Tensor-Centric Warfare I: Tensor Lanchester Equations

We propose the basis for a rigorous approach to modeling combat, specifically under conditions of complexity and uncertainty. The proposed basis is a tensorial generalization of earlier Lanchester-type equations, inspired by the contemporary debate in defence and military circles around how to best utilize information and communications systems in military operations, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). Despite attracting considerable interest and spawning several efforts to develop sound theoretical frameworks for informing force design decision-making, the development of good frameworks for analytically modeling combat remains anything but decided. Using a simple combat scenario, we first develop a tensor generalization of the Lanchester square law, and then extend it to also include the Lanchester linear law, which represents the effect of suppressive fire. We also add on-off control inputs, and discuss the results of a simple simulation of the final model using our small scenario.


Introduction
ing away from fixed and prescriptive theories-while nonetheless conceding that such ideas are possible within sufficiently narrowly constrained problem domains. It is thus sometimes remarked that Clausewitz' focus is strategic rather than tactical. More deeply, this point of view gives a basis that more methodologically focused than theory focused: we are here primarily concerned with the means of solving problems that are effectively unique, messy, under-specified, socially complex, evolving and for which there is generally a scarcity of available data. In other words, decision-making in military matters has all the characteristics of what are often now described as "wicked problems" [9]. 13 Intelligent Control and Automation lem, however, is that NCW presented a discredited though intuitively appealing explanation for exactly how this is to occur, with practical consequences that have proven problematic [10] [11]; its methodological framework held that knowledge is the outcome of collecting and consuming data and that the accuracy of the knowledge is a function of the amount of information and ability to process it. The slightly later EBO notion similarly concludes that information collection, dissemination and processing are crucial, yet arguably overstates the strength and usefulness for prediction of supposed connections between outcomes and ostensible causes. Therefore, instead of the NCW, we are talking about distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance); for one of its examples, see [12].
These observations may be seen in terms of the limitations of classical systems engineering achievements in dealing with wicked problems [9]; the remedy was a methodological shift to essentially the kind of problem-solving approach espoused and exemplified by Clausewitz, but fleshed out later by Popper and subsequent authors [13] [14]. This line of reasoning inspires our approach.
As with most approaches to modeling combat, there can little doubt that connectivity is important, that the availability and quality of information matters; our explanation for why this is so departs from earlier methods in this that the networks at the heart of our model permit ideas to be tested more rapidly and more thoroughly. Thus we have in mind the ideas of problem-solving, where the problem choices are primarily of the wicked variety; it is not information per se that matters, but rather the ability to obtain information that reveals unacceptable error in proposed solutions and problem formulation. The goal of our modeling thus shifts from predicting outcomes of combat directly to questions about system control. In this paper and its successor we present a basis for a novel mathematical approach to modeling war and battle under the conditions of complexity and uncertainty as a step towards overcoming the limitations that have been so challenging to most models of combat outcomes proposed in the past.

Lanchester Equations
In defence operation research community it is well known that classical Lanchester-Osipov combat equations (also called Lanchester-style mass action models [15] 1 We remark that before Lanchester and Osipov, similar mass action models in naval applications were proposed by Chase [18], Fiske [19] and Baudry [20]. where overdot denotes time derivative, and R k and B k denote individual combat-rate coefficients for the Red and Blue forces, respectively (e.g., tank versus tank, concentration of fire).
2) Lanchester linear law for area-unaimed fire: where BR k and RB k denote mixed combat-rate coefficients for Red and Blue forces (e.g., artillery barraging an area without precise knowledge of target locations).
Although similar Lanchester-type models had been extensively used in the past Century, today they are widely regarded as grossly oversimplified representations of modern warfare, at best. This motivates our effort towards a modeling methodology of much higher complexity, including both continuous and discrete spatiotemporal dynamics, as proposed in the present paper 3 .  Peterson [21] and in 1990s by Bracken [22], which reads:

Brief Review of Recent Military Thinking
where the exponents p and q (such that 1 p q α + − =) need to be empirically determined. The conserved quantity in the LPB model (1), separating and integrating. The Lanchester aimed-fire (or, square law) model: and thus to 2 α = ; and the unaimed-fire (or, linear law) model: and thus 0 α = in the LPB model (1) (for more technical details see, e.g. [23] and the references therein).
hierarchically yields separate Command and Control (C2) channels for different military functions, which are then attached to acicular organizations that necessarily centralism planning and coordination to achieve desired effects. The emergence of modern communications and information technology was consequently broadly seen as offering the potential to dissolve such crystalline arrangements in favor of military forces able to fluidly self-organize in rapidly changing situations to both counter threats and take advantage of opportunities, by enabling collaboration directly between elements formerly widely separated by hierarchy.
Other advances including those the fields of telecommunications, robotics, artificial intelligence and autonomous systems have further opened apparent opportunities for collaborative planning, coordination and rapid response; at its core, modern defence thinking seeks to achieve highly distributed C2 arrangements enabled by communications and information systems in which information can be rapidly disseminated while also being protected from outside interdiction and interference. The extreme instantiation of this lies in the idea that by the provision of such a system, together with the training and procedures to utilize it, "information superiority"-the ability to acquire, transport and process more information than the opposition-will deliver superior ability to

From the Air Campaign Scenario to the Combat Tensor
To be able to compare our system with McLemore et al. [24] we need to have the same/similar system input and after computations to compare the outputs. In     A , respectively, are presented, using default graph embeddings in Mathematica ® , in Figure 3 and Figure 4.

