2 The construction

We take as our basic theory $S^1_2$ of Buss [Reference Buss1] (cf. [Reference Krajíček5, Section 9.3]), denoting its language simply L. The language has a canonical interpretation in the standard model $\mathbf{N}$ . The theory is finitely axiomatizable and formalizes smoothly syntax of first-order logic. Language L allows to define a natural syntactic hierarchy $\Sigma ^b_i$ of bounded formulas that define in $\mathbf{N}$ exactly corresponding levels $\Sigma ^p_i$ , for $i \geq 1$ , of the polynomial time hierarchy.

An L-formula $\Psi $ will be identified with the binary string naturally encoding it and $|\Psi |$ is the length of such a string. An L-theory T is thus a subset of ${\{0,1\}^*}$ , a set of L-sentences, and it makes sense to say that it is p-time. It is well-known (and easy) that each r.e. theory has a p-time axiomatization (a simple variant of Craig’s trick; cf. [Reference Craig3]).

If $u,v$ are two binary strings we denote by $u \subseteq _e v$ the fact that u is an initial subword of v. The concatenation of u and v will be denoted simply by $uv$ . Both these relation and function are definable in $S^1_2$ by both $\Sigma ^b_1$ and $\Pi ^b_1$ formulas that are (provably in $S^1_2$ ) equivalent. We shall assume that no formula is a proper prefix of another formula.

Let $T \supseteq S^1_2$ be a first-order theory in language L that is sound (i.e., true in $\mathbf{N}$ ) and p-time. Define function $g_T$ as follows:

1. 1. Given input u, $|u| = n$ , find an L-formula $\Phi \subseteq _e u$ with one free variable x such that $|\Phi | \le \log n$ . (It is unique if it exists.)

   • • If no such $\Phi $ exists, output $g_T(u) := \overline 0 \in {\{0,1\}^{n+1}}$ .

   • • Otherwise go to 2.

2. 2. Put $c: = |\Phi |+1$ . Going through all $w \in {\{0,1\}^{c+1}}$ in lexicographic order, search for a T-proof of size $\le \log n$ of the following sentence ${\Phi ^w}$ :

   (1) $$ \begin{align} \exists y \forall x> y\ \Phi(x) \rightarrow \neg (w \subseteq_e x). \end{align} $$

   • • If a proof is found for all w output $g_T(u) := \overline 0 \in {\{0,1\}^{n+1}}$ .

   • • Otherwise let $w_0 \in {\{0,1\}^{c+1}}$ be the first such w such that no proof is found. Go to 3.

3. 3. Output $g_T(u) := w_0 u_0 \in {\{0,1\}^{n+1}}$ , where $u = \Phi u_0$ .

The infinitude of the complement of the range follows as at most half of strings in ${\{0,1\}^{n+1}}$ are in the range.

Applying the theorem to $A := {\{0,1\}^*} \setminus Rng(g)$ (and using Lemma 2.1) yields the following version of Gödel’s First Incompleteness theorem.

Note that the argument shows that for each formula $\Phi $ defining the complement, some true sentence ${\Phi ^w}$ as in (1) is unprovable in T. The complement of $Rng(g_T)$ is in $\mbox {coNP}$ and that leaves room for the following problem.

The affirmative answer would imply by Theorem 2.2 that $Rng(g_T)$ intersects all infinite ${\mbox {NP}}$ sets and hence $g_T$ solves the proof complexity conjecture mentioned at the beginning of the paper, and thus ${\mbox {NP}} \neq \mbox {coNP}$ . Note that, for each T, it is easy to define even as simple set as

$$ \begin{align*}\{1 u\ |\ u \in {\{0,1\}^*}\} \end{align*} $$

by a formula $\Phi $ such that T does not prove that no string in it starts with $0$ . But in the problem we do not ask if there is one definition leading to unprovability but whether all definitions of the set lead to it.

