For a real closed field $\mathbf{R}$, we use the theory of the refined Bloch group to give a new short proof of the isomorphisms $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\cong K_{3}^{\mathrm{ind}}(\mathbf{R})$ and $H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})\cong H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\oplus K_{3}^{M}(\mathbf{R})^{0}$ for $n\geq3$.

§ INTRODUCTION.

In the paper Homology of classical Lie groups made discrete III ([7]) the author, Chih-Han Sah, proves the injectivity of the map $H_{3}(SL_{2}(F),\mathbb{Z})\rightarrow H_{3}(SL_{3}(F),\mathbb{Z})$ and $H_{3}(SL_{3}(F),\mathbb{Z})\cong H_{3}(SL_{2}(F),\mathbb{Z})\oplus K_{3}^{M}(F)^{0}$ for $F=\mathbb{R}$, $\mathbb{H}$ or $F$ algebraically closed (<cit.> and <cit.> respectively). He states, without giving details, that the result is also valid for any real closed field <cit.>.

In the present article we use the theory of the refined Bloch group ([2]) to give a new proof of this result for real closed fields $\mathbf{R}$, and we deduce the structure of $H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})$, $n\geq3$, in terms of $K$-theory. In fact, all the results are valid for the larger class of real quadratically closed fields (See section $2$ below).

§.§ Some notation

For a field $F$, we let $F^{\times}$ denote the group of units of $F$. For $x\in F^{\times}$ we will let $\langle x\rangle\in F^{\times}/(F^{\times})^{2}$ denote the corresponding square class. Let $R_{F}$ denote the integral group ring $\mathbb{Z}\left[F^{\times}/(F^{\times})^{2}\right]$ of the group $F^{\times}/(F^{\times})^{2}$. We will use the notation $\langle\langle x\rangle\rangle$ for the basis element, $\langle x\rangle-1$, of the augmentation ideal $\mathcal{I}_{F}$ of $R_{F}$.

§ REAL QUADRATICALLY CLOSED FIELDS.

Recall that an ordered field $(\mathbf{R},\geq)$ is real closed if it satisfies the following two properties:

(1)Any positive element $x>0$ in $\mathbf{R}$ has a square root in $\mathbf{R}$; i.e. there exist $a\in \mathbf{R}\setminus\{0\}$ such that $x=a^{2}$.

(2) Any odd degree polynomial with coefficients in $\mathbf{R}$ has a root in $\mathbf{R}$.

In fact, our main results will apply to any ordered field satisfying property $(1)$. For convenience, in this paper we will refer to such fields as real quadratically closed fields

The field of (straightedge-and-compass) constructible real numbers is real quadratically closed but not real closed.

We will frequently use the following easily verified fact

Let $(\mathbf{R},\geq)$ be an ordered field and $\mathbf{R}_{>0}$ denote the  group of positive elements in $\mathbf{R}$ under the multiplication of $\mathbf{R}$.

(i)$\mu_{\mathbf{R}}=\mu_{2}$ and $\mathbf{R}^{\times}=\mu_{2}\oplus\mathbf{R}_{>0}$.

(ii) If $\mathbf{R}$ is real quadratically closed then $\mathbf{R}_{>0}=(\mathbf{R}_{>0})^{2}$

Since $\mathbf{R}_{>0}=(\mathbf{R}_{>0})^{2}$, it follows that $\mathbf{R}^{\times}/(\mathbf{R}^{\times})^{2}=\mu_{2}$ and hence $R_{\mathbf{R}}=\mathbb{Z}[\mu_{2}]=\mathbb{Z}[\langle-1\rangle]$. Futhermore we will need the following observation:

If $\mathbf{R}$ is real quadratically closed then $\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})\cong\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}_{>0})$ which is $2$-divisible

§ REVIEW OF BLOCH GROUPS OF FIELDS.

In this section we will recall the definition and applications of the classical pre-Bloch group $\mathcal{P}(F)$ and the refined pre-Bloch group $\mathcal{RP}(F)$.

§.§ Classical Bloch Group $\mathcal{B}(F)$.

For a field $F$, with at least $4$ elements, the pre-Bloch group or Scissors congruence group, $\mathcal{P}(F)$, is the group generated by the elements $[x]$, with $F^{\times}\setminus\{1\}$, subject to the relations

$R_{x,y}:\; 0=[x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]$,    $x\not=y$.

Let $S^{2}_{\mathbb{Z}}(F^{\times})$ denote the group

$\frac{F^{\times}\otimes_{\mathbb{Z}}F^{\times} }{\langle x\otimes y +y\otimes x\; |\;x,y\in F^{\times}\rangle}$

and denote by $x\circ y$ the image of $x\otimes y$ in $S^{2}_{\mathbb{Z}}(F^{\times})$. The map

$\lambda: \mathcal{P}(F)\rightarrow S^{2}_{\mathbb{Z}}(F^{\times})$,     $[x]\mapsto (1-x)\circ x$

is well-defined, and the Bloch group of $F$, $\mathcal{B}(F)\subset\mathcal{P}(F)$, is defined to be the kernel of $\lambda$.

The Bloch group $\mathcal{B}(F)$ of a general field $F$ is of interest because of the following result of Suslin on $K_{3}^{\mathrm{ind}}$:

Let $F$ be an infinite field, then there is a short exact sequence

where $\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{F},\mu_{F}})$  is the unique nontrivial extension of $\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ by $\mathbb{Z}/2$ when $\mathrm{Char}(F)\not=2$ (and $\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{F},\mu_{F}})=\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ if $\mathrm{Char}(F)=2$) .

§.§ The refined Bloch Group $\mathcal{RB}(F)$.

The refined pre-Bloch group $\mathcal{RP}(F)$, of a field $F$ which has at least $4$ elements, is the $R_{F}$-module with generators $[x]$, $x\in F^{\times}$ subject to the relations $[1]=0$ and

$S_{x,y}:\; 0=[x]-[y]+\langle x\rangle[y/x]-\langle x^{-1}-1\rangle[(1-x^{-1})/(1-y^{-1})]+\langle1-x\rangle[(1-x)/(1-y)]$,    $x,y\not=1$.

From the definitions, it follows that $\mathcal{P}(F)=(\mathcal{RP}(F))_{F^{\times}}$. Let $\Lambda=(\lambda_{1},\lambda_{2})$ be the $R_{F}$-module homomorphism

$\mathcal{RP}(F)\rightarrow \mathcal{I}_{F}^{2}\oplus S^{2}_{\mathbb{Z}}(F^{\times})$

where $\lambda_{1}:\mathcal{RP}(F)\rightarrow \mathcal{I}^{2}_{F}$ is the map $[x]\mapsto \langle\langle1-x\rangle\rangle\langle\langle x\rangle\rangle$ and $\lambda_{2}$ is the composite

$\xymatrix{ \mathcal{RP}(F)\ar[r]&\mathcal{P}(F)\ar[r]^{\lambda} &S^{2}_{\mathbb{Z}}(F^{\times})}$,

where $S^{2}_{\mathbb{Z}}(F^{\times})$ has the trivial $R_{F}$-module structure.

The refined Bloch group of $F$ is the module

$\mathcal{RB}(F):=\mathrm{Ker}(\Lambda: \mathcal{RP}(F)\rightarrow \mathcal{I}_{F}^{2}\oplus S^{2}_{\mathbb{Z}}(F^{\times}  )$.

Furthermore, the refined scissor congruence group of $F$ is the $R_{F}$-module

Thus $\mathcal{RB}(F)=\mathrm{Ker}(\lambda_{2}:\mathcal{RP}_{1}(F)\rightarrow S^{2}_{\mathbb{Z}}(F^{\times}))$.

Now for any infinite field $F$, Suslin shows that there is a natural homomorphism $H_{3}(SL(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$, where $SL(F)=\displaystyle\bigcup_{n\geq1}SL_{n}(F)$ and $K_{3}^{\mathrm{ind}}(F)$ is the indecomposable $K_{3}$ of $F$. It follows that there are induced maps $H_{3}(SL_{n}(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$ for all $n$. Hutchinson and Tao show that $H_{3}(SL_{2}(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$ is surjective for any infinite field <cit.>.

The refined Bloch group is of interest because of the following result on the third homology of $SL_{2}$ over a general field $F$:

There is a complex of $R_{F}$-modules

$\xymatrix{0\ar[r]&\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})\ar[r]&H_{3}(SL_{2}(F),\mathbb{Z})\ar[r]& \mathcal{RB}(F)\ar[r]&0}$,

which is exact except at the middle term where the homology is annihilated by $4$.

Furthermore we have that the following diagram is commutative:

$\xymatrix{0\ar[r]&\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})\ar[r]\ar[d]&H_{3}(SL_{2}(F),\mathbb{Z})\ar[r]\ar[d]& \mathcal{RB}(F)\ar[r]\ar[d]&0\\
0\ar[r]&\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{F},\mu_{F}})\ar[r]&K_{3}^{\mathrm{ind}}(F)\ar[r]& \mathcal{B}(F)\ar[r]&0}$,

We will examine the sequence of Theorem <ref> below in more detail in the case of real quadratically closed fields.

§.§ The elements $\psi_{i}(x)$ in $\mathcal{RP}(F)$

For $x\in F^{\times}$, we define the following elements of $\mathcal{RP}(F)$

$\psi_{1}(x):=[x]+\langle-1\rangle[x^{-1}]$    and   $\psi_{2}(x):=\left\{
\begin{array}{ll}
\langle x^{-1}-1\rangle[x]+\langle1-x\rangle[x^{-1}], & \hbox{$x\not=1$;} \\
0, & \hbox{$x=1$.}
\end{array}
\right.

From the definitions of the the elements $\psi_{i}(x)$, we get that $\langle-1\rangle\psi_{i}(-1)=\psi_{i}(-1)$ for $i\in\{1,2\}$. We review some of the  algebraic properties of the elements $\psi_{i}(x)$:

<cit.> Let $F$ be a field. For $i\in\{1,2\}$, we have:

(1) $\psi_{i}(xy)=\langle x\rangle\psi_{i}(y)+\psi_{i}(x)$, for all $x,y$.

(2) $\langle\langle x\rangle\rangle\psi_{i}(y)=\langle\langle y\rangle\rangle\psi_{i}(x)$, for all $x,y$.

(3) $2\psi_{i}(-1)=0$.

(4) $\psi_{i}(x^{2})=\langle\langle x\rangle\rangle\psi_{i}(-1)$, for all $x$.

(5) $2\psi_{i}(x^{2})=0$ for all $x$ and if $-1\in (A^{\times})^{2}$, then $\psi_{i}(x^{2})=0$ for all $x$.

§.§ The constant $C_{F}$ .

In the section $3.2$ of [4], it is shown that the elements

$C(x)=[x]+\langle-1\rangle[1-x]+ \langle\langle1-x\rangle\rangle\psi_{1}(x) \in \mathcal{RP}(F)$

are constant for a field with at least 4 elements (Lemma 3.5); i.e. $C(x)=C(y)$ for all $x,y\in F^{\times}$. Therefore we have the following definition.

Let $F$ be a field with at least $4$ elements. We will denote by $C_{F}$ the common value of the expression $C(x)$ for $x\in F\setminus\{0,1\}$; i.e.

$C_{F}:=[x]+\langle-1\rangle[1-x]+ \langle\langle1-x\rangle\rangle\psi_{1}(x)$   in   $\mathcal{RP}(F)$.

