On P Versus NP

L. Gordeev

doi:10.20944/preprints202609.1991.v1

Submitted:

20 September 2026

Posted:

23 September 2026

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Abstract

We show that graph-theoretic problem CLIQUE can’t be solved in polynomial time by any deterministic TM. This upgrades the well-known monotone unsolvability and eventually implies $ {P \neq NP} $ as CLIQUE is NP-complete. As compared to familiar Razborov-style arguments we use Boolean logic coupled with standard circuit formalism.

Keywords:

boolean algebra

;

circuit complexity

Subject:

Computer Science and Mathematics - Logic

MSC: 03B05; 68Q05; 68Q15

1. Introduction

We use the following two basic observations 1, 2, where

{CLIQUE}_{m, k}

says that a given graph on

\leq m

vertices has a clique of k vertices.

1.: Computational complexity of arbitrary Boolean circuits is polynomial in that of De Morgan normal (abbr.: DMN) circuits allowing gates $\lor, \land$ and literals $v_{i}, \neg v_{i}$ .
2.: For sufficiently large $m = k^{4}$ , the size of any given DMN circuit solution of ${CLIQUE}_{m, k}$ is exponential in m.

To formalize 2, let $n : = (\binom{m}{2})$ be the total amount of edges of arbitrary graphs under consideration and $π$ be standard 1–1 enumeration thereof. Let $E : = \{ε : [n] \to \{0, 1\}\}$ . Thus the set of all (say, plain) graphs is $G = \{G (ε) : ε \in E\}$ , where $G (ε) = \{π (i) : i \in [n] & ε (i) = 1\}$ . Also let

$C L I Q (ε) : \Leftrightarrow ` ` G (ε) h a s a c l i q u e o f k v e r t i c e s " .$
For any Boolean circuit C (DMN circuit $C^{\pm}$ ) and $ε \in E$ , denote by $C (ε)$ ( $C^{\pm} (ε)$ ), a variable-free Boolean circuit defined by replacing inputs $v_{i} \in C$ and $v_{i}, \neg v_{i} \in C^{\pm}$ ) by $ε (i)$ and $1 - ε (i)$ , respectively. Also let $∥C (ε)∥ \in \{0, 1\}$ and $∥C^{\pm} (ε)∥ \in \{0, 1\}$ denote Boolean values of $C (ε)$ and $C^{\pm} (ε)$ .
${CLIQUE}_{m, k}$ ( $k < m$ ) is called solvable by C, resp. $C^{\pm}$ , iff

$\forall ε (∥C (ε)∥ = 1 \Leftrightarrow C L I Q (ε)), \forall ε (∥C^{\pm} (ε)∥ = 1 \Leftrightarrow C L I Q (ε)) (1)$
To prove 2 we consider disjoint pairs of graphs on $\leq m$ vertices (called double graphs below) $D = 〈G (ε), G^{'}〉$ , for arbitrary $ε \in E$ , such that $G^{'} \subseteq \{π (i) : i \in [n] & ε (i) = 0\}$ , assuming that every $D \in D$ is represented by a DMN circuit

$C_{D}^{\pm} : = \underset{π (i) \in G (ε)}{⋀} v_{i} \land \underset{π (j) \in G^{'}}{⋀} \neg v_{j} .$
Now consider any Boolean circuit C with gates ∨, ∧, ¬, whose inputs are supplied with Boolean constants and variables $v_{i}$ ( $i \in [n]$ ). Corresponding DNM circuit $C^{\pm}$ arises by applying to gates of C the well-known De Morgan rewriting rules 1–4:

1.

$\neg 1 ↪ 0, \neg 0 ↪ 1 .$

2.

$\neg (a \lor b) ↪ \neg a \land \neg b .$

3.

$\neg (a \land b) ↪ \neg a \lor \neg b .$

4.

$\neg \neg a ↪ a .$
It is a folklore that circuit size of $C^{\pm}$ is at most that of C. Namely, circuit structure of $C^{\pm}$ arises by successively replacing $\neg 1$ by 0, $\neg 0$ by 1 and gates ∨ and ∧ occurring in the scope of ¬ in C by the complementary ∧ and ∨, respectively, while deleting gates ¬ in question and all double negations $\neg \neg$ . Then clearly

$\forall ε (∥C (ε)∥ = ∥C^{\pm} (ε)∥) (2)$
Assuming that ${CLIQUE}_{m, k}$ is solvable by C, suppose

$\forall ε (∥C (ε)∥ = 1 \Leftrightarrow C L I Q (ε)) (3)$
Furthermore, we define a crucial DNF expansion $DN (C^{\pm}) \subset D$ such that

$\forall ε (\begin{matrix} ∥C^{\pm} (ε)∥ = 1 \Leftrightarrow \\ (\exists D \in DN (C^{\pm})) ∥C_{D}^{\pm} (ε)∥ = 1 \end{matrix}) (4)$
Having done this we show that the hypothetical equivalence

$\forall ε (\begin{matrix} C L I Q (ε) \Leftrightarrow \\ (\exists D \in DN (C^{\pm})) ∥C_{D}^{\pm} (ε)∥ = 1 \end{matrix}) (5)$

implies that the size of $C^{\pm}$ , and hence that of C, is exponential in m, which eventually implies $P \neq NP$ (in fact, $NP \subseteq P / poly$ ).

Note that

(5)

easily follows from conjunction of

(1)

,

(2)

,

(3)

,

(4)

. That (5) implies exponential complexity of

C^{\pm}

is our crucial technical refinement of the known “monotone” approach. To this end we adapt Razborov-style method of approximations and estimate resulting deviations, a.k.a. “error sets”, via modified Erdos-Rado lemma and double sunflowers with quasi-positive cores

〈G, \emptyset〉

. The sets

DN (C^{\pm})

from (5) are double-graph analogues of plain crude circuits in the “monotone” proofs.

The whole proof is exposed in Chapters 1, 2 below. For obvious reasons it is more technical than standard presentations of the monotone approach that omit logic formalization of the notions involved.

Author contribution statement The author conceived the study, designed the methodology and drafted the initial manuscript.

Acknowledgment I would like to thank René Thiemann from University of Innsbruck who took the time to verify crucial proofs with the theorem prover Isabelle, whose work was extremely helpful in finding flaws and errors in the earlier presentations.

2. Preliminaries

2.1. Basic notations

In the sequel we assume

$\begin{matrix} 2 < ℓ < p < k \leq m^{\frac{1}{4}} and L = {(p - 1)}^{ℓ} ℓ! \end{matrix}$
For any $A, B \subseteq [m]$ let

$\begin{matrix} A * B : = \{\{x, y\} : x \in A & y \in B & x \neq y\} \end{matrix}$

and $A^{(2)} : = A * A$ , where $[m] : = \{1, \dots, m\}$ .

So $|{[m]}^{(2)}| = (\binom{m}{2}) = \frac{1}{2} m (m - 1)$ , where $|S| : = c a r d (S)$ .
For any $X \subseteq {[m]}^{(2)}$ let

$\begin{matrix} v (X) : = \{x \in [m] : (\exists y \in [m]) \{x, y\} \in X\} \end{matrix}$

and $\begin{matrix} ℘_{0} X : = \{Y : Y \subset X\} \end{matrix}$ (the proper subsets of X).
Let $\begin{matrix} F : = \{f : [m] \to [k - 1]\} \end{matrix}$ and for any $f \in F$

$\begin{matrix} C_{f} : = \{\{x, y\} \in {[Dom (f)]}^{(2)} : f (x) \neq f (y)\} \end{matrix}$

2.2. Plain and double graphs

Call $G : = ℘_{0} {[m]}^{(2)}$ the set of graphs (unordered, possibly empty) on (at most) m vertices. For any $\emptyset \neq G \in G$ call pairs $\{x, y\} \in G$ and v $(G)$ the edges and vertices, respectively.
$\begin{matrix} POS : = K : = \{v {(G)}^{(2)} : |v (G)| = k\} \end{matrix}$ and

$\begin{matrix} CLIQ : = \{G \in G : (\exists K \in K) K \subseteq G\} \end{matrix}$

are called positive tests and plain clique problem, respectively.
$\begin{matrix} NEG : = \{C_{f} : f \in F\} \end{matrix}$ and

$\begin{matrix} ACLIQ : = \{G \in G : (\exists H \in NEG) G \subseteq H\} \end{matrix}$

are called negative tests and plain anticliques, respectively.
Pairs of disjoint plain graphs are called double graphs. That is,

$\begin{matrix} D : = \{\begin{matrix} 〈G, H〉 \in G \times G : \\ G \cap H = \emptyset & G \cup H \in ℘_{0} {[m]}^{(2)} \end{matrix}\} \end{matrix}$ is the set of double graphs. Double graph $〈\emptyset, \emptyset〉$ is identified with ∅, while $G$ regarded part of $D$ via $G ∋ G ↪ 〈G, \emptyset〉 \in D$

and/or $G ∋ G ↪ 〈\emptyset, G〉 \in D$ .
For any $D = 〈G, H〉 \in D$ and $X \subseteq D$ let

$\begin{matrix} D^{+} : = G, D^{-} : = H \end{matrix}$ $\in G$ and

$\begin{matrix} X^{+} : = \{D^{+} : D \in X\} \end{matrix}$ $\subseteq G$ and

$\begin{matrix} X^{-} : = \{D^{-} : D \in X\} \end{matrix}$ $\subseteq G$ .
For any $G \in G$ and $D \in D$ let

$\begin{matrix} G \subseteq^{+} D : \Leftrightarrow G \subseteq D^{+} \end{matrix}$ .
$\begin{matrix} {CLIQ}_{2} : = \{D \in D : (\exists K \in POS) K \subset^{+} D\} \end{matrix}$ are called double cliques.
$\begin{matrix} {ACLIQ}_{2} : = \{D \in D : (\exists G \in NEG) G \subseteq^{+} D\} \end{matrix}$ are called double anticliques.

