Set-theoretic duality: A fundamental feature of combinatorial optimisation

The duality between conflicts and diagnoses in the field of diagnosis, or between plans and landmarks in the field of planning, or between unsatisfiable cores and minimal co-satisfiable sets in SAT or CSP solving, has been known for many years. Recent wo

Capturing (Optimal) Relaxed Plans with Stable and Supported Models of Logic Programs

We establish a novel relation between delete-free planning, an important task for the AI Planning community also known as relaxed planning, and logic programming. We show that given a planning problem, all subsets of actions that could be ordered to produce relaxed plans for the problem can be bijectively captured with stable models of a logic program describing the corresponding relaxed planning problem. We also consider the supported model semantics of logic programs, and introduce one causal and one diagnostic encoding of the relaxed planning problem as logic programs, both capturing relaxed plans with their supported models. Our experimental results show that these new encodings can provide major performance gain when computing optimal relaxed plans, with our diagnostic encoding outperforming state-of-the-art approaches to relaxed planning regardless of the given time limit when measured on a wide collection of STRIPS planning benchmarks.Comment: Paper presented at the 39th International Conference on Logic Programming (ICLP 2023), 14 page

Optimal Planning with State Constraints

In the classical planning model, state variables are assigned values in the initial state and remain unchanged unless explicitly affected by action effects. However, some properties of states are more naturally modelled not as direct effects of actions but instead as derived, in each state, from the primary variables via a set of rules. We refer to those rules as state constraints. The two types of state constraints that will be discussed here are numeric state constraints and logical rules that we will refer to as axioms. When using state constraints we make a distinction between primary variables, whose values are directly affected by action effects, and secondary variables, whose values are determined by state constraints. While primary variables have finite and discrete domains, as in classical planning, there is no such requirement for secondary variables. For example, using numeric state constraints allows us to have secondary variables whose values are real numbers. We show that state constraints are a construct that lets us combine classical planning methods with specialised solvers developed for other types of problems. For example, introducing numeric state constraints enables us to apply planning techniques in domains involving interconnected physical systems, such as power networks. To solve these types of problems optimally, we adapt commonly used methods from optimal classical planning, namely state-space search guided by admissible heuristics. In heuristics based on monotonic relaxation, the idea is that in a relaxed state each variable assumes a set of values instead of just a single value. With state constraints, the challenge becomes to evaluate the conditions, such as goals and action preconditions, that involve secondary variables. We employ consistency checking tools to evaluate whether these conditions are satisfied in the relaxed state. In our work with numerical constraints we use linear programming, while with axioms we use answer set programming and three value semantics. This allows us to build a relaxed planning graph and compute constraint-aware version of heuristics based on monotonic relaxation. We also adapt pattern database heuristics. We notice that an abstract state can be thought of as a state in the monotonic relaxation in which the variables in the pattern hold only one value, while the variables not in the pattern simultaneously hold all the values in their domains. This means that we can apply the same technique for evaluating conditions on secondary variables as we did for the monotonic relaxation and build pattern databases similarly as it is done in classical planning. To make better use of our heuristics, we modify the A* algorithm by combining two techniques that were previously used independently – partial expansion and preferred operators. Our modified algorithm, which we call PrefPEA, is most beneficial in cases where heuristic is expensive to compute, but accurate, and states have many successors

Cost-Optimal Planning, Delete Relaxation, Approximability, and Heuristics

Cost-optimal planning is a very well-studied topic within planning, and it has proven to be computationally hard both in theory and in practice. Since cost-optimal planning is an optimisation problem, it is natural to analyse it through the lens of approximation. An important reason for studying cost-optimal planning is heuristic search; heuristic functions that guide the search in planning can often be viewed as algorithms solving or approximating certain optimisation problems. Many heuristic functions (such as the ubiquitious h+ heuristic) are based on delete relaxation, which ignores negative effects of actions. Planning for instances where the actions have no negative effects is often referred to as monotone planning. The aim of this article is to analyse the approximability of cost-optimal monotone planning, and thus the performance of relevant heuristic functions. Our findings imply that it may be beneficial to study these kind of problems within the framework of parameterised complexity and we initiate work in this direction

