Abstract - Government Executive



Toward Universal Laws of Technology Evolution: Modeling Multi-Century Advances in Mobile Direct-Fire SystemsAlexander Kott, U.S. Army CCDC Army Research Laboratory, Adelphi, MD, USA; email: alexander.kott1.civ@mail.milKeywords: History of Technology; Technology Forecast; Allometric Relations; Exponential Trend; Ground Warfare; Military Technology; Weapon Systems; Mobile Direct-Fire WeaponsDisclaimer: The views presented in this paper are those of the author and not of his employer.AbstractThis paper explores the question of whether a single regularity of technological growth might apply to a broad range of technologies, over a period of multiple centuries. To this end, the paper investigates a collection of diverse weapon systems called here the mobile direct-fire systems. These include widely different families of technologies that span the period of 1300–2015 CE: foot soldiers armed with weapons from bows to assault rifles; horse-mounted soldiers with a variety of weapons; foot artillery and horse artillery; towed antitank guns; self-propelled antitank and assault guns; and tanks. The main contribution of this paper is that, indeed, a single, parsimonious regularity describes the historical growth of this extremely broad collection of systems. Multiple, widely different families of weapon systems — from a bowman to a tank — fall closely on the same curve, a simple function of time.?This suggests a general model that unites allometric relations (such as the Kleiber’s Law) and exponential growth relations (such as the Moore’s Law).To this author’s knowledge, no prior research describes a regularity in the temporal growth of technology that covers such widely different technologies and over such a long period of history. This regularity is suitable for technology forecasting, as this paper illustrates with explorations of two systems that might appear 30 years in the future from this writing: a heavy infantryman and a tank. In both cases, the regularity helped lead to nonobvious conclusions, particularly regarding the power of the weapons of such future systems.?Furthermore, this result suggests a possibility – and related research questions – that even broader collections of technology families might evolve historically in accordance with what might be called universal laws of technological evolution.1. Introduction and MotivationA significant and growing body of research points to the existence of strong regularities in the historical evolution of technologies. A particularly well known example of such a regularity is Moore’s Law.[1?4] This law demonstrates empirically that the number of transistors on a computer processor’s chip (a proximate measure of a chip’s performance) has been growing at an approximately regular rate, i.e., increasing by a near-constant percentage every year, for a period of several decades. On a log-linear scale, Moore’s Law is a straight line: a growing number of transistors per chip as a function of years. This regularity – a steady exponential growth – has a practical significance. It allows, for example, an R&D manager to forecast the likely capabilities of future computer chips.Similar regularities apply to many other classes of technological artifacts. (A note on terminology: instead of the term “law”, as in Moore’s Law, I prefer the less ambitious “regularity”. On relations between “law” and “regularity”, see Encyclopaedia Britannica [5] and Swartz.[6]) Often, an exponential growth is observed for a measure of performance or a proximate measure of performance of an artifact class, like in Moore’s Law. In other cases, a measure of productivity or performance per dollar of cost has shown to exhibit a similar exponential growth.[7] Other cases exhibit superexponential regularities, i.e., a performance measure that grows every year by a fraction that is not a constant but rather itself grows over time.[7,8] This growth could be explained by the argument that as society acquires greater knowledge, it is able to accelerate the rate of technological progress.[9]A well-defined measure of performance is often unknown, especially when the performance involves multiple attributes of an artifact. This paper focuses on those military technologies where performance measures have proven difficult to define. In such cases, it may be possible to discover a composite measure that exhibits a regular, parsimoniously described growth over time. In particular, Alexander and Nelson [10] found that a weighted sum of logarithms of certain key attributes (in their specific case, attributes of jet engines) closely correlates with time over a relatively long historical period. In other words, a composite of an artifact’s attributes exhibits a regular exponential growth over time. Kott et al. [11] proposed to generalize the approach of Alexander and Nelson [10] by correlating a composite measure of an artifact’s attributes (which they called the Figure of Regularity [FoR]) with a diverse range of functions that describe the rate at which the FoR grows over time. They illustrated the proposed generalized approach using a case study of infantry small arms, such as bows, crossbows, handgonnes, harquebuses, smoothbore muskets, rifled muskets, repeating rifles, and modern assault rifles, over the period spanning 1200–2015 CE. They explored a range of function – a conventional exponential growth (similar to Moore’s Law), a double exponential function, and others – and found that the evolution of all these small arms can be described by a parsimonious regularity holding over 800 years.Taking the results of Kott et al. [11] as a point of departure, this paper explores the question of whether a single regularity might cover a broad range of technologies. For example, could it apply to a range of technologies far broader than infantry small arms, which are limited to a relatively well-defined, narrow function and a type of users? To this end, I investigated whether a single, unified regularity applies to a collection of diverse weapon systems that I call mobile direct-fire (MDF) systems. This collection includes widely different families of technologies that span the period of 1300–2015 CE: foot soldiers armed with weapons from bows to assault rifles; horse-mounted soldiers with a