Checking Dynamic Consistency of Conditional Hyper Temporal Networks via Mean Payoff Games (Hardness and (pseudo) Singly-Exponential Time Algorithm)

In this work we introduce the \emph{Conditional Hyper Temporal Network (CHyTN)} model, which is a natural extension and generalization of both the \CSTN and the \HTN model. Our contribution goes as follows. We show that deciding whether a given \CSTN or CHyTN is dynamically consistent is \coNP-hard. Then, we offer a proof that deciding whether a given CHyTN is dynamically consistent is \PSPACE-hard, provided that the input instances are allowed to include both multi-head and multi-tail hyperarcs. In light of this, we continue our study by focusing on CHyTNs that allow only multi-head or only multi-tail hyperarcs, and we offer the first deterministic (pseudo) singly-exponential time algorithm for the problem of checking the dynamic-consistency of such CHyTNs, also producing a dynamic execution strategy whenever the input CHyTN is dynamically consistent. Since \CSTN{s} are a special case of CHyTNs, this provides as a byproduct the first sound-and-complete (pseudo) singly-exponential time algorithm for checking dynamic-consistency in CSTNs. The proposed algorithm is based on a novel connection between CSTN{s}/CHyTN{s} and Mean Payoff Games. The presentation of the connection between \CSTN{s}/CHyTNs and \MPG{s} is mediated by the \HTN model. In order to analyze the algorithm, we introduce a refined notion of dynamic-consistency, named $\epsilon$-dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction time $\hat{\varepsilon}$ where a \CSTN/CHyTN transits from being, to not being, dynamically consistent. The proof technique introduced in this analysis of $\hat{\varepsilon}$ is applicable more generally when dealing with linear difference constraints which include strict inequalities.Comment: arXiv admin note: text overlap with arXiv:1505.0082

Incorporating Decision Nodes into Conditional Simple Temporal Networks

A Conditional Simple Temporal Network (CSTN) augments a Simple Temporal Network (STN) to include special time-points, called observation time-points. In a CSTN, the agent executing the network controls the execution of every time-point. However, each observation time-point has a unique propositional letter associated with it and, when the agent executes that time-point, the environment assigns a truth value to the corresponding letter. Thus, the agent observes but, does not control the assignment of truth values. A CSTN is dynamically consistent (DC) if there exists a strategy for executing its time-points such that all relevant constraints will be satisfied no matter which truth values the environment assigns to the propositional letters. Alternatively, in a Labeled Simple Temporal Network (Labeled STN) - also called a Temporal Plan with Choice - the agent executing the network controls the assignment of values to the so-called choice variables. Furthermore, the agent can make those assignments at any time. For this reason, a Labeled STN is equivalent to a Disjunctive Temporal Network. This paper incorporates both of the above extensions by augmenting a CSTN to include not only observation time-points but also decision time-points. A decision time-point is like an observation time-point in that it has an associated propositional letter whose value is determined when the decision time-point is executed. It differs in that the agent - not the environment - selects that value. The resulting network is called a CSTN with Decisions (CSTND). This paper shows that a CSTND generalizes both CSTNs and Labeled STNs, and proves that the problem of determining whether any given CSTND is dynamically consistent is PSPACE-complete. It also presents algorithms that address two sub-classes of CSTNDs: (1) those that contain only decision time-points; and (2) those in which all decisions are made before execution begins

Sound-and-Complete Algorithms for Checking the Dynamic Controllability of Conditional Simple Temporal Networks with Uncertainty

A Conditional Simple Temporal Network with Uncertainty (CSTNU) is a data structure for representing and reasoning about time. CSTNUs incorporate observation time-points from Conditional Simple Temporal Networks (CSTNs) and contingent links from Simple Temporal Networks with Uncertainty (STNUs). A CSTNU is dynamically controllable (DC) if there exists a strategy for executing its time-points that guarantees the satisfaction of all relevant constraints no matter how the uncertainty associated with its observation time-points and contingent links is resolved in real time. This paper presents the first sound-and-complete DC-checking algorithms for CSTNUs that are based on the propagation of labeled constraints and demonstrates their practicality

A Streamlined Model of Conditional Simple Temporal Networks - Semantics and Equivalence Results

