pudlák and Takeuti26]) ptime student S, given $1^{(n)}$ , produces his candidate solution $b_1 \in {\{0,1\}^m}$ . A computationally unlimited teacher T either acknowledges the correctness or she produces a counterexample: $x_1 \in {\{0,1\}^n}$ s.t. $g(x_1) = b_1$ . S then produces his second candidate solution $b_2$ using also $x_1$ , T either accepts it or gives counterexample $x_2$ etc. The requirement is that within a given bound t on the number of rounds S always succeeds. This can be written in a universal way as
The witnessing theorem for ${T_{{\tiny{\mathrm{PV}}}}}$ implies that $\mbox {dWPHP}(g)$ is provable in ${T_{{\tiny{\mathrm{PV}}}}}$ iff (2) holds for some ptime student S and some constant t. We remark that the witnessing for ${T_{{\tiny{\mathrm{PV}}}}} + S^1_2$ yields S–T protocol with polynomially many rounds $t= m^{O(1)}$ ; this relates to the notion of pseudosurjectivity mentioned in the Introduction (the universal statement (2) can be represented by an infinite family of psize tautologies and pseudosurjectivity requires that these tautologies do not have short proofs; cf. [Reference Krajíček15, Reference Krajíček16] for details).
Let us state the conclusion of this discussion formally.
Theorem 2.2. Assume hypothesis (H). Then dWPHP for function ${{\mathbf {tt}}}_{s,k}$ with parameters as in Theorem 2.1 cannot be witnessed by a Student–Teacher computation with ptime Student and constantly many rounds.
Hence, to witness the nonemptiness of the complement of ${{\mathbf {tt}}}_{s,k}$ with parameters as in Theorem 2.1 by a constant round S–T protocol with ptime student would imply arbitrarily high polynomial lower bounds for circuits computing a language in $\mathcal {P}$ .
Recently, Ilango, Li, and Williams [Reference Ilango, Li and Williams8] proved that the dWPHP for the circuit value function $CV$ (cf. Section 4) is not provable in ${T_{{\tiny{\mathrm{PV}}}}}$ by showing that it cannot be witnessed by an S–T computation with parameters as in Theorem 2.2, assuming a couple of hypotheses of a different nature: that $co\mathcal {N}\mathcal {P}$ is not infinitely often in the Arthur–Merlin class AM and a heuristically justified conjecture in cryptography about the security of the indistinguishability obfuscation $i\mathcal {O}$ . Both these hypotheses appear to be accepted by majority of experts (as opposed to hypothesis (H)). However, one may wonder whether the hypothesis that the dWPHP cannot be in general witnessed by a constantround (or even with polynomially many rounds) S–T protocol with a ptime student is not more fundamental, in the sense of being closer to basic concepts, than the hypotheses above used to derive it.
If we manage to extend the unprovability to theory ${T_{{\tiny{\mathrm{PV}}}}} \cup S^1_2$ then we would rule out witnessing by S–T computation with polynomially many rounds. Extending it further to theory ${T_{{\tiny{\mathrm{PV}}}}} \cup T^1_2$ (or equivalently to ${T_{{\tiny{\mathrm{PV}}}}} \cup S^2_2$ ) would rule out witnessing by ptime machines accessing an $\mathcal {N}\mathcal {P}$ oracle. All these statements need to be conditional as they imply (unconditionally) that $\mathcal {P}$ differs from $\mathcal {N}\mathcal {P}$ : if $\mathcal {P} = \mathcal {N}\mathcal {P}$ then this is implied by a true universal statement in the language of ${T_{{\tiny{\mathrm{PV}}}}}$ (saying that a particular ptime algorithm solves SAT) and hence all true universal closures of bounded formulas are equivalent over ${T_{{\tiny{\mathrm{PV}}}}}$ to universal statements which are axioms of ${T_{{\tiny{\mathrm{PV}}}}}$ .
Further note that in the relativized world we have a number of unconditional results about the impossibility to witness dWPHP. As an example let us mention that we cannot witness dWPHP by a nonuniform ptime machine (i.e., using a sequence of polynomial size circuits; cf. [Reference Sipser30]) with an access to an $\mathcal {N}\mathcal {P}^{R}$ oracle where R is the graph of g that g is not a bijection between $[0,a]$ and $[0,2a]$ . Another example is that even if we have oracle access to g and to another function f we cannot witness by a PLS problem with base data defined by ptime machines with oracle access to $f,g$ that g is not a bijection between $[0,a]$ and $[0,2a]$ with f being its inverse map. The interested reader can find these results (and all background) in [Reference Krajíček13, Sections 11.2 and 11.3] and in references given there.
