Complexity Issues in Multiagent Logics

Our previous research presents a methodology of cooperative problem solving for beliefdesire- intention (BDI) systems, based on a complete formal theory called TEAMLOG. This covers both a static part, defining individual, bilateral and collective agent attitudes, and a dynamic part, describing system reconfiguration in a dynamic, unpredictable environment. In this paper, we investigate the complexity of the satisfiability problem of the static part of TEAMLOG, focusing on individual and collective attitudes up to collective intention. Our logics for teamwork are squarely multi-modal, in the sense that different operators are combined and may interfere. One might expect that such a combination is much more complex than the basic multi-agent logic with one operator, but in fact we show that it is not the case: the individual part of TEAMLOG is PSPACE-complete, just like the single modality case. The full system, modelling a subtle interplay between individual and group attitudes, turns out to be EXPTIME-complete, and remains so even when propositional dynamic logic is added to it. Additionally we make a first step towards restricting the language of TEAMLOG in order to reduce its computational complexity. We study formulas with bounded modal depth and show that in case of the individual part of our logics, we obtain a reduction of the complexity to NPTIME-completeness. We also show that for group attitudes in TEAMLOG the satisfiability problem remains in EXPTIMEhard, even when modal depth is bounded by 2. We also study the combination of reducing modal depth and the number of propositional atoms. We show that in both cases this allows for checking the satisfiability in linear time.

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