TO TEMPORAL DE VEHÍCULOS

the constraint Counto ≥ Counto,e, which ensures that the operator is used at least as

often as the constraints trigger. If some effect conditions are mutually exclusive, this constraint can be strengthened further. Say the conditions of e1 and e2 are mutually ex-

clusive, then we can add the constraint Counto≥ Counto,e1+ Counto,e2. The net change

induced by an operator on an atom is more complex with conditional effects, though. A normal form comparable to TNF does not exist for tasks with conditional effects and as mentioned above, an operator can induce multiple (non-deterministic) transitions in the projection to a single variable. In Appendix B.1, we show how net change constraints can be defined for tasks that are not in TNF by distinguishing operators that always produce/consume an atom from those that sometimes produce/consume it. The same technique, applied on the level of conditional effects, can be used to extend net change constraints to conditional effects.

As a side note, we remark that variables like Counto,e may be useful in other con-

straints as well. While the operator-counting framework restricts the set of shared vari- ables to operator-counting variables, this could be extended to include other variables as long as all constraints share the same interpretation of these variables. Given a plan, the value of all shared variables has to be determined and every constraint has to have a solution consistent with these values. In a similar way, the variables Timeo from

delete-relaxation constraints could be shared by interpreting them as an order of the first occurrence of each operator in a given plan. Variables that count the number of times an axiom (Helmert, 2009) triggers would also be interesting for general FDR tasks, but it is less clear how the semantics of axioms could be encoded.

12.4. Extension to Other Planning Formalisms

There are many extensions of classical planning. Level 2 of PDDL 2.1 (Fox and Long, 2003) adds numeric state variables to the finite domain variables in level 1. Numeric planning is often combined with temporal planning (levels 3 and 4 of PDDL 2.1), where executing actions takes time and not all effects are instantaneous. In probabilistic plan- ningthe effect of an action is selected according to a probability distribution (Bellman, 1957; Bertsekas, 1995; Mausam and Kolobov, 2012). The problem of over-subscription planningis to find a path with a limited cost budget that maximizes a utility function over states (Smith, 2004; Aghighi and Jonsson, 2014). Heuristic search has been used in all of these formalisms (e.g. Hoffmann, 2003; Bonet and Geffner, 2003; Coles et al., 2012; Domshlak and Mirkis, 2015).

One option of coming up with admissible heuristic estimates for an extension of classical planning is to ignore the extension and compute an admissible heuristic for the “classical core” of the task. For example, this means projecting away the numeric variables (e.g. Do and Kambhampati, 2001), ignoring time (e.g. Eyerich, Mattmüller, and Röger, 2009) or transforming a non-deterministic task into a deterministic one via determinization (e.g. Yoon, Fern, and Givan, 2007; Teichteil-Königsbuch, Kuter, and

12. Related and Future Work

Infantes, 2010). Such heuristics ignore the aspect of the problem that distinguishes it from classical planning and are therefore usually not that informative. Heuristics that only take the distinguishing part into account (e.g. only reason about the values of numerical variables) would suffer from the same problem. Heuristics that combine both parts are often very specific to the particular planning formalism and cannot directly benefit from new ideas in classical planning.

Operator-counting heuristics might offer a general way to connect other planning formalisms with classical planning heuristics. If aspects of the problem that go beyond classical planning can be represented in terms of linear constraints and connected to the number of times an action is executed, operator-counting constraints can be combined with these constraints. Each set of constraints then focuses on one aspect of the problem but since they mention the same variables, they can interact and strengthen each other. New results for operator-counting heuristics in classical planning can then be easily used in other planning formalisms.

For example, in numeric planning, if a numeric variable is only incremented and decremented by one with operators, is 0 in the initial state and must be at least 5 in the goal, then the number of times an increment operator is used minus the number of times a decrement operator is used must be at least5. The LPRPG heuristic (Coles et al., 2008) uses reasoning like this in an LP to derive a heuristic value. In contrast to operator-counting heuristics, their LP reasons about time steps in a relaxed planning graph (Hoffmann, 2003) but the same idea could be easily combined with operator counting. Constraints of this kind could then be combined with, say, landmarks and the state equation heuristic on the classical part.

In over-subscription planning, adding constraints that limit the cost of a plan to the given budget is straight-forward. If the utility can also be expressed in terms of operator- counting variables, it can be maximized subject to operator-counting constraints and the budget constraint to get an upper bound for the achievable utility.

One recent example of extending operator-counting heuristics to other planning for- malisms is from the area of probabilistic planning. Trevizan, Thiébaux, and Haslum (2017) use occupation measures instead of operator-counting variables. The occupa- tion measure of a policy π for an operator o and a state s is the expected number of times o is executed in s following policy π. They use LP variables for the occupation measures to express the expected number of times atoms are produced and consumed in atomic projections. Their heuristic is directly analogous to the state equation heuris- tic for classical planning. They also show how arbitrary operator-counting constraints can be translated into their framework, so any advances on classical operator-counting heuristics can be directly used for probabilistic planning.

We believe that a similar approach could work for many other planning formalisms as well and plan to work on extensions to numeric planning and over-subscription planning in the future.

13. Summary

We introduced a class of MIP/LP heuristics based on operator-counting constraints that subsumes many existing heuristics, including the state equation, post-hoc optimiza- tion and admissible landmark heuristics as well as the delete-relaxation heuristic. With an analysis of flow heuristics, we also showed that general (explicit-state) abstraction heuristics can be represented in this framework. While operator counting is very gen- eral, not every heuristic can be represented in this framework. Critical path heuristics are not superadditive (i.e. they can benefit from cost partitioning with themselves) and thus cannot be represented as operator-counting heuristics.