Lemma 1.

{CLIQ}_{2} \cap

{ACLIQ}_{2} = \emptyset

. Moreover

|POS|

= (m k)

and

|F| = {(k - 1)}^{m} > |NEG| .

Proof.

This is easily verified (precise estimation of

|NEG|

is unimportant). □

2.3. Basic operations on double graphs

Except for standard set-theoretic operations ∪ and ∩ we consider double union

⋓ : D \times D \to D

and double product

⊙ : ℘ D \times ℘ D \to ℘ D

.

For any $D, E \in D$ let

$D ⋓ E : = \{\begin{matrix} 〈D^{+} \cup E^{+}, D^{-} \cup E^{-}〉, & i f i t & \in D, \\ \emptyset, & e l s e . \end{matrix}$

For any $X, Y \subseteq D$ let

$X ⊙ Y : = \{D ⋓ E : 〈D, E〉 \in X \times Y\}$ $\subseteq D .$

Note that

\emptyset ⊙ Y = X ⊙ \emptyset = \emptyset

. The following conditions easily hold for any

X, Y, X^{'}, Y^{'} \subseteq D .

1.: $X ⊙ Y = Y ⊙ X, X ⊙ (Y ⊙ Z) = (X ⊙ Y) ⊙ Z .$
2.: $X ⊙ (Y \cup Z) = (X ⊙ Y) \cup (X ⊙ Z), X \cup (Y ⊙ Z) \subseteq (X \cup Y) ⊙ (X \cup Z) .$
3.: $X \subseteq X^{'} & Y \subseteq Y^{'} \Rightarrow X ⊙ Y \subseteq X^{'} ⊙ Y^{'} .$

3. Proof proper

3.1. Acceptability

With any given set of double graphs

X

we correlate accepted double tests

AC (X) \subseteq D,

{AC}^{p} (X) \subseteq POS

and negative double coloring

{AC}^{n} (X) \subseteq NEG

. Corresponding sets of accepted double tests, resp. colorings, are as follows.

Definition 1.

For any

X \subseteq D

and

D \in D

let

X ⊢ D

abbreviate

(\exists E \in X) E^{+} \subseteq D^{+}

. Then let:

1.: $AC (X) : = \{D \in D : X ⊢ D\}$ .
2.: ${AC}^{p} (X) : = AC {(X)}^{+} \cap POS$ ,
3.: ${AC}^{n} (X) : = AC {(X)}^{+} \cap NEG$ .

Lemma 2.

Conditions 1–6 hold for any

X, Y \subseteq D

.

1.

AC (\emptyset) = {AC}^{p} (\emptyset) = {AC}^{n} (\emptyset) = \emptyset

.

2.

AC (D) = D, {AC}^{p} (POS) = POS,

3.

{AC}^{n} (NEG) = NEG .

4.

If

X \subseteq Y

then

{AC}^{(p) (n)} (X) \subseteq {AC}^{(p) (n)} (Y)

.

5.

(a): $AC (X \cup Y) = AC (X) \cup AC (Y)$ ,
(b): ${AC}^{p} (X \cup Y) = {AC}^{p} (X) \cup {AC}^{p} (Y)$ ,
(c): ${AC}^{n} (X \cup Y) = {AC}^{n} (X) \cup {AC}^{n} (Y)$ .

6.

(a): $AC (X \cap Y) \subseteq AC (X ⊙ Y) = AC (X) \cap AC (Y)$ ,
(b): ${AC}^{p} (X \cap Y) \subseteq {AC}^{p} (X ⊙ Y) = {AC}^{p} (X) \cap {AC}^{p} (Y)$ ,
(c): ${AC}^{n} (X \cap Y) \subseteq {AC}^{n} (X ⊙ Y) = {AC}^{n} (X) \cap {AC}^{n} (Y)$ .

Proof.

1–5: trivial.

6 (a). It will suffice to prove

AC (X ⊙ Y) = AC (X) \cap AC (Y)

. So suppose

D \in AC (X ⊙ Y)

, i.e.

X ⊙ Y ⊢ D

, i.e.there are

E_{1} \in X

and

E_{2} \in Y

such that

E_{1}^{+} \cup E_{2}^{+} \subseteq D^{+}

, which by

E_{1}^{+} \cup E_{2}^{+} \subseteq D^{+} \Leftrightarrow E_{1}^{+} \subseteq D^{+} & E_{2}^{+} \subseteq D^{+}

yields both

D \in AC (X)

and

D \in AC (Y)

. Suppose

D \in AC (X) \cap AC (Y)

, i.e.

X ⊢ D

and

Y ⊢ D

, i.e. there are

E_{1} \in X

and

E_{2} \in Y

such that

E_{1}^{+} \subseteq D^{+}

and

E_{2}^{+} \subseteq D^{+}

, and hence

E_{1} ⋓ E_{2} \in X ⊙ Y

, which by the same token yields

D \in AC (X ⊙ Y)

.

6 (b), (c) follow analogously. □

3.2. Approximations and deviations

In what follows we revise and generalize conventional monotone approach that was initiated in [1,3,4], further elaborated and presented in textbooks [2,5,6] (et al) and course lecture notes like [7].

We supply operations ∪ and ⊙ on

℘ D

with their approximators ⊔ and ⊓ operating on arbitrary subsets

X \subseteq D

such that for all D from

X

,

|v (D^{+})| \leq ℓ

(thus we approximate only positive parts of double graphs). We define corresponding deviations

\partial_{⊔}^{p}, \partial_{⊔}^{n},

\partial_{⊓}^{p}, \partial_{⊓}^{n}

from ∪ and ⊙ with respect to accepted test graphs and show that these deviations make “small” fractions thereof (Lemmata 10, 11). These deviations are analogous to “error sets” caused by approximations in conventional monotone approach based on the Erdos-Rado lemma [8] (cf. [2,6,7]).

3.2.1. Basic notations and definitions

Let $G^{ℓ} : = \{G \in G : |v (G)| \leq ℓ\}$ and

$D^{ℓ} : = \{D \in D : |v (D^{+})| \leq ℓ\}$ . 1

Let $℘_{L} D : = \{X \subseteq D : |X^{+}| \leq L\}$ and

$℘_{L} D^{ℓ}$ $: =$ $\{X \subseteq D^{ℓ} : |X^{+}| \leq L\}$ .
If $D, E \in D^{ℓ}$ and $X, Y \subseteq D^{ℓ}$ , let

$D ⋓^{ℓ} E : = \{\begin{matrix} D ⋓ E, & i f & i t \in D^{ℓ}, \\ \emptyset & e l s e, \end{matrix}$
$X ⊙^{ℓ} Y : = \{D ⋓^{ℓ} E \in D^{ℓ} : D \in X & E \in Y\} \in ℘ D^{ℓ}$ .
Together with double graphs consider double sets

$S = \{〈A, B〉 : A, B \subseteq [m] & A \cap B = \emptyset\}$ , where for $S = 〈A, B〉 \in S$ we denote A and B by $S^{+}$ and $S^{-}$ , respectively.

Let $S^{ℓ} : = \{S \in S : |S^{+}| \leq ℓ\}$ ,

$℘_{L} S : = \{X \subseteq S : |X^{+}| \leq L\}$ and

$℘_{L} S^{ℓ}$ := $\{X \subseteq S^{ℓ} : |X^{+}| \leq L\}$ , where

$X^{+} = \{S^{+} : S \in X\} .$
For any $G, H \in G$ and $D \in D$ we let

v $(G ∖ ∖ H) : = s c v (G) ∖ s c v (H)$ and

s $(D) : = 〈v (D^{+} ∖ ∖ D^{-}), v (D^{-} ∖ ∖ D^{+})〉 \in S$ .

For any $X \subseteq D$ let s $(X) : = \{s (D) : D \in X\} \subseteq S$ .

Then for any $X \subseteq D$ , $Y \subseteq D^{ℓ}$ and $Z \in ℘_{L} D^{ℓ}$

we get s $(X) \subseteq S$ , s $(Y) \subseteq S^{ℓ}$ and s $(Z) \in ℘_{L} S^{ℓ}$ .

