1 Networks
In the context of this paper, a network – a.k.a. an interaction system – is any set of entities/parameters that we consider as a whole for the following reason: we presume that the changes underwent by the entities in the set are causally related to one another and account for the global system changes that we are interested in.
Remark: An interaction system can be comprised of just one entity (the human body for instance, as it might been seen in the light of a different culture). In this case, to explain the system’s global state changes, the only thing we have is, precisely, the system’s global state changes – and also the system’s environment, i.e. everything that is not the system. So to explain the system’s global state changes by something else than themselves, we must turn to the environment and find an external cause. In other terms, we must add to the system a second entity (or more) that interacts with the original. Under different circumstances, the same object might also be regarded as an interaction system comprised of several interacting entities (e.g. the human body as seen by modern medicine). In this case, the environment is not the only place where explanations can be sought. Finer explanations might also come from considering alternative interactions between the different entities of the system. Gladly, since our present representations of objects can be questioned, in neither case are we condemned to uncovering the same sorts of explanations in the same sorts of ways.
In this paper, we will illustrate the five following observations about interaction systems.
Observation 1:
Some properties that we regard as properties of systems – properties that the systems may have or not have – are actually properties of the way we look at the systems.
Observation 2:
A statement like “There exists no interaction between entity X and entity Y.” has no essential meaning in itself. Its truth value is dependent on the specific level of abstraction from which the system – its entities, the interactions between them – is being looked at and defined.
Observation 3:
Incomplete data is not the only reason for the model of a real interaction system to fail to account for existing interactions between entities of the system. Despite the modeller’s flawless observation and formalisation of the system, some causal relationships between entities might still be intrinsically imperceptible to him/her under his/her current perspective.
Observation 4:
What information a formal object can provide about the real system it is meant to model, and what uninterpretable, nonmodelling information it provides on top of that, does not just depend on the semantics associated to the formalism describing the formal object, it also essentially depends on the relative consistency of those semantics.
Observation 5:
The semantics associated to a formal object can be decisively affected by the history of the object and how our scientific community came to inherit it. And thus, so can the precise definition of the object that we choose among different customary variations of the definition, and the properties it has that we take interest in, and those we don’t.
2 Prototypes of networks
To illustrate those five observations, we are going to use a minimalist mathematical prototype of interaction systems named Boolean Automata Networks (BANs).
I will introduce the formalism of BANs using the BAN represented below in Fig.1.
is comprised of entities, namely , , , , , and , a.k.a. automata and . Each automaton has a variable state in . If denotes the current state of , then denotes the current state of . On the right of Fig.1 are listed six functions . One for each automaton . Function defines the possible variations of ’s state depending on ’s. In state of , ’s state can change if and only if belongs to the set of automata that are unstable in . The state of actually does change if: (i) it can and (ii) is updated. For instance, in state of , if , , and are updated, and , , and aren’t, then transitions to state .
Formally, a BAN is defined as a set of Boolean functions with no mention to automata updates. A BAN is therefore not a dynamical system. This choice of definition is deliberate although nontraditional. In the literature, Automata Networks (ANs) are usually taken to represent dynamical systems ^{[}[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]^{,} [21, 22, 23, 24, 25, 26, 27, 28]^{]}. Their definitions are made to imply specific updating conditions. The notions of causality and time are thereby fused together and a distinction is implied between (1) process of change and (2) (result of) change – the latter being either an approximation of the former or an observed consequence of it. The object of this paper requires to take a less abstract view on the formalism, and not make this semantical distinction, so:
Change is anything that has observable consequence, if only the observable consequence of us noticing it. Process of change is change if it has observable consequence in itself; if it hasn’t, then it’s not something there is anything to be said about.
In agreement with the formalism of BANs, we will assume that in a BAN, all considered changes are represented with the same status. The focus is therefore not so much on automata states, as it is on their changes.
3 Illustrations
Observation 1:
Some properties that we regard as properties of interaction systems – properties that they can have or not have – are actually properties of the way we look at them.
Observation 1 can be illustrated with the BAN properties of monotony and nonmonotony. A BAN is said to be monotone when the following holds for any two of its automata and . If influences , then it always does so in the same way: either always influences positively, or always influences negatively. In mathematical terms this translates to the following where is the state defined by and . By definition, in a monotone BAN, for any such that , i.e. such that the Conjunctive Normal Form of depends on :either , or .
