Métodos alternos de solución de conflictos

Definition 35 (Common Variables). For a set of instantiations ∆, the set of common variables, written T

v∆, comprises variables in S

t∈∆R(t) that occur in the ranges of more than one

member of∆.

Definition 36 (Sustainable Variables). A set of variables vs is sustainable in a problem Π iff Ivs =Gvs.

Theorem 18. Take problem Π, partition vs1..n of D(Π) where the quotient Π/vs1..n(= Π0) is

well-defined, with solution →π0, and consistent instantiations∆. Suppose {_tLΠ0M | _t ∈ ∆}(=

Π) covers Π, and QLT

v∆M∩ D(N(Π0))—i.e., based on Definition 30, the orbits of common

variables from needed assignments—are sustainable inΠ0. Then any concatenation of the plans

{rem-condless(N(tLΠ0M),tL→π0M|_t∈∆}solvesΠ.

Proof. Identify the elements in∆by indices in{1..|∆|}. Let k ∈ {1..|∆|}andΠ_k = tkLΠ0M,

and noteR(_tk) = D(Π_k). Taketi,tj ∈ ∆, wherei, j ∈ {1..|∆|}andi 6= j. We first show

thatΠ_i Π_j. For any vs ∈ vs1..n, ti(vs) = tj(vs) ifti(vs) ∈ R(ti)∩ R(tj). Therefore,

IiΠj = IjΠi andGiΠj = GjΠi, providing the right conjunct in Definition 34. For the left

conjunct, noteD(N(Πj))⊆ D(Πj)andD(Πi)∩ D(Πj) ⊆T_v∆. The sustainability premise— Q(T

v∆)∩ D(N(Π

0_{))is sustainable inΠ}0_{—then provides thatD(Πi)∩ D(N(Π}

j))is sustainable

inΠ_i—i.e.,IiN(Πj) = GiN(Πj). ThusGiN(Πj) = (IjΠi)N(Πj), and we concludeΠi Πj.

Since a plantkL →

π0MsolvestkLΠ0M,(tkLΠ0M).I ⊆Π.I, therefore as per Theorem 17 a solution to Πisrem-condless(N(t1LΠ0M),t1L → π0M)_rem-condless(N(tNLΠ0M),tNL → π0M). In practice we takevs∗=QLT

v∆M∩ D(N(Π/vs1..n)), and augment the goal of the quotient Π/vs1..nby adding(Π/P).Ivs∗. Call the resulting problemΠq and its solution

→

πq. Theorem 18 shows that any concatenation of the plans{rem-condless(tLΠqM,tL→πqM) | _t ∈ ∆}solvesΠ. Thus, the proposed algorithm is sound.

Example 29. Take Π1, vs1..n, t and t0 from Example 27. There is one common variable, {v3} = Tv∆ from the orbit {p1} = QLTv∆M. Here N(Π1/vs1..n) = {p1, p2}, and the or-

bit of the common needed variable is{p1}=QLTv∆M∩ N(Π1)/vs1..n. To solveΠ1via solving Π1/vs1..n, we augment the goal(Π1/vs1..n).G with the assignment top1 in(Π1/vs1..n).I. The

resulting problem, Πq₁, is equal to Π1/vs1..n except that it has the goalsΠq₁.G = {p1, p3}. A

plan for Πq₁ is →πq = ({p1, p2},{p3, p1})(∅,{p1}), and two instantiations of it are tL → πqM = ({v3,v1},{v4,v3})(∅,{v3}) andt0L → πqM = ({v3,v2},{v5,v3})(∅,{v3}). ConcatenatingtL → πqM andt0L→πqMin any order solvesΠ₁.

4.7

Experimental Results

Implemented in C++,1 our approach uses: NAUTY\TRACES by McKay and Piperno (2014) to calculate symmetries; Z3 by De Moura and Bjørner (2008) to find isomorphic subproblems; and the initial-plan search by LAMA by Richter and Westphal (2010) as thebase planner. We limit base plannerruntimes to 30 minutes.

By running our algorithm, we obtained a set of benchmarks with soluble descriptive quotients whose solutions can be instantiated to concrete plans. That set includes439problems, from16IPC benchmarks and4benchmarks from Porco et al. (2013). In all our experimentation we identified

120instances where we were able to confirm that the descriptive quotient does not have a solution 1

Found at the public repositorybitbucket.org/MohammadAbdulaziz/planning.gitin the directory

0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 10000 12000 14000 16000

Count of Actions in Quotient

Count of Actions in Concrete Problem Number of Actions in Problem Description

(a) 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 10000 12000 14000 16000

Count of Propositions in Quotient

Count of Propositions in Concrete Problem Propositions in Problem Descriptions

(b)

Figure 4.4:(a) Scatter-plot comparing the number ofactionsin problem posed by descriptive quotientvs.

concrete problem, with red line plottingf(x) =x. (b) Scatter-plot comparing the number ofstate variables

in problem posed by descriptive quotientvs.concrete problem, with red line plottingf(x) =x.

and the concrete problem does. Figure 4.4 plots the sizes in terms of both, the number of actions and the number of state variables, of concrete and quotient problem descriptions. The plotted data shows that the descriptive quotient can be much smaller than its concrete counterpart. In just over

15%of instances the quotient has less than half the number of actions. Here, we also analyzed what aspects of our approach are computationally expensive. In79%of cases99%of the runtime of our approach is executing the base planner. In96%of cases95%of the runtime of our approach is executing the base planner. Overall,3%of time is spent in instantiation and finding chromatic numbers.

We examined where our approach is comparatively scalable and fast compared to the base planner. For430instances where the base planner and our approach are successful, Figure 4.5a displays the speedup factors where planning via the descriptive quotient is comparatively fast. Overall, for 68% of instances our approach is comparatively fast, and in15%planning via the quotient is at least twice as fast. With few exceptions, instances where our approach is at least twice as fast are from GRIPPER, HIKING, MPRIME, MYSTERY, PARCPRINTER, PIPESWORLD, TPP and VISITALL. In 5 problems from HIKING, 2 from PARCPRINTER, 1 from TETRISand

1 from KCOLOURABILITY, the base planner cannot solve the concrete problem, but can solve the quotient. Planningviathe quotient can also be slow, primarily due to the extra cost of finding symmetries,2and because LAMA is heuristic and occasionally finds that the quotient poses a more challenging problem. Figure 4.5b provides the dual to Figure 4.5a, showing cases where planning directly for the concrete problem is comparatively fast. This is the case in 32%instances, and indeed in2%our approach is at least twice as slow.

Finally, it is worth highlighting the difference between searching for a plan in the state space of the descriptive quotient versus approximate orbit search—i.e., searching for a plan in an approx- imation of the quotient state space—as is done in the state-of-the-art techniques for exploiting symmetries in planning.3 In comparing those approaches, we measure the number of states ex- panded using a breadth-first search. We consider the IPC instancesGRIPPER-20 andMPRIME-21, the largest instances from domainsGRIPPERandMPRIMEreported solved by Pochter et al. (2011) using breadth-first search. Those authors report the number of states expanded to be 60.5K and 438K, respectively. Using the same search to solve the problem posed by our descriptive quotient,

2

If symmetries are given as part of the problem description, or if one resorts to more heuristic methods of discovering them, such as those described by Guere and Alami (2001); Fox and Long (2002), this burden is relieved.

3

Such a comparison is admittedly unfair, because the problem posed by a descriptive quotient is not a bisimulation of the concrete problem.