Definition 2.

A collection of double sets

V = \{V_{1}, \dots, V_{p}\} \subset

S

is called a sunflower with p (different) positive petals

V_{1}^{+}, \dots, V_{p}^{+}

if

V_{1}^{+} \cap V_{2}^{+} = V_{i}^{+} \cap V_{j}^{+}

holds for all

i < j \in [p]

. Then

V_{©} : = 〈V_{1}^{+} \cap V_{2}^{+}, \emptyset〉

is called the core of

V

.

Lemma 3.

Any given

U \subseteq S^{ℓ}

such that

|U^{+}| > L

contains a sunflower

V \subset U

with p positive petals

V_{1}^{+}, \dots, V_{p}^{+}

and core

V_{©} \in S^{ℓ}

.

Proof.

By the original Erdos-Rado lemma [8], since

L = {(p - 1)}^{ℓ} ℓ!

. □

Definition 3

(plucking). Plucking algorithm

℘ D^{ℓ} ∋ Z \mapsto PL (Z) \in ℘_{L} D^{ℓ}

arises by recursion on

|s (Z)|

. If

|s {(Z)}^{+}| \leq L

, let

PL (Z) : = Z

. Otherwise, let

Z_{0} : = Z

, thus

|s {(Z_{0})}^{+}| > L

. By the last lemma with respect to

U : = s (Z_{0}) \subseteq S^{ℓ}

we choose a sunflower of cardinality p,

V = \{V_{1}, \dots, V_{p}\} \subseteq

s

(Z_{0})

with positive petals

V_{1}^{+}, \dots, V_{p}^{+}

and core

V_{©} = 〈V_{1}^{+} \cap V_{2}^{+}, \emptyset〉 \in S^{ℓ}

. Let

Z_{0}^{'} : = \{D \in Z_{0} : (\exists j \in [p]) s (D) = V_{j}\} \subseteq D^{ℓ}

and

D_{©} = 〈D_{©}^{+}, \emptyset〉 \in D^{ℓ}

, where

D_{©}^{+} : = ⋂ \{D^{+} : D \in Z_{0}^{'}\}

, which also yieldss

{(D_{©})}^{+} \subseteq V_{©}^{+}

. Then rewrite

Z_{0}

to

Z_{1}

that arises by replacing every

D \in Z_{0}^{'}

by

D_{©}

. 2 Note that

|s {(Z_{1})}^{+}| \leq |s {(Z_{0})}^{+}| - p + 1 .

If

|s {(Z_{1})}^{+}| \leq L

, let

PL (Z) : = Z_{1}

. Otherwise, if

|s {(Z_{1})}^{+}| > L

, we analogously pass from

Z_{1} \subseteq D^{ℓ}

to

Z_{2} \subseteq D^{ℓ}

. Proceeding this way we eventually arrive at

Z_{q} \subseteq D^{ℓ}

with

|s {(Z_{q})}^{+}| \leq L

and then let

PL (Z) : = Z_{q}

.

Lemma 4.

For any given

Z \in ℘ D^{ℓ}

,

PL (Z) \in ℘_{L} D^{ℓ}

requires

< |s {(Z)}^{+}|

elementary pluckings. That is, if

PL (Z) : = Z_{q}

as above, then

q < |s {(Z)}^{+}|

.

Proof.

Each elementary plucking reduces the number of sets at least by

p - 1

. Hence

q < |s {(Z)}^{+}| {(p - 1)}^{- 1}

< |s {(Z)}^{+}|

. □

Definition 4.

For any

X, Y \in ℘ D^{ℓ}

call the following operations ⊔, ⊓ and sets

X ⊔ Y

,

X ⊓ Y

the approximators and approximations of operations ∪, ⊙ and sets

X \cup Y

,

X ⊙ Y

, respectively, which determine deviations

\partial_{⊔}^{p}, \partial_{⊔}^{n}, \partial_{⊓}^{p}, \partial_{⊓}^{n}

with respect to the accepted tests. 3

1.: $X ⊔ Y : = PL (X \cup Y) \in ℘_{L} D^{ℓ}$ .
2.: $X ⊓ Y : = PL (X ⊙^{ℓ} Y) \in ℘_{L} D^{ℓ}$ .
3.: $\partial_{⊔}^{p} (X, Y) : = {AC}^{p} (X \cup Y) ∖ {AC}^{p} (X ⊔ Y) \subseteq POS .$
4.: $\partial_{⊓}^{p} (X, Y) : = {AC}^{p} (X ⊙ Y) ∖ {AC}^{p} (X ⊓ Y) \subseteq POS .$
5.: $\partial_{⊔}^{n} (X, Y) = {AC}^{n} (X ⊔ Y) ∖ {AC}^{n} (X \cup Y) \subseteq NEG .$
6.: $\partial_{⊓}^{n} (X, Y) : = {AC}^{n} (X ⊓ Y) ∖ {AC}^{n} (X ⊙ Y) \subseteq NEG .$

For

U \subseteq NEG

let

{|U|}^{*} : = |\{f \in F : C_{f} \subseteq U\}|

(functional cardinality of U). In particular

{|NEG|}^{*} = F

. In the sequel we use functional cardinality as our basic measure of the number of negative double tests involved.

3.2.2. Upper bounds

We assume that m is sufficiently large and

k = 2 ℓ^{2}

.

Lemma 5.

For any

D \in D^{ℓ}

let

R_{\subseteq} (D) : = \{f \in F : D^{+} \subseteq C_{f}\}

and

R_{⊈} (D) : = \{f \in F : D^{+} ⊈ C_{f}\} = F ∖ R_{\subseteq} (D)

. Then

|R_{\subseteq} (D)| \geq \frac{1}{2} |F|

and

|R_{⊈} (D)| \leq \frac{1}{2} |F|

. Moreover, for any

D_{1}, \dots, D_{q} \in D^{ℓ}

such that

(\forall i \neq j \in [q]) D_{i}^{+} \cap D_{j}^{+} = \emptyset

it holds

|\overset{q}{⋂_{i = 1}} R_{⊈} (D_{i})| \leq 2^{- q} |F|

.

Proof.

For any

G \in G^{ℓ}

let

R_{\subseteq} (G) : = \{f \in F : G \subseteq C_{f}\}

. This yields by standard monotone arguments

|R_{\subseteq} (G)| \geq \frac{1}{2} |F|

, which for any

D \in D^{ℓ}

implies

|R_{\subseteq} (D)| \geq \frac{1}{2} |F|

and hence

|R_{⊈} (D)| = |F ∖ R_{\subseteq} (D)| \leq \frac{1}{2} |F|

(see Appendix A). To establish the last assertion it will suffice to observe that for any

j \in [q - 1]

, it holds

P [\overset{q}{⋂_{i = j}} R_{⊈} (D_{i})] = P [R_{⊈} (D_{j})] \cdot P [\overset{q}{⋂_{i = j + 1}} R_{⊈} (D_{i})]

where for any

X \subseteq F

we set

P [X] : = |X| {|F|}^{- 1}

(the probability). The latter holds by standard arguments, as

R_{⊈} (D_{1}), \dots, R_{⊈} (D_{q})

are independent events in the space

F

(see also Appendix A). □

Lemma 6.

Let

Z = X \cup Y \in ℘ D^{ℓ}

and

PL (Z) \in ℘_{L} D^{ℓ}

for

X, Y \in ℘_{L} D^{ℓ}

. Then

PL (Z)

requires

< 2 L

elementary pluckings. Moreover

\begin{matrix} \partial_{⊔}^{p} (X, Y) = 0 \end{matrix}

while

\begin{matrix} {|\partial_{⊔}^{n} (X, Y)|}^{*} < 2^{1 - p} L |F| \end{matrix}

.

Proof.

We argue as in the analogous monotone case using Lemmata 7, 9. Let

V = \{V_{1}, \dots, V_{p}\}

\subseteq

s

({(X \cup Y)}_{i})

be the sunflower with positive petals

V_{1}, \dots, V_{p}

and core

V_{©} = 〈V_{1}^{+} \cap V_{2}^{+}, \emptyset〉 \in S^{ℓ}

arising at

i^{t h}

elementary plucking (

i > 0

) and let

D_{©} = 〈D_{©}^{+}, \emptyset〉 \in D^{ℓ}

be the corresponding double graph. Consider

PL (Z)

and corresponding

\partial_{⊔}^{p} (X, Y)

and

\partial_{⊔}^{n} (X, Y)

.

\partial_{⊔}^{p} (X, Y) = \emptyset

is clear as elementary pluckings replace some (plain) graphs by subgraphs and thereby preserve the accepted positive tests.

