Dynamic Consistency of Conditional Simple Temporal Networks via Mean Payoff Games: A Singly-Exponential Time DC-checking

Abstract

Abstract-Conditional Simple Temporal Network (CSTN) is a constraint-based graph-formalism for conditional temporal planning. It offers a more flexible formalism than the equivalent CSTP model of Tsamardinos, Vidal and Pollack, from which it was derived mainly as a sound formalization. Three notions of consistency arise for CSTNs and CSTPs: weak, strong, and dynamic. Dynamic consistency is the most interesting notion, but it is also the most challenging and it was conjectured to be hard to assess. Tsamardinos, Vidal and Pollack gave a doubly-exponential time algorithm for deciding whether a CSTN is dynamicallyconsistent and to produce, in the positive case, a dynamic execution strategy of exponential size. In the present work we offer a proof that deciding whether a CSTN is dynamicallyconsistent is coNP-hard and provide the first singly-exponential time algorithm for this problem, also producing a dynamic execution strategy whenever the input CSTN is dynamically-consistent. The algorithm is based on a novel connection with Mean Payoff Games, a family of two-player infinite games played on finite graphs, well known for having applications in model-checking and formal verification. The presentation of such connection is mediated by the Hyper Temporal Network model, a tractable generalization of Simple Temporal Networks whose consistency checking is equivalent to determining Mean Payoff Games. In order to analyze the algorithm we introduce a refined notion of dynamic-consistency, named -dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction timeε where the CSTN transits from being, to not being, dynamically-consistent. The proof technique introduced in this analysis ofε is applicable more generally when dealing with linear difference constraints which include strict inequalities.

Dynamic Controllability of Conditional Simple Temporal Networks Is PSPACE-complete

“…We prove that, for discrete CSTNs, one can always restrict her attention to discrete execution strategies, whose execution times are expressible with a number of bits at most polynomial in the size of the input. Our proof is a generalization of an argument given in [5]. Lemma 13 (Discrete CSTNs admit discrete strategies).…”

“…For CSTNs, determining the right complexity class of dynamic controllability is still a widely open question. Deciding their weak controllability has been proven to be coNP-complete [2], [5] and, since weak controllability can easily be reduced to a special case of dynamic controllability, then CSTN-DC is at least coNP-hard [5]. A first complete algorithmic solution had been proposed in [6] by reducing CSTN-DC to a time automaton game of high complexity.…”

“…Later, a complete constraint propagation algorithm was achieved with much better performances in practice [7], based on the sound constraint-propagation rules provided in [8], [9], [10] for the more general setting of Conditional Simple Temporal Networks with Uncertainty (CSTNU). In [5], a worst-case upper bound is obtained, thanks to an algorithm which requires singly-exponential time and memory. To the best of our knowledge, no better bounds are known in the literature.…”

Solving Mean-Payoff Games via Quasi Dominions

“…With similar application contexts in mind, [9] and [8] further contribute to that effort, by providing complexity results and practical solutions for the verification and automatic synthesis of reactive systems from quantitative specifications expressed in linear time temporal logic extended with mean-payoff and energy objectives. Further applications to temporal networks have been studied in [16] and [15]. Consequently, efficient algorithms to solve mean-payoff games become essential ingredients to tackle these problems in practice.…”

Instantaneous Reaction-Time in Dynamic-Consistency Checking of Conditional Simple Temporal Networks

“…Dynamic-Consistency (DC) is the most interesting notion, but it is also the most challenging. In order to address the DC-Checking problem, Comin and Rizzi (2015) introduced ε-DC (a refined, more realistic, notion of DC), and provided an algorithmic solution to it. Next, given that DC implies ε-DC for some sufficiently small ε > 0, and that for every ε > 0 it holds that ε-DC implies DC, it was offered a sharp lower bounding analysis on the critical value of the reaction-timeε under which the two notions coincide.…”