A Linguistic Geometry of the Chess Model

In order to discover the inner properties of human expert heuristics, which are successful in computer chess, to investigate these heuristics, improve them and, further, apply to various complex control systems, we develop a formal theory, the so-called Linguistic Geometry. This research includes the development of syntactic tools for knowledge representation and reasoning about the game of chess as about a hierarchical complex system. It relies on the formalization of search heuristics, which allow to decompose this game into the hierarchy of subsystems, and thus solve this problem reducing the search drastically. The hierarchy of subsystems is represented as a hierarchy of formal attribute languages . Syntactic tools generating the hierarchy of languages, the controlled grammars, are introduced in this paper. The performance of the grammar generating paths of pieces is considered in detail. The generation of the entire hierarchy of languages is considered on example of the R.Reti endgame. INTRODUCTION The group led by Prof. Botvinnik attacked the computer chess problem by simulating the method of a human chess master. The main idea was to replace a one-level complex system, as the game of chess use to be seen, by a multi-level hierarchical model that could allow us to break the system into subsystems, to search these subsystems separately and in combinations, and then combine optimal solutions for subsystems as approximately optimal for the entire system. This approach reduced the search drastically, down to a hundred of moves. However, the main difficulty was transferred from the development of efficient search methods to the design of methods for implementation of this decomposition. Basically, the questions are as follows. What are these subsystems? What are the most efficient knowledge representation mechanisms to handle them? Is this decomposition dynamic, i.e., does it change when we move from one position to another during the search? If so, how to generate new subsystems, how to understand that subsystem is obsolete and destroy it, how to regenerate old subsystem by changing it, and, thus, avoid tremendous recomputation. How to organize the search within the subsystem and evaluate the results? These and many other questions have been answered in our research. However, some answers are ambiguous, some questions are still open. Different subsystems were developed. They are called trajectories, zones, chains of trajectories, and so on. The results were presented by Botvinnik (1970, 1975, 1984) with contributions of members of research team. Of course, many interesting advances were made later. The language used for the development of the model and presentation of the results was either the language of plausible discussions (for general ideas) or the lower-level algorithmic language close to programming languages (for the description of implementations). There was a huge gap between these two languages. The model itself could not be expressed adequately. The language of plausible discussions is very ambiguous for such a complex subject, while an algorithmic language is too detailed, and, thus, useless as well. This inadequacy caused many problems, and not only for presentation. Following Russian proverb, often “we could not see a forest behind the trees.” It seems that each new result was not really built into the model but simply added to it violating previous results. The model looked like a collection of bright ideas, complex algorithms, and interesting results. We needed a mathematical skeleton, something like a search tree for conventional search models. For many years I was looking for adequate mathematical tools. Eventually it resulted in the development of a Hierarchy of Formal Languages (Stilman, 1985) that emerged later into a Linguistic Geometry (Stilman, 1992-1993). These formal, expressive tools are intended for investigation and development of all the results achieved in the PIONEER project. In particular, employing these tools I answer in a formal way the questions listed above. Based on that I investigate our later results, develop them and go ahead. These new tools allowed not only to present the results formally, but to prove the correctness of algorithms and data structures. One of the newest results on correctness of computation of the piece planning path is considered in this paper. This research already allowed to demonstrate the errors and correct them in algorithms, which had been implemented as subroutines of the PIONEER program and executed for a long time. These were some principle mistakes in the adjustment of the model in the process of search. It is very unlikely they could be found without formal methods. Some advances were made in the evaluation of the computational complexity of this model. I hope in the future to approach a formal evaluation of the accuracy of the solutions. Actually, employing these tools I hope to answer the most intriguing question: why the PIONEER finds solutions of complex positions, why it fails other cases, and what should be corrected. The application of Linguistic Geometry is far beyond the framework of the chess problem. There are many real-world problems where human expert skills in reasoning about complex systems are incomparably higher than the level of modern computing systems. At the same time there are even more areas where advances are required but human problem-solving skills can not be directly applied. For example, there are problems of planning and automatic control of autonomous agents such as space vehicles, stations and robots with cooperative and opposing interests functioning in a complex, hazardous environment. Reasoning about such complex systems should be done automatically, in a timely manner, and often in a real time. Moreover, there are no high-skilled human experts in these fields ready to substitute for robots (on a virtual model) or transfer their knowledge to them. There is no grand-master in robot control, although, of course, the knowledge of existing experts in this field should not be neglected – it is even more valuable. Due to the special significance of these problems, the quality of solutions must be very high and usually subject to continuous improvement. Thus, it is very important to study human expert reasoning about similar complex systems in the areas where the results are successful, e.g., in computer chess, in order to discover the keys to success, and then apply and adopt these keys to the new, as yet, unsolved problems. It should be considered as investigation, development, and consequent expansion of advanced human expert skills into new areas. The Linguistic Geometry provides formal tools for this investigation and transfer. THEORETICAL BACKGROUND In the 1960’s a formal syntactic approach to the investigation of properties of natural language resulted in the fast development of a theory of formal languages by Chomsky (1963), Ginsburg (1966), Knuth (1968), Rozenkrantz (1969), and others. This development provided an interesting opportunity for dissemination of this approach to different areas. In particular, there came an idea of analogous linguistic representation of images. This idea was successfully developed into syntactic methods of pattern recognition by Fu (1982), Narasimhan (1966), and Pavlidis (1972), and picture description languages by Shaw (1969), Feder (1971), Phaltz and Rosenfeld (1969). The power of a linguistic approach might be explained, in particular, by the recursive nature and expressiveness of language generating rules, i.e., formal grammars. Searching for the adequate mathematical tools formalizing human heuristics of dynamic hierarchy for the game of chess, we have transformed the idea of linguistic representation of complex real-world and artificial images into the idea of similar representation of complex hierarchical systems (Stilman, 1985). However, the appropriate languages should possess more sophisticated attributes than languages usually used for pattern description. They should describe mathematically all of the essential syntactic and semantic features of the system and search, and be easily generated by certain controlled grammars. The origin of such languages can be traced back to the origin of SNOBOL-4 programming language and the research on programmed attribute grammars and languages by Knuth (1968), Rozenkrantz (1969), and Volchenkov (1979). A mathematical environment (a “glue”) for the formal implementation of this approach was developed following the theories of formal problem solving and planning by Fikes (1971), Nilsson (1980), Sacerdoti (1975), and McCarthy, Hayes (1969), and others (Stefik, 1981, Chapman, 1987, McAllester at al, 1991) based on first order predicate calculus. CLASS OF PROBLEMS A class of problems to be studied are problems of optimal operation of a complex system. This system is considered as a twin-set of elements and points(locations) where elements are units moving from one point to another. The elements are divided into two opposite sides; the goal of each side is to attack and destroy opposite side elements and to protect its own. Each side aims to maximize a gain, the total value of opposite elements destroyed and withdrawn from the system. Such a withdrawal happens if an attacking element comes to the point where there is already an element of the opposite side. Obviously, the game of chess is such a problem with pieces substituting for elements and squares of the chess board as points. A formal definition is as follows. A Complex System is the following eight-tuple: < X, P, Rp, {ON} , v , S i, S t, TR>, where X={xi} is a finite set of points; P={pi} is a finite set of elements; P is a union of two non-intersecting subsets P1 and P2; Rp(x ,y) is a set of binary relations of reachability in X (x and y are from X, p from P); ON(p)=x, where ON is a partial function of placement from P into X; v is a function on P with positive integer