There are two main advantages of representing heuristics in a common language like that of operator-counting constraints:

Firstly, operator-counting constraints can be combined arbitrarily. As the main con- tribution in this area, we showed that operator-counting LP heuristics compute an op- timal general operator cost partitioning of their component heuristics. In our exper- iments, the best configuration combines constraints from the state equation heuristic and from optimal cost partitioning for LM-cut landmarks. This configuration solves13 more tasks on the IPC benchmark set than the state-of-the-art LM-cut heuristic. We saw synergy effects between most constraint types where the combination with an LP produced better heuristic estimates than the maximum. Computing the MIP heuristic for the same constraints offers an even stronger method to combine the heuristics. In practice, the combination with an LP already produces high quality heuristic estimates and the additional effort to solve an (NP-hard) MIP does not pay off in most cases.

Secondly, the relationship of different heuristics expressed in a common language is easier to analyze. For example, we show that the state equation heuristic can be seen as cost-partitioned atomic abstractions when isolated values are removed from variable domains. The proof simply compares the generated constraints, instead of having to reason about Petri nets, abstractions and shortest paths at the same time. By further analyzing the constraints, we show that extensions of the heuristic like partial merges or using mutex information correspond to other abstractions and that the safety-based extension of the state equation heuristic cannot improve heuristic accuracy. Similarly, we use the operator-counting representation to show dominance results, for example that hSEQ_{dominates the post-hoc optimization heuristic for atomic projections if there}

are no dead values.

Operator counting in general makes it easy to express declarative knowledge about plans and use off-the-shelf LP solvers to combine this knowledge in an optimal way. We saw that such declarative knowledge can be extracted from a number of different

13. Summary

sources like landmarks or abstractions. For other planning formalism such as numeric planning, oversubscription planning, or probabilistic planning, additional information about the task could be added to encode the specific mechanics of the formalism.

Part III.

14. Introduction to

Potential Heuristics

In this part of the thesis, we introduce a new family of heuristics called potential heuris- tics. Like operator-counting heuristics, potential heuristics are declarative: they fix the mathematical form of the heuristic and allow us to declare properties that restrict the heuristic function. While operator-counting heuristics use properties of plans to restrict the heuristic value, potential heuristics use properties of the heuristic function itself, like consistency and admissibility.

A potential heuristic associates a numerical weight with a set of state features and computes the potential of a state as the sum of all weights of features that are present in the state. A well-known example are evaluation functions in chess (e.g. Lolli, 1763; Shannon, 1950): an approximate value of a position in chess can be computed as

200(K_{− K}0) + 9(Q_{− Q}0) + 5(R_{− R}0) + 3(B_{− B}0+ N _{− N}0) + (P _{− P}0) where K, Q, R, B, N , and P are the number of kings, queens, rooks, bishops, knights, and pawns owned by the player, and K0, . . . , P0 are the number of pieces owned by the opponent. The evaluation function assigns a (positive or negative) value to each piece and includes this value in the sum if the piece is still on the board.

For now, we focus on features that can be expressed as conjunctions of atoms. More general features are conceivable (e.g. disjunctions or general formulas over atoms), and we explore a slightly broader definition in Section 16.2.

Definition 14.1 (feature). Let Π be a planning task with atomsA. A feature of Π is a conjunction of atoms from_{A and the number of conjuncts is its size. A feature f is true} in a states (written as s_{|= f) if all atoms of f are true in s.}

Definition 14.2 (potential function). Let Π be a planning task andF a set of features ofΠ. A weight function is a function w : F → R. The potential function for a weight functionw maps each state s to its potential:

ϕw(s) =X

f ∈F

w(f )[s _{|= f].}

Thedimension of a potential function is the maximal size of one of its features.

We call features of size 1 and 2 atomic and binary, and likewise for potential func- tions of dimension 1 and 2. Potential functions can be used to estimate goal distances,

in which case we call them potential heuristics. They also have other uses, which we discuss in Chapter 18.

Every finite heuristic h for a task with n variables can be described as a potential function with at most n-dimensional features: in the worst case, we can define an n- dimensional feature with weight h(s) for every state s. But there are interesting poten- tial heuristics with lower dimension as well. The STRIPS heuristic (Fikes and Nilsson, 1971) counts the number of unsatisfied goals. It can be seen as an atomic potential heuristic with features _{hV, vi of weight 1 where V is a goal variable and v is not its} goal value. The Manhattan distance for the sliding tile puzzle (Korf, 1985) sums up the Manhattan distance of every tile to its goal location and can be also be seen as an atomic potential function. In this puzzle, a linear conflict (Hansson, Mayer, and Yung, 1992) exists if two tiles have to pass each other in the same row or column. This can be expressed with binary potential heuristics. As a final example, note that the PDB heuristic for a pattern P is a potential heuristic of dimension _{|P | where the features} directly correspond to the abstract states in projection to P and the weights are their abstract goal distances.

15. Admissible and Consistent

Potential Heuristics

The main advantage of potential functions is that their fixed mathematical structure makes it possible to analyze and reason about the heuristics. In this chapter we will show how linear constraints over feature weights can be used to express interesting properties like admissibility and consistency of potential heuristics. The constraints that we introduce characterize admissible and consistent potential heuristics for certain sets of features. This means that every weight function satisfying the constraints induces an admissible potential heuristic. Maximizing a formula that measures how “good” a heuristic is subject to these constraints gives the best admissible potential heuristic according to this criterion. If the criterion can be expressed as linear functions over the weights, an off-the-shelf LP solver can be used to automatically derive the heuristic function. In Section 15.4 we discuss different objective functions.