One can deduce some properties of the element $C_{F}$

Let $F$ be a field. Then

(1)$\langle-1\rangle C_{F}=C_{F}$.

(2)For any field $F$, $C_{F}\in \mathcal{RB}(F)$; i.e. $\Lambda(C_{F})=0$.

Observe that the image of $C_{F}$ in $\mathcal{P}(F)$ is $[x]+[1-x]$ for $x\not=0,1$. We also denote this element of $\mathcal{B}(F)\subset \mathcal{P}(F)$ by $C_{F}$.

§ THE REFINED BLOCH GROUP OF A REAL QUADRATICALLY CLOSED FIELD

In this section we show that $\langle-1\rangle$ acts trivially on $\mathcal{RP}(\mathbf{R})$ where $\mathbf{R}$ is a real quadratically closed field; i.e. $R_{\mathbf{R}}$ acts trivially on $\mathcal{RP}(\mathbf{R})$.

Let $x\in\mathbf{R} $, if $x>0$ then $\psi_{i}(x)=0$ for $i\in\{1,2\}$ in $\mathcal{RP}(\mathbf{R})$.

$\textbf{\emph{Proof}}.$ If $x>0$, then $x=a^{2}$ for some $a\in \mathbf{R}$ and hence $\psi_{i}(x)=\langle\langle a\rangle\rangle\psi_{i}(-1)$ by Proposition <ref> (4). Replacing $a$ by $-a$ if necessary, we can suppose $a>0$. Thus $a$ is a square so that $\langle\langle a\rangle\rangle=0$ and hence $\psi_{i}(x)=0$.

Let $x\in\mathbf{R} $, if $x>0$ then the action of $\langle-1\rangle$ on $[x]$ is trivial i.e. $\langle-1\rangle[x]=[x]$.

$\textbf{\emph{Proof}}.$ Since $x>0$, we have that $x=a^{2}$ for some $a\in \mathbf{R}\setminus\{0\}$. Therefore

$\langle x^{-1}-1\rangle=\langle1-x\rangle\langle x\rangle=\langle1-x\rangle\langle a^{2}\rangle=\langle 1-x\rangle$.

By Lemma <ref>, we have that

$0=\psi_{2}(x)=\langle x^{-1}-1\rangle[x]+\langle1-x\rangle[x^{-1}]=\langle1-x\rangle([x]+[x^{-1}])$.

Multiplying the equation $\langle1-x\rangle([x]+[x^{-1}])=0$ by $\langle1-x\rangle$, we get that $[x]+[x^{-1}]=0$, whence we get that $[x]=-[x^{-1}]$. Similarly applying Lemma <ref> to $\psi_{1}(x)$, we get that $[x]=-\langle-1\rangle[x^{-1}]$. By combining these last two equalities, we get our desired result.

If $x<0$ then $C_{\mathbf{R}}=[x]+[1-x]$ in $\mathcal{RP}(\mathbf{R})$.

$\textbf{\emph{Proof}}.$ If $x<0$ then $1-x>0$. Hence $\langle\langle1-x\rangle\rangle\psi_{1}(x)=0$ and by Proposition <ref>, we get that $\langle-1\rangle[1-x]=[1-x]$. It follows that $C_{\mathbf{R}}=[x]+[1-x]$ in $\mathcal{RP}(\mathbf{R})$.

Let $x\in\mathbf{R} $, if $x<0$ then the action of $\langle-1\rangle$ on $[x]$ is trivial i.e. $\langle-1\rangle[x]=[x]$.

$\textbf{\emph{Proof}}.$ Since $1-x>0$, by Proposition <ref>, we get that $\langle-1\rangle[1-x]=[1-x]$. Therefore from this latter result, Proposition <ref> and Proposition <ref> $(1)$, we obtain that:

$0=C_{\mathbf{R}}-\langle-1\rangle C_{\mathbf{R}}=[x]+ [1-x]-\langle-1\rangle[x]-\langle-1\rangle[1-x]=[x]-\langle-1\rangle[x]$.

It follows that $\langle-1\rangle[x]=[x]$.

For any real quadratically closed field $\mathbf{R}$. The natural map $\mathcal{RP}(\mathbf{R})\rightarrow\mathcal{P}(\mathbf{R})$ is an isomorphism.

$\textbf{\emph{Proof}}.$ $\mathcal{RP}(\mathbf{R})=\mathcal{RP}(\mathbf{R})_{\mathbf{R}^{\times}}\cong \mathcal{P}(\mathbf{R})$ by Proposition <ref> and Proposition <ref>

For any real quadratically closed field $\mathbf{R}$, the natural map $\mathcal{RP}_{1}(\mathbf{R})\rightarrow\mathcal{P}(\mathbf{R})$ is an isomorphism.

$\textbf{\emph{Proof}}.$ Note that the map $\lambda_{1}:\mathcal{RP}(\mathbf{R})\rightarrow \mathcal{I}_{\mathbf{R}}^{2}$ given by $[x]\mapsto \langle\langle1-x\rangle\rangle\langle\langle x\rangle\rangle$ is the zero map for all $x\in \mathbf{R}^{\times}$ since either $x>0$ or $1-x>0$. Hence $\mathcal{RP}_{1}(\mathbf{R})=\mathcal{RP}(\mathbf{R})$.

For any real quadratically closed field $\mathbf{R}$, the natural map $\mathcal{RB}(\mathbf{R})\rightarrow\mathcal{B}(\mathbf{R})$ is an isomorphism.

$\textbf{\emph{Proof}}.$ $\mathcal{RB}(\mathbf{R})=\mathrm{Ker}(\lambda_{2}:\mathcal{RP}_{1}(\mathbf{R})\rightarrow S^{2}_{\mathbb{Z}}(\mathbf{R}^{\times}))=\mathrm{Ker}(\lambda:\mathcal{P}(\mathbf{R})\rightarrow S^{2}_{\mathbb{Z}}(\mathbf{R}^{\times}) )=\mathcal{B}(\mathbf{R})$ by Corollary <ref>.

§ THE THIRD HOMOLOGY OF $SL_{2}(\MATHBF{R})$

We consider the spectral sequence used in section $4$ of [2] to relate $H_{3}(SL_{2}(F),\mathbb{Z})$ to $\mathcal{RB}(F)$.

First we recall two standard subgroups of $SL_{2}(F)$.

\begin{array}{cc}
a & 0 \\
0 & a^{-1} \\
\end{array}
\right)
\;|a\in F^{\times}\right\}$     $B:=\left\{\left(
\begin{array}{cc}
a & b \\
0 & a^{-1} \\
\end{array}
\right)
\;|a\in F^{\times}\;, b\in F\right\}$.

We will now examine this spectral sequence in the case of real quadratically closed fields. We will need the following lemma:

<cit.>If $F$ is an infinite field, then the inclusion $T\rightarrow B$ induces homology isomorphisms

$H_{k}(T,\mathbb{Z})\cong H_{k}(B,\mathbb{Z})$

for all $k\geq0$.

§.§ The Spectral Sequence.

Let $F$ be a field, and $G=SL_{2}(F)$ act (on the left) on $\mathbb{P}^{1}(F)$ by fractional linear transformations. Let $X_{n}$ be the set of ordered $(n+1)$-tuples $(x_{0},\ldots,x_{n})$ of distinct points of $\mathbb{P}^{1}(F)$. $X_{n}$ is a left $G$-set by diagonal action. Let $L_{n}=\mathbb{Z}[X_{n}]$ and let $d_{n}:L_{n}\rightarrow L_{n-1} $ given by

The natural augmentation $\varepsilon:L_{0}\rightarrow \mathbb{Z}$ gives a weak equivalence  $L_{\bullet}\rightarrow \mathbb{Z}$ where $\mathbb{Z}$ is a complex concentrated in degree $0$. It follows that there is a spectral sequence

$E^{1}_{p,q}=H_{p}(G,L_{q})\Rightarrow H_{p+q}(G,\mathbb{Z})$.

In fact $E^{1}_{0,q}=H_{0}(G,L_{q})=(L_{q})_{G}$.

Let $F$ be a field

(1) $SL_{2}(F)$ acts transitively on $X_{1}$. The stabilizer of $(\infty)$ is the subgroup $B$.

(2) $SL_{2}(F)$ acts transitively on $X_{2}$. The stabilizer of $(\infty,0)$ is the subgroup $T$.

Let $F$ be an infinite field, then

(1) $E^{1}_{p,0}\cong H_{p}(G,\mathbb{Z}[G/B])\cong H_{p}(B,\mathbb{Z})\cong H_{p}(T,\mathbb{Z})$.

(2) $E^{1}_{p,0}\cong H_{p}(G,\mathbb{Z}[G/T])\cong H_{p}(T,\mathbb{Z})$.

<cit.> Let $F$ be an infinite field. Let

\begin{array}{ll}
(z-x)(x-y)(z-y)^{-1}, & \hbox{$x,y,z\not=\infty$;} \\
(y-z)^{-1}, & \hbox{$x=\infty$;} \\
z-x, & \hbox{$y=\infty$;} \\
x-y, & \hbox{$z=\infty$.}
\end{array}
\right.

Let $Z_{0}=\emptyset$ and for $n\geq1$, let $Z_{n}$ denote the set of ordered $n$-tuples $[z_{1},\ldots,z_{n}]$ of distinct points of $F^{\times}\setminus\{1\}$. Then for $n\geq2$, there is a isomorphism of $R_{F}-$modules

$E^{1}_{0,n+1}=(L_{n+1})_{G}\longleftrightarrow R_{F}[Z_{n-2}]$

$(x_{0},\ldots,x_{n})\longmapsto\langle\phi(x_{0},x_{1},x_{2})\rangle\left[\frac{\phi(x_{0},x_{1},x_{3})}{\phi(x_{0},x_{1},x_{2})},\cdots,\frac{\phi(x_{0},x_{1},x_{n})}{\phi(x_{0},x_{1},x_{2})} \right]$

<cit.>Let $F$ be an infinite field, the $E^{1}$-page of the spectral sequence $E^{1}_{p,q}=H_{n}(G,L_{\bullet})\Rightarrow H_{n}(G,\mathbb{Z})$ has the form

$\scalebox{0.85}{\xymatrix{&\ar@{-}[ddddddd]&\vdots &\vdots&\vdots&\vdots&\\
&& R_{F}[Z_{2}]\ar[d]^{d^{1}}&R_{F}[Z_{2}]\otimes\mu_{2}\ar[d]^{d^{1}}&\vdots&\vdots&\cdots\\
&& R_{F}[Z_{1}]\ar[d]^{\lambda_{1}}&R_{F}[Z_{1}]\otimes\mu_{2}\ar[d]^{\lambda_{1}\otimes Id}&0&R_{F}[Z_{1}]\otimes \mathbb{Z}/2\mathbb{Z}\ar[d]^{d^{1}}&\cdots\\
&& R_{F}\ar[d]^{\varepsilon}&R_{F}\otimes\mu_{2}\ar[d]^{\varepsilon\otimes H_{1}(\mathrm{inc})}&0&R_{F}\otimes\mathbb{Z}/2\mathbb{Z}\ar[d]^{d^{1}}&\cdots\\
&& \mathbb{Z}\ar[d]^{0}&H_{1}(T,\mathbb{Z})\ar[d]^{C_{\omega}-Id}&H_{2}(T,\mathbb{Z})\ar[d]^{C_{\omega}-Id}&H_{3}(T,\mathbb{Z})\ar[d]^{C_{\omega}-Id}&\cdots\\
&& \mathbb{Z}&H_{1}(T,\mathbb{Z})& H_{2}(T,\mathbb{Z})&H_{3}(T,\mathbb{Z})&\cdots\\
\ar@{-}[rrrrrr]&& &&&&\\
&& &&&&\\}}$

where $C_{\omega}$ denotes the map induced in homology by conjugation $\omega:=\left(
\begin{array}{cc}
0 & -1 \\
1 & 0 \\
\end{array}
\right)
$ and $\varepsilon:R_{F}\rightarrow\mathbb{Z}$ is the augmentation map.