New perspectives on cost partitioning for optimal classical planning

Admissible heuristics are the main ingredient when solving classical planning tasks optimally with heuristic search. There are many such heuristics, and each has its own strengths and weaknesses. As higher admissible heuristic values are more accurate, the maximum over several admissible heuristics dominates each individual one. Operator cost partitioning is a well-known technique to combine admissible heuristics in a way that dominates their maximum and remains admissible. But are there better options to combine the heuristics? We make three main contributions towards this question: Extensions to the cost partitioning framework can produce higher estimates from the same set of heuristics. Cost partitioning traditionally uses non-negative cost functions. We prove that this restriction is not necessary, and that allowing negative values as well makes the framework more powerful: the resulting heuristic values can be exponentially higher, and unsolvability can be detected even if all component heuristics have a finite value. We also generalize operator cost partitioning to transition cost partitioning, which can differentiate between different contexts in which an operator is used. Operator-counting heuristics reason about the number of times each operator is used in a plan. Many existing heuristics can be expressed in this framework, which gives new theoretical insight into their relationship. Different operator-counting heuristics can be easily combined within the framework in a way that dominates their maximum. Potential heuristics compute a heuristic value as a weighted sum over state features and are a fast alternative to operator-counting heuristics. Admissible and consistent potential heuristics for certain feature sets can be described in a compact way which means that the best heuristic from this class can be extracted in polynomial time. Both operator-counting and potential heuristics are closely related to cost partitioning. They offer a new look on cost-partitioned heuristics and already sparked research beyond their use as classical planning heuristics

Symbolic search and abstraction heuristics for cost-optimal planning in automated planning

