Handbook of Constraints Programming——Chapter19 Temporal CSPs-Preliminaries

来源：F.Rossi, P.Van Beek, T. Walsh. Handbook of Constraints Programming. Elsevier, 2006.

Chapter 19

Temporal CSPs

Manolis Koubarakis

 

Reasoning with temporal constraints has been a hot research topic for the last twenty years. The importance of this topic has been recognized in many areas of Computer Science and Artificial Intelligence e.g., planning [4], scheduling [23], natural language understanding [91], knowledge representation [79], spatio-temporal databases and geographical information systems [62], constraint databases [89], medical information systems [102], computer-aided verification [5], multimedia presentations [2] etc.

Temporal reasoning is an area that has greatly benefited by the application of techniques from constraint programming ever since the early papers by James Allen and others [3, 107, 31, 108, 34]. The CSP framework introduced in Chapter 2 of this handbook is immediately applicable for representing and reasoning about temporal information, and so are the algorithms of Chapters 3, 4 and 5. Temporal CSPs have been proved to be a robust framework where general CSP results such as the ones surveyed in Chapters 7 and 8 of this handbook could be applied profitably. Moreover, specific results about temporal CSPs have often provided the motivation for deriving general results about CSPs. Temporal CSPs have been studied in depth, not only because of intellectual curiosity, but mostly due to their importance for applications such as planning, scheduling, temporal databases and others mentioned above. In many cases, the problems studied come straight from the application front and developed solutions are immediately put into practical use.

In this chapter, we survey work on temporal CSPs starting from the papers that appeared in the early nineties [3, 107, 31, 108, 34] and continue with contributions that have been published as recently as last year. We have covered all of the influential works, but due to space, we have sometimes been brief in our presentation. Our presentation is sometimes historical; we hope this will turn out to be useful for the readers. For more information on temporal CSPs and temporal reasoning in general, the reader can read the Handbook of Temporal Reasoning in Artificial Intelligence [41] or the original papers that have appeared in the literature.

The rest of this chapter is organized as follows. Section 19.1 introduces some preliminary concepts of temporal reasoning and temporal CSPs. Section 19.2 introduces the most influential temporal reasoning formalisms based on constraint networks that have been proposed in the literature and relevant algorithmic problems. Then, Section 19.3 discusses efficient constraint satisfaction algorithms for these formalisms. Section 19.4 introduces the application need for more expressive queries over temporal constraint networks (especially queries combining temporal and non-temporal information) and surveys various proposals that address this need. Sections 19.5 and 19.6 introduce the scheme of indefinite constraint databases that is, up to today, the most comprehensive proposal for querying hybrid representations consisting of a relational database component and a constraint network component. In the case of temporal CSPs, the constraint network can be used to store temporal constraints on various temporal objects, and the relational database to store facts referring to these objects. Finally, Section 19.7 concludes the chapter and points out some open problems.

 

19.1 Preliminaries

In this section, we introduce the topic of representing and reasoning about temporal information, and discuss the representational choices that have been made in the temporal reasoning literature. We also introduce some basic concepts of CSPs that will subsequently be used throughout the chapter.

 

19.1.1 Temporal Representation and Reasoning: Basic Concepts

In everyday life, most people are able to communicate their knowledge and understanding of temporal phenomena without any major difficulties. However, quite different intuitions surface as soon as people undertake to construct a formal temporal representation. The literature distinguishes among three approaches for representing temporal phenomena: the change-based approach (exemplified by situation calculus [74] or event calculus [64]), the time-based approach (exemplified by various temporal logics [106]) or temporal database models [62]) and their combination [86]. Research on temporal CSPs adopts a time-based approach to temporal representation and inference. Time is introduced explicitly via an appropriate set of times (called the time structure) and change is manifested when propositions become true or false at different elements of this set. Once one adopts this approach, the time structure must be precisely defined. The relevant issues here are:

• What are the elements of the time structure? Points, intervals or both? Research in temporal CSPs has usually adopted some set of numbers P (e.g., the rationals) to be the set of points and pairs (x, y) ∈ P such that x < y to be the set of intervals. Conventional time unit systems have also been studied (e.g., see the TUS system of [70]).

• Is time totally ordered, partially ordered, branching or cyclic? Research in temporal CSPs usually assumes time to be totally ordered. There has recently been some interesting work on CSPs for other models of time e.g., partially ordered time, branching time etc. [16].

• Is time discrete or dense? The issue here is whether there exists a unit of time which cannot be decomposed. Discrete time is usually considered to be isomorphic to the integers (Z). Proponents of dense time have a choice between rationals (Q) and reals (R). Various kinds of temporal CSPs have been studied that deal nicely with all three cases.

• Is time bounded or unbounded? Time is unbounded when for every element of the time structure there is a “previous” and a “next” element. Temporal CSPs can easily handle both cases.