§.§ Third homology of $SL_{2}$ of a real quadratically closed field

Since $\lambda_{1}=0$, then $E^{2}_{1,2}=\mathrm{Ker}(\varepsilon)\otimes\mu_{2}=\mathcal{I}_{\mathbf{R}}\otimes\mu_{2}\cong\mu_{2}$ (since $\mathcal{I}_{\mathbf{R}}=\mathbb{Z}\langle\langle-1\rangle\rangle\cong\mathbb{Z}$).

Now $H_{1}(T,\mathbb{Z})=T\cong\mathbf{R}^{\times}$ and the map $C_{\omega}$ induces  $x\mapsto x^{-1}$, so $d^{1}=C_{\omega}-\mathrm{Id}:E^{1}_{1,1}=H_{1}(T,\mathbb{Z})\cong \mathbf{R}^{\times}\rightarrow \mathbf{R}^{\times}\cong H_{1}(T,\mathbb{Z})=E^{1}_{1,0}$ is given by $x\mapsto x^{-2}$. Therefore $E^{2}_{1,0}=\mathbf{R}^{\times}/(\mathbf{R}^{\times})^{2}=\mu_{2}$. Also $\mathrm{Ker}(C_{\omega}-\mathrm{Id})=\mu_{2}=\mathrm{Im}(\varepsilon\otimes H_{1}(\mathrm{inc}))$. It follows that $E^{2}_{1,1}=0$.

Now we have that $C_{\omega}$ induces the identity in $H_{2}(T,\mathbb{Z})=\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})$, thus $d^{1}=C_{\omega}-\mathrm{Id}:E^{1}_{2,1}=H_{2}(T,\mathbb{Z})\cong \mathbf{R}^{\times}\wedge \mathbf{R}^{\times}\rightarrow \mathbf{R}^{\times}\wedge\mathbf{R}^{\times}\cong H_{2}(T,\mathbb{Z})=E^{1}_{2,0}$ is the zero map. Hence $E^{2}_{2,1}=E^{2}_{2,0}=\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})$.

By the arguments of <cit.>, we have that $E^{2}_{3,0}=\mathrm{Tor}^{\mathbb{Z}}_{1}(\mu_{\mathbf{R}},\mu_{\mathbf{R}})$.

Therefore the relevant part of the $E^{2}$-page has the form

&\ar@{-}[ddddd]& \mathcal{RP}_{1}(\mathbf{R})\ar[ddr]^{d^{2}}&E^{2}_{1,3}\ar[ddr]^{d^{2}}&0&\vdots\\
&& \ast&\mu_{2}\ar[ddr]^{d^{2}}&0&\vdots\\
&& \ast&0&\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})&\vdots\\
&& \ast&\ast& \bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times}) &\mathrm{Tor}^{\mathbb{Z}}_{1}(\mu_{\mathbf{R}},\mu_{\mathbf{R}})\\
\ar@{-}[rrrrr]&& &&&\\
&& &&&\\}}$

§.§ The calculation of $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})$

Now, $E^{\infty}_{1,2}$ is a subquotient of $E^{2}_{1,2}=\mathcal{I}_{\mathbf{R}}\otimes\mu_{2}\cong\mu_{2}$. So $E^{\infty}_{1,2}=\{1\}$ or $\mu_{2}$.

In <cit.>, it is shown that $E^{\infty}_{0,3}=\mathcal{RB}(F)$  and $E^{\infty}_{0,3}=\mathrm{Tor}^{\mathbb{Z}}_{1}(\mu_{F},\mu_{F})$ for any infinite field $F$. Thus $E^{\infty}_{0,3}=\mathcal{B}(\mathbf{R})$ and $E^{\infty}_{0,3}=\mathrm{Tor}^{\mathbb{Z}}_{1}(\mu_{\mathbf{R}},\mu_{\mathbf{R}})$ respectively.

By Corollary <ref>, we have that $E^{2}_{2,1}=\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})=2\cdot\bigwedge^{2}_{\mathbb{Z}}(\mathbf{R}^{\times})=2\cdot E^{2}_{2,1}$. Now in the section $4.7$ of [2], it was proved that $2\cdot E^{\infty}_{2,1}=0$. Since $E^{2}_{2,1}$ maps onto $E^{\infty}_{2,1}$, it follows that $E^{\infty}_{2,1}=2\cdot E^{\infty}_{2,1}=0$.

Now let $K:=\mathrm{Ker}(H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\rightarrow \mathcal{B}(\mathbf{R}))$. Thus the convergence of the spectral sequence gives us short exact sequences

$\xymatrix{0\ar[r]&K \ar[r]&H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\ar[r]& \mathcal{B}(\mathbf{R})\ar[r]&0     }$

$\xymatrix{0\ar[r]&\mathrm{Tor}^{\mathbb{Z}}_{1}(\mu_{\mathbf{R}},\mu_{\mathbf{R}})  \ar[r]&K\ar[r]& E^{\infty}_{1,2}\ar[r]&0     }$.

In particular, it follows that $|K|=2$ or $4$.

§ THE INDECOMPOSABLE $K_{3}$ OF A REAL QUADRATICALLY CLOSED FIELD

Let $\mathbf{R}$ be a real quadratically closed field, then the natural map $H_{3}(SL_{2}(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$ is an isomorphism.

$\textbf{\emph{Proof}}.$  From the previous remarks and the previous section, we have the following commutative diagram:

$\xymatrix{0\ar[r]&K\ar[r] \ar@{->>}[d]& H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z}) \ar[r]\ar@{->>}[d]& \mathcal{B}(\mathbf{R})\ar[r]\ar[d]^{=}&0 \\

The exactness of the bottom row is the main theorem in Suslin (Theorem <ref> above). The middle vertical arrow is a surjection by <cit.>. Thus the map $K\rightarrow\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{\mathbf{R}},\mu_{\mathbf{R}}})$ is surjective, and since $|K|\leq4$, it follows that it is an isomorphism and the result follows.

From the proof of Corollary <ref>, it follows that the term $E^{\infty}_{1,2}$ of the spectral sequence of above is equal to $\mathcal{I}_{\mathbf{R}}\otimes\mu_{2}\cong\mu_{2}$.

Now, let us recall the definition of the Milnor $K$-theory.

For a field $F$, let $T(F^{\times})=\displaystyle\bigoplus_{n\geq0}(F^{\times})^{\otimes n}$, where $(F^{\times})^{\otimes 0}=\mathbb{Z}$, be the tensor algebra over the $\mathbb{Z}$-module $F^{\times}$. Let $J$ be the two-sided homogeneous ideal in $T(F^{\times})$ generated by elements $a\otimes(1-a)$ with $a\in F^{\times}\setminus\{1\}$.

The Milnor K-theory of a field $F$ is defined to be the graded ring $K_{\bullet}^{M}(F)=T(F^{\times})/J$.

The image of an element $a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n}$ in $K_{n}^{M}(F)$ is denoted by $\{a_{1},\ldots,a_{n}\}$.

It follows that for any real quadratically closed field $\mathbf{R}$ that $K_{n}^{M}(\mathbf{R})=\langle\{-1,\ldots,-1\rangle\oplus K_{n}^{M}(\mathbf{R})^{0}$ where $K_{n}^{M}(\mathbf{R})^{0}:=\langle\{x_{1},\ldots,x_{n}\}\;| x_{i}>0\; \forall i\rangle$ is $2$-divisible.

Note that, it follows that $K_{n}^{M}(\mathbf{R})^{0}=2\cdot K_{n}^{M}(\mathbf{R})$ when $\mathbf{R}$ is real quadratically closed.

For any real quadratically closed field $\mathbf{R}$ and any $n\geq3$, the inclusion $SL_{2}(\mathbf{R})\rightarrow SL_{n}(\mathbf{R})$, induces an injection $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\rightarrow H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})$ and $H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})\cong H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\oplus K_{n}^{M}(\mathbf{R})^{0}$.

$\textbf{\emph{Proof}}.$ By definition, the map $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(\mathbf{R})$ factors through $H_{3}(SL_{3}(\mathbf{R}),\mathbb{Z})$ and hence
$H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\rightarrow H_{3}(SL_{3}(\mathbf{R}),\mathbb{Z})$ is injective.

Furthermore $H_{3}(SL_{3}(\mathbf{R}),\mathbb{Z})\cong H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})$ $\forall n\geq3$ by <cit.>.

For any infinite field $F$ the sequence

$\xymatrix{H_{3}(SL_{2}(F),\mathbb{Z})\ar[r]&H_{3}(SL_{3}(F),\mathbb{Z})\ar[r]&2\cdot K_{3}^{M}(F)\ar[r]&0}$

is exact by <cit.>. Thus, for real quadratically closed fields, we have an exact sequence

$\xymatrix{0\ar[r]&H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\ar[r]&H_{3}(SL_{3}(\mathbf{R}),\mathbb{Z})\ar[r]& K_{3}^{M}(\mathbf{R})^{0}\ar[r]&0}$.

The above sequence is split, since we have a commutative triangle

$\xymatrix{H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\ar[r]\ar[dr]_{\cong}& H_{3}(SL_{3}(\mathbf{R}),\mathbb{Z}) \ar@{->>}[d]\\

where the diagonal arrow is an isomorphism.

§ ACKNOWLEDGEMENT

I thank my supervisor Dr. Kevin Hutchinson for comments and advice on this paper. The work leading to this paper was supported by the Research Demonstratorships programme of University College Dublin.

[1]
Brown K.S., Cohomology of groups, Graduate text in Mathematics 87,
Springer Verlag, New York $(1982)$.

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Hutchinson Kevin. A Bloch Wigner Complex for $SL_{2}$
J.K. Theory 12 $(2013)$ $15-68$.

[3]
Hutchinson Kevin and Liqun Tao The third homology of the special linear group of a fieldJournal of Pure and Applied Algebra 213 $1665-1680$ $(2009)$.

[4]
Hutchinson Kevin. A refined Bloch group and the third homology  of $SL_{2}$ of a fieldJournal of Pure and Applied Algebra 217 $(2013)$ $2003-2035$.

[5]
A. Mazzoleni. A new proof of a Theorem of Suslin
K-Theory 35 $(2005)$ $199-211$.

[6]
Walter Parry and Chih-Han Sah. Third homology of SL(2,R) made discrete. J. Pure Appl. Algebra, $30(2)$:$181$-$209$, $1983$.

[7]
Chin-Han Sah. Homology of classical Lie groups made discrete, III. J. Pure Appl. Algebra,
$56(3)$:$269$-$312$, $1989$.