Now consider

\partial_{⊔}^{n} (X, Y)

. We estimate the total number of fake negative double tests that arise after rewriting

Z_{i - 1} ↪ Z_{i}

involved. Suppose

Z_{i}

is obtained by substituting

D_{©} = 〈D_{©}^{+}, \emptyset〉 = 〈⋂ \{D^{+} : D \in Z_{0}^{'}\}, \emptyset〉 \in D^{ℓ}

, for every

D \in Z_{i - 1}^{'}

, where

Z_{i - 1}^{'} = \{D \in Z_{i - 1} : (\exists j \in [p]) s (D) = V_{j}\}

. Let

|Z_{i - 1}^{'}| = p^{'} \geq p

with

Z_{i - 1}^{'} = \{D_{1}, \dots, D_{p^{'}}\}

. Now let

C_{f} \in NEG

be any fake negative test created by this substitution. I.e.

D_{©}^{+} \subseteq C_{f}

, although for every

t \in [p^{'}]

, we have

D_{t}^{+} ⊈ C_{f}

. Let

D_{t}^{'} : = 〈D_{t}^{+} ∖ D_{©}^{+}, D_{t}^{-}〉 \in D^{ℓ}

. Note that for any

s \neq t \in [p^{'}]

we have

D_{s}^{' +} \cap D_{t}^{' +} = \emptyset \neq D_{t}^{+}

, while s

(D_{©})

contains the only common nodes of

D_{s}^{+}

and

D_{t}^{+}

. Furthermore by Lemma 9 we know that

P [R_{⊈} (D_{t}^{'})] \leq P [R_{⊈} (D_{t})] \leq \frac{1}{2}

holds for every

t \in [p^{'}]

. Summing up, by Lemma 9 we obtain

\begin{matrix} P [\underset{t = 1}{⋂^{p^{'}}} R_{⊈} (D_{t}) \cap R_{\subseteq} (D_{©})] \\ = & P [\underset{t = 1}{⋂^{p^{'}}} (R_{⊈} (D_{t}) \cap R_{\subseteq} (D_{©}))] \\ \leq & P [\underset{t = 1}{⋂^{p^{'}}} R_{⊈} (D_{t}^{'})] \leq 2^{- p^{'}} \\ \leq & 2^{- p} \end{matrix}

Hence with regard to functional cardinality there are less than

\begin{matrix} |\underset{t = 1}{⋂^{p^{'}}} R_{⊈} (D_{t}) \cap R_{\subseteq} (D_{©})| & = \\ P [\underset{t = 1}{⋂^{p^{'}}} R_{⊈} (D_{t}) \cap R_{\subseteq} (D_{©})] |F| & \leq & 2^{- p} |F| \end{matrix}

fake negative tests

C_{f}

created by the replacement

Z_{i - 1} ↪ Z_{i}

. Recall that by Lemma 7 there are

q < L

elementary pluckings involved. This yields

\begin{matrix} \partial_{⊔}^{n} (X, Y) & \subseteq & \overset{q - 1}{⋃_{i = 0}} \partial_{⊔}^{n} {(X, Y)}_{i} for \\ \partial_{⊔}^{n} {(X, Y)}_{i} & : & = {AC}^{n} {(X \cup Y)}_{i + 1} ∖ {AC}^{n} {(X \cup Y)}_{i} . \end{matrix}

Hence

{|\partial_{⊔}^{n} (X, Y)|}^{*} \leq \overset{q - 1}{\sum_{i = 0}} {|\partial_{⊔}^{n} {(X, Y)}_{i}|}^{*} < q 2^{- p} |F| < 2^{1 - p} L |F|

. □

Lemma 7.

Let

X, Y \in ℘_{L} D^{ℓ}

,

X ⊙^{ℓ} Y \in ℘ D^{ℓ}

and

Z = PL (X ⊙^{ℓ} Y) \in ℘_{L} D^{ℓ}

. Thus

|s {(Z)}^{+}| \leq L

and

|s {(X ⊙ Y)}^{+}| \leq L^{2} .

Then

\begin{matrix} |\partial_{⊓}^{p} (X, Y)| < L^{2} (m - ℓ - 1 k - ℓ - 1) \end{matrix}

and

\begin{matrix} {|\partial_{⊓}^{n} (X, Y)|}^{*} < 2^{- p} L^{2} |F| \end{matrix}

.

Proof.

|\partial_{⊓}^{n} (X, Y)| * < 2^{- p} L^{2} |F|

is analogous to the inequality for

\partial_{⊔}^{n} (X, Y)

. Consider

\partial_{⊓}^{p} (X, Y)

. We adapt standard arguments used in the “monotone” proofs (cf. e.g. [2,7]). It is readily seen that deviations can only arise by deleting a

D \cup E \notin D^{ℓ}

for some

D, E \in D^{ℓ}

when passing from

X ⊙ Y

to

X ⊙^{ℓ} Y

(note that

X ⊙ Y

can completely disappear, in which case

PL (X ⊙^{ℓ} Y) = X ⊙^{ℓ} Y = \emptyset

). So suppose

H \in (X ⊙ Y) ∖ D^{ℓ}

. Thus

ℓ < |v (H^{+})| \leq 2 ℓ

. Let us estimate

|K_{H}|

for

K_{H} : = \{K \in POS : H^{+} \subseteq K\}

. Note that

ℓ < |v (H^{+})|

implies that

K_{H}

contains at most

(m - ℓ - 1 k - ℓ - 1)

cliques K. So

|K_{H}| \leq (m - ℓ - 1 k - ℓ - 1)

and

\begin{matrix} \partial_{⊓}^{p} (X, Y) & \subseteq & ⋃ \{K_{H} : H \in (X ⊙ Y) ∖ D^{ℓ}\} \\ \subseteq & ⋃ \{K_{H} : H \in X ⊙ Y\}, \end{matrix}

which by

|s {(X ⊙ Y)}^{+}| \leq L^{2}

and Lemma 7 yields the result. □

3.3. Formalism

We’ll formalize previous considerations in basic De Morgan logic with atomic negation (called DMN logic) over

(m 2)

distinct variables. For any given DMN formula

φ

we define its double graph representation

DN (φ)

and approximation

AP (φ)

augmented with total deviations

\partial^{p} (φ) \subseteq POS

and

\partial^{n} (φ) \subseteq NEG

. Using our estimates on

\partial_{⊔}^{p}, \partial_{⊔}^{n}, \partial_{⊓}^{p}, \partial_{⊓}^{n}

we show that

{AC}^{p} (DN (φ)) = POS

plus

{AC}^{n} (DN (φ)) = \emptyset

infers exponential circuit size of

φ

(cf. Theorem 14 below).

3.3.1. Syntax

In the sequel we let

n : = (m 2) = \frac{1}{2} m (m - 1)

and

π : [n] \overset{1 - 1}{⟶} {[m]}^{(2)}

.

Let $A$ denote Boolean algebra with constants $0, 1$ , operations $\lor, \land$ , atomic negation ¬ and variables $v_{i}$ for any $i \in [n]$ . That is, formulas (abbr.: $φ, σ, τ$ ) are built up from constants and literals $v_{i},$ $\neg v_{i}$ ( $i \in [n]$ ) by positive operations ∨ and ∧. For brevity we also stipulate $1 \lor φ = φ \lor 1 : = 1$ , $0 \land φ = φ \land 0 : = 0$ and $1 \land φ = φ \land 1 = 0 \lor φ = 0 \lor φ = φ \lor 0 : = φ$ . Let $c s (φ)$ denote structural complexity (i.e. circuit size) of $φ$ . 4

De Morgan rules for negation provide length-preserving interpretation of full Boolean algebra.
We define by recursion on $c s (φ)$ two assignments

$\begin{matrix} A & ∋ & φ ↪ DN (φ) \in \{1\} \cup ℘ D and \\ A & ∋ & φ ↪ AP (φ) \in \{1\} \cup ℘_{L} D^{ℓ} \end{matrix}$

that represent DNFs and corresponding approximations of $φ$ , respectively.

1.

$DN (1) = AP (1) : = 1, DN (0) = AP (0) : = \emptyset .$

2.

$DN (v_{i}) = AP (v_{i}) : = \{〈\{π (i)\}, \emptyset〉\} .$

3.

$DN (\neg v_{i}) = AP (\neg v_{i}) : = \{〈\emptyset, \{π (i)\}〉\} .$

4.

$DN (σ \lor δ) : = DN (σ) \cup DN (δ), AP (σ \lor δ) : = AP (σ) ⊔ AP (δ) .$

5.

$DN (σ \land δ) : = DN (σ) ⊙ DN (τ), AP (σ \land δ) : = AP (σ) ⊓ AP (δ) .$

Thus for any $φ = \overset{r}{\underset{i = 1}{⋁}} φ_{i}$ and $ψ = \underset{i \in I}{⋀} v_{i} \land \underset{j \in J}{⋀} \neg v_{j}$ , where $I \cap J = \emptyset$ , we have $DN (φ) =$ $\overset{r}{⋃_{i = 1}} DN (φ_{i})$ and $DN (ψ) = \{〈G, H〉\}$ for $G : = \{π (i) : ı \in I\}$ and $H : = \{π (j) : j \in j\}$ . By the same token, $℘ D = \{DN (φ) : φ \in A\}$ .
For any $φ \in A$ we define total deviations $\partial^{p} (φ)$ and $\partial^{n} (φ)$ as follows, where ${AC}^{p} (1) : = POS$ and ${AC}^{n} (1) : = NEG$ , while ${AC}^{p} (DN (φ))$ and ${AC}^{n} (DN (φ))$ abbreviate ${AC}^{p} (φ)$ and ${AC}^{n} (φ)$ , respectively.

