On the Complexity of Resource-Bounded Logics

Abstract

We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB±ATL is 2exptime-complete by using recent results on alternating VASS (and in exptime when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is expspace-complete. The problem is decidable and of the same complexity for RBTL * , proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in pspace. We also establish that the model checking problem for RB±ATL * , the extension of RB±ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource-bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations.

Perfect half space games

“…• deciding extended multi-dimensional energy games with given initial input [5], • deciding whether a Petri net weakly simulates a finite state system, or satisfies a formula of the µ-calculus fragment defined in [1], and • deciding the model-checking problem for RB±ATL [2]. The second outcome is a rather precise description of the winning strategies for Player 2 in these games.…”

“…These games with arbitrary initial credit are still coNPcomplete as a consequence of [7,Lemma 4]. With given initial credit, they were first proven decidable by Abdulla, Mayr, Sangnier, and Sproston [1], and used to decide both the model-checking problem for a suitable fragment of the µ-calculus against Petri net executions and the weak simulation problem between a finite state system and a Petri net; they also allow to decide the model-checking problem for the resource logic RB±ATL * [2]. As shown by Jančar [11], d-dimensional energy games using 2p priorities can be reduced to 'extended' multi-dimensional energy games of dimension d def = d + p, with complexity upper bounds shown earlier by Brázdil, Jančar, and Kučera [5] to be in (d − 1)-EXPTIME when d ≥ 2 is fixed, and in TOWER when d is part of the input, leaving a substantial complexity gap with the 2-EXPTIME-hardness shown in [10].…”

Parameterised Resource-Bounded ATL

“…From the result that the model-checking problem for RB±ATL is 2EXPTIME-hard (Theorem 3, (Alechina et al 2018)) when r is an input parameter. 2 Note that the proof of Theorem 3 in (Alechina et al 2018) does not require infinite bounds, so the result also holds for RB±ATL without infinite bounds. It is straightforward to reduce the modelchecking problem for RB±ATL to the known initial assignment model-checking problem for ParRB±ATL.…”

“…In other words, it is not possible to extract the values of resource parameters using model-checking procedures for these logics. The only exception is the logic ParRB±ATL * (Parameterised RB±ATL * , Resource Bounded Alternating Time Temporal Logic) introduced in (Alechina et al 2018) that allows concrete values for resource parameters to be synthesised. Using ParRB±ATL * , we can ask what is the minimal amount of energy needed for achieving some goal, or for which values of x and y does it hold that coalition A has a strategy requiring at most x units of resource to enforce some temporal goal, and coalition A does not have a strategy to enforce another temporal goal with y units of resource.…”

“…Using ParRB±ATL * , we can ask what is the minimal amount of energy needed for achieving some goal, or for which values of x and y does it hold that coalition A has a strategy requiring at most x units of resource to enforce some temporal goal, and coalition A does not have a strategy to enforce another temporal goal with y units of resource. The function form of the model-checking problem for ParRB±ATL * (return a set of constraints on the resource variables for which the formula is true) was shown to be decidable in (Alechina et al 2018), but the upper bound on the complexity of the problem was left open.…”