[8]
Serge Lang, Algebra, Graduate text in Mathematics 211,
Springer Verlag, New York $(2002)$.

[9]
Suslin, A. A.: $K_{3}$ of a field and the Bloch group. Proc. Steklov Inst. Math. $183$, no $4$, $217-239$ (1991)


# A Decentralized Spike-based Learning Framework for Sequential Capture in
Discrete Perimeter Defense Problem

Mohammed Thousif1, Shridhar Velhal1, Suresh Sundaram, and Shirin Dora Thousif,
Shridhar Velhal, and Suresh Sundaram are with the Department of Aerospace
Engineering, Indian Institute of Science, Bengaluru, India. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Dora is with
the Department of Computer Science, Loughborough Univerity, United Kingdom.
Email<EMAIL_ADDRESS>
###### Abstract

This paper proposes a novel Decentralized Spike-based Learning (DSL) framework
for the Perimeter Defense Problem (PDP). The PDP in this paper is termed
discrete-PDP (d-PDP), as the circular territory is discretized into multiple
segments. A team of defenders is operating on the perimeter to protect the
circular territory from radially incoming intruders. At first, the d-PDP is
formulated as a spatio-temporal multi-task assignment problem (STMTA). The
problem of STMTA is then converted into a multi-label learning problem to
obtain labels of segments that defenders have to visit in order to protect the
perimeter. The DSL framework uses a Multi-Label Classifier using Synaptic
Efficacy Function spiking neuRON (MLC-SEFRON) network for deterministic multi-
label learning. Each defender contains a single MLC-SEFRON network. Each MLC-
SEFRON network is trained independently using input from its own perspective
for decentralized operations. The input spikes to the MLC-SEFRON network can
be directly obtained from the spatio-temporal information of defenders and
intruders without any extra pre-processing step. The output of MLC-SEFRON
contains the labels of segments that a defender has to visit in order to
protect the perimeter. Based on the multi-label output from the MLC-SEFRON a
trajectory is generated for a defender using a Consensus-Based Bundle
Algorithm (CBBA) in order to capture the intruders. The target multi-label
output for training MLC-SEFRON is obtained from an expert policy. Also, the
MLC-SEFRON trained for a defender can be directly used for obtaining labels of
segments assigned to another defender without any retraining. The performance
of MLC-SEFRON has been evaluated for full observation and partial observation
scenarios of the defender. The overall performance of the DSL framework is
then compared with expert policy along with other existing learning
algorithms. The scalability of the DSL has been evaluated using an increasing
number of defenders.

###### Index Terms:

Perimeter Defence Problem (PDP), Spiking Neural Network(SNN), Multi-label
learning, spatiotemporal task

11footnotetext: Equal contribution

##  I Introduction

Technological advances in sensors and computer vision have enabled
organizations to use autonomous Unmanned Aerial Vehicles (UAVs) for various
applications such as logistics [1], search and rescue [2], agriculture [3],
security and surveillance [4], fire-fighting and defence [5]. The use of UAVs
also pose a threat to the privacy and security of vital infrastructures [6].
UAV-based methods for this purpose have been evaluated for the protection of
critical airspace infrastructures in [7]. These methods employ a team of UAVs,
termed as defenders, that patrol the boundary of critical infrastructures to
prevent intruders from infiltrating them. This problem is termed as the
Perimeter Defense Problem (PDP) [8, 9]. Generally, the defenders are
constrained to operate on the perimeter of the territory. The PDP has been
addressed for different boundary shapes like linear territory [10], conical
territory [11], circular territory [12], and more generally for any convex
territory [8, 9]. A detailed review of existing approaches and challenges for
the PDP are presented in [13, 14].

The differential geometric approaches have been used to compute feasible
regions for defenders. Then using these feasibility constraints, one-to-one
assignments of the defenders to intruders are computed as in [8, 9]. However,
these geometric approaches are limited to one-to-one capture and require full
observation. In [15], the decentralized policy for communication and decision-
making for defenders is obtained from a centralized solution. But this
decentralized solution allows each defender to capture only one intruder. As a
result, the solution obtained does not account for sequential capture, where a
defender captures multiple intruders in a sequence. In [16], a two-stage
adaptive partitioning approach is proposed for protecting circular territories
against multiple radially incoming intruders. The territory is divided into
partitions in the first stage of the partitioning approach. In the second
stage, intruders in each partition are independently assigned to the
defenders. However, the defenders lack cooperation as the defender in each
partition solves the assignments independently.

In [17] and [18], PDP has been formulated as a spatio-temporal Multi-Task
Assignment Problem (STMTA) for convex territories. In this formulation, each
intruder’s arrival time and location on the perimeter are predicted by
defenders. Then to neutralize an intruder, a defender has to reach that
arrival location at the arrival time of that intruder, which represents a
spatio-temporal task. The PDP is converted into an STMTA problem when there
are multiple spatio-temporal tasks for a single defender to handle. The
solution to the STMTA problem is necessary to compute assigned intruders for
each defender. The trajectory, following which assigned intruders will be
neutralized, is computed for each defender using the assignment solution. In
[19], the dynamic programming approach has been presented to solve the STMTA
formulation of PDP. However, due to its exponential computational complexity,
it is not implementable in real-time.

Analytical solutions for STMTA based on the exact time-constrained multiple
Traveling Salesman Problem (mTSP) problem have been developed in [20, 21] for
music-playing robots, in [22] for warehouse automation, and in [17, 18] for
PDP. These approaches involve solving the linear sum assignment problem using
the Hungarian solution for mTSP, which has a computational complexity of order
${(N+M-1)}^{3}$. The aforementioned works based on analytical, geometric, and
numerical approaches are not scalable. Their complexity grows with an increase
in the number of intruders. Hence they are limited to the protection of small
territories. Designing decentralized and scalable strategies with partial
observability is one of the major challenges for real-time implementations of
perimeter defense systems.

Recently, Graph Neural Networks (GNN) are used to develop a learning-based
decentralized approach for solving PDP as an assignment problem [23]. Each
defender is equipped with a vision camera that uses ANN to extract features of
observed intruders and defenders. These features are used as input to GNN. The
GNN communicates with its neighboring defender (i.e. another GNN) to share
features dynamically. The maximum matching algorithm [24] is used as an expert
policy for training the GNN. The GNN has a fixed number of output neurons
representing the closest intruders to a defender (arranged in a specific
order). The choice of the number of input and output channels affects
performance. Also, as the number of input and output channels is hard-coded,
the setting is not generic. However, the GNN approach is only suitable for
one-to-one assignments of defenders to intruders. Thus, there is a need to
develop a generic learning-based scalable solution for one-to-many assignments
of defenders to intruders to protect large territories.

In this paper, a discrete PDP (d-PDP) is considered in which a defender needs
to capture an intruder by visiting the segment of that intruder at the arrival
of that intruder. To the best of the author’s knowledge for the first time in
this paper, the STMTA problem posed by d-PDP is converted into a deterministic
multi-label learning problem using a Spiking Neural Network (SNN). The
conventional multi-label learning algorithms provide a probabilistic solution,
but multi-task assignment problem posed by d-PDP requires a deterministic
solution for better performance. For predicting the multi-label
deterministically a novel multi-label classifier using SNN is developed in
this paper. The input to the SNN is obtained from the spatio-temporal
information about the defenders and intruders. Each defender is trained using
a different SNN, enabling the decentralized operation of the DSL framework.
The DSL framework is executed for different sensing ranges of defenders. If a
defender is capable of sensing the entire perimeter then it is termed a full
observation scenario. Otherwise, it is termed a partial observation scenario.

DSL framework is designed to handle the spatio-temporal problem posed by
d-PDP. As the d-PDP has inherent spatio-temporal nature, the DSL framework
does not require any pre-processing to generate input spikes. The DSL
framework is event-triggered which means that a neuron in the framework emits
a spike when an object (either a defender or intruder) is detected otherwise,
the neuron remains silent. This spiking nature of DSL makes it energy
efficient.

For the deterministic multi-label learning, each label prediction is
formulated as a binary classification problem using Synaptic Efficacy Function
spiking neuRON (SEFRON) [25] network. Hence the classifier is termed a Multi-
Label Classifier using SEFRON (MLC-SEFRON). A 3-layered MLC-SEFRON
architecture is used for this purpose namely input layer, SEFRON layer and
output layer. Each label prediction is evaluated using the response of two
neurons in the SEFRON layer. If one neuron spikes earlier than the other then
the output neuron connected to these two neurons in the SEFRON layer generates
a label $1$ and vice versa. If the response of the output neuron is $1$ then
the defender is assigned to the segment associated with that neuron and vice
versa. Therefore MLC-SEFRON learning predicts the labels in a deterministic
manner. Since the number of intruders exceeds the number of defenders
considered in this paper, the MLC-SEFRON is designed to predict multiple
labels of segments assigned for a single defender. Each defender has one MLC-
SEFRON architecture. The DSL framework is designed such that all the defender
uses the same trained MLC-SEFRON architecture. After training the MLC-SEFRON
network with a single defender, it can be used for obtaining assignments of
other defenders without retraining. Hence the DSL framework is scalable and
can be used for any number of defenders without any extra learning
computations. Once defenders obtain their assigned labels, they may have
conflicts in their trajectories due to the decentralized approach. These
conflicts are resolved to generate the trajectories of each defender using the
Consensus-Based Bundle Algorithm (CBBA). The dataset and expert assignment
labels for training are generated by solving d-PDP using expert policy derived
from the simplified form of [18].

The performance of MLC-SEFRON in training is evaluated using two different
observation scenarios. One is the partial observation scenario and the other
is the full observation scenario of the defender around the perimeter. In a
partial observation scenario, the defender is assumed to have a sensing range
of 150 degrees of the perimeter. Whereas in full observation, the defender has
a sensing range of 360 degrees which covers the entire perimeter. The MLC-
SEFRON trained for one defender can be used for other defenders without
retraining, hence the learning in this paper is decentralized. The performance
of the MLC-SEFRON is evaluated using mutli-label metrics such as $Precision$,
$Recall$, and $F1-Score$ as in [26]. The DSL framework is trained with five
defenders using the MLC-SEFRON algorithm, and then the same is tested. The
results indicate that the DSL framework’s success rate is on par with the
expert solution. The performance of DSL framework is then compared with the
state-of-the-art adaptive partitioning approach [16] and the expert policy.
The results indicate that the DSL framework performs better compared.
Furthermore, the DSL performance is evaluated for different-sized teams of
defenders without retraining, to illustrate the scalability of the DSL
framework. The DSL framework performs at par compared to the centralized
expert policy for the different sizes of defenders to showcase its
generalization performance.

The main contributions of this paper can be summarized as follows:

  1. 1.

The formulation of the d-PDP into the spiking multi-label learning problem
with the help of MLC-SEFRON architecture.

  2. 2.

Development of deterministic MLC-SEFRON learning algorithm for predicting
multiple labels of segments.

  3. 3.

Distributed and scalable learning-based solution for sequential capture in
d-PDP. To the authors’ best knowledge this is the first time in literature, a
learning-based solution is proposed for fewer defenders protecting territory
against multiple intruders by sequential capture.

