Persistence module and barcode

The barcode of this persistence module, illustrated in Figure3.7b, lacks a full length interval, implying that there is no escape path in Figure3.3a.

On the other hand, consider the escape region from Figure 3.5a. The associated sheaf, illustrated in Figure 3.5b, results in a persistence module illustrated in Figure

3.8a, and its barcode illustrated in Figure3.8bcontains a full length interval, implying the fact that there exists an escape path.

(A) Persistence module from sheafF (B) Barcode FIGURE3.8: Persistence module and barcode

3.2

Boolean Pursuit and Evasion

Consider a variation of a pursuit and evasion problem, where the domainDis now a graph or a grid. Assume that the pursuers know their exact coordinates on D. More- over, assume that the pursuers are divided into teams of different colors, say red, blue, and yellow. Specific teamwork is required for a pursuer to be captured. For example, an evader can be captured if both the red and blue pursuers are at the same location as the evader, or if both the blue and yellow pursuers are at the same location.

Such rule for combination of sensors required for capture will be called a capture criterion. Letp = {p1, . . . ,pM}be the collection of different colored team of pursuers.

For the remainder of this section, we will refer to eachp_i, which is a collection of pur- suers belonging to a same colored team, as one pursuer. For example, ifpiis a collection

of red pursuers, we will refer topi as the red pursuer.

Let Pi be a binary variable. One can think of Pi = 1 as indicating that an evader is

sensed by pursuerpi andPi =0 as indicating that the evader is not sensed bypi.

A capture criterion can be written as

C= T1∨T2∨ · · · ∨TK,

where eachT_i = P_i1∧ · · · ∧P_iM denotes the teams of pursuers required for capture. For example, the following capture criterion

C= (PR∧PB)∨(PR∧PY) (3.3)

indicates that capture occurs if both red and blue pursuers are present or if both red and yellow pursuers are present.

For each nodex ∈D, the variablesPiare assigned a value depending on whetherx

is sensed by pursuer pi. LetCxdenote the value of expressionCfor the nodex. Then,

Cx =1 if an evader at nodexis captured by pursuers, andCx =0 if the evader at node

xcan escape.

Given a coverage criterionC, our goal is to determine the nodesx∈ Dthat do not satisfyC, i.e., we want to find the nodes x∈ Dsuch thatCx = 0. Note that¬Cx =1 if

Cx =0 and¬Cx =0 ifCx=1. So let

E=¬C

be the escape criterion. IfEx = 1, then an evader at nodexcan escape, and ifEx = 0,

For the capture criterion C = (PR∧PB)∨(PR∧PY)from Equation 3.3, the corre-

sponding escape criterion is

E= (¬PR∨ ¬PB)∧(¬PR∨ ¬PY).

Since every propositional formula can be written in a conjunctive normal form, we can equivalently write the escape criterion as

E=¬PR∨(¬PB∧ ¬PY).

Since each pursuerpiknows the exact nodes ofDthat are covered bypi, the pursuer

also knows the nodes ofDthat remain undetected by pursuerpi. From each pursuer’s

knowledge of the uncovered region, we can use sheaves and cosheaves to determine if it’s possible for an evader to hide.

3.2.1 Boolean capture via sheaves and cosheaves of sets

In this section, we construct sheaves and cosheaves that allow us to determine if it’s possible for an evader to hide given an escape criterion. We assume that the escape criterion is given in either conjunctive normal form or disjunctive normal form.

Escape criterion in disjunctive normal form

Assume that we are given an escape criterion of the form

E=E1∨E2∨ · · · ∨EK,

where eachEi has the formEi =¬Pi0 ∧ · · · ∧ ¬Pin.

We first define the base space. Given M number of pursuers, letXbe a (M−1)- simplex. Each vertex ofXcorresponds to a pursuerpi, so label such vertex byvpi. For

each n-simplex σ of X, label σ by its vertices. For example, if vpi₀, . . . ,vp_in+1 are the

We will now construct a cosheafSof sets onXthat encodes the region remaining undetected by the pursuers. For each vertex vpi of X, let S(vpi) be the set of nodes

of D that are not detected by the pursuer pi. For each σpi₀,...,p_in+1, letS(σpi0,...,pin+1) =

n+1

T

j=0

S(vp_ij), the set of nodes ofDthat are not detected by any of the pursuerspi0, . . . ,pin+1.

Let the extension maps be the inclusion of sets.

For example, let D be a graph that is covered by three pursuers as illustrated in Figure 3.9. The base space X and the labels of its simplices are illustrated in Figure

3.10a. The cosheafS on X is illustrated in Figure 3.10b. Note that even though Fig- ure3.10bshows the graph D as local sections, the local sections ofSare just the sets corresponding to the colored nodes of the graphD.