Proof compression and NP versus PSPACE. Part 2

TL;DR

This paper provides a complete proof confirming the conjecture that NP equals PSPACE, building upon previous work that addressed proof compression and complexity class relationships.

Contribution

It offers the first complete proof establishing NP = PSPACE, advancing the understanding of proof complexity and computational complexity class equivalences.

Findings

Confirmed NP = PSPACE conjecture

Extended previous partial results to a complete proof

Implications for proof compression and complexity theory

Abstract

We upgrade [1] to a complete proof of the conjecture NP = PSPACE. [1]: L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55-83 (2019)

Equations

$$\framebox{$\left(\rightarrow I\right):\dfrac{\QDATOP{\QDATOP{\left[\alpha\right]}{\vdots}}{\beta}}{\alpha\rightarrow\beta}$}\ ,\quad\framebox{$\ \left(\rightarrow E\right):\dfrac{\alpha\quad\quad\alpha\rightarrow\beta}{\beta}$}$$

$$\left(\rightarrow S\right):\dfrac{\overset{n\ times}{\overbrace{\alpha\quad\cdots\quad\alpha}}}{\alpha\ }\$$

$$\framebox{$\ \dfrac{\dfrac{\beta\quad\quad\gamma\quad\quad\gamma\rightarrow\left(\alpha\rightarrow\beta\right)}{\alpha\rightarrow\beta}}{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)}$}$$

$$\framebox{$\left(\rightarrow I\right)\ \dfrac{\left(\rightarrow S\right)\ \dfrac{\ \left(\rightarrow I\right)\ \dfrac{\beta}{\alpha\rightarrow\beta\ }\quad\left(\rightarrow E\right)\ \dfrac{\gamma\quad\quad\gamma\rightarrow\left(\alpha\rightarrow\beta\right)}{\alpha\rightarrow\beta\ }}{\alpha\rightarrow\beta\ }}{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\ }$}\text{.}$$

$$\framebox{$\dfrac{\dfrac{\dfrac{\beta\,;A=\left\{\beta\right\}}{\alpha\rightarrow\beta:\text{{i}}_{\rightarrow}\,;A=\left\{\beta\right\}\ }\quad\dfrac{\gamma\,;A=\left\{\gamma\right\}\quad\quad\gamma\rightarrow\left(\alpha\rightarrow\beta\right);A=\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}}{\alpha\rightarrow\beta:\text{{e}}_{\rightarrow}\,;A=\left\{\gamma,\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}}}{\alpha\rightarrow\beta\,:\text{{s}}_{\rightarrow};A=\circledS\left(\left\{\beta\right\},\left\{\gamma,\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)}}{\gamma\rightarrow\left(\alpha\rightarrow\beta\right):\text{{i}}_{\rightarrow}\,;A=\circledS\left(\left\{\beta\right\},\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)\ }$}$$

$$\framebox{$\dfrac{\dfrac{\dfrac{\dfrac{\beta\,;A=\left\{\beta\right\}}{\alpha\rightarrow\beta:\text{{i}}_{\rightarrow}\,;A=\left\{\beta\right\}\ }\quad\dfrac{\gamma\,;A=\left\{\gamma\right\}\quad\quad\gamma\rightarrow\left(\alpha\rightarrow\beta\right);A=\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}}{\alpha\rightarrow\beta:\text{{e}}_{\rightarrow}\,;A=\left\{\gamma,\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}}}{\alpha\rightarrow\beta\,:\text{{s}}_{\rightarrow};A=\circledS\left(\left\{\beta\right\},\left\{\gamma,\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)}}{\gamma\rightarrow\left(\alpha\rightarrow\beta\right):\text{{i}}_{\rightarrow}\,;A=\circledS\left(\left\{\beta\right\},\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)\setminus\left\{\gamma\right\}\ }}{\beta\rightarrow\left(\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right):\text{{i}}_{\rightarrow}\,;A=\circledS\left(\left\{\beta\right\},\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)\setminus\left\{\gamma\right\}\setminus\left\{\beta\right\}\ }$}$$

$$\mathcal{A}:\text{v}\left(\partial \right)\ni x\hookrightarrow A\left(x\right)\subseteq \mathrm{FOR}\left(\partial \right)$$

Full text

Proof Compression and NP Versus PSPACE. Part 2 ††margin:

July 2019

L. Gordeev

lew.gordeew@uni-tuebingen.de

**Abstract. **We upgrade [1] to a complete proof of the conjecture NP = PSPACE that is known as one of the fundamental open problems in the mathematical theory of computational complexity. Since minimal propositional logic is known to be PSPACE complete, while PSPACE to include NP, it suffices to show that every valid purely implicational formula $\rho$ has a proof whose weight (= total number of symbols) and time complexity of the provability involved are both polynomial in the weight of $\rho$. As is [1], we use proof theoretic approach – in both sequential and natural deduction forms. Recall that in [1] we considered any valid $\rho$ in question that had (by the definition of validity) a “short” tree-like proof $\pi$ in the Hudelmaier-style cutfree sequent calculus for minimal logic. The “shortness” means that the height of $\pi$, the total number and maximum weight of different formulas occurring in it are all polynomial in the weight of $\rho$. However, the size (= total number of nodes), and hence also the weight, of $\pi$ could be exponential in that of $\rho$. To overcome this trouble we embedded $\pi$ into Prawitz’s proof system of natural deductions containing single formulas, instead of sequents. As in $\pi$, the height, the total number and maximum formula weight of the resulting tree-like natural deduction $\partial_{1}$ are polynomial, whereas the size of $\partial_{1}$ still could be exponential, in the weight of $\rho$. In our next, crucial move, $\partial_{1}$ was deterministically compressed into a “small” dag-like deduction $\partial$ whose horizontal levels contained only mutually different formulas, which made the whole weight polynomial in that of $\rho$. However, $\partial$ required a more complicated verification of the underlying provability of $\rho$. In the present paper we further compress $\partial$ into a desired deduction $\partial_{0}$ that deterministically proves $\rho$ in time and space polynomial in the weight of $\rho$. [Working in a natural deduction calculus is essential because tree-to-dag horizontal compression of $\pi$ merging equal sequents, instead of formulas, is (possible but) insufficient, since the total number of different sequents occurring in $\pi$ might be exponential in the weight of $\rho$ – even assuming that all formulas occurring in sequents are subformulas of $\rho$.]

1 Introduction

Gordeev and Haeusler [1] presented a dag-like version of Prawitz’s [7] tree-like natural deduction calculus for minimal logic, NM*→, and left open a problem of computational complexity of the dag-like provability involved ([1]: Problem 22). In this paper we show a solution that proves the conjecture NP = PSPACE. To explain it briefly first consider plain tree-like provability. Recall that our basic deduction calculus NM→* includes two standard inferences

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and one auxiliary repetition rule \framebox{\left(\rightarrow R\right):\dfrac{\alpha}{\alpha\ }}, where $\left[\alpha\right]$ in $\left(\rightarrow I\right)$ indicates that all $\alpha$-leaves occurring above $\beta$-node exposed are discharged assumptions.

