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**ABSTRACT:**We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt{2r + 1/4+ 3/2). Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.Discrete Mathematics. 06/2013; 313(11):1248–1255. - [Show abstract] [Hide abstract]

**ABSTRACT:**When modelling realistic systems, physical constraints on the resources available are often required. For example, we might say that at most N processes can access a particular resource at any moment, exactly M participants are needed for an agreement, or an agent can be in exactly one mode at any moment. Such situations are concisely modelled where literals are constrained such that at most N, or exactly M, can hold at any moment in time. In this paper we consider a logic which is a combination of standard propositional linear time temporal logic with cardinality constraints restricting the numbers of literals that can be satisfied at any moment in time. We present the logic and show how to represent a number of case studies using this logic. We propose a tableau-like algorithm for checking the satisfiability of formulae in this logic, provide details of a prototype implementation and present experimental results using the prover.Journal of Applied Logic 03/2013; 11(1):30–51. - [Show abstract] [Hide abstract]

**ABSTRACT:**Model checking is a well-established technique for the formal verification of concurrent and distributed systems. In recent years, model checking has been extended and adapted for multi-agent systems, primarily to enable the formal analysis of belief–desire–intention systems. While this has been successful, there is a need for more complex logical frameworks in order to verify realistic multi-agent systems. In particular, probabilistic and real-time aspects, as well as knowledge, belief, goals, etc., are required. However, the development of new model checking tools for complex combinations of logics is both difficult and time consuming. In this article, we show how model checkers for the constituent temporal, probabilistic, and real-time logics can be re-used in a modular way when we consider combined logics involving different dimensions. This avoids the re-implementation of model checking procedures. We define a modular approach, prove its correctness, establish its complexity, and show how it can be used to describe existing combined approaches and define yet-unimplemented combinations. We also demonstrate the feasibility of our approach on a case study.Theoretical Computer Science. 01/2013; 503:61–88. - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper describes an approach to multi-agent classification using an argumentation from experience paradigm whereby individual agents argue for a given example to be classified with a particular label according to their local data. Arguments are expressed in the form of classification rules which are generated dynamically. As such each local database can be conceptualised as an experience repository; and the individual classification rules, generated from this repository, as describing generalisations drawn from this experience. The argumentation process and the supporting data structures are fully described. The process has been implemented in the PISA (Pooling Information from Several Agents) multi-agent framework which is fully described. Experiments indicate that the operation of PISA is comparable with other classification approaches and that, when operating groups or in the presence of noise, PISA outperforms such comparable approaches.Data & Knowledge Engineering. 05/2012; 75:34–57. - [Show abstract] [Hide abstract]

**ABSTRACT:**An alternative to deploying a single robot of high complexity can be to utilize robot swarms comprising large numbers of identical, and much simpler, robots. Such swarms have been shown to be adaptable, fault-tolerant and widely applicable. However, designing individual robot algorithms to ensure effective and correct overall swarm behaviour is actually very difficult. While mechanisms for assessing the effectiveness of any swarm algorithm before deployment are essential, such mechanisms have traditionally involved either computational simulations of swarm behaviour, or experiments with robot swarms themselves. However, such simulations or experiments cannot, by their nature, analyse all possible swarm behaviours. In this paper, we will develop and apply the use of automated probabilistic formal verification techniques to robot swarms, involving an exhaustive mathematical analysis, in order to assess whether swarms will indeed behave as required. In particular we consider a foraging robot scenario to which we apply probabilistic model checking.Robotics and Autonomous Systems. 01/2012; 60:199-213. -
##### Conference Paper: On P/NP Dichotomies for EL Subsumption under Relational Constraints.

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**ABSTRACT:**We consider the problem of characterising relational con-straints under which TBox reasoning in EL is tractable. We obtain P vs. coNP-hardness dichotomies for tabular constraints and constraints imposed on a single reflexive role.Proceedings of the 24th International Workshop on Description Logics (DL 2011), Barcelona, Spain, July 13-16, 2011; 01/2011 - [Show abstract] [Hide abstract]

**ABSTRACT:**We introduce and investigate a natural extension of Dung's well-known model of argument systems in which attacks are associated with a weight, indicating the relative strength of the attack. A key concept in our framework is the notion of an inconsistency budget, which characterises how much inconsistency we are prepared to tolerate: given an inconsistency budget β, we would be prepared to disregard attacks up to a total weight of β. The key advantage of this approach is that it permits a much finer grained level of analysis of argument systems than unweighted systems, and gives useful solutions when conventional (unweighted) argument systems have none. We begin by reviewing Dung's abstract argument systems, and motivating weights on attacks (as opposed to the alternative possibility, which is to attach weights to arguments). We then present the framework of weighted argument systems. We investigate solutions for weighted argument systems and the complexity of computing such solutions, focussing in particular on weighted variations of grounded extensions. Finally, we relate our work to the most relevant examples of argumentation frameworks that incorporate strengths.Artificial Intelligence. 01/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we assess the feasibility of using formal methods, and model checking in particular, within the certification of Unmanned Aircraft Systems (UAS) for civil airspace. We begin by modelling a basic UAS control system in PROMELA, and verify it against a selected subset of the CAA's Rules of the Air using the SPIN model checker. We then refine our UAS control system to incorporate probabilistic aspects, verifying it against the same Rules of the Air using the probabilistic model checker PRISM. This shows how we can measure statistical adherence to such rules. Next we build a similar UAS control system using the autonomous agent language Gwendolen, and verify it against the small subset of the Rules of the Air using the agent model checker AJPF. We introduce more advanced autonomy into the UAS agent and show that this too can be verified. Finally we compare and contrast the various approaches, discuss the paths towards full certification, and present directions for future research.01/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the $k$-attribute model, in which the preference lists are determined by dot products of "preference vectors" with "attribute vectors" and (ii) the $k$-Euclidean model, in which the preference lists are determined by the closeness of the "positions" of the people to their "preferred positions". Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem. We show that counting the number of stable roommate assignments in the $k$-attribute model ($k \geq 4$) and the 3-Euclidean model($k \geq 3$) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class $#RH\Pi_1$. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all of $#RH\Pi_1$. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time.Computing Research Repository - CORR. 12/2010; - [Show abstract] [Hide abstract]

**ABSTRACT:**Rotating spiral and scroll waves (vortices) are investigated in the FitzHugh-Nagumo model of excitable media. The focus is on a parameter region in which there exists bistability between alternative stable vortices with distinct periods. Response functions are used to predict the filament tension of the alternative scrolls and it is shown that the slow-period scroll has negative filament tension, while the filament tension of the fast-period scroll changes sign within a hysteresis loop. The predictions are confirmed by direct simulations. Further investigations show that the slow-period scrolls display features similar to delayed after-depolarization and tend to develop into turbulence similar to ventricular fibrillation (VF). Scrolls with positive filament tension collapse or stabilize, similar to monomorphic ventricular tachycardia (VT). Perturbations, such as boundary interaction or shock stimulus, can convert the vortex with negative filament tension into the vortex with positive filament tension. This may correspond to transition from VF to VT unrelated to pinning.Chaos (Woodbury, N.Y.) 12/2010; 20(4):043136.

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