Traditionally, when BANs model biological (genetic) regulation networks ^{[}[15, 29, 30, 12, 31]^{]}, they are supposed to be monotone, like BAN of Fig.1 and unlike BAN of Fig.2.
Despite BAN being nonmonotone, it is the exact representation of what we see of the monotone BAN in some circumstances.
Imagine that there exists a real system in nature that works exactly like the monotone BAN of Fig.1 does. Call the real system too. Imagine that we human observers of reality are observing in action, and at the time we are doing that, for some reason, parts of are behaving rhythmically: state changes of and are happening at the same frequency with a slight phase offset; everything is exactly as if was systematically updated immediately before is. If we were witnessing each event occurring in and knew that we were, then we would have enough information to build the representation of given in Fig.1. But assume that deliberately, we are considering from a specific level of abstraction (as opposed to considering from the godlike perspective of the Laplacian demon that sees Everything because it is interested in Everything). In other terms, assume we have specific interests, and because of that we are focusing on specific attributes of . Imagine that in the present case, this results in us being unaware of ’s existence.
nb: This does not imply a default in our observation. What it is we are looking at in the entities of system might simply not exist, not make sense, or not be measurable in entity . For instance, ’s state changes might represent rapid decoding of mRNA sequences, while the state changes of the other might represent slower processes such the increase of protein concentrations in the cell during the protein’s synthesis.
Every time we witness change states, just has. While is taking trajectory:
we observers are just seeing:
The BAN description of a system that behaves like this is the BAN description given in Fig.2. Under such circumstances – circumstances that constrain the temporality of events in together with the way we observe those events, the level of abstraction from which we do that, and the temporality of our observations of ’s changes with respect to the temporality of ’s changes – what is given of for us to understand is .
In agreement with Observation 1, this shows that monotony and nonmonotony are not so much properties qualifying the interactions of a system as they are of how we look at it.
In the literature, wherever BANs are considered as standalone mathematical objects, it is customary to restrict the local update functions to a certain class of functions for convenience. A typical example is the restriction to functions that are expressible in terms of a limited number of logical connectors ^{[}[32, 2, 9, 33, 34, 35, 7, 36]^{,} [21, 37, 22, 38]^{]}. And as mentioned above, the ’s are also often restricted to functions that are expressible, on the contrary, without certain connectors such as the (XOR) connector which makes the BAN severely nonmonotone ^{[}[39, 40]^{,} [41, 37]^{]}.
If we want the mathematical understanding we develop about mathematical representations of ’real’ interaction systems to apply and to apply rightly, then we need to understand the meaning of the restrictions we make when we derive this mathematical understanding. Observation 1 shows how important it is to consider thoroughly the way our perspective on a system and our interpretation of its representation are involved in the properties that we build our understanding on.
Observation 2:
A statement like “There exists no interaction between entity X and entity Y.” has no essential meaning in itself. Its truth value is dependent on the specific level of abstraction from which the system – its entities, the interactions between them – is being looked at.
Observation 3:
Incomplete data is not the only reason for the model of a real interaction system to fail to account for existing interactions between entities of the system. Despite the modeller’s flawless observation and formalisation of the system, some causal relationships between entities might still be intrinsically imperceptible to him/her under his/her current perspective.
To illustrate observations 2 and 3, consider the system represented by BAN of Fig.3.
This system is actually three independent systems that we have no reasons to consider as a whole. Automaton in particular, is stuck in state . There is no reason for us to consider as an interacting entity interacting with other entities. Yet in some circumstances, this BAN too is the perfect representation of what is given of system (of Fig.1) for us to understand.
Imagine that entities , , , of happen to be caught in the same rhythm, and for the same sort of reasons as before, we are unaware of entity (e.g. we are looking at interactions between varying concentrations in the cell of different proteins, and are thereby unable to perceive changes affecting genes like ).
In , everything is as if updates were being made in the following periodic order:
Imagine also, that we are observing the system with regularity. But still, we are not there absolutely each and every time something in changes. More precisely, imagine that in the changes of states of entities happen very fast in comparison to the whole duration of a period of updates . Because of the that and because of the regularity of our observations with respect to that, while takes trajectory:
we observers of reality just see:
where :
So under these circumstances, accurately represents all the information we get out of our absolutely flawless observation of under those circumstances. And according to this accurate representation of , in particular there exists no interaction between entities and : .
Traditionally in the Bioinformatics literature ^{[}[42, 43, 44, 45]^{]}, at best only three cases are considered for any two entities and of a real system :