Once one adopts an ontology and a structure for time, one usually turns to another, equally important, consideration: what are the kinds of temporal knowledge that must be represented? There are many kinds of temporal information that are useful in applications:

• Definite temporal information. We have definite temporal information when the time associated with an event or fact is known to be equal to an absolute time i.e., a point or interval on the time line. In other words, the time associated with an event or fact is known to full precision in the desired level of granularity. For example, the sentences “The car was on service throughout March 25th, 1993” and “The car has gone for service every March 25th for the years 1993-2000” give definite temporal information with respect to the time line of the Gregorian calendar. Note that the information in the second sentence is periodic.

• Indefinite or indeterminate temporal information. We have indefinite temporal information when the time associated with an event or fact is either unknown or has not been fully specified. The time associated with an event or fact can be under-specified in various ways [39]:

– The time associated with an event or fact might be specified via a qualitative relationship (different than equality) to some absolute time. As an example, consider the sentence “John became manager after March, 1993”.

– The time associated with an event or fact might be specified via a relationship to the time associated with another event or fact. In this case, the two times can be related through a qualitative, metric (or quantitative) or mixed temporal constraint. For example, consider the statements “The explosion occurred after John left the scene” (qualitative temporal information), “The explosion occurred 5 to 10 minutes after John left the scene” (metric temporal information), and “The explosion occurred 5 to 10 minutes after John left the scene while he was getting into his car” (mixed temporal information).

– The granularity of the system time line does not match the precision to which the time associated with an event or fact is known. As an example, consider storing the information “John was hired on January 25, 1993” in a system with time-stamps in the granularity of a second.

– Dating techniques can be imperfect. All clocks have inherent imprecision. Temporal CSPs are an expressive framework and they can represent all the above types of temporal information.

 

19.1.2 Background on CSPs

The area of temporal CSPs was initiated by James Allen in his seminal paper [3]. Allen proposed to represent qualitative temporal knowledge by interval constraint networks. An interval constraint network (see Figure 19.1) is a directed graph where nodes represents intervals and edges are labelled with vectors (i.e., disjunctions) of the thirteen binary qualitative interval relations presented in [3]. Following [3], many researchers concentrated on CSPs (or, equivalently, constraint networks) as a means for representing and reasoning about temporal knowledge. Their proposals are surveyed in Section 19.2 of this chapter.

In this chapter, the equivalent terms CSP, constraint network and set (conjunction) of constraints will be used interchangeably. We now define formally some of the concepts from the standard CSP literature that we will use in this chapter. We use dom(xi) to refer to the domain of variable xi.

Definition 19.1. Let C be a set of constraints in variables x1, . . . , xn. The solution set of C, denoted by Sol(C), is the following relation:

{(v1, . . . , vn) ∈ dom(x1)×…×dom(xn) : for every c ∈ C, (v1, . . . , vn) satisfies c}. Each member of Sol(C) is called a solution of C.

Definition 19.2. A set of constraints is called consistent or satisfiable if and only if its solution set is non-empty.

We now define the standard concepts of i-consistency, strong i-consistency and global consistency (or decomposability).

Let C be a set of constraints in variables x1, . . . , xn. For any i such that 1 ≤ i ≤ n, C(x1, . . . , xi) will denote the set of constraints in C involving only variables x1, . . . , xi.

Definition 19.3. Let C be a set of constraints in variables x1, . . . , xn and 1 ≤ i ≤ n. C is called i-consistent iff for every i − 1 distinct variables x1, . . . , xi−1, every valuation u ={x1 ← v₁, . . . , x_i−1 ← v_i−1} such that v1 ∈ dom(x1), . . . , vi−1 ∈ dom(xi−1) and u satisfies the constraints C(x1, . . . , xi−1), and every variable xi different from x1, . . . , xi−1, there exists a value vi ∈ dom(xi) such that u can be extended to a valuation u′=u ∪ {xi ← vi} which satisfies the constraints C(x1, . . . , xi−1, xi). C is called strong i-consistent if it is j-consistent for every j, 1 ≤ j ≤ i. C is called globally consistent or decomposable iff it is i-consistent for every i, 1 ≤ i ≤ n.

We now define the standard concept of minimal set of constraints. Minimal sets of constraints are especially important in temporal CSPs because they make explicit all implied binary constraints (e.g., the strictest constraints between the endpoints of an interval or the constraints capturing the strictest qualitative relation between two points etc.). In a constraint network representation of binary constraints, the concept of minimal constraint set is equivalent to the concept of minimal network.

Definition 19.4. A set of constraints C will be called minimal if any instantiation of two variables, which satisfies the constraints involving these variables only, can be extended to a solution of C.

In temporal CSPs, the variables are used to represent time elements (points or intervals), the domains are time structures (usually Z,Q or R for time points, and the set of intervals over Z,Q or R for time intervals), and the constraints represent temporal relationships. Section 19.2 presents various temporal CSP frameworks with appropriate choices for variables, domains and temporal constraints.

The following reasoning problems have been traditionally associated with CSPs:

• Deciding whether a set of constraints is consistent.

• Finding a solution or all the solutions of a consistent constraint set.

• Computing the minimal set of constraints equivalent to a given one.

• Determining if a set of constraints is i-consistent, strong i-consistent or globally consistent.

The above reasoning problems have also been the main focus of algorithms for temporal CSPs proposed in the literature. These algorithms are surveyed in Section 19.3 of this chapter.