Algorithms and Conditional Lower Bounds for Planning Problems

We consider planning problems for graphs, Markov decision processes (MDPs), and games on graphs. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment. We consider two planning problems where there are k different target sets, and the problems are as follows: (a) the coverage problem asks whether there is a plan for each individual target set, and (b) the sequential target reachability problem asks whether the targets can be reached in sequence. For the coverage problem, we present a linear-time algorithm for graphs and quadratic conditional lower bound for MDPs and games on graphs. For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs. Our results with conditional lower bounds establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs and for the sequential reachability problem games on graphs are harder than MDPs and graphs; (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.Comment: Accepted at ICAPS'1

On the equivalence between logic programming semantics and argumentation semantics

This work has been supported by the National Research Fund, Luxembourg (LAAMI project), by the Engineering and Physical Sciences Research Council (EPSRC, UK), grant Ref. EP/J012084/1 (SAsSy project), by CNPq (Universal 2012 – Proc. 473110/2012-1), and by CNPq/CAPES (Casadinho/PROCAD 2011).Peer reviewedPreprin

Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter

A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and co-liveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with $O(D)$ symbolic operations, where $D$ is the diameter of the graph. Additionally we present an approximation algorithm for the graph diameter which requires $\tilde{O}(n \sqrt{D})$ symbolic steps to achieve a $(1+\epsilon)$-approximation for any constant $\epsilon > 0$. This compares to $O(n \cdot D)$ symbolic steps for the (naive) exact algorithm and $O(D)$ symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of Gentilini et al., showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components

Conditionally Optimal Algorithms for Generalized B\"uchi Games

Games on graphs provide the appropriate framework to study several central problems in computer science, such as the verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or B\"uchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized B\"uchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized B\"uchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with $n$ vertices, $m$ edges, and generalized B\"uchi objectives with $k$ conjunctions. First, we present an algorithm with running time $O(k \cdot n^2)$, improving the previously known $O(k \cdot n \cdot m)$ and $O(k^2 \cdot n^2)$ worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with $k_1$ conjunctions in the antecedent and $k_2$ conjunctions in the consequent, and present an $O(k_1 \cdot k_2 \cdot n^{2.5})$-time algorithm, improving the previously known $O(k_1 \cdot k_2 \cdot n \cdot m)$-time algorithm for $m > n^{1.5}$

On the Difference between Assumption-Based Argumentation and Abstract Argumentation

Acknowledgements The first author has been supported by the National Research Fund, Luxembourg (LAAMI project) and by the Engineering and Physical Sciences Research Council (EPSRC, UK), grant ref. EP/J012084/1 (SAsSy project). The second and third authors have been supported by CNPq (Universal 2012 - Proc. no. 473110/2012-1), CAPES (PROCAD 2009) and CNPq/CAPES (Casadinho/PROCAD 2011).Peer reviewedPostprin

On the Difference between Assumption-Based Argumentation and Abstract Argumentation

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