Definition 1

A given (whether tree- or dag-like) NM→**-deduction $\partial$ proves its root-formula $\rho$ (abbr.: $\partial\vdash\rho$) iff every maximal thread connecting the root with a leaf labeled $\alpha$ is closed (= discharged), i.e. it contains a $\left(\rightarrow I\right)$ with conclusion $\alpha\rightarrow\beta$, for some $\beta$. 2. 2.

A purely implicational formula $\rho$ is valid in minimal logic iff there exists a tree-like NM→**-deduction $\partial$ that proves $\rho$; such $\partial$ is called a proof of $\rho$.

Remark 2

Tree-like constraint in the second part of definition is inessential. That is, for any dag-like $\partial\in\,$NM→* with root-formula $\rho$, if $\partial\vdash\rho$ then $\rho$ is valid in minimal logic.*

This is because any given dag-like $\partial$ can be unfolded into a tree-like deduction $\partial^{\prime}$ by straightforward thread-preserving bottom-up recursion. Namely, every node $x\in\partial$ having $n>1$ distinct conclusions should be replaced by $n$ distinct nodes $x_{1},\cdots,x_{n}\in\partial^{\prime}$ with corresponding single conclusions and (identical) premises of $x$. This operation obviously preserves the closure of threads, i.e. $\partial\vdash\rho$ infers $\partial^{\prime}\vdash\rho$.

Formal verification of the assertion $\partial\vdash\rho$ is simple, as follows – whether for tree-like or generally dag-like $\partial$. Every node $x\in\partial$ is assigned, by descending recursion, a set of assumptions $A\left(x\right)$ such that:

$A\left(x\right):=\left\{\alpha\right\}$ if $x$ is a leaf labeled $\alpha$, 2. 2.

$A\left(x\right):=A\left(y\right)$ if $x$ is the conclusion of $\left(\rightarrow R\right)$ with premise $y$, 3. 3.

$A\left(x\right):=A\left(y\right)\setminus\left\{\alpha\right\}$ if $x$ is the conclusion of $\left(\rightarrow I\right)$ with label $\alpha\rightarrow\beta$ and premise $y$, 4. 4.

$A\left(x\right):=A\left(y\right)\cup A\left(z\right)$ if $x$ is the conclusion of $\left(\rightarrow E\right)$ with premises $y,$ $z$.

This easily yields

Lemma 3

Let $\partial\in\,$NM→* (whether tree- or dag-like). Then $\partial\vdash\rho\Leftrightarrow A\left(r\right)=\emptyset$ holds with respect to standard set-theoretic interpretations of $\cup$ and $\setminus$ in $A\left(r\right)$, where $r$ and $\rho$ are the root and root-formula of $\partial$, respectively. Moreover, $A\left(r\right)\overset{?}{=}\emptyset$ is verifiable by a deterministic TM in $\left|\partial\right|$-polynomial time, where by $\left|\partial\right|$ we denote the weight (i.e. total number of symbols) of $\partial$. 111The latter is completely analogous to the well-known polynomial-time decidability of the circuit value problem (see also Appendix).*

Now let us upgrade NM*→* to NM${}_{\rightarrow}^{\flat}$ by adding a new separation rule $\left(\rightarrow S\right)$

[TABLE]

whose identical premises are understood disjunctively: “if at least one premise is proved then so is the conclusion” (in contrast to ordinary conjunctive inference: “if all premises are proved then so is the conclusion”). Note that in dag-like deductions the nodes might have several conclusions (unlike in tree-like ones). The modified assignment $A$ in NM${}_{\rightarrow}^{\flat}$ (that works in both tree-like and dag-like cases) is defined by adding to old recursive clauses 1–4 (see above) a new clause 5 with new separation symbol $\circledS$ :

5. $A\left(x\right)=\circledS\left(A\left(y_{1}\right),\cdots,\,A\left(y_{n}\right)\right)$ if $x$ is the conclusion of $\left(\rightarrow S\right)$ with premises $y_{1},\cdots,y_{n}$.

Claim 4

For any dag-like deduction $\partial\in\,$NM${}_{\rightarrow}^{\flat}$ whose root $r$ is labeled $\rho$, $\rho$ is valid in minimal logic, provided that $A\left(r\right)$ reduces to $\emptyset$ (abbr.: $A\left(r\right)\vartriangleright\emptyset$) by standard set-theoretic interpretations of $\cup$, $\setminus$ and nondeterministic disjunctive valuation $\circledS\left(t_{1},\cdots,\,t_{n}\right):=t_{i}$, for any chosen $i\in\left\{1,\cdots,n\right\}$. Moreover, the assertion $A\left(r\right)\vartriangleright\emptyset$ (that is also referred to as ‘$\partial$ proves $\rho$’) can be confirmed by a nondeterministic TM in $\left|\partial\right|$-polynomial time.

This claim reduces to its trivial NM*→* case (see above). For suppose that $A\left(r\right)\vartriangleright\emptyset$ holds with respect to successive nondeterministic valuation of the occurrences $\circledS$. This reduction determines successive ascending (i.e. bottom-up) thinning of $\partial$ that results in a dag-like $\partial_{0}\in\,$NM${}_{\rightarrow}^{\flat}$, while $A\left(r\right)\vartriangleright\emptyset$ in $\partial$ implies $A\left(r\right)=\emptyset$ in $\partial_{0}$. Since $\left(\rightarrow S\right)$ does not occur in $\partial_{0}$ anymore, we have $\partial_{0}\in\,$NM*→. That is, there is a dag-to-dag operation NM${}_{\rightarrow}^{\flat}$ $\ni\partial\hookrightarrow\partial_{0}\in\,$NM→ that preserves provability, the root-formula and the weight upper bound*. By previous considerations with regard to NM*→* we conclude that $\rho$ is valid in minimal logic, which can be confirmed in $\left|\partial\right|$-polynomial time, as required.

Since minimal logic is PSPACE complete ([9], [10]), in order to arrive at the desired conclusion NP = PSPACE it will suffice to show that for any valid $\rho$ there is a dag-like deduction $\partial\in\,$NM${}_{\rightarrow}^{\flat}$ of $\rho$ satisfying $A\left(r\right)\vartriangleright\emptyset$, whose size and maximal formula length are polynomial in the weight $\left|\rho\right|$ of $\rho$. But this is a consequence of [1]. Recall that in [1] we presented a deterministic tree-to-dag horizontal compression $\left|\rho\right|$-polynomially reducing the weight of a given “large” tree-like deduction of $\rho$ in NM→ that is obtained by embedding a derivation of $\rho$ in a Hudelmaier-style [3] cutfree sequent calculus. This compression resulted in a suitable “small” dag-like deduction frame together with a locally coherent set of maximal threads, in the multipremise expansion of NM→ (called NM${}_{\rightarrow}^{\ast}$). This pair determines a deduction in NM${}_{\rightarrow}^{\flat}$ that admits a fundamental set of chains (see below). In this paper we describe a nondeterministic dag-to-dag horizontal cleansing further converting such NM${}_{\rightarrow}^{\flat}$-deduction into a required “cleansed” deduction satisfying $A\left(r\right)\vartriangleright\emptyset$ (actually $A\left(r\right)=\emptyset$, already in NM*→*).