3 Feasible disjunction property and $\bigvee $ hardness
We shall propose in this section a notion of hardness that is preserved by more constructions (and, in particular, by the construction underlying gadget generators in Section 4) than is the original hardness but is presumably weaker than a stronger notion of iterability (mentioned in the introduction) used in [Reference Krajíček, Glymour, Wang and Westerstahl18].
Definition 3.1. A function $g : {\{0,1\}^*} \rightarrow {\{0,1\}^*}$ that for any $n \geq 1$ stretches all size n inputs to size $m:=m(n)> n$ and such that $g_n$ (the restriction of g to ${\{0,1\}^n}$ ) is computed by size $m^{O(1)}$ circuits is $\bigvee $ hard for a pps P is for any $c \geq 1$ , only finitely many disjunctions
with $n, r \geq 1$ and all $b_i \in {\{0,1\}^m}$ , have Pproof of size at most $m^c$ .
Note that the definition can be formulated equivalently as saying that the set of all valid disjunctions of the form (3) is hard for P.
A pps P has the feasible disjunction property (abbreviated fdp) iff whenever a disjunction $\alpha _0 \vee \alpha _1$ of two formulas having no atoms in common has a Pproof of size s then one of $\alpha _i$ has a Pproof of size $s^{O(1)}$ . The strong fdp is defined in the same way, but the starting disjunction can have any arity r: ${\bigvee }_{i < r} \alpha _i$ . The strong fdp plays a role in analysis of a proof complexity generator in [Reference Krajíček20] (see also [Reference Krajíček22, Section 17.9.2]). It is an open problem [Reference Krajíček22, Problem 17.9.1] whether, for example, Frege or Extended Frege systems have the (strong) fdp. Let us note that Garlík [Reference Garlík6] proved that the proof systems $R(k)$ of [Reference Krajíček14] have no fdp.
Lemma 3.2. Assume a pps P has the strong fdp. Then any generator hard for P is also $\bigvee $ hard for P.
Lemma 3.3. Assume that g is a function stretching size n inputs to size $n+1$ and such that $g_n$ (the restriction of g to ${\{0,1\}^n}$ ) is computed by size $n^{O(1)}$ circuits and is $\bigvee $ hard for a pps P.
Then for all $\delta> 0$ there is $g'$ computed by size $n^{O(1)}$ circuits and stretching size n inputs to size $n + n^{1\delta }$ that is $\bigvee $ hard for P.
Proof Let $g'$ compute g in parallel on $n^c$ many different inputs of size n: it stretches $n^{c+1}$ bits into $n^{c+1} + n^{c}$ bits. As the $\tau $ formulas for $g'$ are disjunctions of the $\tau $ formulas for g, the lemma follows by taking $c \geq 1$ large enough.
A strategic choice: use $\boldsymbol {\bigvee }$ hardness
As it was pointed out in [Reference Krajíček20], for the purpose of proving lengthsofproofs lower bounds for some pps P we may assume w.l.o.g. that P satisfies the strong fdp: otherwise it is not pbounded and we are done. This observation, together with Lemma 3.2, justifies the use of $\bigvee $ hardness rather than mere hardness.
The reader skeptical about the choice may interpret the statements contrapositively as sufficient conditions refuting the strong fdp for a particular pps (cf. Lemma 5.4). In particular, it may happen that no strong pps has the strong fdp: but then we can celebrate as $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ .
4 The gadget generator
The class of gadget generators was introduced in [Reference Krajíček, Glymour, Wang and Westerstahl18] and it is defined as follows. Given any ptime function
define a gadget generator based on f
where
as follows:

1. The input $\overline x \in {\{0,1\}^n}$ is interpreted as $\ell + 2$ strings
$$ \begin{align*}v, u^1, \dots, u^{\ell + 1}, \end{align*} $$where $v \in \{0,1\}^\ell $ and $u^i \in {\{0,1\}^k}$ for all i. 
2. The output $\overline y = Gad_f(\overline x)$ is the concatenation of $\ell + 1$ strings $w^s \in \{0,1\}^{k+1}$ where we put
$$ \begin{align*}w^s\ :=\ f(v, u^s). \end{align*} $$
Clearly we may fix f w.l.o.g. to be the circuit value function $CV_{\ell ,k}(v,u)$ which from a size $\ell $ description v of a circuit (denoted also v) with k inputs and $k+1$ outputs and from $u \in {\{0,1\}^k}$ computes the value of v on u, an element of $\{0,1\}^{k+1}$ .
