A Tractable Generalization of Simple Temporal Networks and Its Relation to Mean Payoff Games

Abstract

Simple Temporal Networks (STNs) are used in many applications, as they provide a powerful and general tool for representing conjunctions of maximum delay constraints over ordered pairs of temporal variables. We introduce Hyper Temporal Networks (HyTNs), a strict generalization of STNs, to overcome the limitation of considering only conjunctions of constraints. In a Hyper Temporal Network a single temporal constraint may be defined as a set of two or more maximum delay constraints which is satisfied when at least one of these delay constraints is satisfied. As in STNs, a HyTN is consistent when a real value can be assigned to each temporal variable satisfying all the constraints. We show the computational complexity for this generalization and propose effective reduction algorithms for checking consistency of HyTNs unveiling the link with the field of Mean Payoff Games. HyTNs are meant as a light generalization of STNs offering an interesting compromise. On one side, as we show, there exist practical pseudo-polynomial time algorithms for checking consistency and computing feasible schedules for HyTNs. On the other side, HyTNs allow to express natural constraints that cannot be expressed by STNs like "trigger off an event exactly δ min after the occurrence of the last event in a set".

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

“…The computational equivalence between checking the consistency of HyTNs and determining the winning regions of MPGs was pointed out in [7], [8]. The tightest worst-case time complexity for solving HyTN-Consistency is expressed by the following theorem, which was proven by resorting to the Value Iteration Algorithm for MPGs [3].…”

“…Hα ← {us 2 } ∪ ∆(s1; s2); 9 wα(us 2 ) ← 0; 10 foreach v ∈ ∆(s1; s2) do 11 wα(vs 1 ) ← − ; 12 α (s1; s2; u) ← tα, Hα, wα ; not play it differently [7], [8]. Moreover, the algorithm always computes integral solution to HyTN/MPGs and, therefore, it always computes rational feasible schedules for the CSTNs given in input.…”

“…Recently, STNs have been generalized into Hyper Temporal Networks (HyTNs) [7], [8] by considering weighted directed hypergraphs, where each hyperarc models a disjunctive temporal constraint called hyper-constraint. The computational equivalence between checking the consistency of HyTNs and determining winning regions in Mean Payoff Games (MPGs) [3], [10], [17] was pointed out as well in [7], [8], where the approach was shown to be robust thanks to extensive experimental evaluations [2], [7], [8]. Mean Payoff Games are a family of two-player infinite games played on finite graphs, well known for having theoretical interest in computational complexity, being it one of the few (natural) problems lying in NP ∩ coNP, as well as various applications in model-checking and formal verification [11].…”

“…The difference, however, is that we propose to map CSTNs on HyTNs/MPGs rather than on DTPs. This makes a relevant difference since the consistency check for HyTNs can be reduced to MPGs determination [7], [8], which is amenable to practical and effective pseudo-polynomial time algorithms (indeed, in several cases the resolution methods for determining MPGs exhibit even a strongly-polynomial time behaviour [1], [2], [4], [8]). To summarize, we obtain an improved upper bound on the theoretical time complexity of the DC-checking for CSTNs (i.e., from 2-EXP to NE ∩ coNE) together with a faster DC-checking procedure, which can be used on CSTNs with a larger number of propositional variables and event nodes.…”

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.…”