3 Down to propositional logic

The reason why the algorithm computing $g_T$ searches for T-proofs of formulas ${\Phi ^w}$ rather than of $\neg {\Phi ^w}$ which may seem more natural is that ${\mbox {NP}}$ sets can be defined by $\Sigma ^b_1$ -formulas $\Phi $ and for those, after substituting a witness for y, ${\Phi ^w}$ becomes a $\Pi ^b_1$ -formula. Hence one can apply propositional translation (cf. [Reference Cook2] or [Reference Krajíček5, Section 12.3]) and hope to take the whole argument down to propositional logic, replacing the incompleteness by lengths-of-proofs lower bounds. There are technical complications along this ideal route, but we are at least able to combine the general idea with a construction akin to that underlying [Reference Krajíček4, Th eorem 2.1]Footnote ² and to prove the following statement.

Note the first statement is by [Reference Krajíček6, Th eorem 2.4] equivalent to the non-existence of a time-optimal propositional proof search algorithm.

Before starting the proof we need to recall a fact about propositional translations of $\Pi ^b_1$ -formulas. For $\Phi (x) \in \Sigma ^b_1$ , $c :=|\Phi |$ and $w \in {\{0,1\}^{c+1}}$ , and $n \geq 1$ let ${\varphi _{n,w}}$ be the canonical propositional formula expressing that

$$ \begin{align*}(|x| = n+1 \wedge \Phi(x)) \rightarrow \neg w \subseteq_{e} x\,. \end{align*} $$

We use the qualification canonical because the formula can be defined using the canonical propositional translation $||\dots ||^{n+1}$ (cf. [Reference Krajíček5, Section 12.3] or [Reference Cook2]) applied to ${\Phi ^w}$ after instantiating first y by $1^{(n)}$ . Formula ${\varphi _{n,w}}$ has $n+1$ atoms for bits of x and $n^{O(1)}$ atoms encoding a potential witness to $\Phi (x)$ together with the p-time computation that it is correct. For any fixed $\Phi $ the size of ${\varphi _{n,w}}$ (with $w \in {\{0,1\}^{c+1}}$ ) is polynomial in n and, in fact, the formulas are very uniform (cf. [Reference Krajíček5, Section 19.1]). We shall need only the following fact.

Lemma 3.2. There is an algorithm ${{\mathbf{transl}}}$ that upon receiving as inputs a $\Sigma ^b_1$ -formula $\Phi $ , $w \in {\{0,1\}^{c+1}} $ where $c := |\Phi |$ and $1^{(n)}$ , $n \geq 1$ , outputs ${\varphi _{n,w}}$ such that

$$ \begin{align*}(|x| = n+1 \wedge \Phi(x)) \rightarrow \neg w \subseteq_{e} x\, \end{align*} $$

is universally valid iff ${\varphi _{n,w}}$ is a tautology. In addition, for any fixed $\Phi $ the algorithm runs in time polynomial in n, for $n> |\Phi |$ .

Proof of Theorem 3.1

To prove the theorem we shall assume that statements 1) and 2) fail and (using that assumption) we construct function h satisfying statement 3). Our strategy is akin in part to that of the proof of [Reference Krajíček4, Th eorem 2.1].

For a fixed $\Phi $ assume that formulas ${\varphi _{n,w}}$ are valid for $n \geq n_0$ . By Lemma 3.2 they are computed by ${{\mathbf{transl}}}(\Phi , w, 1^{(n)})$ in p-time. Hence we can consider the pair $1^{(n)}, w$ to be a proof (in an ad hoc defined proof system) of ${\varphi _{n,w}}$ for $n \geq n_0$

Assuming that statement 1) fails and P is a p-optimal proof system we get a p-time function f that from $1^{(n)}, w$ , $n \geq n_0$ , computes a P-proof $f(1^{(n)}, w)$ of ${\varphi _{n,w}}$ . Let $|f(1^{(n)}, w)| \le n^\ell $ where $\ell $ is a constant (depending on $\Phi $ ). The function that from $n,w,i$ , with $i \le n^\ell $ , computes the i-th bit of $f(1^{(n)}, w)$ is in the computational class $\mbox {E}$ .

