Some Theoretical Results Concerning Automated Game-Playing

Though evaluat ion funct ions have been used in many game-playing and problem-solving programs the aim has genera l ly been to const ruct su i tab le funct ions fo r p a r t i c u l a r s i t u a t i o n s . A formal framework is ou t l ined in which general proper t ies of evaluat ion functions can be s tud ied. An app l i ca t ion of these concepts is made to the study of when decis ion ru les (as exh ib i ted by a players ac tua l p lay) can be represented by evaluat ion funct ions 1 . In t roduc t ion . When computers have been used to p lay games such as chess, checkers e t c . , e x p l i c i t use of eva luat ion funct ions (most f requent ly l i near in form) has usua l ly been made. They have also featured in the so lu t i on of non-game problems. In other cases a dec is ion ru le fo r p lay ing games (and so lv ing problems) has been used, but it is shown in 3 how decis ion ru les are in many cases, equivalent to the use of su i tab le EF's (from now on the abbrev ia t ion EF w i l l be used fo r eva luat ion f unc t i on ) . EF s have been used in many circumstances and in varying ways and l i e at the heart of programming machines to play games and solve problems, and t h e i r importance was stressed as ear l y as 1950 by Shannon. However, on the whole they have been introduced fo r spec i f i c app l ica t ions and t h e i r meaning and proper t ies neglected other than t ha t a pos i t i on or s tate w i t h a higher value is l i k e l y to be a be t t e r one. I t is the purpose of t h i s paper to introduce a formal framework in which a general study of EF's can be made. The f u l l length vers ion of t h i s paper shows how t h i s framework can be used when studying e x i s t i n g techniques and how i t is p o t e n t i a l l y usefu l fo r i n t r o ducing new techniques. In t h i s paper the basic concepts of EF's are presented and then a s ingle app l i ca t i on is choBen fo r f u r the r cons iderat ion, namely the study of when a players ac tua l play can be represented by an EF. The main resu l t s are given in Theorems 3.1.3, 3.2.4 and 3 .2 .2 . 2. Basic Propert ies of Evaluat ion Functions. A game usua l l y consists of a set of d e f i n i t e states together w i t h t r ans i t i ons between them. Consequently i t is na tu ra l to represent a game by means of a d i rec ted grapht (X, P) whose ve r t i ces correspond to the states or p o s i t i o n ! and whose arcs correspond to the possible t r ans i t i ons or moves. However, it may be of importance to consider the development of a game (or the nature of the so lu t i on path in problem-solving) in which case the more usual formulat ion in terms of a (rooted) t ree (V, T) is l i k e l y to be advantageous. (V, T) can be obtained from ( X , P ) by s t a r t i n g at the root (the s t a r t pos i t i on ) and 'growing' the tree according to the successor mapping except t ha t a ver tex is never i d e n t i f i e d w i t h a prev ious ly considered one. (X, P) w i l l be used when the discussion concerns graphs (and For graphtheoret ic concepts see Berge.