Neural Codes and the Factor Complex

TL;DR

This paper introduces the factor complex of a neural code, providing algebraic and combinatorial characterizations of certain classes of codes, enhancing understanding of their structure.

Contribution

It presents the novel concept of the factor complex and uses it to characterize max-intersection-complete and intersection-complete neural codes.

Findings

Factor complex captures intervals and maximal codewords.

Provides algebraic characterization of max-intersection-complete codes.

Offers a new combinatorial characterization of intersection-complete codes.

Abstract

We introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of max-intersection-complete codes, as well as a new combinatorial characterization of intersection-complete codes.

Equations

$$[c,d]:=\{w\in 2^{[n]}\mid c\subset w\subset d\}.$$

$$C^{\prime }:=2^{[n]}\setminus C.$$

$${\varphi }_c:=\prod _{i\in c}{x}_i\prod _{j\in [n]\setminus c}(1-x_j)$$

$$J_C:=⟨{\varphi }_c\mid c\in C^{\prime }⟩.$$

$$J_C=\bigcap _{h=1}^g{P}_h,$$

$$P=⟨\{x_i\mid i\in \sigma \}\cup \{(1-x_j)\mid j\in \tau \}⟩\text{ for }\sigma ,\tau \text{ disjoint subsets of }[n].$$

$$\mathcal{P}(\varphi ):=\prod _{i\in \sigma }{x}_i\prod _{j\in \tau }{y}_j\in S.$$

$$\mathcal{P}(J):=⟨\mathcal{P}(\varphi )\mid \varphi \in \mathrm{CF}(J)⟩\subset S.$$

$$I(\Delta ):=⟨\prod _{i\in \sigma }{x}_i\mid \sigma \notin \Delta ⟩\subset \text{mathbbm}k[x_1,\dots ,x_n].$$

$$I(\Delta )=\bigcap _{\sigma \in \text{Facets}(\Delta )}⟨x_i\mid i\notin \sigma ⟩.$$

$$\overline{[n]}=\{\overline{1},\dots ,\overline{n}\}\text{and}[n]\cup \overline{[n]}=\{1,\dots ,n,\overline{1},\dots ,\overline{n}\}.$$

$$\mathrm{FI}(C):=\bigcap _{h=1}^g\mathcal{P}(P_h).$$

$$J_{C^{\prime }}=⟨(1-x_1)(1-x_3),(1-x_1)(1-x_2),x_2(1-x_3),x_3(1-x_2)⟩=⟨x_2,x_3,1-x_1⟩\cap ⟨1-x_2,1-x_3⟩.$$

$$FI(C^{\prime })=⟨x_2,x_3,y_1⟩\cap ⟨y_2,y_3⟩,$$

$$\displaystyle\begin{array}[]{ccccc}\{\textrm{pseudomonomials~{}in~{}}J_{C^{\prime}}\}&\leftarrow&\{\textrm{intervals~{}in~{}}C\}&\to&\{\textrm{effective~{}faces~{}of~{}}\Delta_{\cap}(C)\}\\ \prod_{i\in c}x_{i}\prod_{j\in[n]\smallsetminus d}(1-x_{j})&\leftmapsto&[c,d]&\mapsto&d\cup\overline{[n]\smallsetminus c}\\ \end{array}$$

$$\displaystyle\begin{array}[]{ccccc}\{\textrm{minimal~{}primes~{}of~{}}I(\Delta(C))\}&\leftarrow&\{\textrm{maximal~{}codewords~{}of~{}}C\}&\to&\left\{\begin{array}[]{c}\textrm{minimal prime-sets}\\ \textrm{of }\Delta_{\cap}(C^{\prime})\end{array}\right\}\\ \langle x_{i}\mid i\in[n]\smallsetminus M\rangle&\leftmapsto&M&\mapsto&\overline{[n]\smallsetminus M}\\ \end{array}$$

$$([n]\setminus \bigcup _{v\in H_F}{B}_v)\not\subset F,$$

$$H_F:=\{v\in [u]|{\overline{B}}_v\subset F\}.$$

$$\overline{[n]\setminus c}=\overline{[n]\setminus \bigcap _{v\in V}{m}_v}=\bigcup _{v\in V}\overline{[n]\setminus m_v}=\bigcup _{v\in V}{\overline{B}}_v.$$

$$c=[n]\setminus \bigcup _{v\in H_F}{B}_v=\bigcap _{v\in H_F}[n]\setminus B_v=\bigcap _{v\in H_F}{m}_v,$$

$$\displaystyle\begin{array}[]{ccccc}\{\text{pseudomonomials~{}in~{}}J_{C^{\prime}}\}&\xleftarrow{\alpha}&\{\text{intervals~{}in~{}}C\}&\xrightarrow{\beta}&\{\text{effective~{}faces~{}of~{}}\Delta_{\cap}(C)\}\\ \prod_{i\in c}x_{i}\prod_{j\in[n]\smallsetminus d}(1-x_{j})&\leftmapsto&[c,d]&\mapsto&d\cup\overline{[n]\smallsetminus c}\\ \end{array}$$

$$\left\langle\{x_{i}\ |\ i\not\in d\cup\overline{[n]\smallsetminus c}\}\cup\{y_{j}\ |\ \bar{j}\not\in d\cup\overline{[n]\smallsetminus c}\}\right\rangle=\big{\langle}\{x_{i}\mid i\in[n]\smallsetminus d\}\cup\{y_{j}\mid j\in c\}\big{\rangle}$$

$$\big{\langle}\{x_{i}\mid i\in[n]\smallsetminus d\}\cup\{(1-x_{j})\mid j\in c\}\big{\rangle}.$$

$$\displaystyle\begin{array}[]{ccccc}\{\text{minimal~{}primes~{}of~{}}I(\Delta(C))\}&\xleftarrow{\gamma}&\{\text{maximal~{}codewords~{}in~{}}C\}&\xrightarrow{\delta}&\left\{\begin{array}[]{c}\text{minimal~{}prime-sets}\\ \text{of~{}}\Delta_{\cap}(C^{\prime})\end{array}\right\}\\ \langle x_{i}\mid i\in[n]\smallsetminus M\rangle&\leftmapsto&M&\mapsto&\overline{[n]\smallsetminus M}\\ \end{array}$$

$$\mathcal{P}(J_{C^{\prime }})=⟨y_1{y}_3,y_1{y}_2,x_2{y}_3,x_3{y}_2⟩=⟨x_2,x_3,y_1⟩\cap ⟨y_2,y_3⟩\cap ⟨x_3,y_1,y_3⟩\cap ⟨x_2,y_2,y_3⟩.$$

$$I(\Delta (C))=⟨\{x_{\sigma }|[\sigma ,[n]]\text{ a maximal interval of }{C}^{\prime }\}⟩.$$

$$\tilde{I}:=⟨x_b\mid [b,[n]]\text{ is a maximal interval of }{C}^{\prime }\text{ and }b\ne a⟩\subset I(\Delta (C)).$$

Full text

1. Introduction

A neural code on $n$ neurons is a subset of $2^{[n]}$, where $[n]=\{1,2,\dots,n\}$; determining which neural codes are convex remains a central open problem in this area. The broadest family of codes known to be convex consists of max-intersection-complete codes, those codes closed under taking intersections of maximal elements [2, 4]. Recently, Curto et al. [4] asked for an algebraic signature for max-intersection-complete codes.