variety of weapons; foot artillery and horse artillery; towed antitank guns; self-propelled antitank and assault guns; and tanks. The main contribution of this paper is that, indeed, a single, parsimonious regularity describes the historical growth of this extremely broad collection of systems. To this author’s knowledge, no prior research describes a regularity – both in historical time and in scale of systems -- that covers such widely different technologies and over such a long period of history. Furthermore, this result suggests a possibility – and related research questions -- that even broader collections of technology families might evolve historically in accordance with what I might call universal laws of technological evolution. The remainder of the paper is organized as follows. Section 2 describes the relevant prior work. Section 3 offers an overview of the approach to the discovery of regularities in the temporal evolution of a technology. Section 4 uses a set of empirical data [12] and the approach introduced in section 3 to derive several alternative models of regularities applicable to the systems under consideration. Section 5 discusses the results obtained in section 4. Section 6 uses the models to illustrate possible applications of such results to the conceptual design of military systems. Finally, section 7 concludes the paper with several recommendations. Figure SEQ Figure \* ARABIC 1. Multiple, widely different families of military technologies (various symbols) fall approximately on the same curve (diamond symbols) when characterized by a suitably selected composite measure, the FoR. Here log(FoR) = -5.14 + 1.45*log(KE*R/SM) + 2.82*log(D) – 2.22*log(C) + 1.08*log(S). The temporal evolution (the diamond symbols) is described by the TEL-B model, log(e(T)) = 2.57E-05* (T-1400)2 + 0.743. See sections 3 and 4 for details.2. Prior WorkA paper on historical developments in military technologies cannot fail to mention a few examples of the histories of weapons and warfare. One such work is Dupuy,[13] which offers a broad, multi-millennia overview of the factors affecting the dynamics of weapons’ evolution. The logic of military revolutions, and their connections with technological developments, are the topics explored by Knox and Murray [14][2001]. Van Creveld [15] discusses how military technology was used throughout the history of warfare and how the nature of warfare was influenced by the technology. As this paper aims at identifying regularities in technological developments over a very long – multiple centuries – period, prior work on long-term forecasts is relevant. Although I am unaware of works that study forecasts for horizons measured in centuries, forecasts over several decades are known. For example, Albright [16] studied forecasts made in 1967 for the year 2000. In 1990s, a number of forecasts appeared with regard to military technologies of 2020; these included Newman,[18] Vickers,[19] and O’Hanlon.[20] Kott and Perconti [21] find that the accuracy of such forecasts can be relatively high. Regularities in technology growth have been observed for many decades and have been a subject of voluminous literature. Specifically, with regard to exponential “laws”, there is a rich literature that discusses such trends, their empirical evidence,[22] and their role in forecasting.[23] Attempts at a theoretical explanation of such regularities also exist. For example, Seamans [24] proposes a mechanism – a sequence of actions and reactions by competitive actors, such as corporations in the world of commerce or states in military competition – that could lead to an exponential trend in technology development.Although regularities in historical growth were noted with respect to multiple technologies,[25,26] a particularly well-known example is Moore’s Law,[1?4] originally postulated for computer chips. The popularity of Moore’s Law is such that all exponential trends in technology development are often called simply a generalized Moore’s Law.[25,27] Exponential laws tend to be the most commonly discussed, but other forms of temporal regularities also apply. In particular, a number of authors argue in favor of superexponential models, where a measure of technology grows with time faster than in an exponential law.[7?9,28] The limitations of exponential and superexponential models have also engendered significant discussion of alternative models. For example, Sood et al. [27] argue that exponential trends as well as other models mask a complex process of technological innovation, and a more in-depth Step and Wait model is better at capturing such complexities. In another example, the use of Data Envelopment Analysis [29,30] for technology forecasting outperforms exponential models. Studies of trends in the advances of a class of devices generally rely on an a priori assumption of a certain measure of performance applicable to the devices.[23,31] In many cases, a single characteristic of a device may exhibit a noticeable exponential trend over time. Often, however, a combination of several characteristics may need to be composed before a useful temporal regularity becomes obvious.[32] Seeking a composite measure of a complex device – a jet engine -- Alexander and Nelson [10] studied a correlation between the date of a technology’s introduction, e.g., the year of appearance of a certain jet engine, and a function that combines several attributes of the engine. The approach of Alexander and Nelson [10] is generalized in Kott et al. [11] who correlate a composite measure of an artifact’s attributes (which they called FoR) with a diverse range of functions that describe the rate at which the FoR grows over time. This paper uses the modeling approach of Kott et al. [11] but differs in its research aim: to determine whether a single regularity might apply to multiple, drastically different classes of systems. To this author’s knowledge, no prior research has attempted to determine whether there exists a regularity – i.e., a consistent temporal trajectory of evolution – that unites weapon systems as dissimilar as a crossbowman and a main battle tank.3. Overview of the Modeling Approach This paper follows the modeling approach of Kott et al.