1.1 Recollection of [1]

Recall that $\rho$ is called dag-like provable in NM${}_{\rightarrow}^{\ast}$ iff there exists a locally correct deduction frame $\widetilde{D}=\!\left\langle D,\text{{s}},\ell^{\text{{f}}}\right\rangle$ with root-formula $\rho=\ell^{\text{{f}}}\left(r\right)$, 222The weight of $\widetilde{D}$ is assumed to be polynomial in that of $\rho$ (see [1]). together with a locally coherent pair $\partial\!=\!\left\langle\widetilde{D},G\right\rangle$ such that $G:\overrightarrow{\text{{e}}}\!\left(D\right)\rightarrow\left\{0,1\right\}$ determines a set of threads that confirms alleged validity of $\rho$, where $\overrightarrow{\text{{e}}}\!\left(D\right)$ denotes the set of edge-chains in $D$. Such $\partial$ is called a dag-like proof of $\rho$ in NM${}_{\rightarrow}^{\ast}$.333Here and below basic notions and notations are imported from [1]. Note that NM${}_{\rightarrow}^{\ast}$-deductions may have arbitrary many premises and/or conclusions. Without loss of generality we assume that $\widetilde{D}$ is horizontally compressed, i.e. $\ell^{\text{{f}}}\!\left(x\right)\neq\ell^{\text{{f}}}\!\left(y\right)$ for all $x\neq y$ on the same level in $D$ ([1]: Ch. 3.1, 3.2). In [1] we also observed that the local correctness of $\widetilde{D}$ is verifiable in $\left|\rho\right|$-polynomial time, although local coherence of $\!\left\langle\widetilde{D},G\right\rangle$ has no obvious low-complexity upper bound, as $\overrightarrow{\text{{e}}}\!\left(D\right)$ is generally exponential. The currently proposed upgrade is based on the fundamental sets of threads, instead of $G$ and $\overrightarrow{\text{{e}}}\!\left(D\right)$, as follows.

1.2 Upgrade in NM${}_{\rightarrow}^{\ast}$

Let $\widetilde{D}=\!\left\langle D,\text{{s}},\ell^{\text{{f}}}\right\rangle$ be a given locally correct deduction frame with root-formula $\rho=\ell^{\text{{f}}}\left(r\right)$, K$\left(D\right)$ be the set of maximal ascending chains (also called threads) consisting of nodes (vertices) $u\in\,$v$\left(D\right)$ connecting root $r$ with leaves. A given set $\mathcal{F}\subset\,$K$\left(D\right)$ is a fundamental set of threads (abbr.: fst) in $\widetilde{D}$ if the following three conditions are satisfied, where for any $\Theta=\left[r=x_{0},\cdots,x_{h\left(D\right)}\right]\in\,$K$\left(D\right)$ and $i\leq h\left(D\right)$ we let $\Theta\!\upharpoonright_{x_{i}}:=\left[x_{0},\cdots,x_{i}\right]$.

$\mathcal{F}$ is dense in $D$, i.e. $\left(\forall u\in\text{{v}}\left(D\right)\right)\left(\exists\Theta\in\mathcal{F}\right)\left(u\in\Theta\right)$. 2. 2.

Every $\Theta\in\mathcal{F}\,$is closed, i.e. its leaf-formula $\ell^{\text{{f}}}\!\left(x_{h\left(D\right)}\right)$ is discharged in $\Theta$. 3. 3.

$\mathcal{F}$ preserves $\left(\rightarrow E\right)$, i.e.

\left.\begin{array}[]{c}\left(\forall\Theta\in\mathcal{F}\right)\left(\forall u\in\Theta\right)\left(\forall v\neq w\in\text{{v}}\left(D\right)\!:\left\langle u,v\right\rangle\!,\left\langle u,w\right\rangle\in\text{{e}}\left(D\right)\wedge v\in\Theta\right)\\ \left(\exists\Theta^{\prime}\in\mathcal{F}\right)\left(w\in\Theta^{\prime}\wedge\Theta\!\upharpoonright_{u}=\Theta^{\prime}\!\upharpoonright_{u}\right)\qquad\quad\qquad\qquad\qquad.\end{array}\right.

Lemma 5

Let $\widetilde{D}$ be as above and suppose that there exists a fst $\mathcal{F}$ in $\widetilde{D}$. Then $\rho$ is dag-like provable in NM${}_{\rightarrow}^{\ast}$.

Proof. Define $G:\overrightarrow{\text{{e}}}\!\left(D\right)\rightarrow\left\{0,1\right\}$ by setting $G\left(\overrightarrow{e}\right)=1$ iff $\Theta\left[\overrightarrow{e}\right]\in\mathcal{F}$, where $\Theta\left[\overrightarrow{e}\right]\in\,$K$\left(D\right)$ contains all nodes occurring in the canonical thread-expansion of $\overrightarrow{e}$. Then $\partial\!=\left\langle\widetilde{D},G\right\rangle$ is a dag-like* *proof of $\rho$. The local coherence conditions 1, 2, 4, 5 (cf. [1]: Definition 6) are easily verified. In particular, 4 follows from the third fst condition with respect to $\mathcal{F}$.

Lemma 6

For any dag-like proof $\left\langle\widetilde{D},G\right\rangle$ of $\rho$ there are $D_{0}\subseteq\,D$, $G_{0}:\overrightarrow{\text{{e}}}\!\left(D_{0}\right)\rightarrow\left\{0,1\right\}$, $\mathcal{F}\subset\,$K$\left(D_{0}\right)$ and a dag-like proof $\left\langle\widetilde{D_{0}},G_{0}\right\rangle$ of $\rho$ such that $\mathcal{F}$ is fst in $\widetilde{D_{0}}$.