A Conditional Simple Temporal Network (CSTN) augments a Simple Temporal Network to include a new kind of time-points, called observation time-points. The execution of an observation time-point generates information in real time, specifically, the truth value of a propositional letter. In addition, time-points and temporal constraints may be labeled by conjunctions of (positive or negative) propositional letters. A CSTN is called dynamically consistent (DC) if there exists a dynamic strategy for executing its time-points such that no matter how the observations turn out during execution, the time-points whose labels are consistent with those observations have all been executed, and the constraints whose labels are consistent with those observations have all been satisfied. The strategy is dynamic in that its execution decisions may react to observations. The original formulation of CSTNs included propositional labels only on time-points, but the DC-checking algorithm was impractical because it was based on a conversion of the semantic constraints into an exponentially-sized Disjunctive Temporal Network. Later work added propositional labels to temporal constraints, and yielded a sound-and-complete propagation-based DC-checking algorithm, empirically demonstrated to be practical across a variety of CSTNs. This paper introduces a streamlined version of a CSTN in which propositional labels may appear on constraints, but not on time-points. This change simplifies the definition of the DC property, as well as the propagation rules for the DC-checking algorithm. It also simplifies the proofs of the soundness and completeness of those rules. This paper provides two translations from traditional CSTNs to streamlined CSTNs. Each translation preserves the DC property and, for any DC network, ensures that any dynamic execution strategy for that network can be extended to a strategy for its streamlined counterpart. Finally, this paper presents an empirical comparison of two versions of the DC-checking algorithm: the original version and a simplified version for streamlined CSTNs. The comparison is based on CSTN benchmarks from earlier work. For small-sized CSTNs, the original version shows the best performance, but the performance difference between the two versions decreases as the number of time-points in the CSTN increases. We conclude that the simplified algorithm is a practical alternative for checking the dynamic consistency of CSTNs

Propagating Piecewise-Linear Weights in Temporal Networks

This paper presents a novel technique using piecewise-linear functions (PLFs) as weights on edges in the graphs of two kinds of temporal networks to solve several previously open problems. Generalizing constraint-propagation rules to accom- modate PLF weights requires implementing a small handful of functions. Most problems are solved by inserting one or more edges with an initial weight of \u3b4 (a variable), then using the modified rules to propagate the PLF weights. For one kind of network, a new set of propagation rules is introduced to avoid a non-termination issue that arises when propagating PLF weights. The paper also presents two new results for determining the tightest horizon that can be imposed while preserving a network\u2019s dynamic consistency/controllability

Dynamic-Consistency Checking for Conditional Simple Temporal Networks: Strengthening the Theoretical Foundations and Presenting a Faster Algorithm

Recent work on Conditional Simple Temporal Networks (CSTNs) has focused on checking the dynamic consistency (DC) property for the case where an execution strategy can react instantaneously to observations. Three alternative semantics for such strategies\u2014IR-dynamic, 0-dynamic, and \u3c0-dynamic\u2014have been presented. However, the most practical DC-checking algorithm has only been analyzed with respect to the IR semantics. Meanwhile, 0-dynamic strategies were shown to permit a kind of circular dependence among simultaneous observations, making them impossible to implement, whereas \u3c0-dynamic strategies prohibit this kind of circularity. Whether IR-dynamic strategies allow this kind of circularity and, if so, what the consequences would be for the above-mentioned DC-checking algorithm remained open questions. This paper makes the following contributions: (1) it shows that IR-dynamic strategies do allow circular dependence and, thus, that the IR semantics does not properly capture instantaneous reactivity; (2) it shows that one of the constraint-propagation rules from the IR-DC-checking algorithm is unsound with respect to the IR semantics; (3) it presents a simpler DC-checking algorithm, called the \u3c0-DC-checking algorithm, that uses half of the rules from the earlier algorithm, and that it proves is sound and complete with respect to the \u3c0-DC semantics; (4) it empirically evaluates the new algorithm. Thus, the paper places practical DC checking for CSTNs in the case of instantaneous reaction on a solid theoretical foundation

Some results and challenges Extending Dynamic Controllability to Agile Controllability in Simple Temporal Networks with Uncertainties