The rest of the paper is organized as follows: Section III presents the
mathematical formulation of d-PDP for circular perimeter as a multi-label
classification problem using SNN. Section IV presents the SNN architecture and
MLC-SEFRON learning algorithm. Section V provides the performance evaluation
results for MLC-SEFRON learning algorithms using multi-label metrics. This
section also presents results on performance comparison of the DSL framework
with expert policy and ablation studies exhibiting scalability. Finally,
Section VI summarises the conclusions from this paper.

Figure 1: Decentralized spike-based Learning (DSL) framework for discrete
perimeter defense problem

.

##  II Related works

This section presents the related works on key techniques used in this DSL-
based solution for d-PDP, namely multi-label learning, and SNNs.

###  II-A Multi-label learning

In a multi-label learning problem, each sample can have associations with more
than one class. The goal of the learning algorithm is to predict all the
classes that a given sample is associated with. Some existing multi-
classification learning algorithms are also used for handling multi-label
learning problems. One of the first approaches for multi-label learning is
performed for text categorization [27]. In [27], the multi-label learning is
used to categorize the text in a news article into multiple topics such as
politics, society, etc., Then in [28], the prediction of each label is
performed using a binary classifier. Therefore multiple binary classifiers are
designed for learning multiple labels, this strategy is termed as binary
relevance strategy.

Advances in the field of deep learning have led to the usage of deep neural
networks for multi-label learning [29, 30, 31]. In [29] and [30], a
backpropagation approach is proposed for multi-label learning. In [31], a
clustering algorithm has been used for multi-label learning Multi-label
learning using deep neural networks requires a large amount of data for
efficient performance. Moreover, the architecture of deep neural networks
consists of many hidden which makes it computationally expensive. Also, each
label is predicted in a probabilistic manner, which decreases confidence in
the prediction. Therefore in the proposed DSL framework, deterministic multi-
label learning using a binary relevance strategy is performed using an SNN
with no hidden layers. The relevant literature on SNNs is presented in the
next section. II-B.

###  II-B Spiking neural networks

Spikes emitted by spiking neurons efficiently embed the spatio-temporal
information present in the input given to them [32]. The detailed discussion
on various existing learning algorithms for SNNs is presented in [33]. Bohte
$et\ al.$ [34] developed a gradient-based weight update strategy for SNNs and
it is termed as SpikeProp. In SpikeProp, linearities are assumed in the
membrane potential of a neuron to evaluate the gradient of a spike. This is
done as the gradient of spikes cannot be evaluated directly due to their
discontinuous nature. In[35], a supervised learning algorithm for weights in
SNN is developed by exploiting spatio-temporal features. Spike Time Dependent
Plasticity (STDP) [36], is another learning technique developed for SNNs that
is motivated by biological mechanisms. In rank-order learning for SNNs
described in [37] and [38], the weight updates are evaluated based on the
spiking rate of the neuron. To increase the performance further evolving
layers are used in SNN architecture by neuron addition or deletion strategies
as in [39, 40].

In [41], SNN architecture with time-varying weights is proposed for
classification. The SNN in [41] is termed as Synaptic Efficacy Function based
leaky-integrate and fire neuRON (SEFRON)[41]. A novel learning algorithm for
weight updates is proposed in SEFRON which distributes the evaluated weight
updates over time. The SEFRON results clearly highlight that the proposed
synapse model, with its time-varying nature, executes binary classification
tasks with high computational power in comparison to other SNN learning
algorithms now in practice. In [25] the SEFRON is extended for multi-class
classification problems. The results in [41] and [25] point out that modeling
weights with time-varying functions in SNN increase the classification
performance of the network. The aforementioned SNN learning algorithm requires
additional encoding mechanisms to convert the real-valued data into spikes.
Whereas the STMTA problem posed by d-PDP has inherent spatio-temporal nature.
Hence STMTA problem can be directly solved using SNNs without any extra
encoding mechanisms. A Multi-Label Classifier using SEFRON (MLC-SEFRON)
network is proposed in this paper to handle multi-task assignment problems.
This is the first time in the literature that an SNN is used to solve a multi-
task assignment problem.

##  III Mathematical formulation of d-PDP as Decentralized Spiking multi-label
learning problem

The mathematical description of discrete PDP (d-PDP) as a spiking multi-label
learning problem is presented initially in this section. Figure 1 shows the
proposed Decentralized Spike-based Learning (DSL) framework. The spatio-
temporal location information of defenders and intruders is represented as
spikes and given as input to the MLC-SEFRON network. The solution of STMTA is
used as an expert for the training of the network to learn assignments as the
labels. The proposed DSL framework employs the decentralized approach, in
which a defender learns its assignments based on the spatio-temporal inputs
from its own perspective. The labels of assigned locations to a defender are
learned using the MLC-SEFRON learning algorithm. The predicted labels are then
post-processed to compute the conflict-free assignments for all the defenders.
This section also describes the generation of defender trajectory using the
predictions of the MLC-SEFRON network.

###  III-A Discrete Perimeter Defense Problem (d-PDP)

This paper considers the multi-player perimeter defense problem in which a
team of defenders protects a given territory from invading intruders. It is
assumed that intruders are moving radially inwards with a constant velocity,
as proposed in [16, 19]. Defenders are restricted to operate only on the
perimeter [8, 9]. Each defender’s trajectory is computed cooperatively such
that it can capture the intruders assigned to it in a cost-effective sequence.

Without loss of generality, we assume a circular territory $\Omega$ of unit
radius centered at origin $O$ whose perimeter $\partial\Omega$ is given as

$\displaystyle\Omega$ $\displaystyle=\left\\{\bm{p}\in\Re^{2}\ \Big{|}\
{\|\bm{p}\|}_{2}\leq 1\right\\},$ (1) $\displaystyle\partial\Omega$
$\displaystyle=\left\\{\bm{p}\in\Re^{2}\ \Big{|}{\|\bm{p}\|}_{2}=1\right\\}$
(2)

where $\bm{p}$ is a point in 2-dimensional plane. This paper considers
discrete PDP in which the perimeter is divided into $N_{s}$ segments
$(s_{1},...,s_{N_{s}})$, of equal arcs as shown in Figure 2.

Let us consider $N$ defenders $\\{D_{1},\cdots,D_{i},\cdots,D_{N}\\}$
operating on the perimeter. The initial position of the defender $D_{i}$ on
the perimeter is given in by $p_{i}^{D}=(r,S^{D}_{i})$. $r$ represents the
distance from the origin ($O$) and is always equal to $1$ as defenders operate
only on the perimeter. $S^{D}_{i}$ represents the segment of defender $D_{i}$.
The motions of the defenders are limited by a maximum angular velocity
$v^{D}$.

Consider $M>N$ intruders $I_{1},\cdots,I_{j},\cdots,I_{M}$ radially moving
towards the perimeter with speed $v^{I}$ . The position $p^{I}_{j}$ of the
$j^{th}$ intruder is represented by $p_{j}^{I}=(r_{j}^{I},S^{I}_{j})$.
$r_{j}^{I}$ represents the distance of the intruder $I_{j}$ from the center of
the territory. $S^{I}_{j}$ represents the segment of intruder $I_{j}$. Note
that the positions of the intruders are initialized outside the territory,
(i.e., $r_{j}^{I}>1$). The kinematic equation of intruder $I_{j}$ is given as,
$-\dot{r}_{j}^{I}=v^{I}$.

If an intruder crosses the perimeter from any of the segments, then it is
considered that the defenders have failed to defend the perimeter. If a
defender is present in a given segment when an intruder enters that segment
then the defender is considered to have captured the intruder. If $S^{D}_{i}$
and $S^{I}_{j}$ represent the segment locations of defender $D_{i}$ and
intruder $I_{j}$ respectively, then capture of the latter is written as

$\displaystyle Q(I_{j})=\left\\{D_{i}\Big{|}r_{j}^{I}(t)=1\ \&\
S^{I}_{j}(t))=S^{D}_{i}(t))\right\\}$ (3)

Figure 2: Discrete Perimeter defense problem. The center of the perimeter is
shown with O, and segments on the perimeter are denoted by
$s_{1},s_{2},s_{3},s_{4},s_{5}$, etc, D denotes the defender and $\psi$
denotes it’s angular position of on the perimeter respectively

.

(a)

(b)

(c)

Figure 3: Spike-based representation of defenders and intruders. Figure 3a
shows the defender locations as
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bullet}$
and intruder locations as
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\times}$.
The partial observation range of defender present in segment $s_{3}$ is shown
by the shaded yellow region. Figure 3b and Figure 3c shows the spike
representation of the intruders and defenders present in shaded yellow region.

Based on the heading and speed of the intruder $I_{j}$, it is possible to
estimate it’s arrival time $t_{j}^{a}$ and location $\bm{p}^{T}_{j}$ on the
perimeter. To defend the perimeter, a defender has to perform the spatio-
temporal task of intercepting the intruder $I_{j}$ at position
$\bm{p}_{j}^{T}$ and at time $t_{j}^{a}$. Each defender needs to capture
multiple intruders as there are more intruders than defenders. Thus, the team
of defenders needs to solve a spatio-temporal multi-task assignment (STMTA)
problem [17, 18].

###  III-B Spike-based representation for d-PDP

The technique developed in this section utilizes the capabilities of SNNs to
represent both spatial and temporal information relevant for d-PDP. Consider
the perimeter has been divided into n segments, each of which makes an angle
of $(360/n)^{\circ}$ at the center. These segments are denoted as
$\\{s_{1},s_{2},\cdots,s_{i},\cdots,s_{n}\\}$. Each defender has observability
of $m$ segments, where $m\leq n$. A defender in segment $s_{i}$, can observe m
segments i.e.
$\left\\{s_{i+\left\lceil\frac{m-1}{2}\right\rceil},\cdots,s_{i-1},s_{i},s_{i+1},\cdots,s_{i+\left\lfloor\frac{m-1}{2}\right\rfloor}\right\\}$.
(All the negative and zero indices of segments are converted to positive
indices in a cyclic fashion. ) For the decentralized setting, the zones are
defined from the perspective of each defender to represent the information.
The zones for defender present in segment $s_{i}$ ( where
$s_{i}\Leftrightarrow z_{\left\lceil\frac{m+1}{2}\right\rceil}$ ) are
$\left\\{z_{m},z_{m-1},\cdots,z_{\left\lceil\frac{m+1}{2}\right\rceil},\cdots,z_{2},z_{1}\right\\}$.

Figure 3a shows a scenario with 5 defenders and 10 intruders for 36 segments
and observation of 15 zones. The perimeter has been divided into 36 segments
each of which makes an angle of $10^{\circ}$ at the center. These 36 segments
are denoted as $\\{s_{1},s_{2},\cdots,s_{36}\\}$. The current positions of
intruders and defenders have been shown using
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\times}$
and
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\bullet}$,
respectively. The radial trajectory of the intruders towards a specific
segment on the perimeter is shown using red dashed lines.

For the partial observation scenario considered in this paper, it is assumed
that defenders have a limited sensing range of $150^{\circ}$, which
constitutes 15 segments. For example, the yellow region in the figure
represents the sensing range for $D_{1}$ which implies that it can perceive
the spatio-temporal information pertaining of both, intruders and defenders.
The segments $\\{s_{10},s_{9},\ldots,s_{33},s_{32}\\}$ segments are referred
as zones of that defender ($D_{1}$), represented by
($z_{15},z_{14},\ldots,z_{2},z_{1}$) as shown in Figures 3b & 3c. Similar to
segments, the zones are also numbered in the anticlockwise direction from
$z_{1}$ to $z_{15}$. Defender is always assumed to be present in $z_{8}$,
which is the central zone in its sensed region. The zones contain the
decentralized information sensed by each defender, on the contrary, segments
contain global information. In other words, segments of the territory are
designed beforehand and do not depend on the positions of defenders.