We would like to check the validity of ${\varphi _{n,w}}$ by checking the P-proof $f(1^{(n)}, w)$ , but we (i.e., the algorithm that will compute h) cannot construct f from $\Phi $ . Here the assumption that statement 2) fails too, i.e., that $E \subseteq {\mbox {P}}/poly$ , will help us. By this assumption $f(1^{(n)}, w)$ is the truth-table ${{\mathbf{tt}}}(D)$ (i.e., the lexico-graphically ordered list of values of circuit D on all inputs) of some circuit with $\log n + c + \ell \log n \le (2+\ell )\log n$ inputs and of size $|D| \le (\log n)^{O(\ell )}$ . In particular, for all $\ell $ (i.e., for all $\Phi \in \Sigma ^b_1$ ) we haveFootnote ³ $|D| \le (\log n)^{\log \log n}$ for $n>> 1$ . Hence it is enough to look for P-proofs among ${{\mathbf{tt}}}(D)$ for circuits of at most this size.

We can now define function $h_P$ in a way analogous to the definition of function $g_T$ . Namely:

1. 1. Given input u, $|u| = n$ , find a $\Sigma ^b_1$ -formula $\Phi \subseteq _e u$ with one free variable x such that $|\Phi | \le \log n$ . (It is unique if it exists.)

   • • If no such $\Phi $ exists, output $h_P(u) := \overline 0 \in {\{0,1\}^{n+1}}$ .

   • • Otherwise go to 2.

2. 2. Put $c: = |\Phi |+1$ . Going through all $w \in {\{0,1\}^{c+1}}$ in lexicographic order, do the following.Using ${{\mathbf{transl}}}$ compute formula ${\varphi _{n,w}}$ . If the computation does not halt in time $\le n^{\log n}$ stop and output $h_P(u) = \overline 0 \in {\{0,1\}^{n+1}}$ . Otherwise search for a P-proof of formula ${\varphi _{n,w}}$ by going systematically through all circuits D with $\le \log n \cdot \log \log n$ inputs and of size $\le (\log n)^{\log \log n}$ until some ${{\mathbf{tt}}}(D)$ is a P-proof of ${\varphi _{n,w}}$ .

   • • If a proof is found for all $w \in {\{0,1\}^{c+1}}$ output $h_P(u) := \overline 0 \in {\{0,1\}^{n+1}}$ .

   • • Otherwise let $w_0 \in {\{0,1\}^{c+1}}$ be the first such w such that no P-proof is found. Go to 3.

3. 3. Output $h_P(u) := w_0 u_0 \in {\{0,1\}^{n+1}}$ , where $u = \Phi u_0$ .

It is clear from the construction that function $h_P$ stretches each input by one bit (and hence the complement of its range is infinite) and that

$$ \begin{align*}Rng(h_P) \cap \{x \in {\{0,1\}^{n+1}}\ |\ \Phi(x)\} \neq \emptyset \end{align*} $$

for any $\Phi (x) \in \Sigma ^b_1$ and $n>> 1$ .

The time needed for the computation of $h_P(u)$ is $O(n)$ for step 1 and for step 2 it is bounded above by

$$ \begin{align*}2^{c+1} \cdot n^{\log n} \cdot 2^{(\log n)^{\log \log n}} \cdot 2^{O((\log n)^{\log \log n})} \le 2^{O((\log n)^{\log \log n})} . \end{align*} $$

The first factor bounds the number of w, the second bounds the time needed to compute ${\varphi _{n,w}}$ , the third bounds the number of circuits D, and the fourth one bounds the time needed to compute ${{\mathbf{tt}}}(D)$ and to check whether it is a P-proof of ${\varphi _{n,w}}$ (this is p-time in $|{{\mathbf{tt}}}(D)|$ ).