All the software tools developed at STILMAN, including LG-RAID, utilize a powerful explainable Artificial Intelligence (AI) approach which is based on a theory called Linguistic Geometry (LG). LG is a mathematical theory for solving classes of adversarial games , the so-called Abstract Board Games (ABG). Roughly, the LG based solution for an ABG may be envisioned as mathematical modeling of action-reaction-counteraction-… relationships between all the entities involved in the game. This modeling includes generation of present and predicted future relationships among the entities (up to the end of the engagement). Contrary to other approaches to wargaming, LG-based AI is not satisfied by immediate actions, reactions, etc. It looks far into the future and bases its decisions regarding the immediate actions of entities not on the current state of the battlespace, but upon the predicted impact of these actions in the future. The model of action-reaction-counteraction-… relationships is represented as a network of pathways and counter-pathways for the friendly and adversarial entities, respectively. These pathways lead to actions as well as the counteractions, to be executed in the future, at the end of each corresponding pathway. Thus, all the friendly and adversarial entities are confined to movements only along those pathways (also, called trajectories) like along rail tracks.
A major real-world example for applying LG-based AI is advanced wargaming. Military experts exposed to LG-controlled wargames have been unanimous by testifying that all movements and actions generated by LG-based AI appear to look as if generated by experienced military personnel and, in many cases, ingenious. The past and present STILMAN’s projects demonstrated that LG-based AI is scalable and capable of satisfying the most sophisticated requirements of military experts. Due to basing every immediate decision on the outcome of the entire mission, LG-based AI treats every military operation (to be analyzed, planned or wargamed) as a whole, no matter how many components it includes and how complex their relationships are. Specifically, to achieve such wholistic capability, LG utilizes its ability to generate trajectories for each entity representing transition of the said entity into the future, while combining physical motions and all the actions the entity undertakes as it moves along the trajectory. Then, using the multitude of such trajectories, LG-based AI represents every multi-echelon and multi-domain operation as an automatic evolution of the entire operation’s battlespace (and hence the mission of each wargame contestant) into the future. During this evolution, all entities concurrently move along their assigned trajectories while conducting their LG-generated actions and while receiving partial or annihilating damage from the opposing entities. Such wholistic representation provides powerful analytical means for extracting the details of this evolution including intelligent Courses of Action (COAs) for all sides in the conflict.
The theory of LG identified a wide subclass of tractable (polynomial) problems inside of a class of those generally considered intractable (exponential) problems. LG generates the best-known solutions for its subclass. Note that for concurrent games conventionally understood optimality does not exist. However, for certain subset of serial games, applying the same tools, LG generates provably optimal solutions. Originally, LG was developed by Dr. Stilman as formalization and abstraction of search heuristics of advanced chess experts. Subsequent research conducted by Dr. Stilman revealed that the game of chess was only a means for discovering efficient warfighting strategies developed by humans over thousands of years of bloody wars, whereas those strategies (rediscovered unconsciously by generations of chess masters) served as actual source of LG. This was confirmed by direct, though theoretical, application of LG to recreation of winning strategies utilized by such giants as Hannibal, Alexander the Great, and Caesar.
Essentially, Linguistic Geometry is not about linguistics or geometry. It is about a set of search problems and an approach for the construction of mathematical models for knowledge representation and reasoning about large-scale multi-agent systems. Many such systems can be modeled as ABGs. This class of games includes the game of chess, chess-like games including those with 3D board and concurrent moves, as well as all kinds of real-world military operations. Within the ABG milieu, the purpose of LG is to construct winning strategies, i.e., strategies to guide participants of the game to reach their goals with minimal losses. Before the advent of LG, finding these strategies required searches through giant game trees, which were intractable due to their size. In contrast, LG replaced such searches by direct construction of the strategies in a way similar to what an advanced human player or a military expert does.
Speaking formally, LG is a language for describing and generating winning strategies for ABGs as well as for reasoning about those strategies. LG includes several sub-languages, each of which encodes a specific form of the same approach:
The two major LG languages, Pictorial LG and Analytic LG, are independent of each other to a certain degree. Indeed, the constructs and algorithms of Analytic LG are entirely abstract and are not “aware” of specific shapes and constructs of Pictorial LG. Likewise, the shapes and constructs of Pictorial LG may be created without awareness of the algorithms of Analytic LG intended to manipulate with those shapes. In contrast with such level of independence, the Front End of LG-RAID directly implements Pictorial LG, whereas the Back End does this with Analytic LG. Having said the above, there are certain correspondences between Pictorial LG and Analytic LG for development and execution of each specific scenario. For example, within the Pictorial LG, a viewable physical trajectory of a pictorial representation of a game entity (e.g., a chess piece or a tank) corresponds to a certain “trajectory” represented and generated within Analytic LG in a most abstract form as a string of symbols. However, this generation, which is done by the formal grammar of trajectories (also completely abstract) must utilize specific parameters associated with this game entity, which, in this example, are the tank’s capability to move and apply weapons and sensors, the tanks current ammunition and fuel loads. For instance, the tank cannot move if it is out of fuel and the fuel resupply is not available. Additionally, it can only move at the speed bound by its capabilities with respect to the specific type of terrain it is traversing.