It was shown in [Reference Krajíček, Glymour, Wang and Westerstahl18] (see also [Reference Krajíček22, Lemma 19.4.6]) that if we replace the hardness of a generator by a stronger condition then it suffices to consider circuits v of size $\le k^{1 + \epsilon }$ , any fixed $\epsilon> 0$ . The proof of this fact in [Reference Krajíček, Glymour, Wang and Westerstahl18] used the notion of iterability mentioned earlier, as it was at hand. However, the same argument gives Theorem 4.1 using the presumably weaker notion of $\bigvee $ hardness from Section 3; the proof in [Reference Krajíček, Glymour, Wang and Westerstahl18] was only sketched, so we give it here. Recall that a pps P simulates Q iff for all $\sigma \in \mbox {TAUT}$ it holds that $s_P(\sigma ) \le s_Q(\sigma )^c$ .
Notation:
In the rest of paper we shall ease on the notation and we will denote the gadget generator $Gad_f$ based on $f = CV_{k^2,k}$ by ${\mbox {Gad}_{sq}} (sq$ stands for square).
Theorem 4.1 (ess. [Reference Krajíček, Glymour, Wang and Westerstahl18]).
Let P be a pps simulating EF and having the following properties. There is $c \geq 1$ such that $:$

• whenever $\sigma \in \mbox {TAUT}$ and $\sigma '$ is obtained from $\sigma $ by substituting for some atoms constants $0$ or $1$ then $s_P(\sigma ') \le s_P(\sigma )^c$ , and

• for all $\alpha , \beta $ : $s_P(\beta ) \le (s_P(\alpha ) + s_P(\alpha \rightarrow \beta ))^c$ .
Assume that there exists a ptime function $g : {\{0,1\}^*} \rightarrow {\{0,1\}^*}$ that stretches all size n inputs to size $m:=m(n)> n$ and is $\bigvee $ hard for P.
Then the gadget generator based on $CV_{k^2,k}$ is $\bigvee $ hard $($ and hence also hard $)$ for P as well.
Proof Assume P and g satisfy the hypotheses of the theorem; w.l.o.g. we may assume that $m(n) = n+1$ . Let $C_k$ be a canonical circuit of size polynomial in k that computes $g_k$ and let $C_k$ be encoded by a string $\lceil C_k \rceil $ of size $\ell \le k^a$ , some constant $a \geq 1$ .
Claim 1. ${\mbox {Gad}_{f}}$ with $f := CV_{k^a,k}$ is $\bigvee $ hard for P.
Note that the $\tau $ formula for ${\mbox {Gad}_{f}}$ and $b = (b^1, \dots , b^t)\in \{0,1\}^{n+1}$ is a tsize disjunction, $t = k^a + 1$ , of $\tau $ formulas for $CV_{k^a,k}$ and $b^i$ , $i \le t$ . Substitute there for (atoms defining) the gadget $v := \lceil C_k \rceil $ . Using that $EF$ has psize proofsFootnote ^{8} of
and $P \geq EF$ , any proof of the original disjunction for ${\mbox {Gad}_{f}}$ is turned into a polynomially longer Pproof of a disjunction of $\tau $ formulas for g, contradicting the hypothesis.
Claim 2. ${\mbox {Gad}_{sq}}$ is $\bigvee $ hard for P.
Note that ${\mbox {Gad}_{f}}$ in Claim 1 is computed in time $O(k^{2a})$ which is $\le n^{2\delta }$ for some $\delta> 0$ . Hence we may perform the same construction as in Claim 1 but using ${\mbox {Gad}_{f}}$ instead of g now.
Note that a circuit of size s can be encoded by $10 s \log s$ bits, so ${\mbox {Gad}_{sq}}$ uses as gadgets circuits of size a little bit less than quadratic. Observe also that ${\mbox {Gad}_{sq}}$ is computed in time smaller than $n^{3/2}$ .
The next statement shows that nonuniformity is irrelevant in the presence of strong fdp. It is proved analogously as Theorem 4.1 by taking for gadgets circuits needed to compute the generator.
Lemma 4.2. Assume a pps P satisfies the hypotheses of Theorem 4.1 and that it admits a $\bigvee $ hard proof complexity generator computed in nonuniform ptime $($ i.e., by psize circuits $)$ . Then ${\mbox {Gad}_{sq}}$ is $\bigvee $ hard for P.