The spatio-temporal information pertaining to intruders and defenders present
in the zones of a defender $D_{1}$ is represented using spike patterns and
shown in Figure 3b and 3c. These spikes are event-triggered based on the
distance of the object (either intruder or defender) from the perimeter. The
time of a given spike in Figure 3b is directly proportional to the arrival
time of the intruder in the corresponding zone. For instance in Figure 3b, the
time of arrival of intruder $I_{1}$ in zone $z_{6}$ at the perimeter is shown
using a spike at $2.15s$. Similarly, a defender in zone $z_{13}$ (i.e. segment
$s_{8}$) is represented using a spike at $0s$.The position of the defender
whose perspective is being considered is always considered at the $8^{th}$
zone (middle zone) in 15 segments observation scenario (partial observation
scenario). The time of the spikes that represent the position of defenders is
always set to $0s$ as defenders are constrained to move along the perimeter.
The spike patterns generated are directly presented as input to the MLC-SEFRON
via the input layer without any extra pre-processing step.

###  III-C Multi-task assignments to multi-label learning

The solution of spatio-temporal multi-task assignments gives the assignments
of defenders to the intruders. In a decentralized setting of defender $D_{i}$,
each assigned intruder can be identified by the corresponding zone. Using the
assignment solution, one can label the assignment of defenders to the zones.
For the labels of defender $D_{i}$, if it is assigned to a zone $z_{j}$, then
$z_{j}$ is labeled as $TRUE$. A defender can have multiple assignments, all
the assigned segments are labeled as $TRUE$. The assignment solution provides
the target labels ($T^{l}$) for all the observed m zones (
$T^{l}\in{\\{1,0\\}}^{m}$). In this way, the assignments of defender $D_{i}$
are converted into multiple deterministic labels for zones of $D_{i}$. One
should note that the centralized solution to the STMTA problem provides the
assignments of all the defenders. In the DSL framework assignments of each
defender’s assignments (with input from that defender’s perspective) are used
to train the individual MLC-SEFRON network. In this way, training is scalable
and independent of specific defenders. The spiking network needs to learn
these labels. The detailed learning algorithm is discussed in section IV.

###  III-D Trajectory generation from MLC-SEFRON output

The suitable trajectory for the team of defenders is generated using zones
assigned by the MLC-SEFRON to each defender. Predicting assigned zones for
each defender in a decentralized manner can lead to a situation where multiple
defenders are assigned to a single zone. Also during SNN prediction, there is
a chance of assigning a defender to a zone where no intruder is present. To
resolve these two issues, a consensus-based bundle algorithm (CBBA) [42] is
used to compute the final task assignments.

Lets consider that $\hat{l}_{j}$ represents the prediction for the $j^{th}$
segments’ assignment to a defender. To exploit the spatial correlation between
neighboring segments, the prediction for the $j^{th}$ segment is updated using
the predictions for the segments in the neighborhood. Further, the final
assignment for a segment is assigned to $0$ if there is no intruder present in
that segment. Based on this, the effective predictions
($\hat{l}^{\text{eff}}_{j}$) for a given segment are given by

$\displaystyle\hat{l}^{\text{eff}}_{j}=\begin{cases}(\hat{l}_{j}+\alpha*\hat{l}_{j+1}+\alpha*\hat{l}_{j-1}),&{\text{Intruder
present in $s_{j}$}}\\\ 0,&{\text{Intruder absent from $s_{j}$}}\end{cases}$
(4)

where $\alpha\in[0,1]$ is the scaling factor that governs the impact of
predictions for neighboring segments on a given segment. If $\alpha=0$, there
is no effect of predictions of neighboring segments.

Every defender computes its own trajectory by arranging the intruders present
in the assigned zones in ascending order of their arrival times. Every
defender computes the trajectory based on their individual effective labels (
$\hat{l}^{\text{eff}}_{j}$ ). It is possible that multiple defenders are
assigned a given segment because the predictions for each defender are
generated independently. To determine final assignments, a distance-based cost
function is used. Among these defenders, each one (say $D_{i}$) bids for the
$s_{j}$ segment with a cost $l_{ij}$, where $l_{ij}$ is the effective distance
$D_{i}$ needs to travel from its current location to $j^{th}$ segment.

$\displaystyle l_{ij}=\hat{l}^{\text{eff}}_{j}*arc(S^{\prime D}_{i}),s_{j})$
(5)

where $S^{\prime D}_{i}$ is the segment location from which defender $D_{i}$
has to start in order to capture the intruder in the $s_{j}$ segment. To fix
the issue of multiple defender assignments to a single $s_{j}$ segment, the
defender which has the minimum cost is considered the winner in the bidding.
Based on this bidding and consensus algorithm final trajectory for each
defender is generated.

##  IV MLC-SEFRON Architecture & Learning Algorithm

In this section a Multi-Label Classifier using SEFRON (MLC-SEFRON) to predict
the labels of multiple zones assigned to a single defender in a deterministic
manner is presented. The MLC-SEFRON architecture is described first, followed
by its learning algorithm.

Figure 4: Architecture of MLC-SEFRON

.

###  IV-A Architecture of MLC-SEFRON

The architecture of MLC-SEFRON for training a defender with a sensing range
having $m$ zones is shown in Figure 4. The input layer of MLC-SEFRON consists
of $2m$ neurons which are used to present the spike patterns illustrated in
Figures 3c & 3b (which represent the spatio-temporal information about
defenders and intruders). The first $m$ neurons are used to represent the
defender information, whereas the last $m$ neurons are used for representing
intruder information. Let $x_{i}=\\{t_{i}\\}$ be the input spike pattern
presented through the $i^{th}$ input neuron, where $t_{i}$ denotes the time of
spike of that neuron. The SEFRON layer in the architecture also consists of
$2m$ spiking neurons ($f_{1},...,f_{2m}$). The weight of the synapse between
$i^{th}$ input neuron and ${2j}^{th}$ neuron in the SEFRON layer is denoted as
$w_{i2j}(t)$. Each weight $w(t)$ connecting the input layer and SEFRON layer
is modeled as a time-varying Gaussian function as described in SEFRON [25].
The output layer consists of $m$ neurons associated with $m$ zones in the
perimeter. The response of $j^{th}$ output neuron depends on the spike
responses of ${2j-1}^{th}$ and ${2j}^{th}$ neurons in the SEFRON layer as
shown below

$\displaystyle y_{j}=\begin{cases}1,&{\text{If
}\hat{t}_{2j-1}<\hat{t}_{2j}}\\\ 0,&{\text{If
}\hat{t}_{2j-1}\geq\hat{t}_{2j}}\end{cases}$ (6)

where $\hat{t}_{2j-1}$ and $\hat{t}_{2j}$ represents the time of first spikes
of ${2j-1}^{th}$ and ${2j}^{th}$ neurons in the SEFRON layer. $y_{j}$
represents the deterministic label predicted for the $j^{th}$ zone in the
perimeter. If $y_{j}$ is $1$ then a defender $D$ is assigned to $j^{th}$ zone
and vice versa if $y_{j}$ is $0$. In a similar fashion, labels are evaluated
for all $m$ neurons in the output layer. If $\hat{C}$ denotes the predicted
multi-label output from the MLC-SEFRON network then it is given as

$\displaystyle\hat{C}=\\{\hat{c}_{1},...\hat{c}_{j},...\hat{c}_{m}\\},\text{where
}\hat{c}_{j}=y_{j}$ (7)

If $\hat{t}_{j}$ denotes the spike time ${j}^{th}$ neuron in the SEFRON layer
then its spike time can be evaluated as

$\displaystyle\hat{t}_{j}=\\{t|v_{j}(t)=\theta_{j}\\}$ (8)

where $\theta_{j}$ and $v_{j}(t)$ denote the potential threshold and potential
of ${j}^{th}$ neuron in the SEFRON layer respectively. The potential
thresholds for all neurons in the SEFRON layer are initialized as described in
[25]. The potential $v_{j}(t)$ is evaluated as

$\displaystyle v_{j}(t)=\sum_{i=1}^{2m}w_{ij}(t_{i})*\epsilon(t-t_{i})$ (9)

where $\epsilon(t-t_{i})$ is the unweighted membrane potential induced at time
$t$ by input spike at $t_{i}$. It is modeled using the spike response function
[34], given as

$\displaystyle\epsilon(s)=\frac{s}{\tau}\exp({1-\frac{s}{\tau}})$ (10)

where $\tau$ is the time constant of the neuron.

The actual assignment prediction $\hat{c}_{j}$ is then compared with expert
assignment $c_{j}$. The expert assignments for each zone are evaluated using
an expert solution (see Section V-A). In the next section, an MLC-SEFRON
learning algorithm is described.

###  IV-B MLC-SEFRON learning algorithm

MLC-SEFRON uses three strategies for learning, namely initialization strategy,
escaped intruder strategy and incorrect assignment strategy. In the
initialization strategy the potential thresholds of the neurons in the SEFRON
layer and weights connected to them are initialized as described in SEFRON
[25]. In escaped intruder strategy weights are updated such that the defender
is assigned to the zone. In incorrect assignment strategy weights are updated
such that the defender is not assigned to the zone. Next, these three
different strategies are explained in detail.

####  IV-B1 Initialization strategy

Let us consider the initialization of weights connecting to ${2j}^{th}$ neuron
in the SEFRON layer and its threshold $\theta_{2j}$. The first input sample
from the dataset which has a desired prediction that a defender should not be
assigned to $j^{th}$ zone is used for this purpose. This initialization is
done as shown below

$\displaystyle w_{i(2j)}(t)$
$\displaystyle=u_{i(2j)}(T_{d})\exp(-\frac{(t-t_{i})^{2}}{2\sigma^{2}})$
$\displaystyle\theta_{2j}$
$\displaystyle=\sum_{i=1}^{2m}u_{i(2j)}(T_{d})\epsilon(T_{d}-t_{i})\quad\forall{j\in\\{1,\cdots,m\\}}$
(11)

where $T_{d}$ is defined as the ideal firing time $T_{d}$ is chosen such that
a neuron in the SEFRON layer can utilize the information present in incoming
spike patterns to make a decision. If $T_{d}$ is chosen close to the start of
the simulation, then MLC-SEFRON is not able to utilize the information present
in spike patterns. If $T_{d}$ is chosen close to the end of the simulation
(i.e., $T$), then MLC-SEFRON takes a longer time to make a decision. Which
leads to the inaccurate assignment prediction for zone $z_{j}$ by MLC-SEFRON.
$T_{d}$ is chosen appropriately as mentioned in [25].

In Equation (11), $u_{i(2j)}(T_{d})$ is defined as the fractional contribution
of $i^{th}$ input neuron for ${2j}^{th}$ neuron in the SEFRON layer to spike
at $T_{d}$.