1.

$\partial^{p} (φ) : = {AC}^{p} (φ) ∖ {AC}^{p} (AP (φ)) .$

2.

$\partial^{n} (φ) : = {AC}^{n} (AP (φ)) ∖ {AC}^{n} (φ) .$

Lemma 8.

For any

σ, δ \in A

the following holds.

1.: $\partial^{p} (σ \lor δ) \subseteq \partial^{p} (σ) \cup \partial^{p} (δ) \cup \partial_{⊔}^{p} (AP (σ), AP (δ)) .$
2.: $\partial^{p} (σ \land δ) \subseteq \partial^{p} (σ) \cup \partial^{p} (δ) \cup \partial_{⊓}^{p} (AP (σ), AP (δ)) .$
3.: $\partial^{n} (σ \lor δ) \subseteq \partial^{n} (σ) \cup \partial^{n} (δ) \cup \partial_{⊔}^{n} (AP (σ), AP (δ)) .$
4.: $\partial^{n} (σ \land δ) \subseteq \partial^{n} (σ) \cup \partial^{n} (δ) \cup \partial_{⊓}^{n} (AP (σ), AP (δ)) .$

Proof.

Straightforward via boolean inclusion

A ∖ B \subseteq (A ∖ C) \cup (C ∖ B)

(cf. Appendix B). □

Lemma 9.

For any

φ \in A

the following conditions hold.

1.: $|\partial^{p} (φ)| < c s (φ) \cdot L^{2} (m - ℓ - 1 k - ℓ - 1) .$
2.: ${|\partial^{n} (φ)|}^{*} \leq c s (φ) \cdot 2^{- p} L^{2} |F|$ .
3.: If ${AC}^{p} (AP (φ)) \neq \emptyset$ then ${|{AC}^{n} (AP (φ))|}^{*} \geq \frac{1}{2} |F|$ .

Proof.

1–2 follows from Lemmata 10, 11 by induction on

c s (φ)

.

3:

{AC}^{p} (AP (φ)) \neq \emptyset

implies

AP (φ) \neq \emptyset

, so there is at least one

D \in AP (φ)

,

∥v (D)∥ \leq ℓ

. Now by Lemma 9,

{|{AC}^{n} (AP (φ))|}^{*} \geq |R_{\subseteq} (D)| \geq \frac{1}{2} |F|

, as

{AC}^{n} {(AP (φ))}^{*} \supseteq R_{\subseteq} (D)

. □

Final assumptions. Assuming $m ≫ 0$ we let

\begin{matrix} m = k^{4}, k = 2 ℓ^{2}, p = ℓ {log}_{2} m, L = {(p - 1)}^{ℓ} ℓ! \end{matrix}

Theorem 1.

Suppose that

{AC}^{p} (φ) = POS

and

{AC}^{n} (φ) = \emptyset

both hold for a given

φ \in A

. Then for sufficiently large m,

c s (φ) > m^{\frac{1}{5} m^{\frac{1}{8}}}

.

Proof.

Consider two cases (cf. Appendix C).

1: Assume

{AC}^{p} (AP (φ)) = \emptyset

. By

{AC}^{p} (φ) = POS

we have

\partial^{p} (φ) = {AC}^{p} (φ) ∖ {AC}^{p} (AP (φ)) = POS

. Hence by Lemma 13 (1),

c s (φ) \cdot (m - ℓ - 1 k - ℓ - 1) L^{2} \geq |\partial^{p} (φ)| = |POS| = (m k) .

Hence

c s (φ) \geq (m k) {(m - ℓ - 1 k - ℓ - 1)}^{- 1} L^{- 2} > {(\frac{m - ℓ}{k})}^{ℓ} L^{- 2} > m^{\frac{1}{5} m^{\frac{1}{8}}}

.

2: Otherwise, assume

{AC}^{p} (AP (φ)) \neq \emptyset

. So

{AC}^{n} (φ) = \emptyset

implies

\partial^{n} (φ) = {AC}^{n} (AP (φ)) ∖ {AC}^{n} (φ) = {AC}^{n} (AP (φ))

. Hence

c s (φ) \cdot 2^{- p} L^{2} |F| \geq {|\partial^{n} (φ)|}^{*} \geq \frac{1}{2} |F|

by Lemma 13 (2, 3) and then

c s (φ) \geq 2^{p - 1} L^{- 2} > m^{\frac{1}{2} m^{\frac{1}{8}}} > m^{\frac{1}{5} m^{\frac{1}{8}}}

. □

3.3.2. Semantics

Definition 5.

Consider variable assignments

VA = \{ε : [n] \to \{0, 1\}\} .

For any

i \in [n]

, literals

v_{i}

,

\neg v_{i}

, formulas

φ_{1}, \dots, φ_{r} \in A

and

ε \in VA

, Boolean values

{∥-∥}_{ε} \in \{0, 1\}

arise as follows.

1.: ${∥1∥}_{ε} : = 1, {∥0∥}_{ε} = 0$ .
2.: ${∥v_{i}∥}_{ε} : = ε (i)$ .
3.: ${∥\neg v_{i}∥}_{ε} : = 1 - {∥v_{i}∥}_{ε} = 1 - ε (i)$ .
4.: ${∥φ_{1}, \lor \dots \lor φ_{r}∥}_{ε} : = max \{{∥φ_{1}∥}_{ε}, \dots, {∥φ_{r}∥}_{ε}\}$ .
5.: ${∥φ_{1} \land \dots \land φ_{r}∥}_{ε} : = min \{{∥φ_{1}∥}_{ε}, \dots, {∥φ_{r}∥}_{ε}\}$ .

Furthermore, for any

D \in D

,

X \subseteq D

we define formulas

F (D)

,

F (X)

and Boolean values

{∥D∥}_{ε}

,

{∥X∥}_{ε}

:

1.: $F (D) : = \underset{π (i) \in D^{+}}{⋀} v_{i} \land \underset{π (j) \in D^{-}}{⋀} \neg v_{j}$ and

$F (X) : = \underset{D \in X}{⋁} F (D)$ .
2.: ${∥D∥}_{ε} : = {∥F (D)∥}_{ε}$ and ${∥X∥}_{ε} : = {∥F (X)∥}_{ε}$ .

Lemma 10.

{∥φ∥}_{ε} = {∥DN (φ)∥}_{ε}

holds for any

φ \in A

and

ε \in VA

.

Proof.

We argue by induction on

c s (φ)

. Consider induction step

φ = σ \land δ

where

DN (σ), DN (δ) \neq \emptyset

. So

DN (φ) = DN (σ) ⊙ DN (δ) = \{D ⋓ E : 〈D, E〉 \in DN (σ) \times DN (δ)\}

, which yields

{∥DN (φ)∥}_{ε} = ⋁ \{{∥D ⋓ E∥}_{ε} : 〈D, E〉 \in DN (σ) \times DN (δ)\}

= ⋁ \{{∥〈D^{+} \cup E^{+}, D^{-} \cup E^{-}〉∥}_{ε} : D \in DN (σ) & E \in DN (δ)\}

.

(We omit possible occurrences of

D ⋓ E = \emptyset

for

(D^{+} \cup E^{+}) \cap (D^{-} \cup E^{-}) \neq \emptyset

, since they imply

{∥〈D^{+} \cup E^{+}, D^{-} \cup E^{-}〉∥}_{ε} = 0

.) So by the induction hypothesis we get

{∥DN (φ)∥}_{ε} = 1

\Leftrightarrow (\exists D \in DN (σ)) (\exists E \in DN (δ)) (\begin{matrix} (\forall π (i) \in D^{+} \cup E^{+}) {∥v_{i}∥}_{ε} = 1 & \\ (\forall π (j) \in D^{-} \cup E^{-}) {∥\neg v_{j}∥}_{ε} = 1 \end{matrix})

\Leftrightarrow (\exists D \in DN (σ)) (\begin{matrix} (\forall π (i) \in D^{+}) {∥v_{i}∥}_{ε} = 1 & \\ (\forall π (j) \in D^{-}) {∥\neg v_{j}∥}_{ε} = 1 \end{matrix}) &

(\exists E \in DN (δ)) (\begin{matrix} (\forall π (i) \in E^{+}) {∥v_{i}∥}_{ε} = 1 & \\ (\forall π (j) \in E^{-}) {∥\neg v_{j}∥}_{ε} = 1 \end{matrix})

\Leftrightarrow {∥DN (σ)∥}_{ε} = 1 = {∥DN (δ)∥}_{ε}

\Leftrightarrow {∥σ∥}_{\hat{ε}} = 1 = {∥δ∥}_{ε}

,

which yields

{∥φ∥}_{ε} = {∥σ \land δ∥}_{ε} = {∥DN (φ)∥}_{ε}

.