It is known that gadget generators (and ${\mbox {Gad}_{sq}}$ in particular) are hard for many proof systems for which we know any superpolynomial lower bound (cf. [Reference Krajíček22]). Our working hypothesis is that the generator ${\mbox {Gad}_{sq}}$ satisfies Conjecture 1.1. But when working with the generator we encounter the same difficulty as in the case of the truthtable generator ${{\mathbf {tt}}}_{s,k}$ : we know nothing nontrivial about circuits of subquadratic size. Furthermore, the experience with lengthsofproofs lower bounds we have so far suggests that it is instrumental to have hard examples with some clear combinatorial structure. Hence to study the hardness of ${\mbox {Gad}_{sq}}$ it may be advantageous to consider gadgets (i.e., subquadratic circuits) of a special form (technically that would be a substitution instance of ${\mbox {Gad}_{sq}}$ ).
One such specific generator was defined in [Reference Krajíček22, pp. 431–432] and denoted $\mbox {nw}_{k,c}$ there; its gadget is essentially a slightly overdetermined system of sparse equations for a generic function h. Namely the gadget consists of:

• $k+1$ sets $J_1, \dots , J_{k+1} \subseteq \{x_1,\dots , x_k\}$ , each of size $1 \le c \le \log k$ ,

• together with $2^c$ bits defining truth table of a Boolean function h with c inputs.
Given gadget v and $u \in {\{0,1\}^k}$ , $f(v,u) \in \{0,1\}^{k+1}$ are the $k+1$ values h computes on values that u gives to variables in sets $J_1, \dots , J_{k+1}$ . This generator for one fixed, nonuniform gadget was the original suggestion for Conjecture 1.1 in [Reference Krajíček16], but the gadget generator construction allows to avoid the nonuniformity and consider generic case.
5 Stretch and the $Kt$ complexity
The main aim of proof complexity generators is to provide hard examples and for this purpose the stretch $n+1$ of g in Conjecture 1.1 suffices (and it yields the shortest $\tau $ formulas). A larger stretch is of interest in a connectionFootnote ^{9} with the truthtable function ${{\mathbf {tt}}}_{s,k}$ discussed earlier.
We may try to limit possible stretch of generators via some considerations involving timebounded Kolmogorov complexity as we touched upon in the Introduction. We shall use Levin’s measure $Kt(w)$ : the minimum value of $d + \log t$ , where program d prints w in time t (cf. [Reference Allender and Watanabe2]). Its advantage over $K^t$ is that it does not require to fix the time in advance. Although a statement like $Kt(w) \geq 2m/3$ can presumably not be expressed by a psize (in m) tautology, certificates for the membership in an $\mathcal {N}\mathcal {P}$ set A such that all $w \in A$ satisfy $Kt(w) \geq 2w/3$ can be interpreted as proofs of $Kt(w) \geq 2m/3$ .
Let us consider a function with an extreme stretch: ${{\mathbf {tt}}}_{s,k}$ with $s = 100 k$ . This generator sends $n = 10 s \log s \le O(\log m \log \log m)$ bits to $m = 2^k$ bits and is computed in time $t = O(s m) < m^{3/2}$ . Hence both $K^t$ and $Kt$ are bounded above on $rng({{\mathbf {tt}}}_{s,k}) \cap {\{0,1\}^m}$ by $O(\log m \log \log m)$ .
Notation (Allender [Reference Allender and Watanabe2]):
For any set $A \subseteq {\{0,1\}^*}$ define function $Kt_A : {\mathbf N}^+ \rightarrow {\mathbf N}^+$ by
if the righthand side is nonempty, and we leave $Kt_A(m)$ undefined otherwise.
Hence we could rule out a generator with the extreme stretch (as in the above ${{\mathbf {tt}}}_{s,k}$ ) being hard for all proof systems if we could find an infinite $\mathcal {N}\mathcal {P}$ set A such that $Kt_A(m) \geq \omega (\log m \cdot \log \log m)$ . Unfortunately the next theorem suggests that this is likely not an easy task. Following Allender [Reference Allender and Watanabe2] we define an $\mathcal {N}\mathcal {E}$ search problem to be a binary relation $R(x,y)$ such that R implicitly bounds $y$ by $2^{O(n)}$ for $x = n$ and which is decidable in time $2^{O(n)}$ (think of y as an accepting computation of an $\mathcal {N}\mathcal {E}$ machine on input x). The search task is: given x, find y such that $R(x,y)$ , if it exists. As an example related to our situation let A be an $\mathcal {N}\mathcal {P}$ set defined by condition
with S a ptime relation, and consider $R(x,y)$ with $y = [y_1, y_2]$ defined by
Note that $y_1 = x$ expresses that the length of $y_1$ is exponential in the length of x.