Mención Internacional en el título de doctorLa Planificación Automática puede ser definida como el problema de encontrar una secuencia de acciones (un plan) para conseguir una meta, desde un punto inicial, asumiendo que las acciones tienen efectos deterministas. La Planificación Automática es independiente de dominio porque los planificadores toman como información inicial una descripción del problema y deben resolverlo sin ninguna información adicional. Esta tesis trata en particular de planificación automática ´optima, en la cual las acciones tienen un coste asociado. Los planificadores óptimos deben encontrar un plan y probar que no existe ningún otro plan de menor coste. La mayoría de los planificadores óptimos están basados en la búsqueda de estados explícita. Sin lugar a dudas, esta aproximación ha sido la dominante en planificación automática óptima durante los últimos años. No obstante, la búsqueda simbólica se presenta como una alternativa interesante. En esta tesis, proponemos dos mejoras ortogonales para la planificación basada en búsqueda simbólica. En primer lugar, estudiamos diferentes métodos para mejorar la computación de la “imagen”, operación que calcula el conjunto de estados sucesores a partir de un conjunto de estados. Posteriormente, analizamos cómo explotar las invariantes de estado para mejorar el rendimiento de la búsqueda simbólica. Estas propuestas suponen una mejora significativa en el desempeño de los algoritmos simbólicos en la mayoría de los dominios analizados. Hemos analizado dos tipos de heurísticas de abstracción con el objetivo de extrapolar las mejoras que se han realizado en la búsqueda explícita durante los últimos años a la búsqueda simbólica. Las heurísticas analizadas son: las bases de datos de patrones (pattern databases, PDBs) y una generalización de estas, mergeand-shrink (M&S). Mientras que las PDBs se han utilizado con anterioridad en búsqueda simbólica, hemos estudiado el uso de M&S, que es más general. En esta tesis se muestra que determinados tipos de heurísticas de M&S (aquellas que son generadas mediante una estrategia de “merge” lineal) pueden ser representadas como BDDs, con un coste computacional polinomial en el tamaño de la abstracción y la descripción del problema; y por lo tanto, pueden ser utilizadas de forma eficiente en la búsqueda simbólica. También proponemos una nueva heurística”symbolic perimeter merge-andshrink” (SPM&S) que combina la fuerza de la búsqueda hacia atrás simbólica con la flexibilidad de M&S. Los resultados experimentales muestran que SPM&S es capaz de superar, no solo las dos técnicas que combina, sino también otras heurísticas del estado del arte. Finalmente, hemos integrado las abstracciones simbólicas de perímetro, SPM&S, en la búsqueda simbólica bidireccional. En resumen, esta tesis estudia diferentes propuestas para planificación óptima basada en Búsqueda simbólica. Hemos implementado diferentes planificadores simbólicos basados en la Búsqueda bidireccional y las abstracciones de perímetro. Los resultados experimentales muestran cómo los planificadores presentados como resultado de este trabajo son altamente competitivos y frecuentemente superan al resto de planificadores del estado del arte.Domain-independent planning is the problem of finding a sequence of actions for achieving a goal from an initial state assuming that actions have deterministic effects. It is domain-independent because planners take as input the description of a problem and must solve it without any additional information. In this thesis, we deal with cost-optimal planning problems, in which actions have an associated cost and the planner must find a plan and prove that no other plan of lower cost exists. Most cost-optimal planners are based on explicit-state search. While this has undoubtedly been the dominant approach to cost-optimal planning in the last years, symbolic search is an interesting alternative. In symbolic search, sets of states are succinctly represented as binary decision diagrams, BDDs. The BDD representation does not only reduce the memory needed to store sets of states, but also allows the planner to efficiently manipulate sets of states reducing the search time. We propose two orthogonal enhancements for symbolic search planning. On the one hand, we study different methods for image computation, which usually is the bottleneck of symbolic search planners. On the other hand, we analyze how to exploit state invariants to prune symbolic search. Our techniques significantly improve the performance of symbolic search algorithms in most benchmark domains. Moreover, the enhanced version of symbolic bidirectional search is one of the strongest approaches to domain-independent planning even though it does not use any heuristic. Explicit-state search planners are commonly guided with admissible heuristics, which optimistically estimate the cost from any state to the goal. Heuristics are automatically derived from the problem description and can be classified into different families according to their underlying ideas. In order to bring the improvements on heuristics that have been made in explicit-state search to symbolic search, we analyze two types of abstraction heuristics: pattern databases (PDBs) and a generalization of them, merge-and-shrink (M&S). While PDBs had already been used in symbolic search, we analyze the use of the more general M&S heuristics. We show that certain types of M&S heuristics (those generated with a linear merging strategy) can be represented as BDDs with at most a polynomial overhead and, thus, efficiently used in symbolic search. We also propose a new heuristic, symbolic perimeter merge-and-shrink (SPM&S) that combines the strength of symbolic regression search with the flexibility of M&S heuristics. Our experiments show that SPM&S is able to beat, not only the two techniques it combines, but also other state-of-the-art heuristics. Finally, we integrate our symbolic perimeter abstraction heuristics in symbolic bidirectional search. The heuristic used by the bidirectional search is computed by means of another symbolic bidirectional search in an abstract state space. We show how, even though the combination of symbolic bidirectional search and abstraction heuristics has an overall performance similar to the simpler symbolic bidirectional blind search, it can sometimes solve more problems in particular domains. In summary, this thesis studies different enhancements on symbolic search. We implement different symbolic search planners based on bidirectional search and perimeter abstraction heuristics. Experimental results show that the resulting planners are highly competitive and often outperform other state-of-the-art planners.Programa Oficial de Doctorado en Ciencia y Tecnología InformáticaPresidente: José Manuel Molina López..- Vocal: Malte Helmert .- Secretario: Andrés Jonsso