Here we answer the question of Curto et al. Our main result, Theorem 1.1 below, gives a characterization for when a code is max-intersection-complete in terms of the canonical form of its neural ideal (Definitions 2.3 and 2.4) and the Stanley–Reisner ideal $I(\Delta(C))$ of its simplicial complex $\Delta(C)$ (Definitions 2.7 and 2.8).

Theorem 1.1**.**

A code $C$ on $n$ neurons is max-intersection-complete if and only if for every non-monomial $\phi$ in the canonical form of the neural ideal of $C$, there exists $i\in[n]$ such that

• (i)

every associated prime of $I(\Delta(C))$ that contains $x_{i}$ also contains $\phi$, and

• (ii)

$(1-x_{i})\mid\phi$.

We remark that Theorem 1.1 can be turned into an algorithm to verify whether a code is max-intersection-complete. This algorithm’s runtime is sub-exponential in the input size, where the input consists of the maximal codewords of a code $C$ as well as its canonical form $\operatorname{{\operatorname{CF}}}(J_{C})$. On the other hand, the known algorithms for computing $\operatorname{{\operatorname{CF}}}(J_{C})$ are exponential. More details on the computational aspects of Theorem 1.1 can be found in Section 6, which also includes an infinite family of codes for which Theorem 1.1 is more efficient at verifying max-intersection-completeness than brute-force checking of intersections of maximal codewords (see Proposition 6.2).

To prove Theorem 1.1, which translates a property of a code to a property of its neural ideal, we introduce a new combinatorial object, the factor complex of a code. This is a simplicial complex that, like the neural ideal but unlike $\Delta(C)$, captures all the combinatorial information in a code $C$. We are therefore able to elucidate the relationships among codes, their factor complexes, and their related ideals (neural ideals and Stanley–Reisner ideals) – and then use these results to characterize being max-intersection-complete in terms of the factor complex. Finally, this combinatorial criterion directly translates into an algebraic criterion, Theorem 1.1 above.

Along the way, we give a new characterization of intersection-complete codes – those codes that are closed under taking intersections of codewords. Our characterization is combinatorial, via the factor complex, in contrast to a prior algebraic characterization through the neural ideal [4]. Indeed, we expect in the future that the factor complex may help us understand more properties of neural codes.

Our work fits into the literature on neural codes as follows. Like previous works, we are motivated by the question of convexity in neural codes [3, 6, 14, 15, 16, 19, 21], with a specific interest in using neural ideals to study convexity [5, 7, 8, 10, 11, 17]. Also, our factor complexes are motivated by the closely related polar complexes introduced recently by Güntürkün et al. [9] (see also [1, 11]).

Outline

This article is organized as follows. Section 2 contains background material, and Section 3 gives our main results. In Section 4, we prove relationships among codes, their factor complexes, and their neural or Stanley-Reisner ideals, and Section 5 relates factor complexes and polar complexes.

Author contributions

The first author is the main contributor to this article.

Acknowledgements

We are grateful to Carina Curto, Luis García-Puente, and Alex Kunin for inspirational discussions. AS was partially supported by NSF grant DMS-1752672. We thank the referee for helpful suggestions which improved this work.

2. Background

Throughout this article, $C$ is a neural code on $n$ neurons, that is, a subset of $2^{[n]}$, where $[n]=\{1,2,\dots,n\}$. Elements of $C$ are called codewords, and may be represented as subsets of $[n]$ or as $n$-tuples of zeros and ones, where a $1$ in position $i$ indicates that $i$ belongs to the codeword.

Given $c\subset d\subset[n]$, the Boolean interval between $c$ and $d$ is

[TABLE]

The complement of a code $C$ on $n$ neurons is the code

[TABLE]

Convention. In this article, we assume that $\emptyset\subsetneq C\subsetneq 2^{[n]}$, so that the neural ideals (defined below) of $C$ and $C^{\prime}$ have primary decompositions.

Definition 2.1**.**

Let $C$ be a code. The intervals of $C$ are the Boolean intervals contained in $C$. The maximal intervals of $C$ are the intervals of $C$ that are maximal with respect to inclusion.

Example 2.2**.**

For the code $C=\{\emptyset,2,3,12,13\}=\{000,010,001,110,101\}$, the maximal intervals are $[\emptyset,2]$, $[\emptyset,3]$, $[2,12]$, and $[3,13]$.

2.1. Neural ideals and the canonical form

The main reference for this section is [5].

We denote by $\mathbb{F}_{2}$ the field with two elements, and let $R=\mathbb{F}_{2}[x_{1},\dots,x_{n}]=\mathbb{F}_{2}[x]$. A pseudomonomial is a polynomial $\prod_{i\in\sigma}x_{i}\prod_{j\in\tau}(1-x_{j})\in R$, where $\sigma,\tau\subset[n]$ are disjoint. A pseudomonomial ideal is an ideal generated by pseudomonomials. If $c\in 2^{[n]}$, the pseudomonomial

[TABLE]

is called the indicator polynomial of $c$.

Definition 2.3**.**

The neural ideal $J_{C}$ of a code $C$ is the (pseudomonomial) ideal generated by the indicator polynomials of its non-codewords; in symbols,

[TABLE]

Note that, using the convention that $n$-tuples of zeros and ones represent codewords, the zero-set of $J_{C}$ is $C$. In other words, the code $C$ and its neural ideal contain the same information. Moreover, any ideal generated by pseudomonomials is the neural ideal of a code [13, Theorem 2.1].