Theorem 5.1 (Allender [Reference Allender and Watanabe2, Corollary 7 and Theorem 8]).
There exists an infinite $\mathcal {N}\mathcal {P}$ set A s.t. $Kt_A(m) = \omega (\log m)$ iff there exists an $\mathcal {N}\mathcal {E}$ search problem s.t. $:$

• $\exists y R(x,y)$ is satisfied for infinitely many x,

• every algorithm running in time $2^{O(n)}$ solves the search problem for a finite number of inputs x only.
Hence ruling out generators with even very large stretch means likely to prove significant computational lower bounds. The following seems to be a natural test question.
Problem 5.2. Is it true that any infinite $\mathcal {N}\mathcal {P}$ set A contains a string $w \in A$ with $Kt(w) < w$ ? That is, is it true that the set $\{w \ \ Kt(w) \geq w\}$ is $\mathcal {N}\mathcal {P}$ immune?
Theorem 5.3.

1. If Problem 5.2 has the negative answer then the range of no ptime generator g stretching n bits to $n + \omega (\log n)$ bits can intersect all infinite $\mathcal {N}\mathcal {P}$ sets.

2. If Problem 5.2 has the affirmative answer then $\mathcal {N}\mathcal {P}$ is a proper subclass of ${\mathcal {E}\mathcal {X}\mathcal {P}}$ .
Proof For the first part note that all strings in the range of $g_n$ (g restricted to ${\{0,1\}^n}$ ) have Ktcomplexity at most $n + O(\log n$ ).
For the second part note that there is a function g computable in time $2^{O(n)}$ such that the range of $g_n$ is the set of $w \in \{0,1\}^{n+1}$ with $Kt(w) \le n$ . We have that $rng(g) \in \mathcal {E}$ and hence ${\{0,1\}^*} \setminus rng(g)$ is also in $\mathcal {E}$ but it cannot be—assuming the affirmative answer to the problem—in $\mathcal {N}\mathcal {P}$ . This implies that $\mathcal {E} \not \subseteq \mathcal {N}\mathcal {P}$ and hence also ${\mathcal {E}\mathcal {X}\mathcal {P}} \not \subseteq \mathcal {N}\mathcal {P}$ . As $\mathcal {N}\mathcal {P} \subseteq {\mathcal {E}\mathcal {X}\mathcal {P}}$ we have $\mathcal {N}\mathcal {P} \subset {\mathcal {E}\mathcal {X}\mathcal {P}}$ .
We would rather like to see the affirmative answer; not only does it have nice corollary by the previous theorem, but it is also in the spirit of a potential reduction of provability hardness to computational hardness discussed after Conjecture 1.1. Note that the problem has the affirmative answer for all $\mathcal {N}\mathcal {P}$ sets defined in the CSP (constraint satisfaction problem) format: if an instance X of size n has a solution, so do instances obtained by taking t disjoint copies (i.e., in disjoint sets of variables) of X, and these have $Kt$ complexity at most $O(n + \log t + \log tn)$ which is less than the size $tn$ of the new instance if $t> 1$ and $n>> 1$ .
Let us consider the stretch of gadget generators. By default it was taken in the definition to be the minimal required stretch, but there are other options. One could use as gadgets circuits that map k bits to $k'$ bits where $k'>> k$ ; for example, $k'=2k$ or $k'=k^2$ (allowing accordingly a bigger size of gadgets, still polynomial in k). The resulting generator would send n bits to approximately $(k'/k)n$ bits which is about $n^{1 + \epsilon }$ for some $\epsilon> 0$ , for $k' = k^2$ .
However, we want to be conservative with requirements on gadgets. Note that the stretch of gadget generators can be influenced also by taking more strings $u^i$ in the construction of $Gad_f$ than is the minimal number needed, i.e., more than $\ell +1$ . In particular, assume we perform the construction of $Gad_f$ but taking $t>> \ell $ strings $u^i$ and $w^i$ . We still want to maintain, as in Theorem 4.1, that the generator is the $\bigvee $ hardest generator; hence we allow only t polynomial in k. Then
For $\ell \le k^{O(1)}$ (as in ${\mbox {Gad}_{sq}}$ ) and taking $t := k^c$ for very large $c \geq 1$ we can arrange that
for as small $\epsilon> 0$ as wanted. Denote the generator which extends the definition of ${\mbox {Gad}_{sq}}$ in this way by ${\mbox {Gad}^c_{sq}}$ .