Proof. Let $\mathcal{F}:=\left\{\Theta:G\left(\overrightarrow{e}\left[\Theta\right]\right)=1\right\}$ for $\overrightarrow{e}\left[\Theta\right]:=\overrightarrow{e_{m}}\in\overrightarrow{\text{{e}}}\!\left(D\right)$ determined by $\Theta$ as specified in [1]: Definition 8. It is readily seen that such $\mathcal{F}$ is a fst in $\widetilde{D}$. The crucial condition 3 follows directly from the corresponding local coherence condition 4 (cf. [1]: Definition 6). Let $D_{0}\subseteq D$ be the minimum sub-dag containing every edge occurring in $\underset{\Theta\in\mathcal{F}}{\bigcup}\Theta$ and let $\widetilde{D_{0}}=\!\left\langle D_{0},\text{{s}},\ell^{\text{{f}}}\right\rangle$ be the corresponding sub-frame of $\widetilde{D}$. Obviously $\widetilde{D_{0}}$ is locally correct. Define $G_{0}:\overrightarrow{\text{{e}}}\!\left(D_{0}\right)\rightarrow\left\{0,1\right\}$ by setting $G\left(\overrightarrow{e}\right)=1$ iff $\Theta\left[\overrightarrow{e}\right]\in\mathcal{F}$, where $\Theta\left[\overrightarrow{e}\right]\in\,$K$\left(D_{0}\right)$ contains all nodes occurring in the canonical thread-expansion of $\overrightarrow{e}$. Then $\partial\!=\left\langle\widetilde{D_{0}},G_{0}\right\rangle$ is a dag-like* *proof of $\rho$. The crucial density of $\mathcal{F}$ in $D_{0}$ obviously follows from definitions of $D_{0}$ and $G_{0}$, as every edge in $D_{0}$ occurs in some thread from $\mathcal{F}$, while for any $\overrightarrow{e}\in\overrightarrow{\text{{e}}}\!\left(D_{0}\right)$ we have $G_{0}\left(\overrightarrow{e}\right)=1$ iff $\Theta\left[\overrightarrow{e}\right]\in\mathcal{F}$.

Together with [1]: Corollaries 15, 20 these Lemmata yield

Corollary 7

Any given $\rho$ is valid in minimal logic iff there exists a pair $\left\langle\widetilde{D},\mathcal{F}\right\rangle$ such that $\widetilde{D}$ is a locally correct deduction frame with root-formula $\rho=\ell^{\text{{f}}}\left(r\right)$ and $\mathcal{F}$ being a fst in $\widetilde{D}$. We can just as well assume that $\widetilde{D}$ is horizontally compressed and its weight is polynomial in that of $\rho$.

Remark 8

We can’t afford $\mathcal{F}$ to be by any means polynomial in $\rho$. However, mere existence of $\mathcal{F}$ enables a nondeterministic polytime verification of $A\left(r\right)\vartriangleright\emptyset$ in the corresponding modified dag-like formalism, as follows.

2 Modified dag-like calculus NM${}_{\rightarrow}^{\mathrm{\flat}}$

As mentioned above, our modified dag-like deduction calculus, NM${}_{\rightarrow}^{\mathrm{\flat}}$, includes inference rules $\left(\rightarrow I\right)$, $\left(\rightarrow E\right)$, $\left(\rightarrow R\right)$, $\left(\rightarrow S\right)$ (see Introduction). $\left(\rightarrow I\right)$, $\left(\rightarrow R\right)$ and $\left(\rightarrow E\right)$ have one and two premises, respectively, whereas $\left(\rightarrow S\right)$ has two or more ones. NM${}_{\rightarrow}^{\mathrm{\flat}}$-deductions are graphically interpreted as labeled rooted regular dags (abbr.: redags, cf. [1]) $\partial=\,\left\langle\text{{v}}\left(\partial\right),\text{{e}}\left(\partial\right)\right\rangle\,$, whose nodes may have arbitrary many parents (conclusions) – and children (premises), just in the case $\left(\rightarrow S\right)$, – if any at all. The nodes ($x$, $y$, $z$, …) are labeled by $\ell^{\text{{f}}}$ with purely implicational formulas ($\alpha$, $\beta$, $\gamma$, $\rho$, …). For the sake of brevity we’ll assume that nodes $x$ are supplied with auxiliary height numbers $h\left(x\right)\in\mathbb{N}$, while all inner nodes also have special labels $\ell^{\text{{n}}}\left(x\right)\in\left\{\text{{i}}_{\rightarrow}\text{{,e}}_{\rightarrow}\text{{,r}}_{\rightarrow}\text{{,s}}_{\rightarrow}\right\}$ showing the names of the inference rules $\left(\rightarrow I\right)$, $\left(\rightarrow E\right)$, $\left(\rightarrow R\right)$, $\left(\rightarrow S\right)$ with conclusion $x$. The roots and root-formulas are always designated $r$ and $\rho:=\ell^{\text{{f}}}\left(r\right)$, respectively. The edges $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)\subset\,$v$\left(\partial\right)^{2}$ are directed upwards (thus $r$ is the lowest node in $\partial$) in which $x$ and $y$ are called parents and children of each other, respectively. The leaves l$\left(\partial\right)\subseteq\,$v$\left(\partial\right)$ are the nodes without children. Tree-like NM${}_{\rightarrow}^{\mathrm{\flat}}$-deductions are those ones whose redags are trees (whose nodes have at most one parent).

Definition 9

A given NM${}_{\rightarrow}^{\mathrm{\flat}}$-deduction $\partial$ is locally correct if conditions 1–2 are satisfied, for any $x,y,z,u\in\,$v$\left(\partial\right)$.

$\partial$* is regular (cf. [1]), i.e.*

1. (a)

if $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$ then $x\notin\,$l$\left(\partial\right)$ and $y\neq r$, 2. (b)

$h\left(r\right)=0$, 3. (c)

if $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle\in\,$e$\left(\partial\right)$ then $h\left(y\right)=h\left(z\right)=h\left(x\right)+1$. 2. 2.

$\partial$* formalizes the inference rules, i.e.*

1. (a)

if $\ell^{\text{{n}}}\left(x\right)=\,$r→* and $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle\in\,$e$\left(\partial\right)$ then $y=z$ and $\ell^{\text{{f}}}\left(y\right)=\ell^{\text{{f}}}\left(x\right)$ [: rule $\left(\rightarrow R\right)$],* 2. (b)

if $\ell^{\text{{n}}}\left(x\right)=\,$i→* and $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle\in\,$e$\left(\partial\right)$ then $y=z$ and $\ell^{\text{{f}}}\left(x\right)=\alpha\rightarrow\ell^{\text{{f}}}\left(y\right)$ for some (uniquely determined) $\alpha$ [: rule $\left(\rightarrow I\right)$],* 3. (c)

if $\ell^{\text{{n}}}\left(x\right)=\,$e→* and $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle,\left\langle x,u\right\rangle\in\,$e$\left(\partial\right)$ then $\left|\left\{x,y,z\right\}\right|=2$ and if $y\neq z$ then either $\ell^{\text{{f}}}\left(z\right)=\ell^{\text{{f}}}\left(y\right)\rightarrow\ell^{\text{{f}}}\left(x\right)$ or else $\ell^{\text{{f}}}\left(y\right)=\ell^{\text{{f}}}\left(z\right)\rightarrow\ell^{\text{{f}}}\left(x\right)$ [: rule $\left(\rightarrow E\right)$],* 4. (d)

if $\ell^{\text{{n}}}\left(x\right)=\,$s→* and $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$ then $\ell^{\text{{f}}}\left(y\right)=\ell^{\text{{f}}}\left(x\right)$ and $\ell^{\text{{n}}}\left(y\right)\neq\,$s→** [: rule $\left(\rightarrow S\right)$].*