The neural ideal $J_{C}$ has a unique irredundant decomposition

[TABLE]

where each $P_{h}$ is a pseudomonomial ideal that is prime [5, Proposition 6.8]. In particular, $J_{C}$ is a radical ideal. We remark that a pseudomonomial ideal $P$ is prime if and only if it is of the form

[TABLE]

Definition 2.4**.**

Let $J\subset R$ be a pseudomonomial ideal. A pseudomonomial in $J$ is minimal if it is minimal with respect to divisibility among all pseudomonomials in $J$. The canonical form of $J$ is the set $\operatorname{{\operatorname{CF}}}(J)$ of all minimal pseudomonomials of $J$.

The canonical form of a pseudomonomial ideal is a generating set for the ideal [5].

Example 2.5** (Example 2.2, continued).**

The complement of the code $C=\{\emptyset,2,3,12,13\}$ is $C^{\prime}=\{1,23,123\}$. Thus, the neural ideal of $C$ is $J_{C}=\langle x_{1}(1-x_{2})(1-x_{3}),~{}x_{2}x_{3}(1-x_{1}),~{}x_{1}x_{2}x_{3}\rangle$, and the canonical form is $\operatorname{{\operatorname{CF}}}(J_{C})=\{x_{1}(1-x_{2})(1-x_{3}),~{}x_{2}x_{3}\}$.

2.2. Polarization and squarefree monomial ideals

Let $S=\mathbb{F}_{2}[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]=\mathbb{F}_{2}[x,y]$.

The idea of using $y_{i}$ to encode $1-x_{i}$ is well known (see, for instance, [12, 20]). In the context of neural ideals, the following construction was introduced in [9].

Definition 2.6**.**

The polarization of a pseudomonomial $\phi=\prod_{i\in\sigma}x_{i}\prod_{j\in\tau}(1-x_{j})\in R$ is

[TABLE]

If $J\subset R$ is a pseudomonomial ideal, the polarization of $J$ is the ideal in $S$ obtained by polarizing the pseudomonomials in the canonical form of $J$, that is,

[TABLE]

Note that the polarization of a pseudomonomial ideal is a squarefree monomial ideal in $S$, that is, an ideal generated by monomials that are not divisible by the squares of the variables (so, ${\mathcal{P}}(J)$ is radical). We recall the relationship between squarefree monomial ideals and simplicial complexes.

Definition 2.7**.**

Let $\Delta$ be a simplicial complex on $[n]$, and let $\mathbbm{k}$ be a field. The Stanley–Reisner ideal of $\Delta$ is

[TABLE]

The ideal $I(\Delta)$ is radical, with prime decomposition

[TABLE]

It follows that $\Delta$ can be recovered from $I(\Delta)$. In fact, (5) can be used to conclude that any squarefree monomial ideal is the Stanley–Reisner ideal of some simplicial complex.

Definition 2.8**.**

The simplicial complex of a code $C$ is $\Delta(C)$, the smallest simplicial complex containing $C$. Its Stanley–Reisner ideal is denoted by $I(\Delta(C))\subset R=\mathbb{F}_{2}[x]$.

It is a fact that $I(\Delta(C))$ is generated by the monomials in $\operatorname{{\operatorname{CF}}}(J_{C})$ [5, Lemma 4.4].

Example 2.9** (Example 2.5, continued).**

For $C=\{\emptyset,2,3,12,13\}$, the simplicial complex $\Delta(C)$ has two facets, $12$ and $13$. The corresponding Stanley–Reisner ideal is $I(\Delta(C))=\langle x_{2}x_{3}\rangle$, which is generated by the unique monomial in the canonical form $\operatorname{{\operatorname{CF}}}(J_{C})=\{x_{1}(1-x_{2})(1-x_{3}),~{}x_{2}x_{3}\}$.

In this article, we work with squarefree monomial ideals in $S=\mathbb{F}_{2}[x,y]$ that arise from polarization. In order to construct their corresponding simplicial complexes, we use $\{1,\dots,n,\overline{1},\dots,\overline{n}\}$ as a vertex set, with the understanding that $x_{i}$ corresponds to $i$, and $y_{i}$ corresponds to $\overline{i}$. If $B\subset[n]$, we denote $\overline{B}=\{\overline{i}\mid i\in B\}$. In particular,

[TABLE]

We always use overline notation to denote subsets of $\overline{[n]}$; this is justified, as any subset of $\overline{[n]}$ is of the form $\overline{B}$ for some $B\subset[n]$.

Remark 2.10**.**

As noted above, the ideals that are associated to codes (the neural ideal $J_{C}$, the ideal $I(\Delta(C))$, and later the factor ideal $\operatorname{{\operatorname{FI}}}(C)$) are radical ideals, that is, they can be expressed as intersections of prime ideals. We emphasize that the sets of associated primes, minimal primes, and primary components of a radical ideal all coincide.

3. Main results

In this section we introduce a new combinatorial tool to study neural codes: the factor complex (Definition 3.1), and state our four main results. Theorems 3.3 and 3.4 summarize the relationships among codes, their factor complexes, and their related ideals (neural ideals and Stanley–Reisner ideals). These results are used to prove Theorems 3.6 and 3.7, which characterize intersection-complete codes and max-intersection-complete codes in two ways: combinatorially and algebraically.

Definition 3.1**.**

Let $C$ be a code on $n$ neurons, and recall the primary decomposition of the neural ideal $J_{C}$ given in (3). The factor ideal of $C$ is obtained by polarizing the components of $J_{C}$, namely,

[TABLE]

The factor complex $\Delta_{\cap}(C)$ of $C$ is the simplicial complex on $[n]\cup\overline{[n]}$ whose Stanley–Reisner ideal is $\operatorname{{\operatorname{FI}}}(C)$. A face of $\Delta_{\cap}(C)$ is defective if it contains neither $i$ nor $\overline{i}$ for some $i\in[n]$ (we think of $i$ as a defect, or flaw); faces that are not defective are called effective. We say that $\overline{B}\subset\overline{[n]}$ is a prime-set of $\Delta_{\cap}(C)$ if $[n]\cup\overline{B}\notin\Delta_{\cap}(C)$, and $\overline{B}$ is furthermore minimal if $\overline{B}$ is minimal with respect to inclusion among prime-sets. Lemma 4.5 gives the reason why we chose this terminology.