Lemma 5.4. Assume that there is an infinite $\mathcal {N}\mathcal {P}$ set A such that for some $\delta> 0$ ,
Assume further that Conjecture 1.1 is true.
Then there is a pps P such that no pps Q simulating P has the strong fdp.
Proof Choose $c \geq 1$ so large that the stretch of ${\mbox {Gad}^c_{sq}}$ is $n^{1\epsilon }$ , $\epsilon = \epsilon (c)$ , where
for $n>> 0$ (taking $c \geq 1$ such that $0 < \epsilon (c) < \delta $ suffices).
Given an infinite $\mathcal {N}\mathcal {P}$ set A satisfying the hypothesis define a pps P to be, say, resolution but accepting also witnesses to the membership of $b \in A$ as proofs of $\tau ({\mbox {Gad}^c_{sq}})_b$ . It is sound as A must be disjoint from the range of ${\mbox {Gad}^c_{sq}}$ .
If Conjecture 1.1 was true for some g and some Q simulating this P, and Q would satisfy the strong fdp, it would follow by Lemma 3.2 that g is $\bigvee $ hard for Q and hence by Theorem 4.1 (modified trivially for ${\mbox {Gad}^c_{sq}}$ ) that ${\mbox {Gad}^c_{sq}}$ is $\bigvee $ hard (and hence also hard) for Q. That is a contradiction with how P was defined.
6 Modifications for proof search hardness
Proof complexity generators, and Conjecture 1.1 in particular, aim primarily at the problem to establish lengthsofproofs lower bounds. It is easy to modify the concept to aim at time complexity of proof search. Essentially this means to replace everywhere in the previous sections $\mathcal {N}\mathcal {P}$ sets by $\mathcal {P}$ sets. To give a little more detail we shall use the definition of a proof search algorithm from [Reference Krajíček24]: it is a pair $(A,P)$ such that A is a deterministic algorithm that finds for every tautology its Pproof. How much time any algorithm $(A,P)$ has to use on a particular tautology is measured by the information efficiency function $i_P : \mbox {TAUT} \rightarrow \mathbf {N}^+$ ; it is an inherently algorithmic information concept. For each pps P there is a timeoptimal $(A_P,P)$ (it has at most polynomial slowdown over any other proof search algorithm) which is also informationoptimal. The reader can find definitions and proofs of these facts in [Reference Krajíček24].
Define a set $S \subseteq \mbox {TAUT}$ to be searchhard for P iff for any $c \geq 1$ algorithm $A_P$ finds a proof of $\sigma $ in time bounded above by $\sigma ^c$ for finitely many formulas $\sigma \in S$ only. Then analogously with the definition of hardness we define g (in the format as in Conjecture 1.1, i.e., ptime stretching each input by one bit) to be searchhard for P iff the set of tautologies $\tau (g)_b$ , $b \notin rng(g)$ , is searchhard for P. It can be shown that the conjecture that there is a uniform generator searchhard for all pps is then equivalent to the following conjecture.
Conjecture 6.1 (Proof search version of Conjecture 1.1). There exist a ptime function g extending each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {P}$ sets. That is, the complement of $rng(g)$ is $\mathcal {P}$ immune.
There are some more facts known about $Kt_A$ measure for sets in $\mathcal {P}$ (note that Theorem 5.1 was about $\mathcal {N}\mathcal {P}$ sets); for example, [Reference Allender and Watanabe2, Theorems 6 and 8] or [Reference Hirahara7, Theorem 3.11]. These results seem to suggest that Conjecture 6.1 may not be any easier to prove than Conjecture 1.1.