[11] called the FoR approach. In this section, a brief overview of the FoR technique follows.Virtually, all works on trends or regularities in technological forecasting require an a priori definition of a measure – usually called a measure of performance – for which the trend is being determined. However, in many cases, a measure of performance is unknown. For example, in the case of direct-fire weapons, I am not aware of an established measure of performance. Multiple studies, e.g., Brown,[33] Stockfisch,[34] and Crenshaw,[35] discuss the difficulties in defining meaningful measures of performance of effectiveness for weapon systems. Therefore, we (the authors of [11]) seek an alternative measure, which we call the FoR, which is a function of a technology’s attributes and exhibits a regularity often found in the temporal evolution of a measure of performance. To this end, we built on the idea of Alexander and Nelson [10] who found a correlation between the dates of a technology’s introduction, e.g., the year of appearance of a certain jet engine, and a function that combines several attributes of the technology. In this way, a measure of performance is no longer required. We generalize the idea of Alexander and Nelson [10] by considering not just one (exponential) but multiple alternative laws of temporal dynamics and performing the search for the most fitting temporal dynamics simultaneously during the search for the FoR model.In the following formulation, T is the time when a particular instance of a technological artifact first appeared; a is the vector of significant attributes that describe an instance of the technological artifact, a = (a1,… an); and D is the set of N observations of technology instances, data points di ∈ D, di = (Ti, ai). Examples of such data, including time and attributes, appear in Table 1. Further, F is a set of models fi,, each of which hypothesizes how attributes a might be combined to form a FoR. Each model fi is a function of a and is parametrized with a set of parameters Qi, such that fi ∈ F, fi = fi(a; Qi), Qi = (qi,1, …qi,m). For example, in section 4.2, we hypothesize that the FoR is a product of terms, each of which is an exponent of an attribute.E is a set of models ei,, each of which describes how the technology’s FoR evolves with time. Each model (for mnemonic purposes, we call this the Temporal Evolution Law [TEL]) ei is a function of time T and is parametrized with a set of parameters Pi, such that ei ∈ E, ei = ei(T; Pi), Pi = (pi,1, …pi,k). Although ei functions may belong to a broad class, for the purposes of this paper, we limit our attention to the exponential and superexponential functions that tend to be mentioned in technological forecasting literature. For example, in section 4.3, we hypothesize three models; the first is e(T)=exp(p2+p1T).Then, informally, the problem of discovering the FoR is to find the function f* (with parameters Q*) such that when applied to the data set D, the resulting points fit closely a function e* (with parameters P*). Let’s state it more formally and using the least-squares approach as an example of goodness of fit. We seekf*Q*, e*P* | f*Q*∈F ∧ e*P*∈E ∧ ?f'Q'∈F ∧ ?e'P'∈E ∧ ?(Ti, ai)∈D : i=1Nf*ai;Q*-e*Ti;P*2 ≤ i=1Nf'ai;Q'-e'Ti;P'2.(1)In section 4, we solve such a problem for a collection of MDF systems. The following steps outline and illustrate our approach. We assume the ranks of sets of E and F are small enough that exhaustive comparisons of all pairs of models e and f are computationally feasible. The solution steps are as follows:Step 1. We use domain-specific expertise to identify the most significant attributes a for the technology in question and collect the historical data set D. For example, in section 4.1, we describe several attributes that reflect the important and broadly applicable properties of MDF systems.Step 2. We hypothesize a set of models F. For example, in section 4.2, we hypothesize a model that describes how the FoR of small arms depends on several attributes, as determined in Step 1. In this particular case, the set F includes only one model. Step 3. Using prior research, literature, and expert knowledge, we hypothesize a set of TELs E . For example, in section 4.3, we hypothesize three models that describe how a FoR of MDF systems might vary over the period of 1300–2015 CE, each model with its own set of parameters. Step 4. We pick a pair of f*, e* and use the least-squares approach (or minimize another appropriate loss function) to find the parameters Q*, P* that minimize i=1N(f*ai;Q*-e*Ti;P*)2 over all N data points in set D. We then iterate over all pairs f*, e*. In section 4.4, for example, we do so for three pairs, where each pair consists of the only model we hypothesized for the FoR as a function of attributes and one of the three models for TELs. Step 5. For each pair of models and their optimal parameters determined in Step 4, we compute one or more characteristics that reflect the goodness of fit. We then assess the overall goodness of fit for all pairs and select the best-fitting pair f *(with parameters Q*) and e* (with parameters P*). Section 4.5 is an example of this step. 4. Data and ModelsIn this section, I describe the details pertaining to the data set, and the specific models and statistical analysis techniques used.4.1 The DataThis paper uses Kott [12] to extract a set of data describing the attributes of 195 weapon systems that first appeared in the years between 1300 and 2015 CE. When referring to a “system”, I include everything that is required for its mobility and ability to deliver effects on hostile targets. For example, a bowman system includes the bow, the human operator of the bow, and the necessary supply of arrows. Similarly, a field artillery cannon system also includes the crew, the horses, and the caisson. The systems differ widely in many respects, but do have two common characteristics. All these systems are (1) ground mobile (i.e., commonly maneuvering on the ground during a battle) and (2) achieve their effects on hostile targets via the kinetic energy of their projectiles, delivered along a line of sight at a relatively flat trajectory. This excludes, for example, medieval artillery, which remained generally static during a battle; heavy artillery that did not commonly maneuver in a ground engagement; indirect-fire artillery; or artillery and missile systems that use explosive shells. Even with these restrictions, the resulting collection of systems was extremely diverse, including lightly armored bowmen; light and heavy armored horse-mounted archers; longbowmen; crossbowmen; foot soldiers with handgonnes and harquebusiers; pistol-armed knights and reiters; musket-armed foot soldiers; soldiers with long rifles, Minié-ball rifled muskets, early breechloaders, repeaters, or modern assault rifles; early modern artillery; pre-Napoleonic and Napoleonic artillery; early rifled and breech-loading artillery; WW1, WW2, and modern artillery; foot, horse, and vehicle-towed artillery; antitank-towed artillery; assault guns; self-propelled antitank guns; and tanks from WW1, WW2, and post-WW2.Table SEQ Table \* ARABIC 1. Examples of data. Full set of data and notes on sources are in Kott.