Example 3.2** (Example 2.9, continued).**

For $C^{\prime}=\{1,23,123\}$, the neural ideal decomposes as follows:

[TABLE]

The factor ideal is therefore

[TABLE]

and so the two facets of the factor complex $\Delta_{\cap}(C^{\prime})$ are $1\bar{2}\bar{3}$ and $123\bar{1}$ (both are effective). The minimal prime-sets of $\Delta_{\cap}(C^{\prime})$ are $\{\bar{2}\}$ and $\{\bar{3}\}$.

Theorem 3.3** (Codes, factor complexes, and neural ideals).**

Let $C$ be a code on $n$ neurons, and $C^{\prime}$ its complement code defined in (1). The following two maps are bijections:

[TABLE]

Moreover, every facet of $\Delta_{\cap}(C)$ is effective, and the following are equivalent:

1. (1)

$[c,d]$* is a maximal interval in $C$,* 2. (2)

$\prod_{i\in c}x_{i}\prod_{j\in[n]\smallsetminus d}(1-x_{j})\in\operatorname{{\operatorname{CF}}}(J_{C^{\prime}})$, and 3. (3)

$d\cup\overline{[n]\smallsetminus c}$* is a facet of $\Delta_{\cap}(C)$.*

Theorem 3.4** (Codes, factor complexes, and Stanley–Reisner ideals).**

Let $C$ be a code on $n$ neurons, with complement code $C^{\prime}$ and factor complex $\Delta_{\cap}(C)$. The following two maps are bijections:

[TABLE]

The proofs of Theorems 3.3 and 3.4 are postponed until Sections 4.1 and 4.2, respectively.

Example 3.5** (Example 3.2, continued).**

According to Theorem 3.3, the facets $1\bar{2}\bar{3}$ and $123\bar{1}$ of $\Delta_{\cap}(C^{\prime})$ correspond to the two maximal intervals of $C^{\prime}$, $[1,1]$ and $[23,123]$, respectively, and also to the two pseudomonomials in $\operatorname{{\operatorname{CF}}}(J_{C})$, namely, $x_{1}(1-x_{2})(1-x_{3})$ and $x_{2}x_{3}$, respectively.

Similarly, Theorem 3.4 implies that the minimal prime-sets $\{\bar{2}\}$ and $\{\bar{3}\}$ of $\Delta_{\cap}(C^{\prime})$ correspond to the minimal primes $\langle x_{2}\rangle$ and $\langle x_{3}\rangle$ of $I(\Delta(C))=\langle x_{2}x_{3}\rangle$ and also to the maximal codewords $13$ and $12$ of $C$.

The following result translates the algebraic characterization of intersection-complete codes from [4] into a new combinatorial criterion.

Theorem 3.6** (Intersection-complete codes).**

Let $C$ be a code on $n$ neurons with neural ideal $J_{C}$, and let $C^{\prime}$ be the complement code of $C$ with factor complex $\Delta_{\cap}(C^{\prime})$. The following are equivalent:

1. (1)

$C$* is intersection-complete,* 2. (2)

every pseudomonomial $\prod\limits_{i\in\sigma}x_{i}\prod\limits_{j\in\tau}(1-x_{j})$ in $\operatorname{{\operatorname{CF}}}(J_{C})$ satisfies $|\tau|\leq 1$, and 3. (3)

every facet $F$ of $\Delta_{\cap}(C^{\prime})$ satisfies $\left|F\cap[n]\right|\geq n-1$.

Proof.

The equivalence between 1 and 2 is [4, Theorem 1.9]. By Theorem 3.3, $\prod\limits_{i\in\sigma}x_{i}\prod\limits_{j\in\tau}(1-x_{j})$ belongs to the canonical form of $J_{C}$ if and only if $F=[n]\smallsetminus\tau\cup\overline{[n]\smallsetminus\sigma}$ is a facet of $\Delta_{\cap}(C^{\prime})$. Thus, the condition $|\tau|\leq 1$ is equivalent to $\left|F\cap[n]\right|\geq n-1$, and so 2 is equivalent to 3. ∎

The following result is an expanded version of Theorem 1.1.

Theorem 3.7** (Max-intersection-complete codes).**

Let $C$ be a code on $n$ neurons with neural ideal $J_{C}$, and let $C^{\prime}$ be the complement code of $C$ with factor complex $\Delta_{\cap}(C^{\prime})$. The following are equivalent:

1. (1)

$C$* is max-intersection-complete,* 2. (2)

for every facet $F$ of $\Delta_{\cap}(C^{\prime})$ that does not contain $[n]$, there exists $i\in[n]$ such that

1. (i)

every minimal prime-set of $\Delta_{\cap}(C^{\prime})$ that contains $\overline{i}$ also contains some $\overline{j}$ such that $\overline{j}\notin F$, and 2. (ii)

$i\notin F$, 3. (3)

for every $\phi\in\operatorname{{\operatorname{CF}}}(J_{C})$ that is not a monomial, there exists $i\in[n]$ such that

1. (i)

every minimal prime of $I(\Delta(C))$ that contains $x_{i}$ also contains $\phi$, and 2. (ii)

$(1-x_{i})\mid\phi$.

Proof.

We begin by proving 2$\Leftrightarrow$3. By Theorem 3.3, $\phi=\prod_{i\in c}x_{i}\prod_{j\in[n]\smallsetminus d}(1-x_{j})\in\operatorname{{\operatorname{CF}}}(J_{C})$ if and only if $F=d\cup\overline{[n]\smallsetminus c}$ is a facet of $\Delta_{\cap}(C^{\prime})$. Furthermore, $\phi$ is a non-monomial exactly when $d\not\supseteq[n]$, if and only if $F$ does not contain $[n]$. Thus, by inspection of $\phi$ and $F$, 22ii is equivalent to 33ii, and so we need only show 22i$\Leftrightarrow$33i.

By Theorem 3.4, the prime ideal $P=\langle x_{j}\mid j\in B\rangle$ is associated to $I(\Delta(C))$ if and only if $\overline{B}$ is a minimal prime-set of $\Delta_{\cap}(C^{\prime})$. Thus, $x_{i}\in P$ exactly when $\overline{i}\in\overline{B}$. Next, it is straightforward to check that $P$ contains $\phi=\prod_{i\in c}x_{i}\prod_{j\in[n]\smallsetminus d}(1-x_{j})$ if and only if $B\cap c\neq\emptyset$. As $\phi$ corresponds to the facet $F=d\cup\overline{[n]\smallsetminus c}$ of $\Delta_{\cap}(C^{\prime})$, it follows that $P$ contains $\phi$ if and only if $\overline{j}\notin F$ for some $\overline{j}\in\overline{B}$. This concludes the proof of 2$\Leftrightarrow$3.