Let us conclude this section by noticing that the fdp can be naturally modified for proof search as well: the modification requires that the time $A_P$ needs on $\alpha _0$ or $\alpha _1$ is bounded above by a polynomial in time it needs on $\alpha _0 \vee \alpha _1$ . However, such a property implies the usual feasible interpolation property. Namely, if $\pi $ is a Pproof of a disjunction
(the disjuncts are not required to have disjoint sets of variables this time) consider disjunction $\beta \vee (\gamma _0 \vee \gamma _1)$ where $\beta $ is a propositional sentence that is the conjunction of $0$ with all bits of $\pi $ . Then $A_P$ (recall from Section 6 that $(A_P,P)$ is timeoptimal) when given this disjunction reads $\pi $ and hence proves $\gamma _0 \vee \gamma _1$ and thus also $\beta \vee (\gamma _0 \vee \gamma _1)$ . By the search version of fdp $A_P$ must find in time polynomial in $\pi $ a proof of $\gamma _0 \vee \gamma _1$ (as $\beta $ is false) and thus also of any instance $\gamma _0(a,y) \vee \gamma _1(a,z)$ (this requires that Pproofs are closed under substitution of constants as in Theorem 4.1). By the new property again algorithm $A_P$ , for each a succeeds on either $\gamma _0(a,y)$ or on $\gamma _1(a,z)$ in time polynomial in $\pi $ . That yields feasible interpolation. This observation means that the proof search variant of fdp cannot hold for any strong proof systems and is subject to the same limitations as is feasible interpolation and, in particular, cannot hold for any strong proof systems unless some standard cryptographic assumptions fail. The reader can find all background in [Reference Krajíček22].
7 Feasibly infinite $\mathcal {N}\mathcal {P}$ sets
Two natural ways how to make Conjecture 1.1 weaker and hence more tractable are either to allow generator g from a larger class of functions than just ptime computable or to restrict the requirement of the finiteness only to a subclass of all $\mathcal {N}\mathcal {P}$ sets. The proof of part 2 of Theorem 5.3 shows that finding $g : {\{0,1\}^n} \rightarrow \{0,1\}^{n+1}$ computable in exponential time would imply $\mathcal {N}\mathcal {P} \subset {\mathcal {E}\mathcal {X}\mathcal {P}}$ (it would be that $rng(g) \in {\mathcal {E}\mathcal {X}\mathcal {P}}$ and hence also ${\{0,1\}^*} \setminus rng(g) \in {\mathcal {E}\mathcal {X}\mathcal {P}}$ , but by choice of g this is not in $\mathcal {N}\mathcal {P}$ ), so such a weakening is definitely interesting although it may not advance proof complexity. In this section we look at how to restrict sensibly the class of $\mathcal {N}\mathcal {P}$ sets in the conjecture.
We have seen one such restriction in the Introduction (classes $Res^P_g$ ). There is, however, another natural restriction of the class of $\mathcal {N}\mathcal {P}$ sets in the conjecture possible. Take a sound theory T whose language extends that of ${T_{{\tiny{\mathrm{PV}}}}}$ consider the class of all $\mathcal {N}\mathcal {P}$ sets A such that the infinitude of A
can be proved in T, representing $y \in A$ by a formula in the language of ${T_{{\tiny{\mathrm{PV}}}}}$ of the form
with $c \geq 1 $ a constant and $A_0$ open and defining a ptime relation. Hence $Inf_A$ is a $\forall \exists $ sentence.
Knowing that a particular T proves $Inf_A$ yields, in principle, nontrivial information about A. For example, if ${T_{{\tiny{\mathrm{PV}}}}}$ proves the sentence then by applying Herbrand’s theorem we get a ptime function f witnessing it. That is, f finds elements of A:
We shall call sets A for which such ptime function f exists feasibly infinite. This remains true (by Buss’s theorem) if ${T_{{\tiny{\mathrm{PV}}}}}$ is augmented by $S^1_2$ . If ${T_{{\tiny{\mathrm{PV}}}}}$ is extended by some stronger bounded arithmetic theory then $Inf_A$ will be witnessed by a specific $\mathcal {N}\mathcal {P}$ search problem attached to the theory. For example, if we add to ${T_{{\tiny{\mathrm{PV}}}}}$ induction axioms for $\mathcal {N}\mathcal {P}$ sets (theory $T^1_2$ ) then $Inf_A$ is witnessed by a PLS problem (by the Buss–K. theorem [Reference Buss and Krajíček4]). The reader can find the bounded arithmetic background in [Reference Krajíček13].
It is easy to see that Problem 5.2 has the affirmative answer for feasibly infinite $\mathcal {N}\mathcal {P}$ sets. Namely applying function $f(x)$ to $x := 1^{(n)}$ produces $y := f(x) \in A$ with $y> n$ but $Kt(y) \le O(\log n)$ . For the conjecture we need to work a bit.
Theorem 7.1. Assume hypothesis (H) from Section 2. Then Conjecture 1.1 holds relative to the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets $:$ there is a generator g whose range intersects every feasibly infinite $\mathcal {N}\mathcal {P}$ set.