[12]Weapon SystemTPMVDRSMHPCSSoldier w/ longbow13000.050053755850.113Soldier w/ crossbow13990.100055751850.113Horse-mounted harquebusier16200.025020010306001.1115Infantryman w/ AK-12 assault rifle20110.00369006007001050.1138-pdr Gribeauval cannon17653.8958390800247315.313517-pdr antitank towed gun19423.4000120015002013120147620Self-propelled gun SU-100194415.60008953000531600500420T-72 main battle tank19723.900017853000842500780345The data exemplified in Table 1 are the following:T is the approximate year in which the weapon was designed or initially employed. In this paper, I limit the period under consideration to 1300–2015 CE. PM is the mass of the projectile. It influences, among other things, the kinetic energy of the projectile, and thereby, the ability to disable the adversary.V is the projectile velocity at the moment of separation from the weapon, e.g., the arrow’s velocity as it exits the bow or the bullet’s velocity as it exits the gun muzzle. D is the maximum effective range, i.e., the distance at which an infantryman can fire the weapon with an acceptable probability of hitting and disabling the targeted adversary. R is the maximum cyclic rate of fire, i.e., the maximum number of projectiles per minute that an infantryman can fire from the weapon. SM is the mass of the system including everything that is directly required for that system to maneuver and operate tactically on the battlefield. In the case of an infantryman, it includes the mass of the person’s body, the armor, and typical equipment, as well as the weight of the weapon(s) and ammunition. In the case of the cavalryman, the mass of the horse is included. In case of towed cannon, the mass of the limber, ready ammunition, horses, and crew are included; caissons with additional ammunition are seen here as a part of the logistic support and are not included. HP is the motive power of the system, i.e., the power directly available to move the system on the battlefield. In the case of an infantryman, this is typically about 0.1 hp, the representative power of a human. For horse-towed artillery, this includes the power of the horses and the crew. For modern systems, it is the engine power of the platform or the towing vehicle (track or tractor). C refers to the crew size – the number of personnel directly serving the system during the engagement. It ranges from one in the case of an infantryman or cavalryman, to as many as 15 in the case of an artillery piece. S is the offroad speed; it characterizes the approximate speed with which the system can maneuver on the broken terrain of a battlefield for a relatively prolonged time as opposed to a short sprint.All these attributes describe a system from the perspective of its function and performance, not its design. For example, these attributes do not specify the caliber of the cannon, the type of crossbow bolt, or the thickness of a tank’s armor. Instead, they treat the system as a black box (i.e., without considering what’s “inside”) and focus on the effects that the system is likely to produce on hostile targets or the demands it places on its operators. Building a model of technology evolution based on performance attributes rather than on design attributes allows us to seek regularities across multiple technology families, increase the temporal span of the investigation, and offer superior capability for longer-term forecasting (compare similar arguments by Koh and Magee [36]). As such, the model I seek in this research — and any forecast produced with the model — tells us nothing directly about the design features that might produce the forecasted values of the performance attributes. Thus, interpretations of the model require caution. Merely increasing the performance attributes does not necessarily lead to a successful system design.Inevitably, with such a diversity of systems and the wide range of the historical period, the data had to come from multiple sources of uneven quality and veracity, with multiple contradictions, incompleteness, and uncertainty. The entire compilation of the data is found in Kott [12] along with numerous notes on sources, assumptions, approximations, and limitations.4.2 Candidate Models of the FoRHaving identified the data, I proceed with the step 2 of the process outlined in Section 3, i.e., I now formulate a set of models F where each model fi describes how the technology’s FoR depends on the attributes a = (PM, V, D, R, SM, HP, C, S). To constrain the scope of this research, I consider a set consisting of a single model:f=q0PMq1Vq2Dq3Rq4SMq5HPq6Cq7Sq8, (2)Here, the model’s parameters Q = (q0, q1, q2, q3, q4, q5, q6, q7, q8). This model is similar to the one used in Alexander and Nelson [10] and Martino.[32] It was also the one used by Kott et al.[11]4.3 Candidate Models of the Temporal Evolution Law The third step of the process presented in section 3 is to hypothesize the set of models E, which describe how the technology’s FoR evolves with time, i.e., the TELs. Each model ei is a function of time T and is parametrized with a set of parameters Pi. Based on the results of Kott et al.,[11] I elect to investigate the following three models.Model TEL-A – ExponentialThe hypothesis for this form of temporal evolution implies that the FoR increases by a constant fraction per unit of time such that the growth de/dT is proportional to the value e achieved at the given time, i.e., dedT=p1eT. Here p1 is a constant parameter. Solving this equation for eT , gives logeT=p2+p1T. This exponential model is the most common form of Moore’s Law [7][25]. It is also the form that underlies the work of Alexander and Nelson [10] and Martino.