Maintaining knowledge about temporal intervals
Communications of the ACM
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Our approach to the representation of time is based on Dinsmore's theory of knowledge partitioning which factors any piece of knowledge into two components, a context and a description and accordingly distributes the knowledge over multiple knowledge bases called spaces each characterized by its context. Each space is a domain for localized reasoning processes and its context bears strict relationship to other spaces. Since a space is characterized by its context, the legal consistency of a space is a property of its context. Thus we define legal contexts as follows: If f is a propositional function that maps a proposition P into a proposition f(P), we can use f as a legal context if and only if it is entailment preserving i.e., (for all P, Q) f(P & (P— Q)) — f(Q). If C is the context of a space s1 with respect to another space s2 then a proposition P true in s1 is equivalent to the proposition C(P) true in s2. This process called context climbing helps retrieve knowledge from one space into another. (Contexts may be temporal or otherwise e.g. “George believes___”, “In 1985___”, “If P then___”, “During the past hour,___” are some legal contexts.) Descriptions having the same context with respect to a space may be stored in the same space. The creation of a unique space with a context C is possible only if the context can be distributed over conjunction, i.e., (for all P, Q) (C(P) & C(Q) — C(P&Q)). If C is a distributive context then propositions P and Q are stored as P, Q in the unique space referred to by C.The idea of temporal spaces (those defined by temporal contexts) is more general than that of history proposed by Hayes. A temporal space contains propositions true at the corresponding time irrespective of their spatial dimensions. Temporal contexts are more general than the tense operators invented by Prior that map propositions true in the past or future into the present. This is because the knowledge partitioning scheme is not restricted to the four tense operators as contexts but allows any entailment preserving propositional function as a context of a temporal space. This scheme is more advantageous than a system which does not make the structural distinction between contexts and descriptions, for instance Allen's system, because such a system does not consolidate information into units that model possible situations. The proposed representation makes no logical distinction between contexts referring to time instants or intervals. One can reason about each space in isolation and use context climbing to transfer the results of the reasoning process and other information to other spaces. The set of 13 mutually exclusive and exhaustive temporal relations formulated by Allen could serve as some of the temporal contexts, since a temporal context specifies the relation of propositions in a space to those in another space. In this sense, then successive context climbing between spaces is analogous to the transitivity of temporal relations given by Allen. Contexts generalize the idea of temporal relations, as they map a proposition into a world existing at a particular time. This scheme avoids combinatorial problems by focussing reasoning processes on restricted domains and by allowing complex temporal inferences in relatively few steps. It allows one to reason about temporal and atemporal knowledge in an integrated framework. This mode of representation removes a great deal of redundancy in both storage and processing since the common context of all descriptions in a space is stored only once and does not take part in local reasoning in the space.