# Group testing via residuation and partial geometries

The motivation for this paper comes from the ongoing SARS-CoV-2 Pandemic. Its goal is to present a previously neglected approach to non-adaptive group testing and describes it in terms of residuated pairs on partially ordered sets. Our investigation has the advantage, as it naturally yields an efficient decision scheme (decoder) for any given testing scheme. This decoder allows to detect a large amount of infection patterns. Apart from this, we devise a construction of good group testing schemes that are based on incidence matrices of finite partial linear spaces. The key idea is to exploit the structure of these matrices and make them available as test matrices for group testing. These matrices may generally be tailored for different estimated disease prevalence levels. As an example, we discuss the group testing schemes based on generalized quadrangles. In the context at hand, we state our results only for the error-free case so far. An extension to a noisy scenario is desirable and will be treated in a subsequent account on the topic.

## Authors

• 5 publications
• 1 publication
10/05/2021

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## I Introduction

During the initial low-prevalence phase of a pandemic like the current SARS-CoV-2 pandemic, where a new pathogen has started spreading, it is helpful to use a technique called group testing in order to identify infected individuals, particularly when testing is complicated or otherwise expensive.

The technique goes back to initial ideas by Dorfman [2] in 1943, whose challenge was to test WWII troops for syphilis based on an insufficient number of tests available. As the rate of infected soldiers was rather small (far below ), it turned out to be possible to significantly reduce the number of tests exploiting the following idea: Test a group of samples in a way that each test is used for a pool of specimen from a variety of samples. Dorfman correctly observed, that it was possible to identify a small number of infected participants out of a larger batch without having to medically check each individual for the infection in question. His method was further refined by Katona [3] who spread the specimen over several test. This method may be considered as a first approach to non-adaptive group testing.s

To organize group tests we follow Katona [3]

and divise a table (a binary matrix) for tests and participants showing which participant’s specimen is contained in which test. For the test, an (unknown) vector of samples is multiplied by the test matrix using the arithmetic of the Boolean semi-field

. The resuling test vector (syndrome) contains information about the infection pattern of the initial vector, and the goal is to reconstruct this pattern from the syndrome.

In this paper, we show how this approach to group testing is closely related to the mathematical discipline of Residuation Theory. Its advantage is, that it comes with a natural choice of decision scheme (syndrome decoder) that generically works for a large class of infection patterns.

Adopting and refining conditions and techniques discussed in [5], we suggest to optimize the quality of the testing task by using incidence matrices of a class of finite geometries. These bear advantages due to certain intrinsic regularity, sparsity, and symmetry properties.

We will state our results merely for the case where the tests are error-free, where an adjustment to noisy scenarios is devoted to a subsequent article. Our approach is clearly not tied to a particular pathogen, nor does the pathogen itself have to be viral. Its novelty lies in the (re-)establishment of an order-theoretic approach in conjunction with the application of finite geometries to design non-adaptive pool testing strategies.

## Ii Preliminaries

### Ii-a Linear Algebra on the Boolean Semifield

In contrast with a set-based combinatorial approach to be found in [5], the Boolean semifield will play a main role in this paper. On the -element set , it is the only non-trivial (yet meaningful) alternative to the binary field , its operations are described in Figure 1.

An interesting aspect of , that appears trivial on a first glance: it comes with an order relation , naturally defined by , and with a negation mapping where and . This involution has the properties and which are well-known as de Morgan’s laws.

The commutative idempotent monoid (equipped with componentwise addition and identity ) may be considered as a -vectorspace with and for all . We have the distributive laws

 λ(x+y) = λx+λy (λ+μ)x = λx+μx

satisfied for all and .

We will naturally extend order and negation to and omit references to where-ever confusion is impossible.

The reader may readily recognize that what we are talking about is known from Boolean Algebra. The partial order on is given via if and only if .

In fact, defining multiplication componentwise with identity on , we obtain the mentioned Boolean Algebra as the -tuple .

