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On Computational Complexity of Automorphism Groups in Classical Planning Proceedings of the Twenty-Ninth International Conference on Automated Planning and Scheduling (ICAPS 2019) On Computational Complexity of Automorphism Groups in Classical Planning Alexander Shleyfman Technion, Haifa, Israel [email protected] Abstract 2008), a rather direct corollary of that result is that com- Symmetry-based pruning is a family of powerful methods for puting the automorphism group of explicitly given state- reducing search effort in planning as heuristic search. Apply- transition graphs of classical planning problems is reducible ing these methods requires first establishing an automorphism to the Graph Isomorphism problem (GI-hard). However, group that is then used for pruning within the search process. since the state-transition graph in classical planning is worst- Despite the growing popularity of state-space symmetries in case exponential in the size of its compact representation, planning techniques, the computational complexity of finding and since the work on symmetries in planning is focused on the automorphism group of a compactly represented planning symmetries derived from such compact representations, the task has not been formally established. In a series of reduc- relevance of this complexity result is rather limited. tions, we show that computing the automorphism group of a The first notion of grounded symmetries for classical grounded planning task is GI-hard. Furthermore, we discuss planing was proposed by Pochter et al. (2011), and then re- the presentations of these symmetry groups and list some of et al. their drawbacks. fined by Domshlak (2012). The definitions presented in these works, practical however they are, were based on the notion of colored graphs, and thus are quite cumbersome Introduction to reason about. Later on, Shleyfman et al. (2015) came up Symmetry breaking is a method for search-space reduction with the notion of structural symmetries that captures pre- that has been well explored across several areas in computer viously proposed concepts, and which can be derived from science, and in particular in classical planning (Starke 1991; the syntax of a planning task in a simple declarative man- Emerson and Sistla 1996; Fox and Long 1999; Rintanen ner. Still, to our knowledge, the complexity of computing the 2003; Pochter, Zohar, and Rosenschein 2011; Domshlak, automorphism group of a grounded planning task remained Katz, and Shleyfman 2012; Gnad et al. 2017). Symmetry- open. based pruning divides the states in the search space into In this work, we present reductions to graphs that estab- orbit-based equivalence classes, which in turn allows for lish a “negative” result that computing automorphism groups exploring only one representative state per such class. Ap- for grounded planning tasks (in both FDR and STRIPS for- plication of this technique to forward search partially cur- malisms) is as hard as for general undirected graphs. Along tails the exponential growth of the search space in the this route, we also show a nontrivial connection between presence of objects with symmetric behavior. Beyond the the automorphism groups of a planning task and its causal state-space pruning, symmetries have been also success- graph, and discuss the presentations of the groups of classi- fully used in classical planning to enhance performance of cal planning tasks, as well as the suitableness of group gen- heuristics (Domshlak, Katz, and Shleyfman 2013; Sievers et erators for representation of the group’s structure and size. al. 2015b), prune redundant operators (Wehrle et al. 2015; Fiser,ˇ Alvaro´ Torralba, and Shleyfman 2019), and even de- Background compose planning tasks (Abdulaziz, Norrish, and Gretton To define a planning task we use the finite-domain repre- 2015). sentation formalism (FDR) (Backstr¨ om¨ and Nebel 1995; In the foundations of all these techniques lies computing Helmert 2006). Each planning task is given by a tuple a subgroup of automorphisms of the state-transition graph Π = hV; A;I;Gi, where V is a set of multivalued vari- of the problem. Considering the complexity of finding auto- ables, each associated with a finite domain D(v). The sets morphism groups of state-transition graphs, Juntilla (2003) of variable/value pairs are written as hvar; vali, and some- showed that the problem of finding symmetry groups in times