State symmetries play an important role in planning and generalized planning. In the first case, state symmetries can be used to reduce the size of the search; in the second, to reduce the size of the training set. In the case of general planning, however, it is also critical to distinguish non-symmetric states, i.e., states that represent non-isomorphic relational structures. However, while the language of first-order logic distinguishes non-symmetric states, the languages and architectures used to represent and learn general policies do not. In particular, recent approaches for learning general policies use state features derived from description logics or learned via graph neural networks (GNNs) that are known to be limited by the expressive power of C_2, first-order logic with two variables and counting. In this work, we address the problem of detecting symmetries in planning and generalized planning and use the results to assess the expressive requirements for learning general policies over various planning domains. For this, we map planning states to plain graphs, run off-the-shelf algorithms to determine whether two states are isomorphic with respect to the goal, and run coloring algorithms to determine if C_2 features computed logically or via GNNs distinguish non-isomorphic states. Symmetry detection results in more effective learning, while the failure to detect non-symmetries prevents general policies from being learned at all in certain domains.

Recently, a simple but powerful language for expressing and learning general policies and problem decompositions (sketches) has been introduced in terms of rules defined over a set of Boolean and numerical features. In this work, we consider three extensions of this language aimed at making policies and sketches more flexible and reusable: internal memory states, as in finite state controllers; indexical features, whose values are a function of the state and a number of internal registers that can be loaded with objects; and modules that wrap up policies and sketches and allow them to call each other by passing parameters. In addition, unlike general policies that select state transitions rather than ground actions, the new language allows for the selection of such actions. The expressive power of the resulting language for policies and sketches is illustrated through a number of examples.

GNN-based approaches for learning general policies across planning domains are limited by the expressive power of $C_2$, namely; first-order logic with two variables and counting. This limitation can be overcomed by transitioning to $k$-GNNs, for $k=3$, wherein object embeddings are substituted with triplet embeddings. Yet, while $3$-GNNs have the expressive power of $C_3$, unlike $1$- and $2$-GNNs that are confined to $C_2$, they require quartic time for message exchange and cubic space for embeddings, rendering them impractical. In this work, we introduce a parameterized version of relational GNNs. When $t$ is infinity, R-GNN[$t$] approximates $3$-GNNs using only quadratic space for embeddings. For lower values of $t$, such as $t=1$ and $t=2$, R-GNN[$t$] achieves a weaker approximation by exchanging fewer messages, yet interestingly, often yield the $C_3$ features required in several planning domains. Furthermore, the new R-GNN[$t$] architecture is the original R-GNN architecture with a suitable transformation applied to the input states only. Experimental results illustrate the clear performance gains of R-GNN[$1$] and R-GNN[$2$] over plain R-GNNs, and also over edge transformers that also approximate $3$-GNNs.

It has been observed that many classical planning domains with atomic goals can be solved by means of a simple polynomial exploration procedure, called IW, that runs in time exponential in the problem width, which in these cases is bounded and small. Yet, while the notion of width has become part of state-of-the-art planning algorithms such as BFWS, there is no good explanation for why so many benchmark domains have bounded width when atomic goals are considered. In this work, we address this question by relating bounded width with the existence of general optimal policies that in each planning instance are represented by tuples of atoms of bounded size. We also define the notions of (explicit) serializations and serialized width that have a broader scope as many domains have a bounded serialized width but no bounded width. Such problems are solved non-optimally in polynomial time by a suitable variant of the Serialized IW algorithm. Finally, the language of general policies and the semantics of serializations are combined to yield a simple, meaningful, and expressive language for specifying serializations in compact form in the form of sketches, which can be used for encoding domain control knowledge by hand or for learning it from small examples. Sketches express general problem decompositions in terms of subgoals, and sketches of bounded width express problem decompositions that can be solved in polynomial time.

We consider the problem of reaching a propositional goal condition in fully-observable nondeterministic (FOND) planning under a general class of fairness assumptions that are given explicitly. The fairness assumptions are of the form A/B and say that state trajectories that contain infinite occurrences of an action a from A in a state s and finite occurrence of actions from B, must also contain infinite occurrences of action a in s followed by each one of its possible outcomes. The infinite trajectories that violate this condition are deemed as unfair, and the solutions are policies for which all the fair trajectories reach a goal state. We show that strong and strong-cyclic FOND planning, as well as QNP planning, a planning model introduced recently for generalized planning, are all special cases of FOND planning with fairness assumptions of this form which can also be combined. FOND+ planning, as this form of planning is called, combines the syntax of FOND planning with some of the versatility of LTL for expressing fairness constraints. A sound and complete FOND+ planner is implemented by reducing FOND+ planning to answer set programs, and its performance is evaluated in comparison with FOND and QNP planners, and LTL synthesis tools. Two other FOND+ planners are introduced as well which are more scalable but are not complete.  