[32]Model TEL-B – “Quadratic Exponential”If the rate at which the FoR increases is a fraction that is not constant, but increases over the time, then a parsimonious hypothesis would be that the fraction is a linear function of time. This suggests the model logeT=p2*T-p12 . Model TEL-C – Piecewise ExponentialThis model is inspired in part by the piecewise model of Nagy et al. [7] and in part by Lienhard [28] who argued that a drastic change in the rate of technological evolution occurred in the year 1832. The model is logeT=p2+p1T for T ≤ 1832 and logeT=p3+p4T for T > 1832; p4 > p1. In the actual fitting of the model, I treated the number 1832 as a parameter and varied its value to seek the best fit. (I found that the optimum value for this parameter was indeed fairly close to 1832 and kept the value of 1832 as a tribute to Lienhard’s [28] original observation.)4.4 Fitting the ModelsIn executing the fourth step of the process, I performed linear regression on the data for each of the three pairs of FoR and TEL models. Each pair consisted of the one FoR model defined in section 4.2 and one of the three TEL models defined in section 4.3. I optimized the best least-squares fit between the values of the TEL model and the FoR model. This yielded the values of parameters qi for the FoR model (generally different for each corresponding temporal evolution model) and parameters pi for each temporal evolution model. While fitting the models, I found that some of the variables and parameters in the FoR model (2) are far less influential than others and can be excluded without reducing the goodness of fit. Eliminating such components of the model and combining parameters with similar values provides a more compact version of Eq. (2) as follows:f=q0(KE*R/SM)q1Dq3Cq7Sq8, (2a)where KE = 0.5*PM*V2 is the kinetic energy of the projectile at the muzzle.The results of the fitting process are depicted in Fig. 1 for the TEL-B model and in Fig. 2 for the TEL-C model. The captions of the figures contain the values of the parameters obtained by fitting. The scale for the values of log(FoR) in these figures was chosen so that log(FoR) = 1.0 at the year 1300 and log(FoR) = 10.0 at the year 2000. This is an arbitrary choice; a different scale could work equally well. (Here and elsewhere in the paper, log denotes logarithm to base 10.) I further characterized the results of the fitting process by the coefficient of determination R2 and the Bayesian Information Criterion (BIC), which includes a penalty for a greater number of parameters.[37] These are summarized in Table 2. Table SEQ Table \* ARABIC 2. Summary of the model fitting results.TEL-ATEL-BTEL-CR2.855.942.952BIC-7.8-110.2-195.54.5 Assessing the Goodness of FitExamination of table 2 suggests that the best combination of R2 (higher is better) and BIC (lower is better) are exhibited by the pair of the FoR model per Eq. (2a) and the TEL-C model. The coefficients of the corresponding formulae and the graphic depiction of the fit are shown in Fig. 2. However, the goodness of fit between the FoR model per Eq. (2a) and the TEL-B model is not far behind (see Fig. 1). For the purposes of further discussion in this paper, I use both the TEL-B and TEL-C models. The goodness of fit associated with the TEL-A model is clearly far worse than the other two, and therefore, I exclude the model from further consideration.Figure SEQ Figure \* ARABIC 2. Similar to Fig. 1, this graph depicts the FoR of widely different systems with fit optimized for the temporal evolution of the TEL-C model. Here log(FoR) = -2.91 + 1.51*log(KE*R/SM) + 1.78*log(D) – 1.71*log(C) + 1.47*log(S). The temporal evolution (diamond symbols) is described by the TEL-B model, log(e(T)) = 4.92 + k(T- 1832), where k=0.0074 for T<1832 and k=0.030 for T>1832.5. Discussion of ResultsHere I discuss results from three perspectives: their domain-specific plausibility; retroactive forecasting; and allometric relations. 5.1 Domain-specific PlausibilityThe parameters obtained for both models (see Figs. 1 and 2) appear plausible, in the following sense. In both models, a primary factor is the group KE*R/SM, essentially the maximum kinetic energy of projectiles that the system can direct at a target per unit of time per unit of mass of the overall system. This group contributes to Eq. (2a) with a positive exponent: 1.45 in the case of TEL-B and 1.51 in the case of TEL-C. In other words, the FoR grows with higher kinetic energy of the projectile and higher rate of fire (both are considered desirable performance attributes for MDF systems), and diminishes with higher system mass (an undesirable attribute, generally related to higher costs and logistics burden). It is plausible that this ratio grows strongly over the years as MDF technology advances. Similarly, the exponents associated with the effective range and speed are positive in both models – which is plausible as these would generally grow in more advanced systems. The exponent of the crew size is negative in both models, as historically more advanced MDF systems would tend toward the more economical use of personnel. With this, one can even attempt to interpret the expression of the FoR as a ratio of “value” (attributes with positive exponents) to “cost” (attributes with negative exponents). Let’s rewrite the Eq. (2a) as follows:f=(q0(KE*R)q1Dq3Sq8)/(SMq1*C-q7), (2b)One can see the group q0(KE*R)q1Dq3Sq8 as “value” because these parameters represent desirable performance attributes that the designer of a MDF system would wish to increase. At the same time, the group SMq1*C-q7 can be seen as “cost” because the designer of a MDF system would wish, all other things being equal, to reduce the weight of the system and minimize the crewing requirements (note that q7 emerges as a negative number in all of the models I consider). Hence, the FoR might be interpreted as value per unit cost.As the technology advances, the developers of the technology – driven by competition – seek to increase the value per unit cost to the maximum level possible for the technological opportunity available at that historical