Simple Temporal Networks with Uncertainty (STNU) are an expressive means to represent temporal constraints, requirements, or obligations. They feature contingent timepoints, which are set by the environment with a specified interval. Dynamic controllability is the current most relaxed notion for checking that the constraints are not in conflict. It requires that a timepoint may only depend on earlier timepoints. Agile controllability extends dynamic controllability by taking into account that a later timepoint might already be known earlier and allowing a timepoint to depend on all timepoints whose value is known before. In this report, we formally introduce the notion of an STNU with oracle timepoints, formally define the notion of agile controllability, and discuss approaches for checking agile controllability

Reducing Dynamic-Consistency (DC) Checking for Conditional Simple Temporal Networks (CSTNs) with Bounded Reaction Times to Standard DC Checking for CSTNs

Recent work on Conditional Simple Temporal Networks (CSTNs) has introduced the problem of checking the dynamic consistency (DC) property for the case where the reaction of an execution strategy to observations is bounded below by some fixed \u3b5 > 0. This paper shows how the \u3b5-DC-checking problem can be easily reduced to the standard DC-checking problem for CSTNs. Given any CSTN S with k observation time-points, the paper defines a new CSTN S0 that is the same as S, except that it includes k new observation time-points. For each observation time-point P? in S that observes some proposition p, the time-point P? in S0 is demoted from an observation time-point to an ordinary time-point; and the job of observing p is taken over by a new observation time-point P0? that is constrained to occur exactly \u3b5 after P?. The paper proves that S is \u3b5-DC if and only if S0 is DC; and shows that the application of the \u3b5-DC- checking constraint-propagation rules to S is equivalent to the application of the corresponding DC-checking constraint-propagation rules to S0. Two versions of these results are presented, depending on whether a dynamic strategy for S0 can react instantaneously or only after some arbitrarily small, positive delay. Finally, the paper demonstrates empirically that the performance of building S0 and DC-checking it is even better than \u3b5-DC-checking the original instance S

Conditional Simple Temporal Networks with Uncertainty and Decisions

A Conditional Simple Temporal Network with Uncertainty (CSTNU) is a formalism able to model temporal plans subject to both conditional constraints and uncertain durations. The combination of these two characteristics represents the uncontrollable part of the network. That is, before the network starts executing, we do not know completely which time points and constraints will be taken into consideration nor how long the uncertain durations will last. Dynamic Controllability (DC) implies the existence of a strategy scheduling the time points of the network in real time depending on how the uncontrollable part behaves. Despite all this, CSTNUs fail to model temporal plans in which a few conditional constraints are under control and may therefore influence (or be influenced by) the uncontrollable part. To bridge this gap, this paper proposes Conditional Simple Temporal Networks with Uncertainty and Decisions (CSTNUDs) which introduce decision time points into the specification in order to operate on this conditional part under control. We model the dynamic controllability checking (DC-checking) of a CSTNUD as a two-player game in which each player makes his moves in his turn at a specific time instant. We give an encoding into timed game automata for a sound and complete DC-checking. We also synthesize memoryless execution strategies for CSTNUDs proved to be DC and carry out an experimental evaluation with Esse, a tool that we have designed for CSTNUDs to make the approach fully automated

Faster Dynamic-Consistency Checking for Conditional Simple Temporal Networks

A Conditional Simple Temporal Network (CSTN) is a structure for representing and reasoning about time in domains where temporal constraints may be conditioned on outcomes of observations made in real time. A CSTN is dynamically consistent (DC) if there is a strategy for executing its time-points such that all relevant constraints will necessarily be satisfied no matter which outcomes happen to be observed. The literature on CSTNs contains only one sound-and-complete DC-checking algorithm that has been implemented and empirically evaluated. It is a graph-based algorithm that propagates labeled constraints/edges. A second algorithm has been proposed, but not evaluated. It aims to speed up DC checking by more efficiently dealing with so-called negative q-loops. This paper presents a new two-phase approach to DC-checking for CSTNs. The first phase focuses on identifying negative q-loops and labeling key time-points within them. The second phase focuses on computing (labeled) distances from each time-point to a single sink node. The new algorithm, which is also sound and complete for DC-checking, is then empirically evaluated against both pre-existing algorithms and shown to be much faster across not only previously published benchmark problems, but also a new set of benchmark problems. The results show that, on DC instances, the new algorithm tends to be an order of magnitude faster than both existing algorithms. On all other benchmark cases, the new algorithm performs better than or equivalently to the existing algorithms