A Unified Theory of Heuristic Evaluation Functions and its Application to Learning

WC prcscnt a characterization of heuristic evaluation functions Hhich unities their trcatmcnt in single-agent problems and twoperson games. ‘l‘hc central result is that a useful heuristic function is one which dctcrmincs the outcome of a search and is invariant along a solution path. ‘I‘his local chnractcrization of heuristics can hc used to predict the cffcctivcncss of given heuristics and to automatically learn useful heuristic functions for problems. In one cxpcrimcnt, a set of rclntivc weights for the different chess pieces was automatically learned. 1. Int reduction Consider the following anomaly. ‘I’hc Manhattan distance heuristic for the Fifteen PuzAo is computed by monsuring the distance along the two-dimensional grid of each tilt from its current position to its goal position, and summing thtic values for each tile. Manhattan distance is a very cffccticc heuristic function for solving the Fifteen l~uz7lc 141. A complctcly analogous heuristic can bc dcfincd in three dimensions for Rubik’s Cube: for each individual movable piccc of the cube. count the nun;hcr of twists rcquircd to bring it to its goal position and orientation. and sum thcsc vducs for each component. ‘I’hrcc dimcnsionzl Manhattan dizlancc. howcvcr, is cffcctivcly worthless as a heuristic function for liubik‘s Cube 151. l’vcn though Rubik’~ Cube is similar to the t-‘iltccn Puy/.lc, the two heuristics arc virtually idcnIical, and i11 both ci\scs the goal is ilchicvcd when the value of th: heuristic is minimized, the hcurislic is very cffcctivc in one USC and usclcss in the other. As another anomalous cxamplc, consider the games of chcckcrs illld OthCll0 with ITliltCriill c0Ullt ilS iIll evaluation filllcti0n. OlhCll0 is a game played on an eight by tight square grid with picccs which arc white on one side and black on the other. t:,ach player altcrniltcly places picccs with his color showing on empty squares. Wliciicvcr ;I phycr rrli\C’CS his picccs i\t IX)th ClldS Of il lint Of his o~~pcmcnt’s picccs. the opponent’s picccs ;IIY flipped over i\nd hcccmc lllc property ol‘thc Wigillill plityCl+. ‘I’hc winner is the player whocc COIOI~ shows on the majority of the picccs iit the end of the gatnc. Material count is ;LII cv,lluation function which sums the number of picccs hclonging to OIIC player and subtracts the total material of the other player. It turns out that lTliltCl3;ll count is a f,tirly successful evaluation function for chcckcrs but rclativcly incffcctivc for OthCll0, CVCn lllougll tllC winiicr is tllC PlilyCr tllilt maximi/.cs his nliltcrial in both GISCS.*