moment. The systems “below the curve” tend to lose competition, because they are too costly for their value. Systems above the curve are rare because they represent exceptionally effective – for their time -- technological solutions that deliver high value per unit cost. Thus, the FoR values tend to cluster around the TEL curve.It should also be noted that the two models of the FoR obtained under the assumptions of TEL-B and TEL-C do exhibit different values of parameters q0, q1, q3, q7, q8. For example, with the TEL-B model, the effective range entered into the FoR is the exponent 2.82, while with the TEL-C model, the exponent is 1.78. Despite these differences, the resulting values of the FoR are fairly similar. In fact, the values of the FoR obtained with the TEL-B and TEL-C models are very closely correlated, R2=0.996. From the perspective of practical technological forecasting, different models can be equally sufficient. For example, section 6 explores possible attributes for two types of MDF systems – a heavy infantryman and a main battle tank – in the year 2050. The value of log(FoR) for such systems in 2050 is forecasted as 11.6 for TEL-B and 11.5 for TEL-C — a difference that is very minor for practical purposes.If the differences between the two models are so slight, it is tempting to select just one of them. However, each model offers its own advantages and disadvantages. TEL-B offers the advantage of parsimony, but raises the question of how long the linear acceleration of dedT with time can proceed. In addition, its goodness of fit is somewhat worse than that of TEL-C. On the other hand, TEL-C presents its own challenges. First, it is less parsimonious – it includes one more parameter as compared to TEL-B. More importantly, it implies a claim that approximately in the first half of the 19th century, a dramatic discontinuity occurred in the rate of technological advance – at least in the technologies of MDF systems. Indeed, visual inspection of the data points in both Figs. 1 and 2 (irrespective of whether the TEL-B or the TEL-C model is used) suggests that a noticeable change in the rate of progress did occur somewhere between 1800 and 1850. To this author’s knowledge, Lienhard [28] was the first to observe and report this discontinuity (which I propose to call the Lienhard Hypothesis), and identify a specific year – 1832 – associated with the discontinuity, based on his study of at least 14 different classes of technology. John H. Lienhard was a professor of mechanical engineering at the University of Houston, and gained fame as the author and host of a long-running, nationwide popular series of broadcasts on the history of technology.[38] The Lienhard Hypothesis has not engendered a follow-on discussion in the literature. Although this paper’s results lend additional credence to this finding, it remains a rather bold hypothesis. To be accepted, it needs broader empirical evidence along with multidisciplinary theoretical underpinnings. I leave that as a topic for future work.With respect to the TEL-A model, it is noteworthy that although it corresponds to the conventional, exponential form of Moore’s Law, and although much evidence supports the broad applicability and validity of Moore’s Law (e.g., Magee et al. [39]), the TEL-A model is significantly inferior to the TEL-B model and especially the TEL-C model. Perhaps this suggests that a constant rate of technology advances – which underpins Moore’s Law – cannot always be applied over a very long period, such as the seven centuries covered in this study. Perhaps a piecewise sequence of Moore’s Law, such as the TEL-C model (which the TEL-B model also approximates), is more likely to describe the evolution of a technology over a long history. 5.2 Retroactive Forecast Recognizing that a key use of such models is for technology forecasting, let us consider how well the models would have worked for hypothetical analysts of the past centuries. Suppose an analyst in the year 1700 used the approach of this paper to fit the TEL-B model using only the data available between 1300 and 1700, and then predicted the evolution of the FoR for the next 300 years. The results are shown in Fig. 3. The dots of the parabolic graph describe the predicted growth of the FoR over the years. The scattered dots reflect the actual systems that appeared between 1700 and 2015. These, of course, would not be known to the analyst of year 1700. Similarly, Fig. 4 assumes that a hypothetical analyst of the year 1800 would build the model using the data available between 1300 and 1800, and then use it the model to forecast the growth of the FoR between 1800 and 2015. Finally, Fig. 5 describes a similar retroactive forecast for a hypothetical analyst performing this process in the year 1900. In all three cases, despite the very long horizon of the forecasts – up to 300 years – the forecasted trends are broadly consistent with the actual data.Figure 3. A hypothetical analyst-forecaster in 1700 finds the optimal parameters pi and qi to fit the TEL-B model and FoR (the dots prior to 1700) and then uses the pi parameters to extend the forecast (parabolic graph dots post 1700). The scattered dots post 1700 show the actual systems with the FoR that uses the same qi parameters.Figure 4. Similar to Fig. 3, a hypothetical analyst of 1800 performs the fit for data known prior to 1800 (the dots to the left of 1800). The data to the right of 1800 depict the forecast (the dots of the parabolic graph) and the actual data (the scattered dots).Figure 5. Similar to Fig. 3 and 4, a hypothetical analyst of 1900 uses the data available up to 1900 to generate forecast. The dots to the right of 1900 compare the forecast (the dots of the parabolic graph) and actual data(the scattered dots).5.3 Allometric PerspectiveAllometry is another useful perspective for interpreting these results. Broadly speaking, allometry is the study of how attributes of an organism (or a system) depend on its scale. Often, a universal relation exists between the scale of the organism and its various attributes, applicable across multiple organisms of widely different scales. Commonly observed universal laws in biology are of the form Y = a·Mb, where Y is an attribute or measure of the organism, a is called an allometric coefficient, b is a constant exponent, and M is the mass of the organism. For example, the Kleiber’s Law states that for the vast majority of animals – from tiny mouse to huge elephant -- the organism’s metabolic rate scales approximately to the 3/4 power of the organism’s mass, and the data for all such organisms fall on the same curve. The allometric coefficient can depend on time. For example, the growth rate of an organism is proportional to the mass of the organism to the constant power of 3/4; however the allometric coefficient changes as a function of the organism’s time from birth.