Proof The proof is a special case of the construction from [Reference Krajíček23]. We shall show that generator ${{\mathbf {tt}}}_{s,k}$ with $s = s(k) := 2^{k/2}$ satisfies the statement.
Let A be a feasibly infinite $\mathcal {N}\mathcal {P}$ set as it is witnessed by a ptime function f. Let $d \geq 1$ be the constant from (H) and put $m' := m^{1/(3d)}$ where $m := f(1^{(n)})$ and $n>> 1$ , and put also $k := \log m$ .
Define the function $\hat f$ that has $m' + k$ variables and on inputs $1^{(m')}$ and $i \in {\{0,1\}^k}$ computes the ith bit of $f(1^{(n)})$ ; it is a ptime function.
Take a circuit $\hat C(z,i)$ that computes $\hat f$ of size guaranteed by hypothesis (H) and define new circuit C by substituting $1^{(m')}$ for z in $\hat C$ and leaving only the k variables for bits of i. Note that C has size $O((m' + k)^d) < 2^{k/2}$ . Further, by its definition, ${{\mathbf {tt}}}_{s,k}(C) = f(1^{(n)})$ ; i.e., $rng({{\mathbf {tt}}}_{s,k}) \cap A \neq \emptyset $ .
Corollary 7.2. Assume hypothesis (H) from Section 2. Then there exists a model $\mathbf M$ of ${T_{{\tiny{\mathrm{PV}}}}}$ in which Conjecture 1.1 holds $:$ there is a ptime generator g such that for any standard $\mathcal {N}\mathcal {P}$ set $A\ ($ i.e., defined without parameters from $\mathbf M)$ it holds $:$
Proof Take the function g from Theorem 7.1. The statement $rng(g)\cap A = \emptyset $ is universal for any $A \in \mathcal {N}\mathcal {P}$ , so it is true in the standard model $\mathbf N$ iff it is true in all models of ${T_{{\tiny{\mathrm{PV}}}}}$ . It thus suffices to show that ${T_{{\tiny{\mathrm{PV}}}}}$ together with all sentences $\neg Inf_A$ for these sets A is consistent.
Assume not; then the Compactness theorem and the fact that a finite number of $A_i$ are all disjoint from $rng(g)$ iff their union is imply that for some $\mathcal {N}\mathcal {P}$ set A such that $rng(g)\cap A = \emptyset $ theory ${T_{{\tiny{\mathrm{PV}}}}}$ proves $Inf_A$ . But then it is feasibly infinite and that contradicts Theorem 7.1.
8 Concluding remarks
I think that it is fundamental for the development of the theory to make a progress on the original problem of the unprovability of dWPHP for ptime functions in $S^1_2$ discussed in the Introduction. For a start we may try to show the unprovability in ${T_{{\tiny{\mathrm{PV}}}}}$ (or some of its extension as mentioned at the end of Section 2) under a more mainstream hypothesis than is (H) and more theoretically fundamental than are those used in [Reference Ilango, Li and Williams8]. Note that this presumably requires a different function than ${{\mathbf {tt}}}_{s,k}$ we used in Section 2: by remarks before and after Theorem 2.1 the unprovability of dWPHP for this function implies $\mathcal {E} \subseteq Size(2^{o(k)})$ which contradicts the hypothesis that $\mathcal {E} \not \subseteq Size(2^{\epsilon k})$ for some $\epsilon>0$ which is—in the eyes of many complexity theorists at least—considered plausible.
However, in my view a real progress will result only from unconditional results. For reasons discussed in the nexttolast paragraph of Section 2 to have a chance to succeed we need to leave theory ${T_{{\tiny{\mathrm{PV}}}}}$ aside and work with theories PV or $S^1_2$ . This implies that an argument cannot rely just on witnessing theorems as they do not change if ${T_{{\tiny{\mathrm{PV}}}}}$ is added. The problem becomes essentially propositional and it is exactly this what led in [Reference Krajíček15, Reference Krajíček16] to the notions of freeness and pseudosurjectivity (of generators for EF) mentioned in Section 2: to show that a ptime generator has this property is essentially equivalent to the unprovability of dWPHP for it in PV or $S^1_2$ , respectively (cf. [Reference Krajíček15, Section 6] and [Reference Krajíček16]).
Acknowledgments
I thank Igor C. Oliveira (Warwick U.) and Jan Pich (Oxford U.) for discussions about the topic. I am indebted to the two anonymous referees for their detailed comments and suggestions.