Lex-Partitioning: A New Option for BDD Search

For the exploration of large state spaces, symbolic search using binary decision diagrams (BDDs) can save huge amounts of memory and computation time. State sets are represented and modified by accessing and manipulating their characteristic functions. BDD partitioning is used to compute the image as the disjunction of smaller subimages. In this paper, we propose a novel BDD partitioning option. The partitioning is lexicographical in the binary representation of the states contained in the set that is represented by a BDD and uniform with respect to the number of states represented. The motivation of controlling the state set sizes in the partitioning is to eventually bridge the gap between explicit and symbolic search. Let n be the size of the binary state vector. We propose an O(n) ranking and unranking scheme that supports negated edges and operates on top of precomputed satcount values. For the uniform split of a BDD, we then use unranking to provide paths along which we partition the BDDs. In a shared BDD representation the efforts are O(n). The algorithms are fully integrated in the CUDD library and evaluated in strongly solving general game playing benchmarks.Comment: In Proceedings GRAPHITE 2012, arXiv:1210.611

Merge-and-Shrink Task Reformulation for Classical Planning

The performance of domain-independent planning systems heavily depends on how the planning task has been modeled. This makes task reformulation an important tool to get rid of unnecessary complexity and increase the robustness of planners with respect to the model chosen by the user. In this paper, we represent tasks as factored transition systems (FTS), and use the merge-and-shrink (M&S) framework for task reformulation for optimal and satisficing planning. We prove that the flexibility of the underlying representation makes the M&S reformulation methods more powerful than the counterparts based on the more popular finite-domain representation. We adapt delete-relaxation and M&S heuristics to work on the FTS representation and evaluate the impact of our reformulation

Automatic Configuration of Benchmark Sets for Classical Planning

The benchmarks from previous International Planning Competitions are commonly used to evaluate new planning algorithms. Since this set has grown organically over the years, it has several flaws: it contains duplicate tasks, unsolvable tasks, trivially solvable domains, and domains with modelling errors. Also, diverse domain sizes complicate aggregating results. Most importantly, however, the range of task difficulty is very small in many domains. We propose an automated method for creating benchmarks that solves these issues. To find a good scaling in difficulty, we automatically configure the parameters of benchmark domains. We show that the resulting benchmark set improves empirical comparisons by allowing to differentiate between planners more easily

Additive Pattern Databases for Decoupled Search

Abstraction heuristics are the state of the art in optimal classical planning as heuristic search. Despite their success for explicit-state search, though, abstraction heuristics are not available for decoupled state-space search, an orthogonal reduction technique that can lead to exponential savings by decomposing planning tasks. In this paper, we show how to compute pattern database (PDB) heuristics for decoupled states. The main challenge lies in how to additively employ multiple patterns, which is crucial for strong search guidance of the heuristics. We show that in the general case, for arbitrary collections of PDBs, computing the heuristic for a decoupled state is exponential in the number of leaf components of decoupled search. We derive several variants of decoupled PDB heuristics that allow to additively combine PDBs avoiding this blow-up and evaluate them empirically

Efficient Evaluation of Large Abstractions for Decoupled Search: Merge-and-Shrink and Symbolic Pattern Databases

Abstraction heuristics are a state-of-the-art technique to solve classical planning problems optimally. A common approach is to precompute many small abstractions and combine them admissibly using cost partitioning. Recent work has shown that this approach does not work out well when using such heuristics for decoupled state space search, where search nodes represent potentially large sets of states. This is due to the fact that admissibly combining the estimates of several heuristics without sacrificing accuracy is NP-hard for decoupled states. In this work we propose to use a single large abstraction instead. We focus on merge-and-shrink and symbolic pattern database heuristics, which are designed to produce such abstractions. For these heuristics, we prove that the evaluation of decoupled states is NP-hard in general, but we also identify conditions under which it is polynomial. We introduce algorithms for both the general and the polynomial case. Our experimental evaluation shows that single large abstraction heuristics lead to strong performance when the heuristic evaluation is polynomial

Symbolic Planning with Axioms

Axioms are an extension for classical planning models that allow for modeling complex preconditions and goals exponentially more compactly. Although axioms were introduced in planning more than a decade ago, modern planning techniques rarely support axioms, especially in cost-optimal planning. Symbolic search is a popular and competitive optimal planning technique based on the manipulation of sets of states. In this work, we extend symbolic search algorithms to support axioms natively. We analyze different ways of encoding derived variables and axiom rules to evaluate them in a symbolic representation. We prove that all encodings are sound and complete, and empirically show that the presented approach outperforms the previous state of the art in costoptimal classical planning with axioms.This work was supported by the German National Science Foundation (DFG) as part of the project EPSDAC (MA 7790/1-1) and the Research Unit FOR 1513 (HYBRIS). The FAI group of Saarland University has received support by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science)

A Theory of Merge-and-Shrink for Stochastic Shortest Path Problems

The merge-and-shrink framework is a powerful tool to construct state space abstractions based on factored representations. One of its core applications in classical planning is the construction of admissible abstraction heuristics. In this paper, we develop a compositional theory of merge-and-shrink in the context of probabilistic planning, focusing on stochastic shortest path problems (SSPs). As the basis for this development, we contribute a novel factored state space model for SSPs. We show how general transformations, including abstractions, can be formulated on this model to derive admissible and/or perfect heuristics. To formalize the merge-and-shrink framework for SSPs, we transfer the fundamental merge-and-shrink transformations from the classical setting: shrinking, merging, and label reduction. We analyze the formal properties of these transformations in detail and show how the conditions under which shrinking and label reduction lead to perfect heuristics can be extended to the SSP setting

Estudio y aplicación de algoritmos de búsqueda al juego del Risk

La Inteligencia Artificial (IA), es un área de la informática que intenta emular comportamientos inteligentes en sistemas informáticos. Con el objetivo de avanzar en esta línea es común realizar sistemas que resuelvan juegos. Los juegos son interesantes porque tienen un dominio acotado es decir, sus reglas están bien definidas y delimitan con total exactitud qué se puede hacer y qué no. Se ejecutan en un entorno controlado por lo que no pueden ocurrir eventos imprevistos durante una partida. Permiten una evaluación objetiva del rendimiento del sistema, midiendo el porcentaje de victorias/derrotas. Un ejemplo bastante clásico en este sentido son los juegos de ajedrez que se han desarrollado durante varias décadas y, a día de hoy, ya se puede considerar que han alcanzado el nivel de un gran maestro, por lo que en los últimos años se ha avanzado un nivel más, afrontando juegos de mayor dificultad. Este proyecto se sitúa en este marco al pretender desarrollar una IA para el juego del Risk. Este juego resulta muy interesante por las siguientes características: Aleatoriedad en las acciones: el resultado de algunas acciones de los jugadores es no determinista, lo que dificulta el cálculo del estado resultante. Varios jugadores: pueden jugar de 2 a 6 jugadores por lo que hay que tener en cuenta las estrategias de cada uno de ellos y hace que dañar a otro jugador pueda no ser bueno (puede beneficiar más a un tercer jugador). Factor de ramificación muy alto: En cada turno el jugador debe decidir las acciones a tomar y, dado que puede realizar cualquier número de ataques que desee, el número de opciones resulta inmanejable. Estas características unidas hacen inviables las técnicas de búsqueda utilizadas en el ajedrez ya que, el factor de ramificación es demasiado extenso y las situaciones con más de dos jugadores dificultan la poda del árbol de búsqueda.Ingeniería en Informátic

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