[40] In this paper’s case, unlike in ontogenetic allometry, the allometric coefficient increases with the passage of time, as the progress of technology helps obtain greater performance from the same mass of an artifact. Indeed, a cannon of 1950 can produce far greater fire power than a 1650 cannon of the same weight.Similarly, this case reveals a universal relation between the scale of a MDF system, its “level of maturity” (i.e., the point in the history of technology when the system appeared), and a certain composite measure of its performance. This universal relation applies to a wide range of MDF systems, from a medieval bowman to a modern tank, just like the Kleiber’s Law applies from mouse to elephant. To clarify, let’s rewrite the equations in Fig.2 as an allometric equation Y = a(T)·Mb, or log(Y) = log(a(T)) + b·log(M):-2.91 + 1.51·log(KE·R/SM) + 1.78·log(D) – 1.71·log(C) + 1.47·log(S) = 4.92 + k·(T-1832),where k = 0.0074 for T <= 1832 and k = 0.03 for T > 1832.Rearranging,log(KE·R) + 1.18·log(D) – 1.13·log(C) + 0.97·log(S) = (5.19 + k·(T-1832)) + log(SM), where k = 0.0049 for T <= 1832 and k = 0.02 for T > 1832.In this allometric equation the left side is the log of the maximum firepower KE·R (here I simply use “firepower” to refer to the product KE·R ) that the system can deliver at a target per minute, with corrections for effective range D, crew size C and speed S. On the right is the log of the allometric coefficient (5.19 + k·(T-1832)) which in this case is a function of the year T when the system appeared, plus the log of the system mass SM. Figure 6 shows the plot of all MDF systems in the dataset under consideration – from medieval bowmen to modern tanks and riflemen – and they all fall close to the same universal allometric curve. Figure 7 shows a similar plot where the value on the y-axis is the firepower KE·R that the system can deliver at a target per minute, with no additional corrections. Figure 6 Multiple MDF systems of 1300-2015 CE fall on the same universal curve of allometric nature. The horizontal axis is the log of system mass plus a function of year T per model TEL-C, specifically log(SM)+k·(T-1832)+3.26, where k = 0.0049 for T <= 1832 and k = 0.02 for T > 1832. The vertical axis is the log of firepower KE·R, adjusted for range, speed, and crew size. R2=0.99. Figure 7. This graph is similar to the previous one, but the Y-axis depicts firepower without any adjustments. The slope value is between 2/3 and 3/4; these tend to be typical for allometric relations.The above discussion inspires a general model that unites allometric relations (such as the Kleiber’s Law) and exponential growth relations (such as the Moore’s Law):(1/A)·(dA/dT) = a·(dG/dT), Where T is time, A ≡ Q/Zb, Q(T) and Z(T) are characteristics of technological artifacts (either directly measurable attributes or composite measures), G(T) is a function describing the growth of A over time (likely reflecting the growth of relevant technological knowledge over time); a and b are constant coefficients.Integrating,logA = logQ – b·logZ = a·G + c, where c is the integration constant.Consider two common cases:If G = T and A is a measure of the artifacts’ performance, then logA = a·T + c, which amounts to the Moore’s Law.If G = 0 and Z is a measure of the artifact’s scale, then logQ = b·logZ + c, which amounts to the Kleiber’s Law. In a more general case, dG/dT is an arbitrary function of time, and Q, Z are arbitrarily complex functions of the artifacts’ attributes. In the case depicted in Fig. 7, for example, Q is the firepower KE·R; Z is the mass of the system, constant b = .69, and G is the piecewise linear function of time T given in caption of Fig. 6. The most significant prediction of the proposed model is that allometric relations may involve a function of the time when an artifact was developed or organism evolved. 6. Illustrative ApplicationsA model such as Fig. 1 can be useful in forecasting of future developments or assessing proposed developments of future weapon systems. In the following two subsections, I illustrate this point by considering two hypothetical systems: (1) an infantryman equipped with an exoskeleton in the year 2050 and (2) a main battle tank in the year 2050. Using the TEL-B model (Fig. 1) for the sake of concreteness, I estimate the potential quantitative characteristics of these two systems. This discussion is country-agnostic; I do not imply that any particular country would or should develop such systems. It should be stressed that these are merely rough illustrative sketches and are not intended to propose or advocate such systems, and are not necessarily accurate portrayals of such potential systems.6.1 Sketch A: a Heavy Infantryman of 2050Inspired by Allgeyer,[41] I explore here a hypothetical heavy infantryman of 2050, equipped with an exoskeleton along with an armor suit and a weapon significantly heavier than are common presently. In particular, let’s ask the following question: in 2050, what power of the main weapon will be required of these systems in order for them to meet the forecasted level of the FoR?The value of log(FoR) in 2050 is projected to be 11.6 (see Fig. 1). Assuming I wish the heavy infantryman of 2050 to possess at least log(FoR) = 11.6, and using Eq. (2a) and Fig. 1, I obtainlog(FoR) = log(q0) + q1*log(KE*R/SM) + q3*log(D) + q7*log(C) + q8*log(S) = 11.6, orlog(KE) = (1/q1)*(11.6 – (log(q0) + q1*log(R/SM) + q3*log(D) + q7*log(C) + q8*log(S))).The crew size C is 1, as I am considering here a single infantryman. The mass of the exoskeleton itself is assumed 25 kg (compare to the range of 50–66 lb suggested in Gorgey.