NM${}_{\rightarrow}^{\ast}$ is easily embeddable into NM${}_{\rightarrow}^{\mathrm{\flat}}$. Namely, consider a locally correct NM${}_{\rightarrow}^{\ast}$-deduction frame $\widetilde{D}=\!\left\langle D,\text{{s}},\ell^{\text{{f}}}\right\rangle$. The corresponding locally correct dag-like NM${}_{\rightarrow}^{\mathrm{\flat}}$-deduction $\partial$ arises from $D$ by ascending recursion on the height. 444For brevity we omit $h$, as every $h\left(x\right)$ is uniquely determined by $x$. The root and basic configurations of types $\left(\rightarrow I\right)$, $\left(\rightarrow E\right)$, $\left(\rightarrow R\right)$ in $\widetilde{D}$ should remain unchanged. Furthermore, if $u$ is a node having several groups of premises in $D$, i.e. $\left|\text{{s}}\left(u,D\right)\right|>1$ (cf. [1]) then in $\partial$ we separate these groups via $\left(\rightarrow S\right)$ with $\left|\text{{s}}\left(u,D\right)\right|$ identical premises; thus for example NM${}_{\rightarrow}^{\ast}$-configuration in $\widetilde{D}$

[TABLE]

goes to this NM${}_{\rightarrow}^{\mathrm{\flat}}$ -configuration in $\partial$

[TABLE]

Corresponding $\ell^{\text{{f}}}$- and $\ell^{\text{{n}}}$-labels are induced in an obvious way. Note that the weight of $\partial$ is linear in that of $\widetilde{D}$. 555Recall that according to [1] we can just as well assume that $\widetilde{D}$ is horizontally compressed and its weight is polynomial in that of $\rho$.

Now suppose that there is a fst $\mathcal{F}$ in a chosen NM${}_{\rightarrow}^{\ast}$-deduction frame $\widetilde{D}$, and let $\mathcal{F}^{\mathrm{\flat}}$ be the image of $\mathcal{F}$ in $\partial$. It is readily seen that $\mathcal{F}^{\mathrm{\flat}}$ is also a dense and $\left(\rightarrow E\right)$ preserving set of closed threads in $\partial$ (see NM${}_{\rightarrow}^{\ast}$-clauses 1–3 in Ch. 1.2). That is, $\mathcal{F}^{\mathrm{\flat}}$ is a dense set of closed threads in $\partial$ such that for every $\Theta\in\mathcal{F}^{\mathrm{\flat}}$ and $\left(\rightarrow E\right)$-conclusion $x\in\Theta$, $\ell^{\text{{n}}}\left(x\right)=\,$e*→*, with premises $y$ and $z$, if $y\in\Theta$ then there is a $\Theta^{\prime}\in\mathcal{F}^{\mathrm{\flat}}$ such that $z\in\Theta^{\prime}$ and $\Theta$ coincides with $\Theta^{\prime}$ below $x$.

2.1 Modified dag-like provability

We formalize in NM${}_{\rightarrow}^{\mathrm{\flat}}$ the modified assignment $\mathcal{A}:\partial\ni x\hookrightarrow A\left(x\right)\subseteq\mathrm{FOR}\left(\partial^{\flat}\right)$.

Definition 10** (Assignment $\mathcal{A}$)**

Let $\partial^{\mathrm{\flat}}$ be any locally correct dag-like NM${}_{\rightarrow}^{\mathrm{\flat}}$-deduction. Assign nodes $x\in\partial$ with terms $A\left(x\right)$ by descending recursion 1–5.

$A\left(x\right):=\left\{\alpha\right\}$* if $x$ is a leaf and $\ell^{\text{{f}}}\left(x\right)=$ $\alpha$.* 2. 2.

$A\left(x\right):=A\left(y\right)$* if $\ell^{\text{{n}}}\left(x\right)=\text{{r}}_{\rightarrow}$ and $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$.* 3. 3.

$A\left(x\right):=A\left(y\right)\setminus\left\{\alpha\right\}$* if $\ell^{\text{{n}}}\left(x\right)=\text{{i}}_{\rightarrow}$, $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$ and $\ell^{\text{{f}}}\left(x\right)=\alpha\rightarrow\ell^{\text{{f}}}\left(y\right)$.* 4. 4.

$A\left(x\right):=A\left(y\right)\cup A\left(z\right)$* if $\ell^{\text{{n}}}\left(x\right)=\text{{e}}_{\rightarrow}$ and $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle\in\,\text{{e}}\left(\partial\right)$.* 5. 5.

$A\left(x\right):=\circledS\left(A\!\left(y_{1}\!\right),\cdots,\,A\!\left(y_{n}\!\right)\right)$* if $\ell^{\text{{n}}}\left(x\right)\!=\!\text{{s}}_{\rightarrow}$ and $\left(\forall i\!\in\!\left[1,n\right]\right)\left\langle x,y_{i}\right\rangle\!\in\,$e$\left(\partial\right)$.*

Definition 11** (Nondeterministic reduction)**

Let $\partial$ and $\mathcal{A}$ be as above, $r$ the root of $\partial$, $S$ a set of formulas occurring in $\partial$. We say that $A\left(r\right)$ reduces* to $S$ (abbr.: $A\left(r\right)\!\vartriangleright$ $\!S$) if $S$ arises from $A\left(r\right)$ by successive (in a left-to-right direction) substitutions $A\left(u\right)=\circledS\left(A\left(v_{1}\right),\cdots,\,A\left(v_{n}\right)\right):=A\left(v_{i}\right)$, for a fixed chosen $i\in\left\{1,\cdots,n\right\}$ and for any occurrence $A\left(u\right)$ in $A\left(w\right)$ and hence in $A\left(w^{\prime}\right)$, for every $w^{\prime}$ below $w$, provided that $u$ is a premise of $w$ such that $\ell^{\text{{n}}}\left(u\right)=\text{{s}}_{\rightarrow}$,666This operation is graphically interpreted by deleting $u$ along with $v_{j}$ for all $j\neq i$.   while using ordinary set-theoretic interpretations of $\cup$ and $\setminus$. We call $\partial$ a modified dag-like proof of $\rho=\ell^{\text{{f}}}\left(r\right)$ (abbr.: $\partial\vdash\rho$) if $A\left(r\right)\vartriangleright\emptyset$ holds.*

Example 12

Previously shown configuration yields a $\partial$ such that $\partial\nvdash\rho$ :

[TABLE]

where $\ell^{\text{{n}}}\left(r\right)=\,$i→, $\ell^{\text{{f}}}\left(r\right)\!=\!\rho\!=\!\gamma\rightarrow\left(\alpha\rightarrow\beta\right)$ and $A\left(r\right)\!=\!\circledS\left(\left\{\beta\right\},\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right)$. Note that $A\left(r\right)\vartriangleright\left\{\beta\right\}$ and $A\left(r\right)\vartriangleright\left\{\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}$, although $A\left(r\right)\ntriangleright\emptyset$.