Basis of induction and case

φ = σ \lor τ

are trivial. □

Definition 6.

For any

X, Y \subseteq D

we let

{VA}_{0} : = \{ε_{0} \in VA : |ε^{- 1} (1)| \leq (\binom{k}{2})\}

and define two equivalences ∼ and

\sim_{0}

:

\begin{matrix} X \sim Y : \Leftrightarrow (\forall ε \in VA) {∥X∥}_{ε} = {∥Y∥}_{ε}, \\ X \sim_{0} Y : \Leftrightarrow (\forall ε \in {VA}_{0}) {∥X∥}_{ε} = {∥Y∥}_{ε} . \end{matrix}

Obviously ∼ in stronger than

\sim_{0}

.

Lemma 11.

Suppose

φ \in A

satisfies

DN (φ) \sim_{0} {CLIQ}_{2}

. Then

{AC}^{p} (φ) = POS

and

{AC}^{n} (φ) = \emptyset

. So by Theorem 14,

c s (φ) > m^{\frac{1}{5} m^{\frac{1}{8}}}

for sufficiently large m. By Lemma 16, the latter holds for any

φ \sim_{0} {CLIQ}_{2}

.

Proof. Suppose

DN (φ) \sim_{0} {CLIQ}_{2}

. We show that

{AC}^{p} (φ) = POS

. Let

K = \{π (i) : i \in S\} \in POS

, which yields

{∥{CLIQ}_{2}∥}_{ε} = 1

for

ε \in {VA}_{0}

with

ε (i) : = \{\begin{matrix} 1, & if & i \in S, \\ 0, & else, \end{matrix}

So

{∥DN (φ)∥}_{ε} = 1

holds by the assumption, and hence there exists

D \in DN (φ)

satisfying

{∥D∥}_{ε} = 1

for

D^{+} = \{π (i) : i \in T\}

. But then for any

i \in T

we get

ε (i) = 1

, which yields

T \subseteq S

and hence

D \subseteq^{+} K

. So

POS \subseteq {AC}^{p} (φ) \subseteq POS

. Thus

{AC}^{p} (φ) = POS

, as required.

Suppose there is a

C_{f} \in {AC}^{n} (φ)

, i.e. there exists

E \in DN (φ)

with

E \subseteq^{+} C_{f}

for

E^{+} = \{π (a_{i}) : i \in S\}

. Define

ε \in {VA}_{0}

as above. Then

{∥E∥}_{ε} = 1

and hence

{∥DN (φ)∥}_{ε} = 1

. Hence

∥{CLIQ}_{2}∥ = 1

holds by the assumption, and therefore there exist

K \in POS

and

D \in D

such that

K \subseteq D^{+}

and

{∥D∥}_{ε} = 1

. But arguing as above this would imply

D^{+} \subseteq E^{+}

and hence

K \subseteq E^{+} \subseteq C_{f}

, which contradicts Lemma 1. Thus

{AC}^{n} (φ) = \emptyset

, as required. □

3.4. General Boolean case

Let $B$ denote full Boolean algebra with constants $1, 0$ , operations $\lor, \land, \neg$ and variables $v_{1}, \dots, v_{n}$ . Note that $A \subset B$ .
Arbitrary Boolean formulas $φ \in B$ are convertible to equivalent DMN formulas $φ^{*} \in A$ that arise by applying as long as possible De Morgan rewriting rules 1–4 :

1.: $\neg 1 ↪ 0, \neg 0 ↪ 1 .$
2.: $\neg (σ \lor τ) ↪ \neg σ \land \neg τ .$
3.: $\neg (σ \land τ) ↪ \neg σ \lor \neg τ .$
4.: $\neg \neg σ ↪ σ .$

It is a folklore that circuit size of

φ^{*}

is at most that of

φ

. Namely, circuit structure of

φ^{*}

arises by successively replacing

\neg 1

by 0,

\neg 0

by 1, gates ∨ and ∧ occurring in the scope of ¬ in

φ

by complementary gates ∧ and ∨, respectively, while deleting gates ¬ in question and all double negations

\neg \neg

. Note that ¬ may occur in

φ^{*}

only in literals

\neg v_{i}

, if at all.

Semantics in $B$ is defined as in $A$ with respect to variable assignments $ε \in VA$ (cf. Definition 15).

Lemma 12.For any

φ \in B

and

ε \in VA

we have

c s (φ^{*}) \leq c s (φ)

and

{∥φ∥}_{ε} = {∥φ^{*}∥}_{ε} = {∥DN (φ^{*})∥}_{ε} .

Proof.

{∥φ∥}_{ε} = {∥φ^{*}∥}_{ε}

holds by trivial induction on

{∥φ^{*}∥}_{ε} = {∥DN (φ^{*})∥}_{ε}

, while

{∥φ^{*}∥}_{ε} = {∥DN (φ^{*})∥}_{ε}

follows from Lemma 16. □

Theorem 2.Suppose that

φ \in B

provides a solution of

{CLIQUE}_{m, k}

in full Boolean logic and semantics involved. Then for sufficiently large

m = k^{4}

,

c s (φ)

is exponentially large in m.

Proof. Without loss of generality assume that double graphs are represented by arbitrary pairs of plain graphs

D = D (ε) = 〈G (ε), G^{'}〉

such that

G (ε) = \{π (i) : i \in [n] & ε (i) = 1\}

and

G^{'} \subseteq \{π (i) : i \in [n] & ε (i) = 0\}

, for any chosen

ε \in VA

. For brevity we also write

D^{+}

and

D^{-}

for

G (ε)

and

G^{'}

, respectively, and denote by

D

the set of all

D = D (ε)

for

ε \in VA

.

Consider Boolean circuits C whose open sources are assigned with Boolean constants and variables

v_{i}

(

i \in [n]

), and let

C (ε)

designate corresponding variable-free Boolean circuits that are obtained by substituting

ε (i)

for all

v_{i}

. Let

∥C (ε)∥ \in \{0, 1\}

denote the Boolean value of

C (ε)

.

Now suppose that there exists a C such that for every

ε \in VA

,

C (ε)

returns “true” iff

G (ε)

contains a subgraph from

POS

. In our formalism this yields

(\forall ε \in VA) (∥C (ε)∥ = 1 \Leftrightarrow C L I Q (ε)) (3)

(cf. Introduction), provided that C corresponds to Boolean formula

φ \in B

.

Furthermore let

C^{\pm}

denote a DMN circuit corresponding to DMN formula

φ^{*}

. That is,

C^{\pm}

has circuit structure of

φ^{*}

whose open sources are assigned with literals

v_{i}

and/or

\neg v_{i}

occurring in

φ^{*}

. For any

ε \in VA

,

C^{\pm} (ε)

will designate the corresponding variable-free Boolean circuit obtained by substituting

ε (i)

for all

v_{i}

, and let

∥C^{\pm} (ε)∥ \in \{0, 1\}

denote the Boolean value of

C^{\pm} (ε)

. By Lemma 19, this yields

(\forall ε \in VA) ∥C (ε)∥ = ∥C^{\pm} (ε)∥ (2)

and

(\forall ε \in VA) (\begin{matrix} ∥C^{\pm} (ε)∥ = 1 \Leftrightarrow \\ (\exists D \in DN (C^{\pm})) ∥C_{D}^{\pm} (ε)∥ = 1 \end{matrix}) (4)

which together with

(3)

implies

(\forall ε \in VA) (\begin{matrix} C L I Q (ε) \Leftrightarrow \\ (\exists D \in DN (C^{\pm})) ∥C_{D}^{\pm} (ε)∥ = 1 \end{matrix}) (5)

(cf. Introduction). Moreover, we prove another crucial equivalence

(\forall ε \in VA) (C L I Q (ε) \Leftrightarrow {∥{CLIQ}_{2}∥}_{ε} = 1) (6)

⊳

{∥{CLIQ}_{2}∥}_{ε} = 1

\Leftrightarrow (\exists D \in D) (\exists K \in POS) (K \subseteq^{+} D & {∥D∥}_{ε} = 1)

\Leftrightarrow (\exists D \in D) (\exists K \in POS)

(\begin{matrix} K \subseteq D^{+} & \\ {∥\underset{π (i) \in D^{+}}{⋀} v_{i} \land \underset{π (j) \in D^{-}}{⋀} \neg v_{j}∥}_{ε} \\ = 1 \end{matrix})

\Leftrightarrow (\exists D \in D) (\exists K \in POS)

(\begin{matrix} K \subseteq D^{+} & \\ \begin{matrix} (\forall π (i) \in D^{+}) ε (i) = 1 & \\ (\forall π (j) \in D^{-}) ε (j) = 0 \end{matrix} \end{matrix})

\Rightarrow (\exists D \in D) (\exists K \in POS) (K \subseteq D^{+} \subseteq G (ε))

\Rightarrow C L I Q (ε)

