9,842 research outputs found

    A Logic Programming Approach to Knowledge-State Planning: Semantics and Complexity

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    We propose a new declarative planning language, called K, which is based on principles and methods of logic programming. In this language, transitions between states of knowledge can be described, rather than transitions between completely described states of the world, which makes the language well-suited for planning under incomplete knowledge. Furthermore, it enables the use of default principles in the planning process by supporting negation as failure. Nonetheless, K also supports the representation of transitions between states of the world (i.e., states of complete knowledge) as a special case, which shows that the language is very flexible. As we demonstrate on particular examples, the use of knowledge states may allow for a natural and compact problem representation. We then provide a thorough analysis of the computational complexity of K, and consider different planning problems, including standard planning and secure planning (also known as conformant planning) problems. We show that these problems have different complexities under various restrictions, ranging from NP to NEXPTIME in the propositional case. Our results form the theoretical basis for the DLV^K system, which implements the language K on top of the DLV logic programming system.Comment: 48 pages, appeared as a Technical Report at KBS of the Vienna University of Technology, see http://www.kr.tuwien.ac.at/research/reports

    Push-Pull Block Puzzles are Hard

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    This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve, settling an open question by Zubaran and Ritt. Push-pull block puzzles are a type of recreational motion planning problem, similar to Sokoban, that involve moving a `robot' on a square grid with 1×11 \times 1 obstacles. The obstacles cannot be traversed by the robot, but some can be pushed and pulled by the robot into adjacent squares. Thin walls prevent movement between two adjacent squares. This work follows in a long line of algorithms and complexity work on similar problems. The 2D push-pull block puzzle shows up in the video games Pukoban as well as The Legend of Zelda: A Link to the Past, giving another proof of hardness for the latter. This variant of block-pushing puzzles is of particular interest because of its connections to reversibility, since any action (e.g., push or pull) can be inverted by another valid action (e.g., pull or push).Comment: Full version of CIAC 2017 paper. 17 page

    Placing Arrows in Directed Graph Drawings

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    We consider the problem of placing arrow heads in directed graph drawings without them overlapping other drawn objects. This gives drawings where edge directions can be deduced unambiguously. We show hardness of the problem, present exact and heuristic algorithms, and report on a practical study.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    The Complexity of Reasoning with FODD and GFODD

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    Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a knowledge representation that is useful in mechanizing decision theoretic planning in relational domains. GFODDs generalize function-free first order logic and include numerical values and numerical generalizations of existential and universal quantification. Previous work presented heuristic inference algorithms for GFODDs and implemented these heuristics in systems for decision theoretic planning. In this paper, we study the complexity of the computational problems addressed by such implementations. In particular, we study the evaluation problem, the satisfiability problem, and the equivalence problem for GFODDs under the assumption that the size of the intended model is given with the problem, a restriction that guarantees decidability. Our results provide a complete characterization placing these problems within the polynomial hierarchy. The same characterization applies to the corresponding restriction of problems in first order logic, giving an interesting new avenue for efficient inference when the number of objects is bounded. Our results show that for Σk\Sigma_k formulas, and for corresponding GFODDs, evaluation and satisfiability are Σkp\Sigma_k^p complete, and equivalence is Πk+1p\Pi_{k+1}^p complete. For Πk\Pi_k formulas evaluation is Πkp\Pi_k^p complete, satisfiability is one level higher and is Σk+1p\Sigma_{k+1}^p complete, and equivalence is Πk+1p\Pi_{k+1}^p complete.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof

    Answer Set Planning Under Action Costs

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    Recently, planning based on answer set programming has been proposed as an approach towards realizing declarative planning systems. In this paper, we present the language Kc, which extends the declarative planning language K by action costs. Kc provides the notion of admissible and optimal plans, which are plans whose overall action costs are within a given limit resp. minimum over all plans (i.e., cheapest plans). As we demonstrate, this novel language allows for expressing some nontrivial planning tasks in a declarative way. Furthermore, it can be utilized for representing planning problems under other optimality criteria, such as computing ``shortest'' plans (with the least number of steps), and refinement combinations of cheapest and fastest plans. We study complexity aspects of the language Kc and provide a transformation to logic programs, such that planning problems are solved via answer set programming. Furthermore, we report experimental results on selected problems. Our experience is encouraging that answer set planning may be a valuable approach to expressive planning systems in which intricate planning problems can be naturally specified and solved

    Complexity of Determining Nonemptiness of the Core

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    Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can do things more efficiently. However, motivating the agents to abide to a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory. However, the computational questions around this constraint have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise general representation for games in characteristic form that relies on superadditivity, and show that it allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is NP\mathcal{NP}-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition), by showing that if these are given, the problem becomes tractable in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains NP\mathcal{NP}-complete even when the collaborative possibilities are given

    Trains, Games, and Complexity: 0/1/2-Player Motion Planning through Input/Output Gadgets

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    We analyze the computational complexity of motion planning through local "input/output" gadgets with separate entrances and exits, and a subset of allowed traversals from entrances to exits, each of which changes the state of the gadget and thereby the allowed traversals. We study such gadgets in the 0-, 1-, and 2-player settings, in particular extending past motion-planning-through-gadgets work to 0-player games for the first time, by considering "branchless" connections between gadgets that route every gadget's exit to a unique gadget's entrance. Our complexity results include containment in L, NL, P, NP, and PSPACE; as well as hardness for NL, P, NP, and PSPACE. We apply these results to show PSPACE-completeness for certain mechanics in Factorio, [the Sequence], and a restricted version of Trainyard, improving prior results. This work strengthens prior results on switching graphs and reachability switching games.Comment: 37 pages, 36 figure
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