Internacional

Algorithms for finding a path or the shortest path between two vertices in a digraph were devised since the middle of last century. The earliest algorithm that we are aware of, the Bellman-Ford algorithm Bellman (1958), finds the shortest path between two vertices in a time linear in the product of the edges and vertices of a digraph. Many algorithms were introduced after that like the ones in Dijkstra (1959); Fredman and Tarjan (1987); Thorup (2003). All of those algorithms assume that the digraph is represented explicitly, and also they are all at least linear in both, the number of vertices and the number of edges. Accordingly a naive application of these algorithms to finding paths in state spaces of factored systems would be infeasible.

Compositional Planning and Model Checking

The compositional approach was employed to find solutions to AI planning and model check- ing problems in work like Knoblock (1994); Williams and Nayak (1997); Berezin et al. (1998); Case et al. (2009); Kroening (2006); Amir and Engelhardt (2003); Brafman and Domshlak (2006); Kelareva et al. (2007); Guere and Alami (2001); Helmert et al. (2014); Sievers et al. (2015). In those works, different types of abstractions and structures were used, and only the state space of a projected system is searched for a path. For example, abstractions based on projection were used in Knoblock (1994); Williams and Nayak (1997), which exploited acyclicity in variable depen- dencies. Another approach for compositional path search, known as factored planning, is used by Amir and Engelhardt (2003); Brafman and Domshlak (2006); Kelareva et al. (2007), in which the factored systemδis abstracted into multiple abstractions, referred to as “factors”. These fac- tors are obtained by an abstraction that, given a partition of the set of state variables vs1..n, it

produces a set of sets of actions, one pervsi ∈ vs1..n, such that the set of actions refers only to

variables invsi. This partition of state variables is based on a tree decomposition of an undirected

version of the variable dependency graph.

Exploiting Symmetries

Symmetries that occur in factored transition systems have for some time been exploited for effi- cient search for paths between states, both by AI planning and model checking communities. The quintessential planning scenario which exhibits symmetries is GRIPPER. This comprises a robot whose left and right grippers can be used interchangeably in the task of moving a set ofNindistin- guishable packages from a source location to a goal location. Intuitively the left and right grippers are symmetric because if we changed their names, by interchanging the terms left and right in the problem description, we are left with an identical problem. Packages are also interchangeable and symmetric.

One method to exploit symmetry is to perform search in a minor of the digraph modelling the state space, referred to as aquotient system, which corresponds to a (sometimes exponentially) smaller bisimulation of the system at hand. Planning in a quotient system, a statexis represented by acanonicalelement from itsorbit, the set of states which are symmetric tox. Giving integer labels to packages in GRIPPER, when the search encounters a state where the robot is holding 1 package using 1 gripper, this is represented using the canonical state where, for example, the left gripper is holding package with identity “1”. Orbit search explores the quotient system by

§4.2 Results 51

simulating actions from known canonical states, and then computing the canonical representa- tions of resultant states. That canonicalisation step requires the solution to theconstructive orbit problemClarke et al. (1998) which isNP-hardEugene (1993). A key weakness, is that for each state encountered by the search an intractable canonicalisation operation is performed. This is mitigated in practice by using approximate canonicalisation. By forgoing exact canonicalisation, one encounters a much larger search problem than necessary. For a GRIPPER instance with 42 packages, the breadth-first orbit search with approximate canonicalisation by Pochter et al. (2011) reportedly performs60.5K state expansion operations, far more than necessary. This method of exploiting symmetry was firstly used in solving model checking problems, which is reviewed by Wahl and Donaldson (2010). Also, this method was recently adapted for planning and studied by Pochter et al. (2011), Domshlak et al. (2012, 2013) and Shleyfman et al. (2015). Related work about state-based planning in equivalence classes of symmetric states includes Guere and Alami (2001); Fox and Long (1999, 2002).

Following the seminal work by Crawford et al. (1996) and Brown et al. (1996), when search- ing for a path (in the context of AI planning)via constraint satisfaction, known symmetries are exploited by:

(i) including symmetry breaking constraints, either directly as part of the problem expres- sion Joslin and Roy (1997); Rintanen (2003), or

(ii) otherwise dynamically as suggested by Miguel (2001) as part of anogoodmechanism. InGRIPPER, we can statically require that if no package is yet retrieved, then any retrieval targets package 1 with the left gripper. Dynamically, having proved that no 3-step plan exists retrieving package 1 with the left gripper, then no plan of that length exists retrieving packagei6= 1using either gripper. Searching using the proposed dynamic approach is quite weak, as symmetries are only broken as nogood information becomes available. A weakness of both approaches is that problems expressed as CSPs include variables describing the state of the transition system at different plan steps. Existing approaches do not break symmetries across steps, and can therefore waste effort exploring partial assignments that express executions which visit symmetric states at different steps.