We set up notation needed to prove 1$\Leftrightarrow$2. Let $\overline{B}_{1},\overline{B}_{2},\dots,\overline{B}_{u}$ be the minimal prime-sets of $\Delta_{\cap}(C^{\prime})$. By Theorem 3.4, the maximal codewords of $C$ are $m_{1}=[n]\smallsetminus B_{1},\dots,m_{u}=[n]\smallsetminus B_{u}$.

We claim that 2 is equivalent to the following: **

1. (2’)

for every facet $F$ of $\Delta_{\cap}(C^{\prime})$ that does not contain $[n]$,

[TABLE]

where

[TABLE]

Indeed, condition ($\star$ ‣ 1) states that there exists $i\in[n]$ such that $i\notin F$ and $\overline{i}$ is not in any minimal prime-set $\overline{B}_{v}\subset\{\overline{1},\overline{2},\dots,\overline{n}\}$ for which $\overline{B}_{v}\subset F$. This latter condition exactly matches 22i. Hence, our claim holds, and we may complete this proof by showing 1$\Leftrightarrow$1.

$(\Leftarrow)$ We prove the contrapositive. Suppose that the intersection of maximal codewords $c=\bigcap\limits_{v\in V}m_{v}$ (for some $\varnothing\neq V\subset[u]$) is not in $C$, that is, $c\in C^{\prime}$. By Theorem 3.3, $c\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C^{\prime})$. Note that

[TABLE]

Let $F$ be a facet of $\Delta_{\cap}(C^{\prime})$ containing $c\cup\overline{[n]\smallsetminus c}$. It follows from (6) that $F$ contains the union of minimal prime-sets $\bigcup\limits_{v\in V}\overline{B}_{v}$, which implies that $F$ does not contain $[n]$ (as, otherwise, each $\overline{B}_{v}\cup[n]$ is contained in $F$ and hence is a face of $\Delta_{\cap}(C^{\prime})$, contradicting the fact that $B_{v}$ is a prime-set). Since $F\supset\overline{[n]\smallsetminus c}=\bigcup\limits_{v\in V}\overline{B}_{v}$, we have that $V\subset H_{F}$. Therefore, $[n]\smallsetminus\bigcup\limits_{v\in H_{F}}B_{v}\subset[n]\smallsetminus\bigcup\limits_{v\in V}B_{v}=c$, where the equality comes from (6). We conclude that $F$ is a facet of $\Delta_{\cap}(C^{\prime})$ not containing $[n]$ such that $([n]\smallsetminus\bigcup\limits_{v\in H_{F}}B_{v})\subset c\subset(c\cup\overline{[n]\smallsetminus c})\subset F$.

$(\Rightarrow)$ Suppose $C$ is max-intersection-complete. Let $F$ be a facet of $\Delta_{\cap}(C^{\prime})$ that does not contain $[n]$. Set $c:=[n]\smallsetminus\bigcup\limits_{v\in H_{F}}{B_{v}}$. Our goal is to show that $c\not\subset F$.

We accomplish this by proving two facts. First, that $c\cup\overline{[n]\smallsetminus c}$ is not a face of $\Delta_{\cap}(C^{\prime})$, and second, that $\overline{[n]\smallsetminus c}=\bigcup\limits_{v\in H_{F}}\overline{B}_{v}$. The first fact implies that $c\cup\overline{[n]\smallsetminus c}\not\subset F$ and the second yields $\overline{[n]\smallsetminus c}\subset F$. Our desired relation $c\not\subset F$ will then follow.

For the first fact, recall that $[n]\smallsetminus B_{v}=m_{v}$. Therefore,

[TABLE]

so $c$ is the intersection of maximal codewords. As $C$ is max-intersection-complete, $c\in C$, and thus $c\not\in C^{\prime}$. Now Theorem 3.3 implies that $c\cup\overline{[n]\smallsetminus c}\not\in\Delta_{\cap}(C^{\prime})$.

For the second fact, $\overline{[n]\smallsetminus c}=\overline{[n]\smallsetminus([n]\smallsetminus\bigcup\limits_{v\in H_{F}}B_{v}})=\overline{\bigcup\limits_{v\in H_{F}}B_{v}}=\bigcup\limits_{v\in H_{F}}\overline{B}_{v}.$ ∎

Example 3.8** (Example 3.5, continued).**

The code $C=\{\emptyset,2,3,12,13\}$ is neither intersection-complete nor max-intersection-complete (as $1=12\cap 13\notin C$). We can read this information from Theorems 3.6 and 3.7, as follows. For non-intersection-completeness, this can be seen in two ways: first, the pseudomonomial $x_{1}(1-x_{2})(1-x_{3})$ is in the canonical form of $J_{C}$, and, second, the intersection of the facet $1\bar{2}\bar{3}$ with $123$ has size 1, rather than 2 or 3.

For non-max-intersection-completeness, recall that the minimal prime-sets of $\Delta_{\cap}(C^{\prime})$ are $\{\bar{2}\}$ and $\{\bar{3}\}$ (equivalently, the minimal primes of $I(\Delta(C))$ are $\langle x_{2}\rangle$ and $\langle x_{3}\rangle$). Now, $1\bar{2}\bar{3}$ is a facet of $\Delta_{\cap}(C^{\prime})$ that does not contain 123, but for $i\in\{1,2,3\}$, either part 22i of Theorem 3.7 is violated (when $i=2,3$) or part 22ii is violated (when $i=1$). Alternatively, $\operatorname{{\operatorname{CF}}}(J_{C})$ contains the non-monomial $x_{1}(1-x_{2})(1-x_{3})$, but for $i\in\{1,2,3\}$, either part 33i of Theorem 3.7 is violated (when $i=2,3$) or part 33ii is violated (when $i=1$). Thus, $C$ is not max-intersection-complete.

4. Factor complexes, neural ideals, and codes

In this section, we prove Theorems 3.3 and 3.4.

4.1. Proof of Theorem 3.3

We wish to prove that the following maps are bijections:

[TABLE]

The fact that $\alpha$ is a bijection is straightforward from [5, Lemma 5.7]. To show that $\beta$ is a bijection, we need to better understand the factor ideal and factor complex of $C$.

