A Generalization of Sleep Sets Based on Operator Sequence Redundancy

Pruning techniques have recently been shown to speed up search algorithms by reducing the branching factor of large search spaces. One such technique is sleep sets, which were originally introduced as a pruning technique for model checking, and which have recently been investigated on a theoretical level for planning. In this paper, we propose a generalization of sleep sets and prove its correctness. While the original sleep sets were based on the commutativity of operators, generalized sleep sets are based on a more general notion of operator sequence redundancy. As a result, our approach dominates the original sleep sets variant in terms of pruning power. On a practical level, our experimental evaluation shows the potential of sleep sets and their generalizations on a large and common set of planning benchmarks. Introduction Depth-first search methods such as IDA∗ (Korf 1985) have difficulty in domains with many alternative paths to the same state. Pruning techniques aim to overcome this by identifying operators that need not be applied at a given search node because doing so is certain to produce a state that can be reached by a different path that is no more costly than the current one. This can greatly reduce the search time without sacrificing the optimality of the solution found. One such pruning technique is sleep sets (Godefroid 1996; Wehrle and Helmert 2012). Sleep sets use the commutativity of operators as the basis for pruning decisions. A different technique, move pruning (Holte and Burch 2014), is based on a much richer notion of operator sequence redundancy than commutativity, but is limited to pruning relatively short operator sequences whereas sleep sets can prune arbitrarily long sequences. Consequently, move pruning and sleep sets are incomparable in terms of pruning power: there exist operator sequences that one of them can prune, but the other cannot, and vice versa (Holte and Burch 2014). The first contribution of this paper is an evaluation and comparison of sleep sets and move pruning on a large set of benchmark problems from the international planning competitions. We show that both pruning methods substantially improve IDA∗’s search time and that neither method dominates the other in terms of search time. The second conCopyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. tribution is that we provide several generalizations of sleep sets based on the richer notion of operator sequence redundancy used by move pruning. These generalizations strictly dominate the original concept of sleep sets in terms of pruning power, while preserving completeness and optimality of tree search algorithms like IDA∗. We show that in domains where the more general kinds of redundancy exist, search time is substantially reduced by using the more general methods. Our experiments also show that, in contrast to generalized sleep sets and move pruning, only the basic sleep set method is faster than IDA∗ in terms of total time (i. e., search time + preprocessing time) due to its relatively low computational overhead. Formal Preliminaries As in SAS (Bäckström and Nebel 1995) and PSVN (Holte, Arneson, and Burch 2014), we define a search problem using finite domain variables. States are represented as a mapping of each variable to a value of its domain. Operators are state transformers, consisting of a precondition and an effect. In this paper, we consider SAS operators, where preconditions and effects both consist of variable/value pairs (indicating the required variable values in the preconditions, and the new variable values after applying the operator, respectively). The empty operator sequence is denoted ε. If A is a finite operator sequence then |A| denotes the length ofA (the number of operators in A, |ε| = 0), cost(A) is the sum of the costs of the operators in A (cost(ε) = 0), pre(A) is the set of states to which A can be applied, and A(s) is the state resulting from applying A to state s ∈ pre(A). We assume the cost of each operator is non-negative. A path from state s to state t is an operator sequence A such that s ∈ pre(A) and A(s) = t. A prefix of A is a nonempty initial segment of A (A1...Ak for 1 ≤ k ≤ |A|) and a suffix is a nonempty final segment ofA (Ak...A|A| for 1 ≤ k ≤ |A|). A left-pointing arrow over an operator sequence denotes the prefix consisting of all operators in the sequence except the last. If |A| ≥ 1,←− A = A1 . . . A|A|−1;←−ε is defined to be ε. If O is a total order on operator sequences, B >O A indicates that B is greater than A according toO. Throughout this paper we identify an operator o with the operator sequence of length 1 that consists of only o. Definition 1. A total order, O, on operator sequences is nested if ε O A implies XBY >O XAY for all A,B,X, and Y . Definition 2. A length-lexicographic orderO is a total order on operator sequences based on a total order of the operators o1 O A iff either |B| > |A|, or |B| = |A| and ob >O oa, where ob and oa are the leftmost operators where B and A differ (ob in B, oa in the corresponding position in A). Every length-lexicographic order is a nested order. A total order on operators induces a length-lexicographic order on operator sequences. We will use the same symbol (O) to refer to both the order on operators and the lengthlexicographic order on operator sequences it induces. Definition 3. Given a nested order, O, on operator sequences, for any pair of states s, t define min(s, t) to be the least-cost path from s to t that is smallest according to O. min(s, t) is undefined if there is no path from s to t. Throughout this paper, the sequence of operators comprisingmin(s, t) is o1o2 . . . o|min(s,t)|. The prefix ofmin(s, t) consisting of the first i operators is denoted Pi (P0 = ε), and Qi denotes the suffix of min(s, t) starting at the i operator (Q|min(s,t)|+1 = ε). Hence, min(s, t) = PiQi+1 = Pi−1oiQi+1 for all i, 1 ≤ i ≤ |min(s, t)|. Note that oi is the last operator in Pi.