We say, the elements be linearly independent, if

 bi≰∑j≠ibj,for all 1≤i≤ℓ.

The Hamming distance is defined as the function

 δ:B2×B2⟶N,(x,y)↦{1:x≠y,0:otherwise.

Following general conventions, we extend this function additively to and obtain that

 δ(x,y)=|{i∈{1,…,n}∣xi≠yi}|.

In a similar fashion we define the Hamming weight such that for all . As common in coding theory, we observe , and, at least as important as the previous,

 δ(x,y)=w(x+y)−w(x⋅y).

The Hamming disk of radius centered in will be denoted by

 Bδ(x,d):={z∈Bn2∣δ(x,z)≤d}.

Particularly the disks and will play a prominent role in our discussion.

### Ii-B Two conditions

The following concepts are slightly deviating from what is standard in the literature. For more information see [5] and references given there.

###### Definition II.1

Let be an -matrix over the semifield . For a natural number consider the following two properties:

d–Rev:

If and , then

 xH=yHwill implyx=y.
d–Dis:

For any and a set of rows of every row of not contained in satisfies .

Property d–Rev is a logically stronger version of -separability introduced in [5], which says, that the mapping is injective on . Property d–Dis is called -disjunctness in [5], and one of its immediate consequences is that any set of rows of is linearly independent in the sense defined above. We will show that conditions d–Rev and d–Dis are equivalent.

###### Theorem II.2

Property d–Dis implies property d–Rev.

###### Proof:

Assume, the -matrix satisfies d–Dis, and let and be given, such that . If , then w.l.o.g. we may assume that there exists such that . If not, then as well, and we only need to interchange the roles of and . We conclude that row of satisfies which comes from assumption d–Dis. In contrast, , a contradiction. This proves d–Rev for .

###### Theorem II.3

Property d–Rev implies property d–Dis.

###### Proof:

Assume, satisfies d–Rev, and let be a sequence of rows of matrix , where . Then there is such that . For we set , where is the vector with only non-zero entry exactly in the th position. If , then which forces by d–Rev, a contradiction, because . Hence , which is what condition d–Dis asserts.

### Ii-C Residuated Mappings

In this section we present the main conceptual novelty of this paper. This will ease the discussion of group testing in that an efficient decision scheme is presented under quite general assumptions.

###### Definition II.4

Let and be two partially ordered sets. For mappings and , the pair is called a residuated pair, if there holds

 f(x)≤y⟺x≤g(y),for all x∈A and y∈B.

We will briefly collect basic facts about residuated pairs. For a source, the reader is referred to [1].

Fact 1:

may be called a residuated mapping, if there is , such that is a residuated pair. The mapping is then uniquely determined by . Dually, is uniquely determined by which is called the residual of .

Fact 2:

and are monotone mappings, and there holds and . Conversely, if two monotone mappings and satisfy and , then they will form a residuated pair.

Fact 3:

and . For this reason, the mappings and form closure and kernel operators, respectively. We observe that for all and for all .

Fact 4:

The mapping restricts to a bijection between the sets of closed elements in and kernel elements in ; this restriction is inverted by the restriction of to the set of kernel elements in .

Fact 5:

If and are complete lattices, then is residuated if and only if is preserving suprema, i.e.  for all . Accordingly is a residual mapping if and only if it preserves infima, meaning for all .

Fact 6:

Any residuated mapping can be canonically represented by an matrix with entries in , where for all . The representation of the residual mapping is the subject of the following theorem.

###### Theorem II.5

Let be a residuated mapping represented by the -matrix . Then the residual mapping is given by the assignment .

###### Proof:

In the following we will need to carefully observe the effects of the negation-operator: they will turn sums into products and products into sums. According to our list of facts, we only have to check if is monotone, and if and .

To begin, we compute

 [g(y)]i = ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∑j≤k¯¯¯¯¯yjHij=∏j