$u_{i(2j)}(T_{d})=\frac{\delta w(T_{d}-t_{i})}{\sum_{i=1}^{2m}\delta
w(T_{d}-t_{i})}$ (12)

where $\delta w(.)$ is defined as the STDP weight update [36]. It is computed
as

$\delta w(s)=\begin{cases}A_{+}\exp(-\frac{s}{\tau_{+}}),\text{if}\hskip
2.84544pts\geq 0\\\ -A_{-}\exp(\frac{s}{\tau_{-}}),\text{if}\hskip
2.84544pts<0\end{cases}$ (13)

where $A_{+},A_{-}$ are the maximum weight changes allowed and
$\tau_{+},\tau_{-}$ are the time constants for STDP. Similarly for
initializing the threshold of ${2j-1}^{th}$ neuron and weights connected to
it, the first input sample which has a desired prediction that a defender
should be assigned to $j^{th}$ zone is used. In this manner, the weights
connected to all neurons in the SEFRON layer and their thresholds are
initialized. The values of constants such as $T,\sigma,\tau_{+}$ and
$\tau_{-}$ are set as in [25].

####  IV-B2 Escaped intruder strategy

This strategy is used when a defender is not assigned to a zone in which an
intruder is present. Given by

If $\hat{c}_{j}\neq c_{j}$ and $\hat{t}_{2j-1}\geq\hat{t}_{2j}$

In this case, the misclassifcation occurs because ${2j}^{th}$ neuron spikes
earlier than ${2j-1}^{th}$ neuron in the SEFRON layer respectively. To fix
this issue the weights connected to these neurons are updated such that they
spike at desired firing times denoted by ${t^{d}_{2j-1}}$ and ${t^{d}_{2j}}$
respectively, which are given by

$\displaystyle t^{d}_{2j-1}$
$\displaystyle=\begin{cases}\text{$\hat{t}_{2j-1}$},&{\text{if}\
\hat{t}_{2j-1}<T_{d}}\\\ \text{$T_{d}$},&{\text{if}\ \hat{t}_{2j-1}\geq
T_{d}}\end{cases}$ $\displaystyle t^{d}_{2j}$
$\displaystyle=\begin{cases}\text{$\hat{t_{2j}}$},&{\text{if}\
\hat{t_{2j}}\geq(t^{d}_{2j-1}+T_{m})}\\\
\text{$t^{d}_{2j-1}+T_{m}$},&{\text{if}\
\hat{t_{2j}}<(t^{d}_{2j-1}+T_{m})}\end{cases}$ (14)

where $T_{m}$ is defined as the margin threshold required between two neurons
in the SEFRON layer for better generalization. If $T_{m}$ is set very small
value (i.e., $0$), then MLC-SEFRON is not able to classify the highly
overlapped samples. Also, if $T_{m}$ is set to a high value then MLC-SEFRON
takes a longer time to converge. $T_{m}$ is chosen appropriately as described
in [25].

Then weight update strategy as described in [25] is used to update the weights
connected to ${2j-1}^{th}$ and ${2j}^{th}$ neurons in the SEFRON layer such
that they spike at ${t^{d}_{2j-1}}$ and ${t^{d}_{2j}}$ respectively.

Without loss of generality the weight update for $k^{th}$ neuron in the SEFRON
layer with a desired firing time $t^{d}_{k}$ is described further. The error
function $e$ to evaluate the update is given as

$\displaystyle
e=\frac{\theta_{k}}{\hat{V_{k}}(t^{d}_{k})}-\frac{\theta_{k}}{\hat{V_{k}}(\hat{t_{k}})}$
(15)

The error function $e$ is designed as above because it can be multiplied
directly with the fractional contribution $u$ to get the required update
value. If $\Delta w_{ik}$ denotes the required weight update then it is given
as

$\displaystyle\Delta w_{ik}=\lambda\hskip 2.84544ptu_{ik}(t^{d}_{k})\hskip
2.84544pte$ (16)

where $\lambda$ is the learning rate. In Equation (15), $\hat{V_{k}}(\hat{t})$
is defined as the potential required for $k^{th}$ neuron in the SEFRON layer
to spike at $\hat{t}$.

$\displaystyle\hat{V_{k}}(\hat{t})=\sum_{i=1}^{2m}u_{ik}(\hat{t})\hskip
2.84544pt\epsilon(\hat{t}-t_{i})$ (17)

The update $\Delta w_{ik}$ is modulated using a gaussian function and then the
existing weight is updated as given below

$\displaystyle w_{ik}(t)\leftarrow w_{ik}(t)+(\Delta
w_{ik}\exp(-\frac{(t-t_{i})^{2}}{2\sigma^{2}}))$ (18)

The weights of ${2j-1}^{th}$ and ${2j}^{th}$ neurons are updated using
Equations 15 to 18 to resolve the misclassification.

####  IV-B3 Incorrect assignment strategy

This strategy is used when a defender is wrongly assigned to a zone. Given by

If $\hat{c}_{j}\neq c_{j}$ and $\hat{t}_{2j-1}<\hat{t}_{2j}$

In this case, the misclassifcation occurs because ${2j-1}^{th}$ neuron spikes
earlier than ${2j}^{th}$ neuron in the SEFRON layer. To fix this issue the
${t^{d}_{2j-1}}$ and ${t^{d}_{2j}}$ are evaluated as

$\displaystyle t^{d}_{2j}$
$\displaystyle=\begin{cases}\text{$\hat{t_{2j}}$},&{\text{if}\
\hat{t_{2j}}<T_{d}}\\\ \text{$T_{d}$},&{\text{if}\ \hat{t_{2j}}\geq
T_{d}}\end{cases}$ $\displaystyle t^{d}_{2j-1}$
$\displaystyle=\begin{cases}\text{$\hat{t}_{2j-1}$},&{\text{if}\
\hat{t}_{2j-1}\geq(t^{d}_{2j}+T_{m})}\\\
\text{$t^{d}_{2j}+T_{m}$},&{\text{if}\
\hat{t}_{2j-1}<(t^{d}_{2j}+T_{m})}\end{cases}$ (19)

To make actual assignment $\hat{c}_{j}$ similar to that of desired assignment
$c_{j}$ the weights connected to ${2j-1}^{th}$ and ${2j}^{th}$ neurons in the
SEFRON layer respectively are updated using Equations 15 to 18.

To summarize the MLC-SEFRON learning algorithm is used to predict the labels
of multiple segments assigned to a defender in a deterministic manner. The
obtained labels of segments that are assigned to a defender are used for its
trajectory generation.

##  V Simulation results

This section presents the results of performance evaluation and comparison of
the proposed DSL framework with the existing STMTA formulation from [18].
Initially in this section, the dataset is generated using the expert (STMTA)
approach for the proposed DSL framework of d-PDP is discussed. Then the
performance of the proposed MLC-SEFRON for training a single defender is
studied using multi-label metrics such as $Precision$, $Recall$, and
$F1-score$ [26]. Finally, the success percentage achieved by the team of
defenders in capturing the intruders is evaluated and compared with the expert
approach. The simulations are carried out using Matlab 2022a, in a Windows 10
system with 16GB memory, 6 cores, and a 3.2GHz machine.

###  V-A Expert policy-based data generation

A circular perimeter is divided into 36 segments. Two different scenarios are
considered for generating a dataset one is a full observation scenario in
which a defender senses all 36 segments. The other is a partial observation
scenario, in which a defender senses 15 segments of the perimeter. The dataset
is generated with a team of 5 defenders, placing them randomly along the
perimeter. The intruders are spawned using Poisson distribution with an
arrival rate of 4 over a period of 8 seconds. The intruders are placed
randomly using a uniform distribution over all segments. The dataset is
generated using random Monte Carlo runs, with at most one intruder spawned in
a segment in each run. The d-PDP is formulated as the spatio-temporal multi-
task assignment (STMTA) problem as described in [17, 18]. This STMTA problem
is solved by an expert policy similar to approaches presented in[18, 43]. The
detailed steps of expert policy are given below. The expert policy computes
the assignments by minimizing the distance traveled by the defenders along the
perimeter while maximizing the number of captures. The cost function and the
optimization problem used by an expert policy are discussed below.

####  V-A1 Cost function

The defender needs to capture the intruder either as a first task from its own
location or as a subsequent task after capturing another intruder. The main
difference is the starting point for the execution of the task. For the first
task, the defender will start from its own location, and for subsequent tasks,
it starts from the previously captured intruder. The cost is defined as the
arc distance that needs to be traveled by the defender along the perimeter to
capture an intruder. If this traveling distance is infeasible due to the
velocity limit, the cost is selected as the large value ( denoted by $\kappa$)
to (avoid or at least) minimize such assignments. If it requires traveling in
negative time, the cost is selected as infinite to avoid assignments. Here,
$d(\mathchoice{\vbox{ \hbox{\leavevmode\resizebox{15.58682pt}{0.0pt}{{\char
62\relax}}} \nointerlineskip\hbox{\set@color$\displaystyle AB$}}}{\vbox{
\hbox{\leavevmode\resizebox{15.58682pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle AB$}}}{\vbox{
\hbox{\leavevmode\resizebox{10.91075pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle AB$}}}{\vbox{
\hbox{\leavevmode\resizebox{7.7934pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle AB$}}})$ denotes arclength
along the arc AB. Mathematically cost function is given as,

$\displaystyle c^{f}_{ij}=\begin{cases}d(\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{22.6914pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{22.6914pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.45297pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{13.21407pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
S_{i}^{D},S^{I}_{j}$}}})&\text{if }\dfrac{d(\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{22.6914pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{22.6914pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.45297pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle S_{i}^{D},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{13.21407pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
S_{i}^{D},S^{I}_{j}$}}})}{V_{max}^{D}}\leq t^{a}_{j}\\\ \kappa&\text{
otherwise}\end{cases}$ (20) $\displaystyle
c^{s}_{kj}=\begin{cases}d(\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{17.28325pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{14.04434pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
S_{k}^{T},S^{I}_{j}$}}})&\text{if }\dfrac{d(\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{17.28325pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{14.04434pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
S_{k}^{T},S^{I}_{j}$}}})}{V_{max}^{D}}\leq t^{a}_{j}-t^{a}_{k}\\\
\kappa&\text{if }\dfrac{d(\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.85379pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{17.28325pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle S_{k}^{T},S^{I}_{j}$}}}{\vbox{
\hbox{\leavevmode\resizebox{14.04434pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
S_{k}^{T},S^{I}_{j}$}}})}{V_{max}^{D}}>t^{a}_{j}-t^{a}_{k}\\\ \infty&\text{if
}t^{a}_{j}<t^{a}_{k}\end{cases}$ (21) $\displaystyle\qquad\text{for
}i\in\mathcal{I}=\\{1,2,\cdots,N\\},\quad j\in\mathcal{J}=\\{1,2,\cdots,M\\},$
$\displaystyle\qquad\qquad k\in\mathcal{K}=\\{1,2,\cdots,M-1\\}$

####  V-A2 Optimization Problem

The formulated optimization problem is a linear sum assignment problem where
decision variable $\delta^{f}_{ij}$ is used to decide whether a defender
$D_{i}$ is assigned to task $T_{j}$ as a first task or not. Another decision
variable $\delta^{s}_{kj}$ is used to decide whether a defender will be
assigned to task $T_{j}$ just after the task $T_{k}$ or not. Mathematically,
the optimization problem is given as