Lemma 4.1**.**

Let $C$ be a code with neural ideal $J_{C}$, and let $\phi$ be a pseudomonomial. Then $\phi\in J_{C}$ if and only if ${\mathcal{P}}(\phi)\in\operatorname{{\operatorname{FI}}}(C)$.

Proof.

Recall the decomposition $J_{C}=\bigcap^{g}_{h=1}P_{h}$ from (3). Hence, $\phi\in J_{C}$ if and only if $\phi\in P_{h}$ for all $h$. Given the form (4) of each component $P_{h}$, it is straightforward to check that $\phi\in P_{h}$ is equivalent to ${\mathcal{P}}(\phi)\in{\mathcal{P}}(P_{h})$. Thus, as $\operatorname{{\operatorname{FI}}}(C)=\bigcap\mathcal{P}(P_{h})$, the desired result follows. ∎

Our next results shows how to use the factor complex of a code to read off its codewords.

Lemma 4.2**.**

Let $C$ be a code on $n$ neurons. Then $c\in 2^{[n]}$ is a codeword of $C$ if and only if $c\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C)$.

Proof.

By [5, Lemma 3.2], $c\in C$ if and only if $\phi_{c}=\prod_{i\in c}x_{i}\prod_{j\notin c}(1-x_{j})\notin J_{C}$. This is equivalent to ${\mathcal{P}}(\phi_{c})\notin\operatorname{{\operatorname{FI}}}(C)$ by Lemma 4.1. Since $\operatorname{{\operatorname{FI}}}(C)$ is the Stanley–Reisner ideal of $\Delta_{\cap}(C)$, we have that ${\mathcal{P}}(\phi_{c})\notin\operatorname{{\operatorname{FI}}}(C)$ exactly when $c\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C)$, which concludes the proof. ∎

We now extend Lemma 4.2 to show how to extract the intervals of $C$ from its factor complex.

Lemma 4.3**.**

(Interval-Face Correspondence) Let $C$ be a code on $n$ neurons, and let $c,d\in 2^{[n]}$. Then $[c,d]\subset C$ if and only if $d\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C)$.

Proof.

$(\Leftarrow)$ Suppose $d\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C)$, and let $w\in[c,d]$. Then $w\cup\overline{[n]\smallsetminus w}\subset d\cup\overline{[n]\smallsetminus c}$ is a face of $\Delta_{\cap}(C)$ and thus $w\in C$ by Lemma 4.2.

$(\Rightarrow)$ We now assume that $d\cup\overline{[n]\smallsetminus c}$ is not a face of $\Delta_{\cap}(C)$ and show that $[c,d]$ is not an interval of $C$. As $\operatorname{{\operatorname{FI}}}(C)$ is the Stanley–Reisner ideal of $\Delta_{\cap}(C)$, the decomposition (5) implies that the ideal

[TABLE]

is not associated to $\operatorname{{\operatorname{FI}}}(C)$, and therefore the following ideal is not associated to $J_{C}$:

[TABLE]

Thus, as $\operatorname{{\operatorname{CF}}}(J_{C})$ is a generating set for $J_{C}$, there exists a pseudomonomial $\phi=\prod_{i\in\sigma}x_{i}\prod_{j\in\tau}(1-x_{j})$ in $\operatorname{{\operatorname{CF}}}(J_{C})$ that is not in the ideal (7), and so $\sigma\subset d$ and $\tau\subset[n]\smallsetminus c$. Note that the indicator pseudomonomial $\phi_{c\cup\sigma}$ is in $J_{C}$, as it is divisible by $\phi$. We conclude that $\sigma\cup c\in[c,d]\smallsetminus C$, and so $[c,d]\not\subset C$. ∎

We can now better understand the facets of $\Delta_{\cap}(C)$.

Lemma 4.4**.**

Let $C$ be a code on $n$ neurons. Every facet of $\Delta_{\cap}(C)$ is effective.

Proof.

By (5), the facets of $\Delta_{\cap}(C)$ correspond to associated primes of $\operatorname{{\operatorname{FI}}}(C)$, which are polarizations of associated primes of $J_{C}$. Since the latter primes cannot contain both $x_{\ell}$ and $1-x_{\ell}$, it follows that the former primes cannot contain both $x_{\ell}$ and $y_{\ell}$, which concludes the proof. ∎

Proof of Theorem 3.3.

By [5, Lemma 5.7], the map $\alpha$ is a bijection, and the correspondence between minimal pseudomonomials and maximal intervals follows from the fact for any two intervals $M_{1}$ and $M_{2}$ of $C$, we have $M_{1}\subset M_{2}$ if and only if $\alpha(M_{2})\mid\alpha(M_{1})$. By Lemma 4.3, plus the fact that effective faces have the form $d\cup\overline{[n]\smallsetminus c}$ for some $c\subset d$, the map $\beta$ is also a bijection. Lemma 4.4 states that all facets of $\Delta_{\cap}(C)$ are effective, and thus for each facet $F$ we have $F=\beta(M)$ for some interval $M$ of $C$. The correspondence between facets and maximal intervals then follows from the fact that for intervals $M_{1}$ and $M_{2}$ of $C$, we have $M_{1}\subset M_{2}$ if and only if $\beta(M_{1})\subset\beta(M_{2})$. ∎

4.2. Proof of Theorem 3.4

We wish to show that the maps

[TABLE]

are bijections. The main step is to understand the relationship between the prime-sets of $\Delta_{\cap}(C^{\prime})$ and the associated primes of $I(\Delta(C))$.

Lemma 4.5**.**

Let $C$ be a code on $n$ neurons with complement code $C^{\prime}$. A subset $\overline{B}\subset\overline{[n]}$ is a prime-set of $\Delta_{\cap}(C^{\prime})$ if and only if $\langle x_{i}\mid i\in B\rangle$ contains $I(\Delta(C))$. Consequently, $\overline{B}$ is a minimal prime-set of $\Delta_{\cap}(C^{\prime})$ if and only if $\langle x_{i}\mid i\in B\rangle$ is a minimal prime of $I(\Delta(C))$.

Proof.

