Model-checking for Resource-Bounded ATL with production and consumption of resources

Abstract

Several logics for expressing coalitional ability under resource bounds have been proposed and studied in the literature. Previous work has shown that if only consumption of resources is considered or the total amount of resources produced or consumed on any path in the system is bounded, then the model-checking problem for several standard logics, such as Resource-Bounded Coalition Logic (RB-CL) and Resource-Bounded Alternating-Time Temporal Logic (RB-ATL) is decidable. However, for coalition logics with unbounded resource production and consumption, only some undecidability results are known. In this paper, we show that the model-checking problem for RB-ATL with unbounded production and consumption of resources is decidable but EXPSPACE-hard. We also investigate some tractable cases and provide a detailed comparison to a variant of the resource logic RAL, together with new complexity results.

On the Complexity of Resource-Bounded Logics

“…. Note also that the above condition is slightly different from the one in [5] but equivalent. We have decided to use our notation in order to more easily show the relationships with VASS decision problems.…”

“…This definition also differs slightly from that given in [5]; the notion of b-strategy in [5] is not relative to a state and therefore the equality should hold for all states.…”

“…The logic RB±ATL was introduced in [4,5], and extends ATL [6] with resources. RB±ATL * extends RB±ATL to allow path formulae to be any LTL-like formula (see Section 6 for a complete formal definition).…”

“…The idle action was introduced in [4,5], where motivations for requiring a distinguished 0-cost action can be found (in Section 6.5 below, we explain why the idle action is not essential for decidability). Given a coalition A ⊆ Agt and a state s, a joint action by A is a map f : A → Act such that for all agents a ∈ A, we have f(a) ∈ act(s, a).…”

“…Similarly to ATL, the language of RB±ATL includes G rather than the release operator R, even though R is not expressible in ATL [35] (hence also not in RB±ATL). The main reason for the choice of operators is consistency with ATL as defined in [6] and RB±ATL as defined in [4], and the fact that this choice of operators appears to be sufficient to describe properties of interest in verification problems. An example property that can be expressed in RB±ATL is 'Agents a and b have a strategy which requires at most 100 units of energy to reach a position where a can stay in orbit forever without requiring any additional energy, and while they are reaching this position they can also always abort the mission, again with no energy requirement'.…”