To obtain an analogous dag-like proof of (say) $\rho^{\prime}:=\beta\rightarrow\left(\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right)$ we’ll upgrade $\partial$ to such $\partial^{\prime}$ :

[TABLE]

and let $\circledS\left(\left\{\beta\right\},\left\{\gamma,\gamma\rightarrow\left(\alpha\rightarrow\beta\right)\right\}\right):=\left\{\beta\right\}$. Then $A\left(r\right)\vartriangleright\emptyset$ , i.e. $\partial^{\prime}\vdash$ $\rho_{1}$ holds.

Lemma 13

Every modified dag-like proof of $\rho$ is convertible to a dag-like NM→-proof of $\rho$.

Proof. Let $\partial$ be a given NM${}_{\rightarrow}^{\mathrm{\flat}}$-proof of $\rho$. Its NM*→-conversion is defined by a simple ascending recursion, as follows. Each time we arrive at a $w$ whose premise $u$ is a conclusion of $\left(\rightarrow S\right)$, we replace this $u$ by its chosen premise that is “guessed” by a given nondeterministic reduction leading to $A\left(r\right)\vartriangleright\emptyset$ – alternatively, we replace this $\left(\rightarrow S\right)$ by the corresponding repetition $\left(\rightarrow R\right)$. It is readily seen that the resulting dag-like deduction $\partial_{0}$ with the same root-formula $\rho$ is locally correct and $\left(\rightarrow S\right)$-free, and hence it belongs to NM→. Obviously this conversion preserves a given assignment $x\hookrightarrow A\left(x\right)$. Also note that $A\left(r\right)\vartriangleright\emptyset$ in $\partial$ infers $A\left(r\right)=\emptyset$ in $\partial_{0}$, and hence $\partial_{0}$ proves $\rho$ in NM→*.

Lemma 14

Let $\widetilde{D}$ be any locally correct deduction frame in NM${}_{\rightarrow}^{\ast}$ with root-formula $\rho$ that admits some fst. There exists a dag-like NM→-proof of $\rho$ whose weight does not exceed that of $\widetilde{D}$.

Proof. Let $\partial$ be the NM${}_{\rightarrow}^{\mathrm{\flat}}$-deduction of $\rho$ induced by $\widetilde{D}$ and $\mathcal{F}$ any fst in $\widetilde{D}$. Furthermore, let $\mathcal{F}^{\mathrm{\flat}}$ be the image of $\mathcal{F}$ in $\partial$ (see above). We’ll show that $\mathcal{F}^{\mathrm{\flat}}$ determines successive left-to-right $\circledS$-eliminations $\circledS\left(\!A\left(y_{1}\right),\cdots,\,A\left(y_{n}\right)\!\right)\!\hookrightarrow\!A\left(y_{i}\right)$ inside $A\left(r\right)$ leading to required reduction $A\left(r\right)\vartriangleright\emptyset$. These eliminations together with a suitable sub-fst $\mathcal{F}_{0}^{\mathrm{\flat}}$ $\subseteq\mathcal{F}^{\mathrm{\flat}}$ arise as follows by ascending recursion along $\mathcal{F}^{\mathrm{\flat}}$. Let $x$ with $\ell^{\text{{n}}}\left(x\right)=\text{{e}}_{\rightarrow}$ be a chosen lowest conclusion of $\left(\rightarrow E\right)$ in $\partial$, if any exists. By the density of $\mathcal{F}^{\mathrm{\flat}}$, there exists $\Theta\in\mathcal{F}^{\mathrm{\flat}}$ with $x\in\Theta$; we let $\Theta\in\mathcal{F}_{0}^{\mathrm{\flat}}$. Let $y$ and $z$ be the two premises of $x$ and suppose that $y\in\Theta$. By the third fst condition there exists a $\Theta^{\prime}\in\mathcal{F}^{\mathrm{\flat}}$ with $z\in\Theta^{\prime}$ and $\Theta\!\upharpoonright_{x}=$ $\Theta^{\prime}\!\upharpoonright_{x}$; so let $\Theta^{\prime}\in\mathcal{F}_{0}^{\mathrm{\flat}}$ be the corresponding “upgrade”of $\Theta$. In the case $z\in\Theta$ we let $\Theta^{\prime}:=\Theta$. Note that $\Theta\!\!\upharpoonright_{x}$ determines substitutions $A\left(u\right)=\circledS\left(A\left(v_{1}\right),\cdots,\,A\left(v_{n}\right)\right):=A\left(v_{i}\right)$ in all parents of $\left(\rightarrow S\right)$-conclusions $u$ occurring in both $\Theta$ and $\Theta^{\prime}$ below $x$ (cf. Definitions 10, 11), if any exist, and hence also $\circledS$-eliminations $A\left(u\right)\hookrightarrow A\left(v_{i}\right)$ in the corresponding subterms of $A\left(r\right)$. The same procedure is applied to the nodes occurring in $\Theta$ and $\Theta^{\prime}$ above $x$ under the next lowest conclusions of $\left(\rightarrow E\right)$; this yields new “upgraded” threads $\Theta^{\prime\prime},\Theta^{\prime\prime\prime},\cdots\in\mathcal{F}_{0}^{\mathrm{\flat}}$ and $\circledS$-eliminations in the corresponding initial fragments of $A\left(r\right)$. We keep doing this recursively until the list of remaining $\circledS$-occurrences in $\Theta\in\mathcal{F}_{0}^{\mathrm{\flat}}$ is empty. The final “cleansed” $\circledS$-free form of $A\left(r\right)$ is represented by a set of formulas that easily reduces to $\emptyset$ by ordinary set-theoretic interpretation of the remaining operations $\cup$ and $\setminus$, since every $\Theta\in\mathcal{F}_{0}^{\mathrm{\flat}}$ involved is closed. That is, the correlated “cleansed” deduction $\partial_{0}$ is a locally correct dag-like deduction of $\rho$ in the $\left(\rightarrow S\right)$-free fragment of NM${}_{\rightarrow}^{\mathrm{\flat}}$, and hence it belongs to NM*→; moreover the set of ascending threads in $\partial_{0}$ is uniquely determined by the remaining rules $\left(\rightarrow R\right)$, $\left(\rightarrow I\right)$, $\left(\rightarrow E\right)$ (cf. analogous passage in the previous proof). Now by the definition these “cleansed” ascending threads are all included in $\mathcal{F}_{0}^{\mathrm{\flat}}$ and hence closed with respect to $\left(\rightarrow I\right)$. 777These threads may be exponential in number, but our nondeterministic algorithm runs on the polynomial set of nodes. This yields a desired reduction $A\left(r\right)\vartriangleright\emptyset$. Hence $\partial_{0}$ proves $\rho$ in NM→*. Obviously the weight of $\partial_{0}$ does not exceed the weight of $\widetilde{D}$.

