Processing disjunctions in temporal constraint networks

Abstract

The framework of Temporal constraint Satisfaction Problems (TCSP) has been proposed for representing and processing temporal knowledge. Deciding consistency of TC-SPs is known to be intractable. As demonstrates in this paper, even local consistency algorithms like path-consistency can be exponential due to the fragmentation problem. We present t wo new polynomial approximation algorithms, Upper-Lower-Tightening (ULT) and Loose-Path-Consistency (LPC), which are ecient y et eective in detecting inconsistencies and reducing fragmentation. The experiments we performed on hard problems in the transition region show that LPC is the superior algorithm. When incorporated within backtrack search LPC is capable of improving performance by orders of magnitude. Problems involving temporal constraints arise in various areas such as temporal databases [6], diagnosis [11], scheduling [22, 2 1 ], planning [16], common-sense reasoning [25] and natural language understanding [1]. Several formalisms for expressing and reasoning about temporal constraints have been proposed; interval algebra [2], point algebra [29], Temporal Constraint Satisfaction Problems (TCSP) [7] and the model of combined quantitative and qualitative constraints [17, 12].The two t ypes of Temporal Constraint Networks that have emerged are qualitative [2] and quantitative [7]. In the qualitative model, variables are time intervals and the constraints are qualitative. In the quantitative model, variables represent time points and the constraints are metric. Subsequently, these two t ypes were combined into a single model [17,12]. In this paper we build upon the model proposed by [17], whose variables are either points or intervals and involves three types of constraints: metric point-point and qualitative pointinterval and interval-interval.Answering queries in constraint processing reduces to the tasks of determining consistency, computing a consistent scenario and computing the minimal network. When time is represented by rational numbers 1 , deciding consistency is in NP-complete [7,17]. For qualitative networks, computing the minimal network is in NP-hard [10,7]. In both qualitative and quantitative models, the source of complexity stems from allowing disjunctive relationships between pairs of variables. Such constraints often arise in many applications, as demonstrated by the following example:

The Fusion of Fuzzy Temporal Plans: Managing Uncertainty and Time in Decentralized Command and Control Systems

“…A solution to a STP problem prescribes values to event/activity time variables in order to satisfy all temporal constraints defined in the network. As STP focuses on nondisjunctive temporal constraint (single time interval), TCSP deals with general disjunctive temporal constraints (multiple intervals) (Schwalb & Dechter, 1997;. Despite this apparent weakness, STP proves to be quite valuable in many practical application domains trading-off problem modeling complexity and tractability (polynomial time solution computation).…”

Temporal Constraint Satisfaction Problems

“…Aunque el peor de los casos en esta aproximación es también O(n 3 k e ) en la práctica permite utilizar técnicas de mejora del backtracking [12,13]. También se proponen: 1) el uso de algoritmos de camino-consistencia, bien como aproximación a la red mínima, bien como preproceso para mejorar la aplicación del algoritmo con retroceso; 2) el uso de algoritmos que explotan las características topológicas de la red [34].…”

“…Sin embargo, cuando el rango R es muy grande o cuando el dominio es el conjunto de números reales, aplicar PC-2 es problemático y puede llegar a ser impracticable porque el número de intervalos crece exponencialmente, lo que se conoce como fragmentación. Para evitar este problema Schwalb y Dechter [34] proponen algoritmos polinomiales de aproximación, semejantes al de camino consistencia, para decidir la consistencia y calcular la red mínima, llamados Upper-Lower Tightening (ULT) y Loose Path-Consistency (LPC).…”

Propagating temporal relations of intervals by matrix

“…To confront this problem, traditional models are designed to improve IA. For example, a number of researches have been focused on constraint satisfactory problems (Allen and Koomen 1983;Haddawy 1996;Ladkin and Maddux 1994;Meiri 1996;Mouhoub, Charpillet, and Haton 1998;Nebel 1997;Pirri and Reiter 1995;Schwalb, Kask, and Dechter 1994;Vilain and Kautz 1986) and solving tractable subclasses of Allen' s interval algebra (Nebel and Burckert 1995;Schwalb and Dechter 1997;Schwalb 1997;Tolba, Charpillet, and Haton 1991;Vila and Reichgelt 1996). However, propagating temporal relations is still non-tractable in general (Nebel and Buckert 1995;Nebel 1997) .…”