$\displaystyle\min_{\delta^{f}_{ij}\ \delta^{s}_{kj}}$
$\displaystyle\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}c^{f}_{ij}\delta^{f}_{ij}+\sum_{k\in\mathcal{K}}\sum_{j\in\mathcal{J}}c^{s}_{kj}\delta^{s}_{kj}\
\ $ (22) $\displaystyle{\rm s.\ t.}\ $
$\displaystyle\delta^{f}_{ij}\in\\{0,1\\}\qquad\forall(i,j)\in{\mathcal{I}\times{\mathcal{J}}}$
(22a)
$\displaystyle\delta^{s}_{kj}\in\\{0,1\\}\qquad\forall(k,j)\in{\mathcal{K}\times{\mathcal{J}}}$
(22b)
$\displaystyle\sum_{i\in\mathcal{I}}\delta^{f}_{ij}+\sum_{k\in{\mathcal{K}}}\delta^{s}_{kj}=1\quad\forall
j\in{\mathcal{J}}$ (22c)
$\displaystyle\sum_{j\in{\mathcal{J}}}\delta^{f}_{ij}\leq 1\quad\forall
i\in\mathcal{I}$ (22d)
$\displaystyle\sum_{j\in{\mathcal{J}}}\delta^{s}_{kj}\leq 1\quad\forall
k\in{\mathcal{K}}$ (22e)

The solution to the optimization problem gives the assignments of the defender
to intruders. This assignment solution is used as a ground truth for the
supervised learning problem. As the expert solution is sub-optimal and it is
obtained using a fixed team of defenders, its success rate cannot be 100%
i.e., some of the intruders will penetrate the perimeter. Therefore for
generating the dataset, intruders who enter the perimeter are sequentially
removed to obtain feasible assignments of remaining intruders. The sequential
removal of these infeasible intruders is not the optimal method 111One needs
to solve the combinatorial optimization problem (to decide which tasks should
be removed) for computing the optimal solution. Computing the optimal solution
is outside the focus of this paper.. The obtained solution will be sub-
optimal, so we call it an expert solution.

(a)

(b)

Figure 5: Multi-label performance evaluation. Figure 5a shows the results of
performance evaluation of MLC-SEFRON algorithm for a single defender in a
partial observation scenario. Whereas Figure 5b shows the results of
performance evaluation of MLC-SEFRON algorithm for a single defender in a full
observation scenario. Purple line shows the $Recall$, blue line shows the $Fl-
score$ and red line shows the $Precision$ along various zones respectively

The dataset is generated for each defender in a decentralized fashion. While
generating the dataset for a particular defender, the locations of the
intruder and the defender are considered from the perspective of that
particular defender. These spatio-temporal representations are carried out as
described in section III-B. The desired output for a zone is labeled as
assigned if the expert solution assigns a defender to that zone; otherwise, it
is labeled as unassigned. If there are more intruders than defenders, the
expert (STMTA) solution assigns some defenders to multiple zones.

The optimization problem in STMTA formulation of d-PDP minimizes the total
distance traveled by a team of defenders while capturing the maximum number of
intruders. Therefore in this solution, defenders are more likely assigned to
the neighboring zones than the further zones to minimize the distance
traveled. As the trainee defender is always located in the central zone, the
intruders closer to these zones are more likely to be assigned when compared
with intruders in the zones farther away from the central zone. Technically,
the dataset has a long-tail problem, where the number of assigned labels is
lower for the further away zones than that of the central zones.

A dataset is generated for a team of five defenders by solving 10000 scenarios
of d-PDP using the expert solution. A total of 50000 samples are generated in
this dataset, out of which only 10000 are used for training the MLC-SEFRON.
The dataset has higher number samples in which trainee defender is assigned to
the central zone when compared with the samples assigned to other zones. This
is due to the presence of trainee defender in the central zone. Thus the
generated dataset using expert policy is unbalanced due to the long-tail
problem. To balance this dataset the minority class samples of the dataset are
over-sampled using the synthetic minority class over-sampling technique [44].
This oversampled dataset is used for training. The remaining 40000 samples of
the dataset are used for the testing. The results are presented for the
testing dataset.

###  V-B Performance evaluation of MLC-SEFRON

In this section, the results of the performance evaluation of MLC-SEFRON in
predicting assigned zones for a single defender are presented. The weights
obtained after training a single defender are directly used for evaluating the
performance of other defenders. Thus the training in the DSL framework is
decentralized. Multi-label performance metrics such as $Precision$, $Recall$,
and $F1-Score$ are used for this performance evaluation. Figure 5 shows the
zone-wise variation of these multi-label metrics for full and partial
observational scenarios. Figure 5a displays the $Precision$, $Recall$, and
$F1-Score$ plots evaluated using the MLC-SEFRON for all 15 zones in partial
observation scenario. It can be illustrated that performance measures are
greater for central zones and subsequently decline for zones farther from
central zones are considered. Figure 5b displays the multi-label performance
metrics of the MLC-SEFRON for full observation scenario. The high $Recall$
performance in the central zone is due to more assigned samples in the dataset
for these zones. Whereas the high $Recall$ performance in the farther zones is
due to a higher number of un-assigned samples in the dataset for these zones.

###  V-C Performance evaluation of DSL framework for d-PDP

The performance of the team of defenders is evaluated using the success
percentage. The success percentage ($S_{p}$) is defined as

$\displaystyle S_{p}=\frac{\text{number of intruders captured}}{\text{total
number of intruders}}\times 100$ (23)

Table I: Success percentage comparisons for STMTA and DSL  |  | Expert policy
---
(centralized)
DSL |  | DSL with
---
neighboring
effect
| Learning
---
efficiency
of DSL
| Full
---
observation
85.0425 | 74.5195 | 82.8825 | 97.50
| Partial
---
observation
85.3439 | 72.5642 | 79.4924 | 93.14

The table I shows the success comparison of DSL framework and expert (STMTA)
policy. The STMTA solution is used as a training dataset and hence STMTA is
expected to perform better than the DSL framework The success of the STMTA is
$84.95\%$, where as SNN gives success of $74.51\%$ for the full observation
scenario i.e. each defender observes all 36 segments and acts accordingly. The
performance of the DSL frameworkimproves up to $82.88\%$ by considering the
effect of neighbors in post-processing. Similar results are obtained for the
case of partial observations. Here, defenders can sense only 15 segments but
the auction of tasks is done in a centralized way. The STMTA (expert) policy
has a success of $85.34\%$ and the DSL framework provides a success of
$72.56\%$, and this can be improved to $79.49\%$ by adding the effect of
neighbors in post-processing.

The proposed DSL framework with post-processing obtains $97.5\%$ and $93.14\%$
success performance compared to the STMTA (expert) solution for the full and
partial observation scenarios, respectively.

(a)

(b)

Figure 6: Scalability study of proposed DSL framework with different-sized
team of defenders (a)Partial observation scenario (b)Full observation scenario

###  V-D Comparison study

An adaptive partitioning (AP) [16] approach has been proposed for PDP for
sequential capture of radially incoming intruders. The arrival distribution of
the intruders is estimated to partition the perimeter into sectors; each
defender assigned to these partitions independently operates to capture
intruders sequentially. Although both AP and DSL are designed for different
settings, we are comparing them to show their efficacy. The naive and adaptive
partitioning is used for the PDP, and the remaining approaches are used for
discrete PDP. The dataset is generated with Poisson distribution with
$\lambda=4$ and at most one intruder in each segment.

Table II shows the performance comparison for success percentage for adaptive
partitioning, expert policy, and DSL framework. Note that, in [17], a
comparison study showed that STMTA performs better than adaptive partitioning
in the same setting.

Table II:  Average percentage and standard deviation of captured intruders for
uniform arrival distribution of intruders, over 1000 runs for Naive, adaptive
partitioning (AP), expert policy, and DSL framework.

Success percentage ($\mu\pm 3\sigma$)) Naive (full observation) $64.9\pm 3.2$
AP (full observation) $64.9\pm 3.2$ Expert (full observation) $85.0425\pm
4.8555$ DSL (full observation) $82.8825\pm 2.5617$ Expert (partial
observation) $85.3439\pm 4.8887$ DSL (partial observation) $79.4924\pm 2.3139$

###  V-E Scalability of proposed DSL framework for d-PDP

In the proposed DSL framework once a defender is trained using MLC-SEFRON, the
same network can be used for the different-sized team of defenders. Re-
training is not required for changes in team size. Also the training of the
defender is independent of the team size of defenders. Hence, the proposed DSL
framework is said to be distributed and scalable.

In this paper, the MLC-SEFRON is trained with a dataset generated using STMTA
formulation proposed in [18], considering a team of five defenders. All
different sized-team of defenders in the DSL framework will use the same
network (which is trained with a team size of five). The expert STMTA solution
is computed (based on the actual number of defenders) using centralized
computation. Figure 6 shows the performance of the proposed DSL framework with
different team sizes. One can observe that, the solution obtained from the
STMTA approach and the DSL framework is comparable. Both the solutions show
similar characteristics; as the defenders’ team size increases the success
rate increases. One should note that the expert used for training (i.e. the
solution of STMTA) is sub-optimal. The performance of DSL framework
asymptotically matches the success rate of the expert. With the increases in
the number of defenders, the difference between the success rates reduces.
Hence, the proposed DSL framework is scalable.

One can observe that, the DSL framework performs better compared to the expert
approach for team sizes of 3 and 4 in a partial observation scenario. It is an
interesting result where the DSL solution outperforms the expert. The DSL has
learned to solve the assignment problem and not just the given expert solution
(which is sub-optimal). Investigating the reasons for this observation is out
of the scope of this paper. If the sensing range of the defender changes then
the number of neurons in the MLC-SEFRON network of the DSL framework also
changes. The proposed DSL framework for d-PDP is not scalable in terms of the
sensing range of the defender.

##  VI Conclusion

In this paper, a novel Decentralized Spike-based Learning (DSL) framework for
the discrete Perimeter Defense Problem (d-PDP) is proposed. The circular
territory is divided into multiple segments and hence the PDP is termed as
d-PDP. Initially, the d-PDP is formulated as a Spatio-Temporal Multi-Task
Assignment problem (STMTA). Then this STMTA problem is converted into
deterministic multi-label learning using Multi-Label Classifier using Synaptic
Efficacy Function spiking neuRON (MLC-SEFRON) network. For decentralized
training, each defender is trained with a separate MLC-SEFRON network. The
spatiotemporal information of the defenders and intruders in the territory is
converted as spikes without any extra pre-processing step. These spikes are
given as input to the MLC-SEFRON network. The labels of segments assigned to
the defender are obtained from the output of the MLC-SEFRON network The
trained weights of one MLC-SEFRON network are directly used for obtaining
labels of assigned segments from other MLC-SEFRON networks without any
retraining. Using these labels the trajectories for defenders are then
obtained with the help of the Consensus-Based Bundle Algorithm (CBBA). The
performance of MLC-SEFRON is evaluated for full and partial observation
scenarios of the defenders. The multi-label performance metrics obtained for
MLC-SEFRON are observed to follow the input data which is obtained from the
expert policy. The performance results show that the proposed DSL framework
performs $93.14\%$ and $97.5\%$ compared to the expert policy in partial and
full observation scenarios, respectively. The DSL performance evaluated for
different numbers of defenders indicates that it is efficiently scalable.
Future work will explore improving the performance of the proposed MLC-SEFRON
and its application for various multi-label learning as well as STMTA
problems. The presented d-PDP dataset will be a good benchmark for testing the
SNN learning algorithms due to its inherent spatio-temporal nature.

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