\Rightarrow (\exists K \in POS) (\begin{matrix} K \subseteq G (ε) & \\ \begin{matrix} (\forall π (i) \in G (ε)) ε (i) = 1 & \\ (\forall π (j) \in \emptyset) ε (j) = 0 \end{matrix} \end{matrix})

\Rightarrow (\exists K \in POS) (\begin{matrix} K \subseteq^{+} D (ε) : = 〈G (ε), \emptyset〉 &, \\ \begin{matrix} (\forall π (i) \in D {(ε)}^{+}) ε (i) = 1 & \\ (\forall π (j) \in D {(ε)}^{-}) ε (j) = 0 \end{matrix} \end{matrix})

\Leftrightarrow (\exists D \in D) (\exists K \in POS) (K \subseteq^{+} D & {∥D∥}_{ε} = 1)

\Leftrightarrow {∥{CLIQ}_{2}∥}_{ε} = 1 ⊲

which together with

(5)

implies

(\forall ε \in VA) (\begin{matrix} {∥{CLIQ}_{2}∥}_{ε} = 1 \Leftrightarrow \\ (\exists D \in DN (C^{\pm})) ∥C_{D}^{\pm} (ε)∥ = 1 \end{matrix}) (7)

It remains to observe that

(7)

is a circuit representation of Lemma 18. Since C and

C^{\pm}

are respectively isomorphic to

φ

and

φ^{*}

, this completes the proof of Theorem. □

Corollary 1.It holds

NP ⊈ P / poly

. In particular

P \neq NP

.

Proof. Boolean circuit complexity is quadratic in derterministic time (cf. e.g. [2]: Proposition 11.1, [5]: Theorem 9.30). Hence the assertion easily follows from Theorem 20 as

{CLIQUE}_{m, k}

is a NP problem. □

3.5. Application

Denote by

A_{0}^{+}

positive (monotone) subalgebra of

A

whose formulas are built up from variables and constants by positive operations ∨ and ∧. Thus CNF and/or DNF formulas

φ \in A_{0}^{+}

do not include negated variables.

Theorem 3.There is no polynomial time algorithm f converting arbitrary CNF formulas

φ \in A_{0}^{+}

into equivalent DNF formulas

f (φ) \in A_{0}^{+}

.

Proof. Suppose that for every

ε : [n] \to \{0, 1\}, ({∥φ∥}_{ε} = 1 \Leftrightarrow {∥f (φ)∥}_{ε} = 1 \Leftrightarrow {∥\neg f (φ)∥}_{ε} = 0)

. Thus

φ \in SAT \Leftrightarrow f (φ) \in SAT \Leftrightarrow \neg f (φ) \notin TAU

. Suppose that the size of

f (φ)

is polynomial in that of

φ

. Note that

\neg f (φ) \in B

is equivalent to CNF formula

{(\neg f (φ))}^{*} \in A

whose size is roughly the same as that of

f (φ)

, and hence polynomial in the size of

φ

. 5 Also note that the validity problem

{(\neg f (φ))}^{*} \in^{?} TAU

is solvable in polynomial time. Hence so is the dual problem

φ \in^{?} SAT

, which by the NP completeness of

SAT

yields

P = NP

, – a contradiction. □

Appendix A. On Lemma 9

Let

\emptyset \neq G \in G^{ℓ}

and

R_{\subseteq} (G) = \{f \in F : G \subseteq C_{f}\}

. To estimate

|R_{\subseteq} (G)|

we calculate the probability that a coloring function

f \in F

is in

R_{\subseteq} (G)

, i.e. every pair of nodes

x, y

connected by an edge in G is colored differently by

f (x) \neq f (y) < k

. Therefore to color every next node in v

(G)

we have to choose an arbitrary color among those not previously used. This yields the probability at least

\begin{matrix} \frac{k - 1}{k - 1} \times \frac{k - 2}{k - 1} \times \dots \times \frac{k - 1 - |v (G)|}{k - 1} > \\ {(\frac{k - 1 - |v (G)|}{k - 1})}^{|v (G)|} \geq {(1 - \frac{ℓ}{k - 1})}^{ℓ} > \\ {(1 - \frac{ℓ}{k})}^{ℓ} > {(1 - \frac{1}{2 ℓ})}^{ℓ} ⟶ \frac{1}{\sqrt{e}} > \frac{1}{2}, \\ as k = 2 ℓ^{2} ⟶ \infty . \end{matrix}

Hence

|R_{\subseteq} (G)| > \frac{1}{2} |F| = \frac{1}{2} {(k - 1)}^{m}

, for sufficiently large k. Now consider

R_{⊈} (G) = \{f \in F : G ⊈ C_{f}\} = F ∖ R_{\subseteq} (G)

and make an obvious conclusion

|R_{⊈} (G)| = |F| - |R_{\subseteq} (G)| \leq \frac{1}{2} |F| = \frac{1}{2} {(k - 1)}^{m}

. Consequently, for any

D \in D^{ℓ}

we have

|R_{\subseteq} (D)| = |\{f \in F : D^{+} \subseteq C_{f}\}| > |R_{\subseteq} (D^{+})| > \frac{1}{2} |F| = \frac{1}{2} {(k - 1)}^{m}

, and hence

|R_{⊈} (D)| \leq \frac{1}{2} |F| = \frac{1}{2} {(k - 1)}^{m}

.

Generally, for any

X \subseteq F

we set

R_{\subseteq} (X : G) : = \{f \in X : G \subseteq C_{f}\}

and

R_{⊈} (X : G) : = \{f \in X : G ⊈ C_{f}\}

. Then analogously

|R_{\subseteq} (X : G)| \geq \frac{1}{2} |X|

and

|R_{⊈} (X : G)| \leq \frac{1}{2} |X|

, provided that

|X (x)| = k - 1

holds for any

x \in

v

(G)

, where

X (x)

abbreviates

\{f (x) : f \in X\}

. Furthermore, for any

D \in D^{ℓ}

we set

R_{⊈} (X : D) : = \{f \in X : D^{+} ⊈ C\}

and obtain

|R_{⊈} (X : D)| \leq \frac{1}{2} |X|

, if

|X^{+} (x)| = k - 1

for any

x \in

v

(D^{+})

. Note that

R_{⊈} (F : D) = R_{⊈} (D)

.

Consider any collection

D_{1}, \dots, D_{q} \in D^{ℓ}

,

(\forall i \neq j \in [q]) D_{i}^{+} \cap D_{j}^{+} = \emptyset

. Then

|\overset{q}{⋂_{i = 1}} R_{⊈} (D_{i})| \leq 2^{- q} |F|

will easily follow from

(\forall j \in [q - 1]) (|\overset{q}{⋂_{i = j}} R_{⊈} (X : D_{i})| \leq 2^{- q} |X|) (*)

provided that

X \subseteq F

satisfies

|X^{+} (x_{i})| = k - 1

for all

x_{i} \in

v

(D_{i}^{+})

,

i \in [q]

. Now

(*)

is proved as follows by induction on q.

Basis :

q = 2

. Since

D_{1}^{+} \cap D_{2}^{+} = \emptyset

, for any

x_{1} \in

v

(D_{i}^{+})

,

x_{2} \in

v

(D_{2}^{+})

we have

|R_{⊈} (X : D_{1}) (x_{2})| = |X (x_{2})|

and

|X (x_{1})| = |X (x_{2})| = k - 1

. This yields

\begin{matrix} |R_{⊈} (X : D_{1}) \cap R_{⊈} (X : D_{2})| = |R_{⊈} (R_{⊈} (X : D_{1}) : D_{2})| \\ \leq \frac{1}{2} |R_{⊈} (X : D_{1})| \leq \frac{1}{4} |X| \end{matrix}

Induction step. By the same token we obtain

\begin{matrix} |\overset{q}{⋂_{i = j}} R_{⊈} (X : D_{i})| = |\overset{q - 1}{⋂_{i = j}} R_{⊈} (X : D_{i}) \cap R_{⊈} (X : D_{q})| \\ = |R_{⊈} (\overset{q - 1}{⋂_{i = j}} R_{⊈} (X : D_{i}) : D_{q})| \leq \frac{1}{2} |\overset{q - 1}{⋂_{i = j}} R_{⊈} (X : D_{i})| \\ \leq 2^{- q} |X| . \end{matrix}

Appendix B. Proof of Lemma 12

Use Lemma 3 and inclusion

A ∖ B \subseteq (A ∖ C) \cup (C ∖ B)

.

1.

