Classical planning asks for a sequence of operators reach-ing a given goal. While the most common case is to compute a plan, many scenarios require more than that. However, quantitative reasoning on the plan space remains mostly unexplored. A fundamental problem is to count plans, which relates to the conditional probability on the plan space. Indeed, qualitative and quantitative approaches are well-established in various other areas of automated reasoning.

We present the first study to quantitative and qualitative reasoning on the plan space. In particular, we focus on polynomially bounded plans. On the theoretical side, we study its complexity, which gives rise to rich reasoning modes. Sincecounting is hard in general, we introduce the easier notion of facets, which enables understanding the significance of operators. On the practical side, we implement quantitative reasoning for planning. Thereby, we transform a planning task into a propositional formula and use knowledge compilationto count different plans. This framework scales well to largeplan spaces, while enabling rich reasoning capabilities suchas learning pruning functions and explainable planning.

Many planning applications require not only a single solution but benefit substantially from having a set of possible plans from which users can select according to preferences. Surprisingly, planning research has primarily focused on quickly finding single plans for decades. Only recently have researchers started to investigate plan enumeration by top-k planning, offering more flexibility to the user. But simply enumerating the k best plans is far from targeted due to the time-consuming nature of enumeration, likely feeding many similar plans to the user, and forcing the user to define filters beforehand. In fact, in extensive search spaces, enumeration is hardly practical. We present an approach and a tool called PlanPilot to navigate solution spaces of planning tasks iteratively and interactively. We build on answer-set programming (ASP) to restrict the plan space. To that end, we employ facets, which are meaningful actions that appear in some, but not all plans. Enforcing or forbidding such facets allows for navigating even large plan spaces while ensuring desired properties quickly and step by step. 

Many planning applications require not only a single solution but benefit substantially from having a set of possible plans from which users can select, for example, when explaining plans. For decades, research in classical AI planning has primarily focused on quickly finding single plans. Only recently have researchers started to investigate preferences, enumerate plans by top-k planning, or count plans to reason about the plan space. Unfortunately, reasoning about the plan space is computationally extremely hard and feeding many similar plans to the user is hardly practical. To circumvent computational shortcomings while still being able to reason about variability in plans, faceted actions have been introduced very recently. These are meaningful actions that can be used by some plan but are not required by all plans. Enforcing or forbidding such facets allows for navigating even large plan spaces while ensuring desired properties quickly and step by step. In this paper, we illustrate an industrial challenge where reasoning with facets enables targeted plan space navigation. We present an approach to handle large plan spaces iteratively and interactively and present a tool that we call PlanPilot. 

Multiple Sequence Alignment (MSA) is a fundamental problem in computational biology that is used to understand the evolutionary history of protein, DNA, or RNA sequences. An optimal alignment for two sequences can efficiently be found using dynamic programming, but computing optimal alignments for more sequences continues to be a hard problem. A common method to solve MSA problems is A* search with admissible heuristics, computed from subsets of the input sequences. In this paper, we consider MSA from the perspective of cost partitioning and relate the existing heuristics for MSA to uniform cost partitioning and post-hoc optimization, two well-known techniques from the automated planning literature. We show that the MSA heuristics are bounded by uniform cost partitioning and that post-hoc optimization yields strictly dominating heuristics. For a common benchmark set of protein sequences and a set of DNA sequences, we show that the theoretical dominance relations between the heuristics carry over to practical instances.

Multiple Sequence Alignment (MSA) is a fundamental prob-lem in computational biology that is used to understand theevolutionary history of protein, DNA, or RNA sequences. Anoptimal alignment for two sequences can efficiently be foundusing dynamic programming, but computing optimal align-ments for more sequences continues to be a hard problem. Acommon method to solve MSA problems is A∗ search with admissible heuristics, computed from subsets of the input se-quences. In this paper, we consider MSA from the perspectiveof cost partitioning and relate the existing heuristics for MSAto uniform cost partitioning and post-hoc optimization, twowell-known techniques from the automated planning litera-ture. We show that the MSA heuristics are bounded by uni-form cost partitioning and that post-hoc optimization yieldsstrictly dominating heuristics. For a common benchmark setof protein sequences and a set of DNA sequences, we showthat the theoretical dominance relations between the heuris-tics carry over to practical instances

Automated problem reformulation is a common technique in classical planning to identify and exploit problem structures. Decoupled search is an approach that automatically decomposes planning tasks based on their causal structure, often significantly reducing the search effort. However, its broad applicability is limited by the need for specialized algorithms. In this paper, we present an approach that embodies decoupled search for non-optimal planning through a novel task transformation. Specifically, given a task and a decomposition, we create a transformed task such that the state space of the transformed task is isomorphic to that of decoupled search on the original task. This eliminates the need for specialized algorithms and allows the use of various planning technology in the decoupled-search framework. Empirical evaluation shows that our method is empirically competitive with specialized decoupled algorithms and favorable to other related problem reformulation techniques. 

Matrix multiplication is a fundamental operation of linear algebra, with applications ranging from quantum physics to artificial intelligence. Given its importance, enormous resources have been invested in the search for faster matrix multiplication algorithms. Recently, this search has been cast as a single-player game. By learning how to play this game efficiently, the newly-introduced AlphaTensor reinforcement learning agent is able to discover many new faster algorithms. In this paper, we show that finding matrix multiplication algorithms can also be cast as a classical planning problem. Based on this observation, we introduce a challenging benchmark suite for classical planning and evaluate state-of-the-art planning techniques on it. We analyze the strengths and limitations of different planning approaches in this domain and show that we can use classical planning to find lower bounds and concrete algorithms for matrix multiplication.

Restricted Tasks (RT) are a special case of numeric planning characterized by numeric conditions that involve one numeric variable per formula and numeric effects that allow only the addition of constants. Despite this, RTs form an expressive class whose planning problem is undecidable. The restricted nature of RTs often makes problem modeling awkward and unnecessarily complicated. We show that this can be alleviated by compiling mathematical operations that are not natively supported into RTs using macro-like action sequences. With that, we can encode many features found in general numeric planning such as constant multiplication, addition of linear formulas, and integer division and residue. We demonstrate how our compilations can be used to capture challenging mathematical problems such as the (in)famous Collatz conjecture. Our approach additionally gives a simple undecidability proof for RTs, and the proof shows that the number of variables needed to construct an undecidable class of RTs is surprisingly low: two numeric and one propositional variable.

Numeric planning is known to be undecidable even under severe restrictions. Prior work has investigated the decidability boundaries by restricting the expressiveness of the planning formalism in terms of the numeric functions allowed in conditions and effects. We study a well-known restricted form of Hoffmann's simple numeric planning, which is undecidable. We analyze the complexity by imposing restrictions on the causal structure, exploiting a novel method for bounding variable domain sizes. First, we show that plan existence for tasks where all numeric variables are root nodes in the causal graph is in PSPACE. Second, we show that for tasks with only numeric leaf variables the problem is decidable, and that it is in PSPACE if the propositional state space has a fixed size. Our work lays a strong foundation for future investigations of structurally more complex tasks. From a practical perspective, our method allows to employ heuristics and methods that are geared towards finite variable domains (such as pattern database heuristics or decoupled search) to solve non-trivial families of numeric planning problems.