It has been recently shown that general policies for many classical planning domains can be expressed and learned in terms of a pool of features defined from the domain predicates using a description logic grammar. At the same time, most description logics correspond to a fragment of $k$-variable counting logic ($C_k$) for $k=2$, that has been shown to provide a tight characterization of the expressive power of graph neural networks. In this work, we make use of these results to understand the power and limits of using graph neural networks (GNNs) for learning optimal general policies over a number of tractable planning domains where such policies are known to exist. For this, we train a simple GNN in a supervised manner to approximate the optimal value function $V^{*}(s)$ of a number of sample states $s$. As predicted by the theory, it is observed that general optimal policies are obtained in domains where general optimal value functions can be defined with $C_2$ features but not in those requiring more expressive $C_3$ features. In addition, it is observed that the features learned are in close correspondence with the features needed to express $V^{*}$ in closed form. The theory and the analysis of the domains let us understand the features that are actually learned as well as those that cannot be learned in this way, and let us move in a principled manner from a combinatorial optimization approach to learning general policies to a potentially, more robust and scalable approach based on deep learning.

Recently Bonet and Geffner have shown that first-order representations for planning domains can be learned from the structure of the state space without any prior knowledge about the action schemas or domain predicates. For this, the learning problem is formulated as the search for a simplest first-order domain description D that along with information about instances I_i (number of objects and initial state) determine state space graphs G(P_i) that match the observed state graphs G_i where P_i = (D, I_i). The search is cast and solved approximately by means of a SAT solver that is called over a large family of propositional theories that differ just in the parameters encoding the possible number of action schemas and domain predicates, their arities, and the number of objects. In this work, we push the limits of these learners by moving to an answer set programming (ASP) encoding using the CLINGO system. The new encodings are more transparent and concise, extending the range of possible models while facilitating their exploration. We show that the domains introduced by Bonet and Geffner can be solved more efficiently in the new approach, often optimally, and furthermore, that the approach can be easily extended to handle partial information about the state graphs as well as noise that prevents some states from being distinguished.

We consider the problem of reaching a propositional goal condition in fully-observable non-deterministic (FOND) planning under a general class of fairness assumptions that are given explicitly. The fairness assumptions are of the form A/B and say that state trajectories that contain infinite occurrences of an action a from A in a state s and finite occurrence of actions from B, must also contain infinite occurrences of action a in s followed by each one of its possible outcomes. The infinite trajectories that violate this condition are deemed as unfair, and the solutions are policies for which all the fair trajectories reach a goal state. We show that strong and strong-cyclic FOND planning, as well as QNP planning, a planning model introduced recently for generalized planning, are all special cases of FOND planning with fairness assumptions of this form which can also be combined. FOND+ planning, as this form of planning is called, combines the syntax of FOND planning with some of the versatility of LTL for expressing fairness constraints. A new planner is implemented by reducing FOND+ planning to answer set programs, and the performance of the planner is evaluated in comparison with FOND and QNP planners, and LTL synthesis tools.

Generalized planning is concerned with the computation of general policies that solve multiple instances of a planning domain all at once. It has been recently shown that these policies can be computed in two steps: first, a suitable abstraction in the form of a qualitative numerical planning problem (QNP) is learned from sample plans, then the general policies are obtained from the learned QNP using a planner. In this work, we introduce an alternative approach for computing more expressive general policies which does not require sample plans or a QNP planner. The new formulation is very simple and can be cast in terms that are more standard in machine learning: a large but finite pool of features is defined from the predicates in the planning examples using a general grammar, and a small subset of features is sought for separating "good" from "bad" state transitions, and goals from non-goals. The problems of finding such a "separating surface" while labeling the transitions as "good" or "bad" are jointly addressed as a single combinatorial optimization problem expressed as a Weighted Max-SAT problem. The advantage of looking for the simplest policy in the given feature space that solves the given examples, possibly non-optimally, is that many domains have no general, compact policies that are optimal. The approach yields general policies for a number of benchmark domains.

It has been observed that in many of the benchmark planning domains, atomic goals can be reached with a simple polynomial exploration procedure, called IW, that runs in time exponential in the problem width. Such problems have indeed a bounded width: a width that does not grow with the number of problem variables and is often no greater than two. Yet, while the notion of width has become part of the state-of-the-art planning algorithms like BFWS, there is still no good explanation for why so many benchmark domains have bounded width. In this work, we address this question by relating bounded width and serialized width to ideas of generalized planning, where general policies aim to solve multiple instances of a planning problem all at once. We show that bounded width is a property of planning domains that admit optimal general policies in terms of features that are explicitly or implicitly represented in the domain encoding. The results are extended to much larger class of domains with bounded serialized width where the general policies do not have to be optimal. The study leads also to a new simple, meaningful, and expressive language for specifying domain serializations in the form of policy sketches which can be used for encoding domain control knowledge by hand or for learning it from traces. The use of sketches and the meaning of the theoretical results are all illustrated through a number of examples.