Operation $\partial\hookrightarrow\partial_{0}$ is referred to as horizontal cleansing (cf. Introduction). Together with Remark 2 and Corollary 7 these lemmata yield

Corollary 15

Any given $\rho$ is valid in minimal logic iff it is provable in NM→* by a dag-like deduction $\partial_{0}$ whose weight is polynomial in that of $\rho$ and such that $\partial_{0}\vdash\rho$ can be confirmed by a deterministic TM in $\left|\rho\right|$-polynomial time. 888See Appendix for a more exhaustive presentation.*

Theorem 16

$\mathbf{PSPACE}\subseteq\mathbf{NP}$* and hence $\mathbf{NP=PSPACE}$.*

Proof. Minimal propositional logic is PSPACE-complete (cf. e.g. [5], [9], [10]). Hence $\mathbf{PSPACE}\subseteq\mathbf{NP}$ directly follows from Corollary 15. Note that in contrast to [1] here we use nondeterministic arguments twice. First we “guess” the existence of Hudelmaier-style cutfree sequential deduction of $\rho$ that leads (by deterministic compression) to a “small” natural deduction frame $\widetilde{D}$ that is supposed to have a fst $\mathcal{F}$. Having this we “guess” the existence of  “cleansed” modified subdeduction that confirms in $\left|\rho\right|$-polynomial time the provability of $\rho$ with regard to $\left\langle\widetilde{D},\mathcal{F}\right\rangle$.

Corollary 17

$\mathbf{NP=Co}$-$\mathbf{NP}$ and hence the polynomial hierarchy collapses to the first level.

Proof. $\mathbf{NP\!=\!PSPACE}$ implies $\mathbf{Co}$-$\mathbf{NP\!=\!Co}$-$\mathbf{PSPACE\!=\!PSPACE\!=\!NP}$ (see also [6]).

3 Appendix: rough complexity estimate

3.1 Dag-like proof system NM*→*

We regard NM*→* as NM${}_{\rightarrow}^{\flat}$ without separation rule $\left(\rightarrow S\right)$. Moreover, without loss of generality we suppose that dag-like NM*→*-deductions $\partial$ of root-formulas $\rho$ have polynomial total number of vertices $\left|\text{{v}}\left(\partial\right)\right|=\mathcal{O}\left(\left|\rho\right|^{4}\right)$ while the lengths (weights) of formulas and the height numbers involved are bounded by $2\left|\rho\right|$ and $\left|\text{{v}}\left(\partial\right)\right|$, respectively (cf. [1]).

Let $\mathrm{LC}\left(\partial\right)$ and $\mathrm{PROV}\left(\partial\right)$ be abbreviations for ‘$\partial$* is locally correct*’ and ‘$\partial$ proves $\rho$’, respectively, and let $\mathrm{PROOF}\left(\partial\right):=\mathrm{LC}\left(\partial\right)\&\,\mathrm{PROV}\left(\partial\right)$. We wish to validate the assertion $\mathrm{PROOF}\left(\partial\right)$ in polynomial time (and space) by a suitable deterministic TM $M$. For technical reasons we choose a formalization of $\partial$ in which edges are redefined as pairs $\left\langle parent,child\right\rangle$.

Input: $a:=2\left|\rho\right|$, $b:=\left|\text{{v}}\left(\partial\right)\right|$ and the list $t$ consisting of tuples $t\left(x\right)=\left[x,y_{1},y_{2},h,h_{1},h_{2},\chi,\gamma,\beta_{1},\beta_{2}\right]$, for every $x\leq b$, where $\chi\in\left\{\text{{r}}_{\rightarrow},\text{{i}}_{\rightarrow},\text{{e}}_{\rightarrow},\text{{l}}\right\}$ and $h,h_{1},h_{2}\leq b$ are natural numbers (the heights, in binary) while $x,y_{1},y_{2}\leq b$ and $\gamma,\beta_{1},\beta_{2}\leq a$ are natural numbers (in binary) which encode nodes and formulas, respectively, occurring in $\partial$. Thus the total length (weight) of $t$ is $\mathcal{O}\left(\left|\rho\right|^{4}\log\left|\rho\right|\right)<\mathcal{O}\left(\left|\rho\right|^{5}\right)$. $\mathrm{LC}\left(\partial\right)$ and $\mathrm{PROV}\left(\partial\right)$ are formalized as follows.

3.2 Local correctness

We observe that $\mathrm{LC}\left(\partial\right)$ is equivalent to conjunction of the following conditions 1–8 on $t\left(x\right)$ under the assumptions: ‘$x$ is parent of $y$’, $h:=h\left(x\right)$, $h_{1}:=h\left(y_{1}\right)$, $h_{2}:=h\left(y_{2}\right)$, $\chi:=\ell^{\text{{n}}}\left(x\right)$, $\gamma:=\ell^{\text{{f}}}\left(x\right)$, $\beta_{1}:=\ell^{\text{{f}}}\left(y_{1}\right)$ and $\beta_{2}:=\ell^{\text{{f}}}\left(y_{2}\right)$.

If $x=x^{\prime}$ then $t\left(x\right)=t\left(x^{\prime}\right)$. 2. 2.

If $t\left(x\right)=\left[x,y_{1},y_{2},h,h_{1},h_{2},\chi,\gamma,\beta_{1},\beta_{2}\right]$ and $x^{\prime}=y_{i}$ ($i\in\left\{1,2\right\}$) with $t\left(x^{\prime}\right)=\left[x^{\prime},y_{1}^{\prime},y_{2}^{\prime},h^{\prime},h_{1}^{\prime},h_{2}^{\prime},\chi^{\prime},\gamma^{\prime},\beta_{1}^{\prime},\beta_{2}^{\prime}\right]$, then $h^{\prime}=h_{i}$ and $\gamma^{\prime}=\beta_{i}$. 3. 3.

If $x=r$ then $h=0$ and $\chi\neq$ l. 4. 4.

If $\chi=\,$l then $y_{1}=y_{2}=h_{1}=h_{2}=\beta_{1}=\beta_{2}=0$ [: case $x\in\,$l$\left(\partial\right)$]. 5. 5.

If $\chi\neq$ l then $h_{1}=h_{2}=h+1$. 6. 6.

If $\chi=\,$r*→* then $y_{2}=\beta_{2}=0$ and $\gamma=\beta_{1}$ [: rule $\left(\rightarrow R\right)$]. 7. 7.

If $\chi=\,$i*→* then $y_{2}=\beta_{2}=0$ and $\gamma=\alpha\rightarrow\beta_{1}$ for some $\alpha$ [: rule $\left(\rightarrow I\right)$]. 8. 8.