By definition, $\overline{B}$ is a prime-set of $\Delta_{\cap}(C^{\prime})$ if and only if $[n]\cup\overline{B}$ is not a face of $\Delta_{\cap}(C^{\prime})$. Equivalently, every facet of $\Delta_{\cap}(C^{\prime})$ of the form $F=[n]\cup\overline{[n]\smallsetminus c}$ satisfies $B\cap c\neq\varnothing$. By Theorem 3.3, $F=[n]\cup\overline{[n]\smallsetminus c}$ is a facet of $\Delta_{\cap}(C^{\prime})$ if and only if the monomial $\prod_{i\in c}x_{i}$ belongs to $\operatorname{{\operatorname{CF}}}(J_{C})$. Also, $B\cap c\neq\varnothing$ if and only if $\prod_{j\in c}x_{j}\in\langle x_{i}\mid i\in B\rangle$. Now the result follows, because the monomials in $\operatorname{{\operatorname{CF}}}(J_{C})$ generate $I(\Delta(C))$. ∎

Proof of Theorem 3.4.

The map $\gamma$ is a bijection, by (5) and the fact that maximal codewords of $C$ are facets of $\Delta(C)$, and $I(\Delta(C))$ is its Stanley–Reisner ideal. Given that $\gamma$ is a bijection, Lemma 4.5 shows that $\delta\circ\gamma^{-1}$ is a bijection, and so, $\delta$ is a bijection, completing the proof. ∎

5. The factor complex and the polar complex

In this section, we explore the relationship between the factor complex and the polar complex introduced in [9]. For a code $C$, the polar complex, denoted by $\Delta_{\mathcal{P}}(C)$, is the simplicial complex whose Stanley–Reisner ideal is ${\mathcal{P}}(J_{C})$, the polarization of the neural ideal of $C$. The ideal ${\mathcal{P}}(J_{C})$ is the polar ideal of $C$.

We first show in an example that polar and factor complexes associated to a code are, in general, not the same.

Example 5.1** (Example 3.8, continued).**

For the code $C^{\prime}=\{1,23,123\}$, we polarize the neural ideal $J_{C^{\prime}}=\langle(1-x_{1})(1-x_{3}),~{}(1-x_{1})(1-x_{2}),~{}x_{2}(1-x_{3}),x_{3}(1-x_{2})\rangle$ to obtain the polar ideal

[TABLE]

It follows that the set of facets of the polar complex $\Delta_{{\mathcal{P}}}(C^{\prime})$ is $\{1\bar{2}\bar{3},123\bar{1},12\bar{2},13\bar{3}\}$. Thus, the polar complex has 2 more facets than the corresponding factor complex (recall Example 3.2).

On the other hand, the polar ideal and the factor ideal (and their corresponding complexes) share many features. A first observation is that ${\mathcal{P}}(J_{C})\subset\operatorname{{\operatorname{FI}}}(C)$ by construction and Lemma 4.1. Furthermore, Lemma 4.1 is valid when we replace $\operatorname{{\operatorname{FI}}}(C)$ by ${\mathcal{P}}(J_{C})$ [9, Theorem 3.2], and consequently Lemma 4.2 holds for $\Delta_{\mathcal{P}}(C)$. Lemma 4.3 also is valid for $\Delta_{\mathcal{P}}(C)$ [9, Corollary 5.2].

As Example 5.1 illustrates, $\operatorname{{\operatorname{FI}}}(C)$ strictly contains ${\mathcal{P}}(J_{C})$ in general. A larger ideal makes for a smaller simplicial complex. The following result explains the relationship between $\Delta_{\cap}(C)$ and $\Delta_{\mathcal{P}}(C)$.

Proposition 5.2**.**

For every code $C$, the factor complex $\Delta_{\cap}(C)$ is the subcomplex of the polar complex $\Delta_{\mathcal{P}}(C)$ whose facets are the effective facets of $\Delta_{\mathcal{P}}(C)$.

Proof.

Lemma 4.4 states that all facets of $\Delta_{\cap}(C)$ are effective, and ${\mathcal{P}}(J_{C})\subset\operatorname{{\operatorname{FI}}}(C)$ implies that $\Delta_{\cap}(C)\subset\Delta_{{\mathcal{P}}}(C)$. So, it suffices to show that every effective facet of $\Delta_{{\mathcal{P}}}(C)$ is a face of $\Delta_{\cap}(C)$. By [9, Corollaries 5.2 and 5.3], the effective facets of $\Delta_{\mathcal{P}}(C)$ are of the form $d\cup\overline{[n]\smallsetminus c}$ where $[c,d]$ is a maximal interval of $C$. Now apply Lemma 4.3. ∎

The key difference between the factor complex and the polar complex of a code is that the latter can have defective facets. While these facets hold useful information about quotient codes, as shown in [9], the structure of the smaller factor complex is more convenient for our purposes here.

6. Computational Considerations

The main result of this article, Theorem 1.1, gives a new method for checking whether a code is max-intersection-complete (Algorithm 1 below). In this section we provide an infinite family $\mathfrak{F}$ of codes for which this method is more efficient at checking max-intersection-completeness than the natural brute-force approaches.

In order to analyze the runtime of our proposed algorithm, we write it explicitly below. Correctness follows directly from Theorem 1.1 and the correspondence between maximal codewords of $C$ and minimal primes of $I(\Delta(C))$ in Theorem 3.4.

Remark 6.1**.**

We point out that Algorithm 1 requires $\operatorname{{\operatorname{CF}}}(J_{C})$ as part of its input, but the brute-force methods below do not. For this reason, a complete runtime analysis of Algorithm 1 requires knowing the complexity of computing canonical forms, which is not currently well understood. The canonical form algorithm given in [18] is easily seen to be exponential in the number of neurons. A faster procedure for finding $\operatorname{{\operatorname{CF}}}(J_{C})$ would be very desirable, and would have implications beyond this article.

We now define $\mathfrak{F}$ to be the family of all neural codes $C$ satisfying the following properties:

• (i)

The number of maximal intervals of $C^{\prime}$ is at most $n$, the number of neurons of $C$.

• (ii)

There exists a maximal interval $[c,d]$ of $C^{\prime}$ with $d\neq[n]$ and $|d\smallsetminus c|=n/2$.

• (iii)

There exists a maximal interval $[a,[n]]$ of $C^{\prime}$, where $a$ contains $n/2$ neurons.

• (iv)

For every maximal interval of $C^{\prime}$ that has the form $[b,[n]]$, if $a\neq b$ then $a\cap b=\emptyset$.

• (v)

$C^{\prime}$ contains at most $\log_{2}(n)$ maximal intervals of the form $[b,[n]]$.