Regarding the relationship with the real world, for LG-based AI’s predictions and guidance to be meaningful, there must be a 1-to-1 correspondence between the real-world entities (to be present in the missions and the battlespace to which LG-RAID is to be applied for planning, wargaming, battle management, etc.) and the collection of Pictorial LG entities to be used within LG-RAID-simulated battlespace. Moreover, the real-world parameters of the real entities must match as closely as possible the parameters of the corresponding Pictorial LG entities such as capability to move and apply weapons and sensors, ammunition and fuel loads, etc. Values of all such parameters could be automatically ingested from the standard military databases. After such an ingest or a manual input, for some parameters a military subject matter expert (SME) must determine which numerical values (out of the provided spread) could be used for a specific operation and which should be corrected. On the other hand, the generation of trajectories and LG zones within Analytic LG must match the application of military doctrines and mission parameters, associated with each real-world contestant, to each type of their military units being modeled by LG. For example, for the US forces, battalion’s heavy weapons must not be moved ahead of the infantry; Rules of Engagement (ROE) may forbid the involved US forces to fire in the direction of a school or a hospital unless being fired upon; certain opposing irregular insurgent forces must not go beyond their area of interest, etc. Finally, elements of Pictorial LG must be transparent to the users, whereas the evolution of the mission and the battlespace initiated by Analytic LG must be understandable to the users from the point of view of advancing the mission goals, compliance with ROE, potential enemy threats, etc.
LG-based AI provides the unmatched capability for generating intelligent evolution of ABGs into the future. To solve a problem with LG-based AI, we must represent this problem as a specific ABG, i.e., introduce ABG for the given problem (instantiate the general ABG). Subsequently, we treat any instantiation of each abstract notion included into the ABG milieu as a constraint on potential arbitrary choices from the set of objects permitted for such instantiation by the general definitions. From the standpoint of ABG, the needed constraints may be of two types, major constraints and additional constraints. The major constraints include specific board, pieces, and relations of reachability (i.e., movement and action patterns). In most of the applications, the additional constraints usually constitute a much larger set of constraints. They may include the game order (i.e., the order of making moves during the game), such as concurrency (vs. alternating moves), various mission definitions, doctrinal and timing constraints, etc. (Roughly speaking, the major constraints are related to instantiation of the notions explicitly defined within the standard definition of ABG, also known as Complex System, whereas the additional constraints are related to instantiation of other notions that may impose limitations on the state transition system associated with the instantiated ABG.) From the Analytic LG point of view, as related to its algorithms, the shape of the board cells, the size of the board, the specific configuration of the reachability patterns as well as other constraints do not matter in the sense that these instantiations do not change the algorithms intrinsic to Analytic LG and hence the corresponding software may not be modified by the inputs. However, these specifics impact the input data for the LG-based AI, and, certainly, the resulting answer, i.e., the generated strategy.
Historically, introductions of the classes of ABG happened in the following order.
For convenience, we will contemplate the introduction of various constraints as the only means for creating ABGs and classes of ABGs. Below, we consider several specific types of constraints and their impact.
The purely general mathematical LG-based AI is “invisible” to the user of the software, i.e., Analytic LG includes just pure formulas. For example, all the components of the general definition of ABG (susceptible to application of LG-based AI), such as Abstract Board, Pieces, Relations of Reachability, and Game States are defined as abstract objects. These objects cannot be visualized unless instantiated. Despite “invisibility”, this formal approach to LG permitted us to prove various theorems of correctness of algorithms for generating LG constructs like trajectories, zones, and translations and formally evaluate their run time. To evaluate the accuracy of solution as well as performance of LG-based AI for a specific problem and effectively control its execution by human subject matter experts (presumed to be non-mathematicians), we must make it “visible” to the users, i.e., apply Pictorial LG. A way to accomplish that is to introduce visualizable ABG instantiations employing appropriate constraints, i.e., ABGs that the experts would be able to visually relate to human experience. We are lucky that most of the problems that have been studied so far for applicability of LG-based AI permit the creation of such visible ABGs. These ABGs include classes 1-3 of ABGs described above.
Chess-like ABGs (Class 2) were introduced and investigated, primarily, to provide a bridge from Chess ABGs to the intended Wargame ABGs (Class 3). However, Chess-like ABGs include a subclass, where ABG’s main components like chess board (with square cells), Chess pieces and rules were “simplified” in comparison with those of the game of Chess. The purpose of that introduction was to provide strategic transparency of the new ABG while removing certain “unimportant” Chess specificities. We call this removal “erasing the particulars”. For example, for the Reti1-like ABGs, such removal included changing the winning condition from checkmate for each side to winning the race to capture the enemy’s immobile piece before the other side would. Another change included removal of the Pawn promotion. The rest of the classic Reti endgame environment such as classic 8×8 board, very small number of mobile pieces and, certainly, serial alternating game order were preserved. This kind of “simplification” provided opportunity to apply visually transparent formal reasoning (at the level of Analytic LG) to the pictorial images representing visualized instantiation of the Reti-like ABG. This application permitted construction of the provably optimal strategy for the Reti-like ABGs employing standard LG tools.
Every ABG from classes 2 and 3 may include multiple additional constraints. For example, Chess-like ABGs may have various complex reachabilities, various levels of concurrency (including total concurrency) and an increased number of pieces. The Wargame ABGs, usually, have the richest sets of additional constraints. They may include multiple terrain features and force structures. It is the specificity of those constraints which permit us focusing the execution of LG-based AI in the desired direction when applying to Wargame ABGs. Various fine changes in control over this desired direction may be caused by varying some of the constraints. When introducing more additional constraints for Wargame ABGs we, primarily, aim to enhance strategic accuracy, although sometimes, to see the overall strategy, we may even need to sacrifice physical fidelity.
Note that the same Analytic LG is capable of solving all the above classes of ABGs and that each said solution is rendered to be explainable and visually transparent using appropriate Pictorial LG languages, which take all the used constraints into account.