The Combat Tensors for the Red and Blue Forces
As a "soft" introduction to dynamics of vector and tensor fields on battlespace manifolds, we define here the Combat-tensors, as the following matrix products (i.e., tensor contractions):

TCW Battlespace
A set of all active and controllable degrees-of-freedom (DOFs) of an arbitrary complex system comprises the configuration manifold for that system (see [25] for more technical details). For example, an nD configuration manifold for a humanoid robot is the set of all its movable joint angles. Following this fundamental manifold prescription, any battlespace (see [26] and the references therein) in TCW can be formally defined as the battle-manifold. In case of a very large battle-manifold M n , it can be approximated with n » , where n, the total number of DOFs, can be in millions (using computational framework outlined in the Appendix).
Complex warfighting dynamics on such battle-manifolds is naturally defined as an interplay of spatiotemporal vector and tensor fields flowing on them. For defining tensor expressions, we will use the abstract tensor notation with Einstein's summation convention upon repeated indices; see [27]- [32].
On any battle-manifold M n we can observe a dynamic interplay of various Actors, all defined by various vector and tensor fields, depending on their complexity.
Simpler Actors are formally defined as spatiotemporal vector-fields, ( )  commutes with any other 2nd-order tensor field of the same covariance on the same battle-manifold M n (e.g. x )-they can be added together as linear machines: T S = ± ± ±   All these tensor fields are spatiotemporal dynamical objects governed by tensor equations, similar to the elastic stress-strain relation:
Similar expansions (though larger) hold for battle-manifolds of any dimensions and can be derived using the fast tensor package xTensor [34] for Mathemati-ca®.
Assuming, for simplicity, the coordinate independence ( In the general case of explicit coordinate dependence ( , we would be actually dealing with the set of the first-order nonlinear PDEs, which would all require spatial discretization (e.g., using the Method of Lines, as implemented in Mathematica), after which the above mentioned ODE-solvers can be used again.
The same computational algorithms will apply, in both cases (ODEs and PDEs), also for the extended tensor Lanchester equations, formulated as follows.

Adding the Lanchester Linear Law
Next, to include the Lanchester linear law Equation (3) into Equation (6), while keeping their covariance (so that each term represents a vector-field), we need to extend them with quadratic terms of the Lanchester unaimed-fire equations (linear law) as: 5 Red : , where the fourth-order tensors ab cd F and ab cd G represent more complex, strategic, 5 The Basic Red and Blue tensor combat Equations (6)-(7) are valid for any linear/flat manifold M 9 .
In case of a strongly nonlinear/curved manifold M 9 , they would need the additional connection coefficients (i.e., Christoffel symbols)-which can be neglected for our purpose, as an unnecessary over-complication. With this view in mind, in the following sections we will introduce more complex dynamics and nonlinear control concepts into Equations (6)- (7), without introducing any geometric connection (e.g., Levi-Civita connection on Riemannian manifolds)-which can always be added to the battlespace system as additional nonlinear complexity (see [30]). Intelligent Control and Automation tactical and operational, Red and Blue capabilities, which can be defined either as the outer products of various matrices from [24], or composed as triple tensor sums: The basic Red and Blue tensor combat Equation (7) are implemented in Mathematica as the initial value problem for the following temporal vector-fields: where the 2nd-order Red and Blue combat-tensors  (4) and (5)   A sR a a t t a n b n A sample simulation of the basic tensor combat Equation (7) is performed in Mathematica (see Figures 5-7) for 10 time units (to match the simulations given in [24]) and random initial conditions.

Interpretation of Dynamical Simulations
The focus of our interpretation is the Red-Blue dynamics phase plot in Figure 7. This illustrates nine instances of engagement between the Red and Blue forces, confirming a fully engaged scenario. Figure 5 and Figure 6 are indicators of the outcome of the engagements: Blue is clearly winning in seven instances (note the exponential-like growth seen in Figure 6). On the other hand, Red is only winning in two of the engagements. It should be noted that one of the engagements seems to be border line or not clear, which is indicative of the uncertainty of the outcome of this specific instance of engagement. The warfare uncertainty will be addressed in [25].    In contrast, the output from [24] presented in their Chart 1 (Blue) and Chart 2 (Red) does not show the actual dynamics of the simulation, but rather the statistical inference from that simulation. Their main point is the "kill" of the aircraft, which they plotted along the same time axes (10 units) that we are using for the simulation. The 9 conflict points in our phase plane (Figure 7) are the points of potential "kill"; by relating to the Red and Blue time evolutions (in Figure 5 and Figure 6) we can also infer who was "killed" or "degraded" at that same time point, as compared to their Chart 1 and Chart 2.
Based on this interpretation, we can see that our proposed tensor framework is capable of addressing the similar questions as those addressed by [24]. In the subsequent paper [25] we will extend this tensor Red-Blue dynamics to model warfare uncertainty.

Adding Bang-Bang Control Actions
For the purpose of recasting the combat-dynamics Equations (7) into a control system, we will add to both Red and Blue forces simple-and-strong bang-bang

( ) ( ) Intelligent Control and Automation
A sample simulation of the bang-bang controlled tensor combat Equation (9) is performed in Mathematica (see  for 10 time units and random initial conditions. From Figures 8-10 we can see that adding strong bang-bang control inputs to tensor combat equations completely changes the natural combat-dynamics behavior-control actions have the overall flattening effect. Even if the control inputs have lower amplitudes (e.g., 5 instead of 10) the outcome would be qualitatively similar: both the time-plots and the phase plot would be flattened out. From these computational observations we can infer that adding artificial control inputs to natural Red-Blue combat dynamics does not make real sense, because in reality the Red and Blue forces mutually control each other.

Conclusion and Future Work
We have presented the basic development of the tensor-centric warfare (TCW),