Note that $\mathfrak{F}$ is infinite, since the number of neurons has not been fixed. We emphasize that a code $C\in\mathfrak{F}$ is given as the maximal intervals of $C^{\prime}$. This information is equivalent to knowing $\operatorname{{\operatorname{CF}}}(J_{C})$. Thus, for codes in $\mathfrak{F}$, the issue raised in Remark 6.1 is avoided. Finally, it can be checked that $\mathfrak{F}$ contains infinitely many max-intersection-complete codes, and infinitely many codes which are not max-intersection-complete.

We compare Algorithm 1 to two brute-force methods for checking max-intersection-completeness:

Brute Force 1: Take all possible intersections of maximal codewords of $C$, and check whether all are contained in $C$.

Brute Force 2: For every $\sigma\in C^{\prime}$, compute $c_{\sigma}$, the intersection of all maximal codewords of $C$ that contain $\sigma$. Then check whether $c_{\sigma}=\sigma$.

Proposition 6.2**.**

For every code $C$ in $\mathfrak{F}$, Brute Force 1 and Brute Force 2 are exponential in the number of neurons, while Algorithm 1 is sub-exponential in the number of neurons.

Proof.

We begin by showing that the number of maximal codewords of any $C\in\mathfrak{F}$ is at least $n/2$ and at most $n^{\log_{2}(n)}$. Recall that these maximal codewords are in bijection with the minimal primes of $I(\Delta(C))$ (Theorem 3.4), and also that

[TABLE]

(Recall that for $\sigma\subseteq[n]$, we use the notation $x_{\sigma}$ to denote the monomial $\prod_{i\in\sigma}x_{i}$.)

The monomial generators of $I(\Delta(C))$ in (8) satisfy the following:

• ($\ast$)

there is a generator $x_{a}$ of degree $n/2$ (from condition (iii)),

• ($\ast\ast$)

if $x_{b}\neq x_{a}$ is a generator of $I(\Delta(C))$, then $\gcd(x_{a},x_{b})=1$ (from condition (iv)), and

• ($\ast\ast\ast$)

there are at most $\log_{2}(n)$ generators (from condition (v)).

We calculate the upper bound by observing that every minimal prime $P$ of $I(\Delta(C))$ has a generating set $G_{P}\subset\{x_{1},x_{2},\dots,x_{n}\},$ with every monomial in (8) divisible by at least one $x_{i}\in G_{P}$. It follows that the number of ways to choose some divisor $x_{i}$ from each generator of $I(\Delta(C))$ is an upper bound on the number of minimal primes. This upper bound is the product of the degrees of the monomial generators of $I(\Delta(C))$, which in turn is bounded above by $n^{N_{\text{mon}}}$, where $N_{\text{mon}}$ is the number of monomials in $CF(J_{C})$. By ($\ast\ast\ast$) there are at most $\log_{2}(n)$ such monomials, so the number of minimal primes – and thus the number of maximal codewords of $C$ – is at most $n^{\log_{2}(n)}$.

For the lower bound, we first note that by ($\ast$) there is a monomial generator $x_{a}$ of $I(\Delta(C))$ that has degree $n/2$. If $I(\Delta(C))=\langle x^{a}\rangle$, then $I(\Delta(C))$ has $n/2$ minimal primes. If $I(\Delta(C))$ strictly contains $\langle x^{a}\rangle$, then let $\widetilde{P}$ be a minimal prime of the following nonzero ideal:

[TABLE]

For every $x_{i}$ that divides $x_{a}$, we claim that $P_{i}=\langle x_{i}\rangle+\widetilde{P}$ is a minimal prime of $I(\Delta(C))$. By construction, $P_{i}$ contains $I(\Delta(C))$. If $Q\subsetneq P_{i}$ is another prime monomial ideal, either $x_{i}\not\in Q$ or there exists $x_{j}\in\widetilde{P}\setminus Q$. In the first case, condition ($\ast\ast$) implies that $x_{a}\not\in Q$. In the second case, by ($\ast\ast$) and the fact that $\widetilde{P}$ is a minimal prime of $\widetilde{I}$, it follows that $\widetilde{I}\not\subset Q$. In both cases $Q$ does not contain $I(\Delta(C))$, and consequently $P_{i}$ is a minimal prime of $I(\Delta(C))$. As a distinct minimal prime $P_{i}=\langle x_{i}\rangle+\widetilde{P}$ arises from each of the $n/2$ divisors $x_{i}$ of $x_{a}$, the number of minimal primes – and also the number of maximal codewords of $C$ – is at least $n/2$.

Having found the upper and lower bounds on the number of maximal codewords of a code $C\in\mathfrak{F}$, we now use these bounds to analyze the brute-force methods and Algorithm 1.

As there are at least $n/2$ maximal codewords, Brute Force 1 checks at least $2^{n/2}$ intersections of maximal codewords, and so is exponential in the number of neurons.

Next, Brute Force 2 checks whether each codeword of $C^{\prime}$ is contained in each maximal codeword of $C$. So, the runtime will be at least the number of codewords of $C^{\prime}$ times the number of maximal codewords of $C$. There are at least $n/2$ maximal codewords and, by condition (ii), at least $2^{n/2}$ elements of $C^{\prime}$. Thus, the runtime is at least $(n/2)*2^{n/2}$, and so is exponential in $n$.

For Algorithm 1, First Loop iterates over the maximal codewords of $C$ (of which there are at most $n^{\log_{2}(n)}$), and the runtime of each iteration is at most $n$. So, the runtime of First Loop is $O(n^{1+\log_{2}(n)})$. The runtime for the subsequent part of the algorithm is the product of the number of iterations of the Outer Loop, the number of iterations of the Middle Loop, and the runtime of the Inner Loop. Since the Outer Loop iterates over a subset of $\operatorname{{\operatorname{CF}}}(J_{C})$, by Theorem 3.3 and condition (i) there are at most $n$ such iterations. Since the Middle Loop iterates over the neurons, there are at most $n$ iterations of this loop. Finally, the Inner Loop iterates over the number of minimal primes of $I(\Delta(C))$, of which there are at most $n^{\log_{2}(n)}$. Checking to see if $x_{s}$ is in some minimal prime $P$ takes at most $n$ steps (check each generator of $P$) and checking to see if any $x_{r}\in P$ divides $\phi$ takes at most $n^{2}$ steps (compare each generator of $P$ with each divisor of $\phi$). Thus the runtime of Inner Loop is at most $n^{3+\log_{2}(n)}$. We conclude that the combined runtime of the Outer, Middle, and Inner Loops is $O(n^{5+\log_{2}(n)})$, which, it is straightforward to check, is sub-exponential in $n$. ∎