Foundations of Linguistic Geometry: Complex Systems and Winning Strategies

both the branching factor and the computation time for the systems based on LG were several times smaller than those for the competition (Stilman, 1994a). Note that although LG has a discrete nature, it has been applied to some continuous systems as well through an ad hoc discretization of the latter (Stilman, 19941995). The Linguistic Geometry (LG) approach to discrete systems was introduced B. Stilman in early 80s. It employed competing/cooperating agents for modeling and controlling of discrete systems. The approach was applied to a variety of problems with huge state spaces including control of aircraft, battlefield robots, and chess. One of the key innovations of LG is the use of almost winning strategies, rather than truly winning strategies for the playing agents. There are many cases where the winning strategies have so high time complexity that they are not computable in practice, whereas the almost winning strategies can be applied and they beat the opposing agent almost guaranteed. Independently of LG the idea of competing/ cooperating agents was employed in the late 80s by A. Nerode, A. Yakhnis, and V. Yakhnis (NYY) within their approach to modeling concurrent systems and, more recently, within the “Strategy Approach to Hybrid Systems” developed for continuous systems by A. Nerode, W. Kohn, A. Yakhnis, and others. In order to enlarge the range of applications of Stilman's results as well as to understand why they work, we introduce a new notion of a multi-agent graph-game. Since the applications of Linguistic Geometry sometimes require modeling of agents making simultaneous moves, the new notion was designed to accommodate such behavior of players. The computational power of LG is based on a utilization of human expert heuristics which were highly successful in a certain class of complex control systems. However, in contrast to the other approaches based on the idea of “two competing agents”, these heuristics made the mathematical essence of LG extremely complex and thus less obvious. After the present investigation we expect to extend the application domain of LG to the area of verification of concurrent systems (through the NYY approach) and to the operating systems supporting persistent truly concurrent objects. Presently an approach to the latter area is being developed (Yakhnis-Yakhnis, in preparation). It is partially based of the NYY ideas. We also expect that the Linguistic Geometry would have more extensive applications to continuous systems via the “Strategy Approach to Hybrid Systems” developed by A. Nerode, W. Kohn, A. Yakhnis (NKY), and others (Kohn-Nerode, 1993; Nerode-Remmel-Yakhnis,1993; Nerode-Yakhnis, 1992). This is possible since the latter approach eventually converts continuous systems into discrete ones.