referred as facts.A state s is a full variable assign- Petri nets is equivalent to the graph automorphism prob- ments which maps each variable v 2 V to some value in lem. Given the established connections between reachabil- its domain, i.e. s(v) 2 D(v). For V ⊆ V, s[V ] denotes ity problem in Petri nets and classical planning (Bonet et al. the partial assignment (also referred as a partial state) of Copyright c 2019, Association for the Advancement of Artificial s over V . Initial state I is a state. The goal G is a par- Intelligence (www.aaai.org). All rights reserved. tial assignment. Let p be a partial assignment. We denote 428 by vars(p) ⊆ V the subset of variables on which p is Structural Symmetries defined. For two partial assignments p and q, we say that p satisfies q, if vars(q) ⊆ vars(p), and p[v] = q[v] for The second ingredient we need was introduced by Shleyf- all v 2 vars(q), this is denoted by p j= q. A is a fi- man et al. (2015). This subsection defines the notion of nite set of actions, where each action is represented by a structural symmetries, which captures previously proposed triplet hpre(a); eff(a); cost(a)i of precondition, effect, and concepts of symmetries in classical planning. In short, struc- cost, where pre(a) and eff(a) are partial assignments to V, tural symmetries are relabelling of the FDR of a given plan- and cost(a) 2 R0+. In this work we assume all actions are ning task Π. Variables are mapped to variables, values to of unit-cost, unless stated otherwise. An action a is appli- values (preserving the hvar; vali structure), and actions are cable in a state s if s j= pre(a). Applying a in s changes mapped to actions. In this work, we follow the definition of the value of all v 2 vars(eff(a)) to eff(a)[v], and leaves s structural symmetries for FDR planning tasks as defined by unchanged elsewhere. The outcome state s0 is denoted by Wehrle et al. (2015). For a planning task Π = hV; A;I;Gi, s a . let P be the set of Π’s facts, and let PV := ffhv; di j d 2 JSK denotes the set of all states of Π. We say that action D(v)g j v 2 Vg be the set of sets of facts attributed to each sequence π is a plan, if it begins in I, ends in sG s.t. sG j= G, variable in V. We say that a permutation σ : P [A! P [A and each action in π is iteratively applicable, i.e. for each is a structural symmetry if the following holds: a 2 π holds that s j= pre(a ) and s a = s . The i i−1 i i−1 i i 1. σ(P ) = P , cost of a plan is defined as cost(π) = P J costK (a ). An V V ai2π i optimal plan, is a plan of a minimal cost. Here, since we 2. σ(A) = A, and, for all a 2 A, σ(pre(a)) = pre(σ(a)), assumed unit-cost domains, an optimal plan is a plan of the σ(eff(a)) = eff(σ(a)), and cost(σ(a)) = cost(a). shortest length. The state space of Π is denoted TΠ. A directed graph is a pair hN; Ei where N is the finite 3. σ(G) = G. set of vertices, and E ⊆ N 2 is the set of edges, where each edge is an ordered pair of vertices. Aloop (sometimes re- We define the application of σ to a set X by σ(X) := ferred as self-loop) is a directed edge from a vertex to itself. fσ(x) j x 2 Xg, where σ is applied recursively up to the In what follows, we will consider only simple graphs, i.e. level of action labels and facts. For example, let s be a partial state, since s can be represented a set of facts. Applying σ to graphs with no loops and no parallel edges. A directed graph 0 directed acyclic graph s will result in a partial state s , s.t. for all facts hv; di 2 s it with no cycles will be called (DAG). 0 0 0 0 0 0 An undirected graph is a pair hN; Ei where N, once holds that σ(hv; di) = hv ; d i 2 s and s [v ] = d . again, is the set of vertices, and E ⊆ fe ⊆ N j jej = 2g is A set of structural symmetries Σ for a planning task Π the set of edges. induces a subgroup Γ of the automorphism group Aut(TΠ), Let G = hN; Ei be a (un-)directed graph, and let σ be which in turn defines an equivalence relation over the states a permutation over the vertices N. We say that σ is a graph S of Π. Namely, we say that s is symmetric to s0 iff there automorphism (or just an automorphism, if this is clear from exists an automorphism σ 2 Γ such that σ(s) = s0. The the context) when (n; n0) 2 E iff (σ(n); σ(n0)) 2 E. The group of all structural symmetries of Π will be denoted by definition of a graph automorphism for an undirected graph Aut(Π). is almost the same, where fn; n0g 2 E iff fσ(n); σ(n0)g 2 E.
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