Another method to exploit symmetries is via abstracting the factored system at hand. Some authors perform search in the state space of acounter abstraction, as surveyed by Wahl and Don- aldson (2010). That approach treats a transition system isomorphic to the quotient system, but it avoids solving the NP-hard constructive orbit problem. Symmetries were also explored in fac- tored planning in the context ofmerge-and-shrinkheuristics Helmert et al. (2014). Sievers et al. (2015) developed a symmetry guidedmergingoperation which yields relatively compact heuristic models, improving the scalability of that approach. Guere and Alami (2001) propose to plan via ashape-graph, a compact description of the problem state space in which states are represented by equivalence classes of symmetric states. As well as planning by searching in that graph, using the diameter of theshape-graphGuere and Alami are able to calculate tight upper bounds for the highly symmetricGRIPPERandBLOCKS-WORLDdomains.

4.2

Results

We consider the AI planning problem defined on factored transition systems, i.e. given an initial stateI ∈_U(δ)and a set of goal statesG, is there an action sequence that reaches one of the goal states if it is executed at the initial state.

Our contribution is a new abstraction of factored transition systems, the result of which we call adescriptive quotient. We provide conditions under which searching for a path between the initial state and a goal state (i.e. a plan) in the digraph modelling the state space of the descritptive quotient can be used to synthesise a path between the initial state and some goal state in the original system. The first condition is that the partition used to obtain the descriptive quotient is induced by the symmetry relation between state variables (a.k.a. orbits of a subgroup of the automorphism group). The second condition is that the descriptive quotient is isomorphic to a “sub-system” of the original factored system. Informally, those two conditions mean that planning via descriptive quotients is a way to exploit “repetitive symmetries” in a factored system, where by repetitive symmetries we mean that the factored system is constituted by a union of isomorphic sub-systems.

Based on that we provide a novel procedure for domain-independent planning with symme- tries. Following, e.g., Pochter et al., in a first step we infer knowledge about the symmetries be- tween state variable. Then departing from existing approaches, our second step uses that knowl- edge to obtain a quotient of the concrete factored system. Called the descriptive quotient, this describes any element in the set of isomorphic subsystems which abstractly model the concrete system. Third, we invoke a planner once to solve the small reachability problem posed by that descriptive quotient. In the fourth and final step, a concrete plan is synthesized by concatenating instantiations of that plan for each isomorphic subproblem.

The non-existence of a plan for the descriptive quotient does not exclude the possibility of a plan in the concrete system. Although sound, in that respect our approach is incomplete. Having an optimal plan for the quotient does not guarantee optimality in the concatenated plan. Aside from computing a plan for the descriptive quotient, the computationally challenging steps occur in preprocessing:

(i) Identification of state variable symmetries from the original description, a problem as hard as graph isomorphism, which is not known to be tractable, and

(ii) Computing an appropriate set of subsystems isomorphic to the quotient.

We introduce the general version of the latter problem: for an undirected graphG and a partition of its verticesP, is the quotientG/P isomorphic to a subgraph ofG, such that the morphism maps every set of vertices fromP to one of its members? We show that this problem is NP-complete. We also show that ifP is a set oforbitsof a symmetry group forG, then there is set of morphisms fromG/P toGthat covers all the vertices ofG, and whose size is logarithmic in the vertices ofG. Unlike existing approaches, our search for a plan does not need to reason about symmetries between concrete states and the effects of actions on those. Plan search can be performed by an off-the-shelf SAT/CSP system, in which casesymmetry breakingconstraints are not required. Al- ternatively, using a state-based planner we avoid repeated (approximate) solution to the intractable canonicalisation problem, a clear bottleneck of recent planning algorithms. In this respect, our approach is similar to searching in a counter abstraction, as surveyed by Wahl and Donaldson (2010). Also, viewing our approach as one that decomposes a problem into subproblems, it is re- lated to factored planning Amir and Engelhardt (2003); Brafman and Domshlak (2006); Kelareva et al. (2007).

The most important advantage of searching in the digraph modelling the state space of the descriptive quotient is that it is a minor that can be very compact relative to the digraph modelling the state space of the original factored system. For instance, a counter abstraction has a state space which can be exponentially larger than that of a descriptive quotient—i.e., the descriptive quotient will model 1 object, whereas the quotient transition system modelsN symmetric objects. Also,

§4.3 Planning Problems and Additional Notation 53