Entity really impacts on entity , possibly indirectly, and the model of formalises this through the arc .

Entity really has no influence on entity and the model accounts for this through the absence of arc .

Entity really impacts on entity but the experimental data collected upstream by biologists has failed to evidence this fact about reality. As a consequence, the theory is failing to represent it: arc is accidentally missing from ’s interaction graph .
In agreement with Observation 2, the example of BAN of Fig.3 modelling system of Fig.1 shows that Case 2 doesn’t make any sense at all beyond the ‘reality’ of a specific level of abstraction (e.g. the one at which concentrations of proteins in the cell are meaningful and visible, and a gene isn’t).
In agreement with Observation 3, the example also evidences there can be other reasons – different from the “unfortunate data deficit” underpinning Case 3 – for the representation of a system to fail to account for interactions between entities.
Now, consider again – the BAN of Fig.3 representing precisely what we see of the system of Fig.1 under the conditions described above. Under the conditions described above, if one entity of were to change pace, slightly slow down for instance, even if only momentarily, then might at some point take state . The oscillations of might even spread to . Or, if and had been locked in state until then, might unlock which in turn might generate the irrevocable effect of allowing to take state . With our perspective on , none of this would fit with what we know. Worse, we would be essentially unable to foresee and even understand any of it if it happened.
Observation 4:
What information a formal object can provide about the real system it is meant to model, and what uninterpretable, nonmodelling information it provides on top of that, does not just depend on the semantics associated to the formalism describing the formal object, it also essentially depends on the relative consistency of the semantics.
Observation 5:
The semantics associated to a formal object can be decisively affected by the history of the object and how our scientific community came to inherit it. And thus, so can the precise definition of the object that we choose among different customary variations of the definition, and the properties it has that we take interest in, and those we don’t.
The last two observations can be illustrated with the notion of “synchronism”.
A surprisingly great many occidental modellers of biological regulation networks confuse synchronism in BANs with the parallel update schedule (PUS) of BANs ^{[}[46, 47, 48, 49, 50, 51, 52, 8, 53, 54, 55, 56, 57, 58, 59]^{]}. The PUS is the update schedule originally used and made sense of by McCulloch and Pitts in their seminal BANs ^{[}[60]^{]}. The PUS forces a BAN to systematically update all its automata so that , the BAN transitions from to . When this makes some automata react more quickly than we would like them to, intermediary automata can simply be added as it was originally done in the McCulloch and Pitts BANs. Asynchronism, to which the PUS is wrongly opposed, is the update constraint that rules out the possibility of updating more than one automaton in . Nonasynchronism, a.k.a synchronism is the possibility of updating more than one in . Formally, it is expressed by: (more than automata can change states in ).
Such great tenacity for such a coarse confusion can only be explained by the fact that wherever it is made, it does not matter, or at least, it is not made to matter.A widespread confusion is nonetheless still is a confusion. Since it is widespread, in agreement with Observation 5, it is much more likely to be the legacy of a community blind spot induced by inherited semantics, than to be the responsibility of individual err.And indeed, the blind spot around synchronism seems to be a natural effect of the constant diverse historical reprocessing of BANs and of the way sense is made out of them ^{[}[60, 5, 61, 62, 63, 64]^{]}. Extensive interdisciplinary interest in BANs has been reassigning modelling responsibilities to BANs faster then it has been answering questions about what BANs can, and actually do formalise.
The confusion between the parallel update schedule and the notion of synchronism results in:

The neglect of all intermediary updating possibilities that neither rule out synchronism altogether, nor rule out asynchronism altogether (around in the case of BAN of Fig.4), and

Research being confined to frameworks in which synchronism is never considered independently of the other very strong characteristic features of the PUS (e.g. determinism, periodicity) for the reason that in those frameworks, it cannot be.
The disregard synchronism owes in particular to its misguided association with determinism, is aggravated by two assumptions commonly used to motivate an asynchronous updating in some communities that are interested in modelling genetic regulation with (B)ANs ^{[}[65]^{]}:

Simultaneity in nature is highly fortuitous, and

Simultaneity in nature maps bijectively onto synchronism in (B)ANs.
The notion of simultaneity implied in these assumptions requires a notion of “objective time” to make sense. A priori, in BANs, it doesn’t. At least not spontaneously. What synchronism in BANs conveys is the absence of a causal relation:
Synchronously possible events are events that don’t need one another to occur.
So the relation of synchronism relates possible events without specifying anything about how these events are otherwise related – that is, how they are related otherwise than by the relation created by their synchronous possibility. In particular, knowing only that events A and B can occur synchronously, means: not knowing what effect the occurrence of A might have on B and vice versa^{1}^{1}1This absence of information conveyed by the relation of synchronism is a typical example of absence of information traditionally getting outshined by specialised knowledge and the assumptions inspired by specialised knowledge. If anything, what this absence of information represents is ”wriggle room” (see [66] Section 9). And making sense out of it calls for the careful attention of Computer Science with its fundamental ability to soundly manipulate representations of information and systems, much more than it is calls for the other natural sciences’ specialised detailed knowledge about the complexity and diversity of real life systems in need of modelling.. Forcing asynchronism in BANs – i.e. forcing asynchronism in , despite possibly being true in some – contradicts the fact mentioned above (“In BANs synchronously possible events are events that don’t need one another to occur”). To rule out the possibility of the synchronous occurrences of possible events by imposing asynchronous updatings, is to assume that the occurrence of any event prevents the occurrence of all other that are also possible – or else that the model conveys the causality that we expect very poorly. In an asynchronous BAN, automata that find themselves synchronously unstable are therefore necessarily automata that have the ability of influencing each other. For the BAN of Fig.4, this means that 23256 arcs are missing from . This inconsistency between what we want the BAN to mean and what we want its constituent updating schedule to mean squarely dismantles the intrinsically discrete modelling capacity of the mathematical formalism of BANs and severely stakes the relevance of any information we might subsequently draw out of it about a real system. In agreement with Observation 4, this shows that overloading formalism with incompatible semantics is generally not innocuous.
4 Conclusion
The observations and examples given above call for a shift of attention from specifics and realism to definition and consistency. They show the need for us to systematically endeavour to refine and update our scientific views so that instead of speaking of theory and formalism as opposed to reality and nature, we rather speak of objects that are abstractions of one another in a sense of the term “abstraction” that we can actually formally explicit.
A characteristic strength (and beauty) of science is its ability and tradition of tackling problems and questions through many different angles. Science does not especially aspire at a onedimensional history of sciencemaking. Different perspectives currently upheld by different contemporary scientific communities may coexist. They don’t need to mutually invalidate one another since science doesn’t especially need a single consistent “survivor scientific perspective” to be selected in the end. So if sciencemaking presently makes sense in itself, then the same way, having different scientific communities upholding different scientific perspectives on the same objects must presently make sense in itself too. We have yet to study and explicit the general coherence there must thereby be in the present coexistence of those perspectives^{2}^{2}2Incidentally, this differs significantly from what the academic trend of interdisciplinarity is actually proposing. As far as Computer Science/Biology collaborations around BANs go, interdisciplinary operates as disciplineconcatenation, prejudicially opposing a conglomerate notion of “fundamental/theoreticalness” to a notion of “appliedness”. From it ensues an effective separation between specialists’ contributions: the work of one starts where the work of the other ends. The fatalistic view on information lack captured by Case 3 on Page 3 – information lack assumed to be necessarily unfortunate and accidental – is bound to consolidate under such circumstances: the validity of the models built by computer scientists for biologists is conditional to the quality and completeness of data that computer scientists have no say on. Under such circumstances, making the interests of distinct academic fields coincide rather than punctually relay each other, becomes a matter of educating the affiliates of one discipline to the specialised knowledge of the other, or else inventing a new discipline circumscribed to their intersection. Interdisciplinarity’s proposition therefore draws its sense out of the fact that specialised academic knowledge reinforces interdisciplinary separation. This paper means to emphasise issues that draw their meaning before any differences between disciplines need be considered.. This paper suggests that the minimalist formalism of BANs may be of value in that.
Science has been concentrating on a certain kind of information, emerging directly from objects, from the explicit statements we make about them. The BAN examples above suggest that if we want to rely on the representations we make for ourselves of the objects we take interest in, then we need to have practical indepth understanding of how a piece of information’s meaning and formalisation relate to one another.The five observations of this paper undercut the assumption that there are representations that are fundamentally more “objective” – as in more “accurate”, more “complete” or more “realistic” – than others. In that, they undercut the necessity of restricting ourselves to looking for new information where beacons the information we already have precisely as we presently represent it and precisely as we presently interpret it.
Let us assume the following. (1) There is such a thing as changes of perspective that makes sense with respect to sciencemaking. In other terms, there are differences in perspectives that are meaningful in science. (2) A change of perspectives on object O, from perspective A to perspective B, does not systematically result from acquiring new information about O under A and simply increasing the set of features we take into account when we consider O. In other terms, the difference between two perspectives on the same object is not necessarily information explicitly expressed about that object. And if it’s not then it raises the following question about the new information we get on O as a result of seeing O with perspective B instead of perspective A. Other than waiting for a fortuitous change of perspective, concentrating on clearing data deficits in the meantime at the risk of ending up mislead by a series of questions and answers that only are relevant under a perspective that isn’t … How do we access this new information on O given that it is not (fully) dependent on what we know of O under A?
This paper suggests that a first step to answer this question is to tame implicit information of the kind that explains how BANs and of Fig.2 and Fig.3 can both be flawless representations of the BAN of Fig.1, and more generally understand the possibilities and the limits of what one given formal representation can actually represent.
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