\partial^{p} (σ \lor τ) =

{AC}^{p} (DN (σ) \cup DN (τ)) ∖ {AC}^{p} (AP (σ) ⊔ AP (τ))

\begin{matrix} \subseteq {AC}^{p} (DN (σ) \cup DN (τ)) ∖ {AC}^{p} (AP (σ) ⊔ {AC}^{p} (AP (τ))) \\ \cup {AC}^{p} (AP (σ) ⊔ {AC}^{p} (AP (τ))) ∖ {AC}^{p} (AP (σ) ⊔ AP (τ)) \end{matrix}

\begin{matrix} = {AC}^{p} (DN (σ) \cup {AC}^{p} (τ)) ∖ {AC}^{p} (AP (σ) \cup {AC}^{p} (AP (τ))) \\ \cup {AC}^{p} (AP (σ) \cup {AC}^{p} (AP (τ))) ∖ {AC}^{p} (AP (σ) ⊔ AP (τ)) \end{matrix}

\begin{matrix} \subseteq {AC}^{p} (σ) ∖ {AC}^{p} (AP (σ)) \cup {AC}^{p} (τ) ∖ {AC}^{p} (AP (τ)) \\ \cup {AC}^{p} (AP (σ)) \cup {AC}^{p} (AP (τ)) ∖ A C^{p} (AP (σ) ⊔ AP (τ)) \end{matrix}

= \partial^{p} (σ) \cup \partial^{p} (τ) \cup \partial_{⊔}^{p} (AP (σ), AR (τ)) .

2.

\partial^{p} (σ \land τ) =

{AC}^{p} (DN (σ) ⊙ DN (τ)) ∖ {AC}^{p} (AP (σ) ⊓ AP (τ))

\begin{matrix} \subseteq {AC}^{p} (DN (σ) ⊙ DN (τ)) ∖ {AC}^{p} (AP (σ) \cap {AC}^{p} (AP (τ))) \\ \cup {AC}^{p} (AP (σ) \cap {AC}^{p} (AP (τ))) ∖ {AC}^{p} (AP (σ) ⊓ AP (τ)) \end{matrix}

\begin{matrix} = {AC}^{p} (σ) \cap {AC}^{p} (τ) ∖ {AC}^{p} (AP (σ) \cap {AC}^{p} (AP (τ))) \\ \cup {AC}^{p} (AP (σ) \cup {AC}^{p} (AP (τ))) ∖ {AC}^{p} (AP (σ) ⊓ AP (τ)) \end{matrix}

\begin{matrix} \subseteq {AC}^{p} (σ) ∖ {AC}^{p} (AP (σ)) \cup {AC}^{p} (τ) ∖ {AC}^{p} (AP (τ)) \\ \cup \partial_{⊓}^{p} (AP (σ), AR (τ)) \end{matrix}

= \partial^{p} (σ) \cup \partial^{p} (τ) \cup \partial_{⊓}^{p} (AP (σ), AR (τ)) .

3.

\partial^{n} (σ \lor τ) =

{AC}^{n} (AP (σ) ⊔ AP (τ)) ∖ {AC}^{n} (DN (σ) \cup DN (τ))

\begin{matrix} \subseteq {AC}^{n} (AP (σ) ⊔ AP (τ)) ∖ {AC}^{n} (AP (σ)) \cup {AC}^{n} (AP (τ)) \\ \cup {AC}^{n} (AP (σ)) \cup {AC}^{n} (AP (τ)) ∖ {AC}^{n} (DN (σ) \cup DN (τ)) \end{matrix}

\begin{matrix} = {AC}^{n} (AP (σ) ⊔ AP (τ)) ∖ {AC}^{n} (AP (σ)) \cup {AC}^{n} (AP (τ)) \\ \cup {AC}^{n} (AP (σ)) \cap {AC}^{n} (AP (τ)) ∖ [{AC}^{n} (σ) \cup {AC}^{n} (τ)] \end{matrix}

\begin{matrix} \subseteq \partial_{⊔}^{n} (AP (σ), AR (τ)) \cup A C^{n} (AP (σ)) ∖ {AC}^{n} (σ) \\ \cup {AC}^{n} (AP (τ)) ∖ {AC}^{n} (τ) \end{matrix}

= \partial_{⊔}^{n} (AP (σ), AR (τ)) \cup \partial^{n} (σ) \cup \partial^{n} (τ) .

4.

\partial^{n} (σ \land τ) =

{AC}^{n} (AP (σ) ⊓ AP (τ)) ∖ {AC}^{n} (DN (σ) ⊙ DN (τ))

\begin{matrix} \subseteq {AC}^{n} (AP (σ) ⊓ AP (τ)) ∖ {AC}^{n} (AP (σ)) \cap {AC}^{n} (AP (τ)) \\ \cup {AC}^{n} (AP (σ)) \cap {AC}^{n} (AP (τ)) ∖ {AC}^{n} (DN (σ) \cap DN (τ)) \end{matrix}

\begin{matrix} = {AC}^{n} (AP (σ) ⊓ AP (τ)) ∖ {AC}^{n} (AP (σ)) \cap {AC}^{n} (AP (τ)) \\ \cup {AC}^{n} (AP (σ)) \cup {AC}^{n} (AP (τ)) ∖ [{AC}^{n} (σ) \cap {AC}^{n} (τ)] \end{matrix}

\begin{matrix} \subseteq \partial_{⊓}^{n} (AP (σ), AR (τ)) \cup {AC}^{n} (AP (σ)) ∖ {AC}^{n} (σ) \\ \cup {AC}^{n} (AP (τ)) ∖ {AC}^{n} (τ) \end{matrix}

= \partial_{⊓}^{n} (AP (σ), AR (τ)) \cup \partial^{n} (σ) \cup \partial^{n} (τ) .

Appendix C. Basic (In)equalities

We have

k = m^{\frac{1}{4}} = 2 ℓ^{2}, p = ℓ {log}_{2} m, L = {(p - 1)}^{ℓ} ℓ!

, where

m ≫ 0 .

So

ℓ = \frac{1}{\sqrt{2}} m^{\frac{1}{8}}

, and hence

Body Math

ℓ! \approx \sqrt{2 π ℓ} {(\frac{ℓ}{e})}^{ℓ} = \sqrt{\sqrt{2} π m^{\frac{1}{8}}} {(\frac{m^{\frac{1}{8}}}{\sqrt{2} e})}^{\frac{1}{\sqrt{2}} m^{\frac{1}{8}}}

< m^{\frac{1}{16} + \frac{1}{8 \sqrt{2}} m^{\frac{1}{8}}} < m^{\frac{1}{11} m^{\frac{1}{8}}}

,

m ≫ 0 .

So

\begin{matrix} ℓ! < m^{\frac{1}{11} m^{\frac{1}{8}}} \end{matrix}

, while

\begin{matrix} {log}_{2} m < m^{α} \end{matrix}

for any chosen

α > 0

.

Now

p = ℓ {log}_{2} m < m^{\frac{1}{11} + α} < m^{\frac{1}{10}}

, and hence

\begin{matrix} {(p - 1)}^{ℓ} < p^{ℓ} < m^{\frac{1}{10 \sqrt{2}} m^{\frac{1}{8}}} < m^{\frac{1}{14} m^{\frac{1}{8}}} \end{matrix}

, while

\begin{matrix} 2^{p} = m^{ℓ} = m^{\frac{1}{\sqrt{2}} m^{\frac{1}{8}}} \end{matrix}

.

Thus

L = {(p - 1)}^{ℓ} ℓ! < m^{\frac{1}{14} m^{\frac{1}{8}}} m^{\frac{1}{11} m^{\frac{1}{8}}} < m^{\frac{3}{50} m^{\frac{1}{8}}}

,

and hence

\begin{matrix} L^{2} < m^{\frac{3}{25} m^{\frac{1}{8}}} \end{matrix}

. Moreover

{(\frac{m - ℓ}{k})}^{ℓ} = {(\frac{m - \frac{1}{\sqrt{2}} m^{\frac{1}{8}}}{m^{\frac{1}{4}}})}^{\frac{1}{\sqrt{2}} m^{\frac{1}{8}}} > m^{\frac{1}{2 \sqrt{2}} m^{\frac{1}{8}}}

,

and hence:

\begin{matrix} {(\frac{m - ℓ}{k})}^{ℓ} L^{- 2} > \frac{m^{\frac{1}{2 \sqrt{2}} m^{\frac{1}{8}}}}{m^{\frac{3}{25} m^{\frac{1}{8}}}} > m^{\frac{1}{5} m^{\frac{1}{8}}} \end{matrix}

\begin{matrix} 2^{p - 1} L^{- 2} > \frac{1}{2} \frac{m^{\frac{1}{\sqrt{2}} m^{\frac{1}{8}}}}{m^{\frac{3}{25} m^{\frac{1}{8}}}} > m^{\frac{1}{2} m^{\frac{1}{8}}} > m^{\frac{1}{5} m^{\frac{1}{8}}} \end{matrix}

.

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1	Note that $G \in G^{ℓ}$ implies $\sqrt{2 \|G\|} < \frac{1}{2} (1 + \sqrt{1 + 8 \|G\|}) \leq \|v (G)\| \leq 2 ℓ .$
2	This operation will be referred to as elementary plucking.
3	We write ∂ instead of $δ$ used in [3]−[4].
4	More precisely, $cs (φ)$ is the total number of pairwise distinct subterms of (including) $φ$ .
5	The difference between plain (linear) and circuit length is inessential for CNF and/or DNF formulas under consideration.

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