[42] The payload possible with the current generation of exoskeletons may reach 90 kg (e.g., ), which I assume by 2050 will increase by 50% to 135 kg. Adding 67 kg of body mass for the soldier, I arrive to a total system mass SM of 227 kg. The 135-kg payload will have to include batteries (or fuel) for powering the exoskeleton, the suit of armor, the weapon, a combat load of ammunition, and other appropriate soldier combat equipment. The offroad speed S is assumed comparable to that of the present-day infantry, on the order of 4 km/hr. The maximum rate of fire is assumed on the order of 700 rounds per minute, and the effective range is assumed to be 2000 m.Under these assumptions, using the previous equation for log(KE) and parameters qi given in Fig. 1,log(KE) = (1/1.45)*(11.6 - (-5.14 + 1.45*log(700/227) + 2.82*log(2000) - 2.22*log(1) + 1.08*log(4))) = 4.19, orKE = 15490 J.If I were to assume that the weapon of the heavy infantryman of 2050 will evolve from today’s families of infantry weapons, then I note that the value of 15490 J approaches the levels of muzzle KE typical for modern 12.7-mm machine guns. Therefore, one possibility is that the heavy infantryman of 2050 might wield a descendant of today’s relatively light heavy machines guns like QJZ-89 [43, 44] or Kord.[45] It is also possible, however, that an entirely different technology will emerge capable of delivering comparable levels of energy. 6.2 Sketch B: a Main Battle Tank of 2050Here I take the inspiration from Steeb et al. [46] in exploring a tank concept with 2-person crew and 55 tons of weight. Again, I wish to achieve the log(FoR) of at least 11.6 projected in 2050. Conservatively extrapolating historical trends for main battle tanks, I assume a maximum effective range D = 5000 m, a rate of fire R up to 10 rounds per minute, and an offroad speed S of 45 km/hr. Then, similarly to calculation in the previous section, I obtainlog(KE) = (1/1.45)*(11.6 - (-5.14 + 1.45*log(10/55000) + 2.82*log(5000) – 2.22*log(2) + 1.08*log(45))) = 7.32orKE = 20.9 MJ.What might be the weapon of 2050 that could deliver such level of muzzle KE? One way to obtain a higher muzzle KE – although not necessarily practical or easy to accommodate in a tank – is to increase the caliber of the gun. For example, Steeb et al. [46] estimated that a tank gun of 135-mm caliber, based roughly on 1990s technologies, would deliver about 20 MJ of muzzle KE. This resonates with recent developments in Germany and France (130 and 140 mm, respectively.[47] If the crew size and mass of the tank are assumed comparable to present Western practices (3–4 person crew and a mass of about 70 tons), the muzzle KE required to achieve log(FoR) = 11.6 would grow significantly. This may imply a gun caliber of perhaps 152–155 mm (compare Gao [48]). Of course, one cannot exclude the possibility that alternative technological solutions with similarly high levels of muzzle KE – or comparable effects on the target -- might become viable by 2050. I close this section by emphasizing once again that these two sketches are merely illustrative explorations of FoR-based forecasting, not recommendations. 7. Conclusions and RecommendationsIt is found that a parsimonious regularity describes how a composite of attributes grows over multiple centuries, characterizing uniformly a set of weapon systems technologies.?Multiple, widely different families of weapon systems — from a bowman to a tank – fall closely on the same curve, a simple function of time, similar to Moore’s Law. Furthermore, because these systems vary dramatically in scale – and yet fall on the same curve – this regularity also reveals strong allometric relation in these technologies, and motivates a general model that unites allometric relations (such as the Kleiber’s Law) and exponential growth relations (such as the Moore’s Law).The results of the paper suggest that long-term technology evolution may not fit well to a single exponential curve. Some degree of superexponential behavior – a change in the rate of technology evolution – is likely to occur. A piecewise exponential model, such as one of the models explored in this paper, could be a strong candidate for describing the evolution of technology over multiple centuries. Unlike a similar regularity applicable to a relatively narrow set of infantry firearms technologies,[11] this paper covers an exceptionally broad diversity of systems. Furthermore, the period over which this regularity holds is exceptionally long, at least 700 years, far longer than any such laws reported in literature.?This regularity is suitable for technology forecasting, as this paper illustrates with explorations of two systems that might appear in the next 30 years from this writing: a heavy infantryman and a tank. In both cases, the regularity helped lead to nonobvious conclusions, particularly regarding the power of the weapons in such future systems.?This research should continue in a number of directions.?In particular,?theoretically grounded techniques are needed to support the optimal choice of attributes composing the FoR. For example, in the case of the MDF systems considered in this paper, further explorations may be desired with respect to hard-to-define attributes like effective range, offroad speed, and level of protection. The appropriate choice of temporal evolution law also presents a number of questions. What is the theoretical basis for the acceleration (and possibly deceleration) of the rates at which military technologies develop at various historical periods? What are the limits of such acceleration? Could these dynamics be expressed as a general predictive theory of technology evolution? What are the mechanisms – as opposed to a mere correlation – of the varying rates of evolution of military technologies in history??Was there really a major discontinuity in the technology development rate in the first half of the 19th century (the Lienhard Hypothesis that the present results support), and if so, why?Finally, the extent of the generality for a given law, i.e., how many different technologies it might be possible to describe with a single curve, requires a range of empirical investigations and an appropriate theoretical treatment. Perhaps, these would lead to what might be called universal laws of evolution for very broad types of technologies. 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