If $\chi=\,$e*→* then $\beta_{2}=\beta_{1}\rightarrow\gamma$ [: rule $\left(\rightarrow E\right)$].

We assume that $\mathrm{LC}\left(\partial\right)$ is validated by a TM $M$ in polynomial time (and space). The verification of conditions 1–8 requires $\mathcal{O}\left(\left|\rho\right|^{5}\right)$ iterations of basic queries $\chi\overset{?}{=}\chi^{\prime}$, $u\overset{?}{=}v$, $\delta\overset{?}{=}\sigma$, $\left(\exists?\alpha\right)\gamma=\alpha\rightarrow\beta$ for $\chi,\chi^{\prime}\in\left\{\text{{r}}_{\rightarrow},\text{{i}}_{\rightarrow},\text{{e}}_{\rightarrow},\text{{l}}\right\}$, $u,v\leq b$ and $\beta,\gamma,\delta,\sigma\leq a$ that are solvable in $\mathcal{O}\left(\left|\rho\right|\right)$ time (note that $\alpha\rightarrow\beta=\,\rightarrow\!\alpha\beta$ in the Łukasiewicz prefix notation). Summing up there is a deterministic TM $M$ that verifies $\mathrm{LC}\left(\partial\right)$ in $\mathcal{O}\left(\left|\rho\right|^{5}\cdot\left|\rho\right|\right)=\mathcal{O}\left(\left|\rho\right|^{6}\right)$ time and $\mathcal{O}\left(\left|\rho\right|^{5}\right)$ space.

3.3 Assignment $\mathcal{A}$

A given locally correct NM*→*-deduction $\partial$ determines an assignment

[TABLE]

that is defined by the following recursive clauses 1–4.

$A\left(x\right):=\left\{\alpha\right\}$ if $x\in\,$l$\left(\partial\right)$ and $\ell^{\text{{f}}}\left(x\right)=$ $\alpha$, 2. 2.

$A\left(x\right):=A\left(y\right)$ if $\ell^{\text{{n}}}\left(x\right)=\,$r*→* and $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$. 3. 3.

$A\left(x\right):=A\left(y\right)\setminus\left\{\alpha\right\}$ if $\ell^{\text{{n}}}\left(x\right)=\text{{i}}_{\rightarrow}$, $\ell^{\text{{f}}}\left(x\right)=\alpha\rightarrow\ell^{\text{{f}}}\left(y\right)$ and $\left\langle x,y\right\rangle\in\,$e$\left(\partial\right)$. 4. 4.

$A\left(x\right):=A\left(y\right)\cup A\left(z\right)$ if $\ell^{\text{{n}}}\left(x\right)=\text{{e}}_{\rightarrow}$ and $\left\langle x,y\right\rangle,\left\langle x,z\right\rangle\in\,$e$\left(\partial\right)$.

$\mathcal{A}$ is defined by recursion of the length $\left|\text{{v}}\left(\partial\right)\right|=\mathcal{O}\left(\left|\rho\right|^{4}\right)$. Recursion steps produce (say, sorted) lists of formulas $A\left(x\right)\subseteq\left\{\ell^{\text{{f}}}\left(y\right):y\in\text{{l}}\left(\partial\right)\right\}$, $x\in$ v$\left(\partial\right)$, $\left|A\left(x\right)\right|\leq$ $\left|\text{{v}}\left(\partial\right)\right|$ using set-theoretic unions $A\cup B$ and subtractions $A\setminus\left\{\alpha\right\}$. Each recursion step requires $\mathcal{O}\left(\left|\rho\right|\cdot\left|\text{{v}}\left(\partial\right)\right|\right)=\mathcal{O}\left(\left|\rho\right|^{5}\right)$ steps of computation. This yields upper bound $\mathcal{O}\left(\left|\rho\right|^{4}\cdot\left|\rho\right|^{5}\right)=\mathcal{O}\left(\left|\rho\right|^{9}\right)$ for $A\left(r\right)\overset{?}{=}\emptyset$. Thus $\mathrm{PROV}\left(\partial\right)$ is verifiable by a suitable deterministic TM $M$ in $\mathcal{O}\left(\left|\rho\right|^{9}\right)$ time and $\mathcal{O}\left(\left|\rho\right|\right)$ space. Hence by the above estimate of $\mathrm{LC}\left(\partial\right)$ we can safely assume that $\mathrm{PROOF}\left(\partial\right)$ is verifiable by $M$ in $\mathcal{O}\left(\left|\rho\right|^{9}\right)$ time and $\mathcal{O}\left(\left|\rho\right|^{5}\right)$ space.

Conclusion 18

There exist polynomials $p,q,r$ of degrees $5,9,5$, respectively, and a deterministic boolean-valued TM $M$ such that for any purely implicational formula $\rho$ the following holds: $\rho$ is valid in minimal logic iff there exists a $u\in\left\{0,1\right\}^{p\left(\left|\rho\right|\right)}$ such that $M\left(\rho,u\right)$ yields $1$ after $q\left(\left|\rho\right|+\left|u\right|\right)$ steps of computation in space $r\left(\left|\rho\right|+\left|u\right|\right)$. Analogous polynomial estimates of the intuitionistic and/or classical propositional and even quantified boolean validity are easily obtained by familiar syntactic interpretations within minimal logic (cf. e.g. [4], [8], [10]).

Remark 19

Recall that $\mathrm{PROV}\left(\partial\right)\!$ is equivalent to the assertion that maximal threads in $\partial$ are closed. This in turn is equivalent to a variant of non-reachability assertion: ‘$r$ is not connected to any leaf $z$ in a subgraph of $\partial$ that is obtained by deleting all edges $\left\langle x,y\right\rangle$ with $\ell^{\text{{n}}}\left(x\right)=\,$i→** and $\ell^{\text{{f}}}\left(x\right)=\ell^{\text{{f}}}\left(z\right)\rightarrow\ell^{\text{{f}}}\left(y\right)$’, which we’ll abbreviate by $\mathrm{PROV}_{1}\!\left(\partial\right)$. Now $\mathrm{PROV}_{1}\!\left(\partial\right)$ is verifiable by a deterministic TM in $\mathcal{O}\left(\left|\text{{v}}\left(\partial\right)\right|\cdot\left|\text{{e}}\left(\partial\right)\right|\right)=\mathcal{O}\left(\left|\rho\right|^{12}\right)$ time and $\mathcal{O}\left(\left|\rho\right|\cdot\left|\text{{v}}\left(\partial\right)\right|\right)=\mathcal{O}\left(\left|\rho\right|^{5}\right)$ space (cf. e.g. [6]). However this does not improve our upper bound for $\mathrm{PROOF}\left(\partial\right)$. Actually there are known much better estimates of the reachability problem (cf. e.g. [11], [2]), but at this stage we are not interested in more precise analysis.

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