Common Misconceptions Concerning Heuristic Search

This paper examines the following statements about heuristic search, which are commonly held to be true: More accurate heuristics result in fewer states being expanded by A* and IDA*. A* expands fewer states than any other equally informed algorithm that finds optimal solutions. Any admissible heuristic can be turned into a consistent heuristic by a simple technique called . In search spaces whose operators all have the same cost A* with the heuristic function for all states, , is the same as breadth-first search. Bidirectional A* stops when the forward and backward search frontiers meet. The paper demonstrates that all these statements are false and provides alternative statements that are true. Introduction Heuristic search is one of the pillars of Artificial Intelligence. Sound knowledge of its fundamental results and algorithms, A* (Hart, Nilsson, & Raphael 1968) and IDA* (Korf 1985), is requisite knowledge for all AI scientists and practitioners. Although its basics are generally well understood, certain misconceptions concerning heuristic search are widespread. In particular, consider the following assertions about heuristic search: If admissible heuristic is more accurate than admissible heuristic fewer states will be expanded if A* and IDA* use than if they use . A* is optimal, in the sense of expanding fewer states than any other equally informed algorithm that finds optimal solutions. Any admissible heuristic can be turned into a consistent heuristic by a simple technique called ! "# $ . In search spaces whose operators all have the same cost A* with the heuristic function &%('*),+.for all states, ' , is the same as breadth-first search. Bidirectional A* stops when the forward and backward search frontiers meet. Although these statements are intuitively highly plausible and are widely held to be true, as a matter of fact, they are all false. The aim of this paper is to demonstrate how they fail with simple counterexamples and to provide alternatives to these statements that are true. The falsity of the above statements, and the corrections given, have all have been reported previously, but usually in specialized publications. The contributions of this paper are to draw attention to them, to bring them together in one widely accessible document, and to give simple counterexamples. Background, Terminology, and Notation This section briefly reviews the terminology, notation, and essential facts needed in the remainder of the paper. It is not a full tutorial on heuristic search. A state space is a set of states, a successor relation defining adjacency between states, and a cost function, / 01'2 %(' 3! !) , defining the cost of moving from state ' to adjacent state . A* and IDA* are algorithms for finding a best (least-cost) path from any given state, '2 4 5* , to a predetermined goal state, 6 0 7 . Both make use of three functions, 6 , and 8 . 6 %9':) is the cost of the best known path from '2 4 5* to state ' at the current stage of the search. &%('*) , the heuristic function, estimates the cost of a best path from state ' to 6;0 7 . 8<%('*)=+>6?%('*) @A &%('*) is the current estimate of the minimum cost of reaching 6;0 7 from '2 4 5* with a path passing through ' . The true minimum cost of a path from '2 4 5* to 6 0 7 is denoted 8CB . All heuristic functions are non-negative and have %D6 0 7()E+>. Heuristic &%('*) is admissible if, for every state ' , it does not overestimate the cost of a best path from ' to 6 0 7 . A* and IDA* are guaranteed to find a least-cost path from 'F 4 5* to 6 0 7 if &%('*) is admissible. %9':) is consistent (p. 83, (Pearl 1984)) if for every two states1, ' and , &%('*)HGI/ 01'2 %(' 3! !)C@A &%D !) (1) A consistent heuristic is guaranteed to be admissible. A* maintains a list of states called J,KMLON . On each step of its search A* removes a state in J,KMLPN with the smallest Pearl (1984) showed that restricting to be a neighbour of produces an equivalent definition that is easier to verify in practice and has an intuitive interpretation: in moving from a node to its neighbour must not decrease more than Q increases. 8 -value and “expands” it, which means marking the state as “closed”, R computing its successors, and putting each successor in J,KMLON if it has not previously been generated or if this path to it is better than any previously computed path to it. A* terminates as soon as 6 0 7 has the smallest 8 -value in J,KMLON . When A* is executed with a consistent heuristic, the 8 -values of the nodes it expands as search progresses form a monotone non-decreasing sequence. This is not true, in general, if A* is executed with an inconsistent heuristic. IDA* does a series of cost-bounded depth-first searches. When searching with a specific cost bound S , state ' is ignored if 8<%('*)UTVS . If 8<%('*)UGWS the successors of ' are searched in a depth-first manner with the same cost bound. IDA* terminates successfully as soon as 6 0 7 is reached by a path whose cost is less than or equal to S . If search does not terminate successfully with the current value of S , the smallest 8 -value exceeding S seen during the current iteration is used as S ’s new value for a depth-first search that begins afresh from '2 4 5* . Better Heuristics Can Result in More Search One admissible heuristic, , is defined to be “better than” (or “dominate”) another, , if for all states, ' , %('*)XG Y %('*) and there exist one or more states for which %9'*)[Z %('*) . Is A* guaranteed to expand fewer states when it is given the better heuristic ? Although intuition urges the answer “yes”, and there do exist provable connections between the accuracy of a heuristic and the number of node expansions A* will do (Dinh, Russell, & Su 2007), the true answer is much more complex (pp. 81-85, (Pearl 1984)). Even when the heuristics are consistent, the answer is not an unequivocal “yes” because with the better heuristic A* might expand arbitrarily more states that have 8<%('*)E+>8CB .