Temporal Constraints Time and Time Again: The Many Ways to Represent Time James F Allen

Overview Various representation techniques – Based on Dating Schemes –Constraint Propagation Approaches –Duration based approaches –Temporal Knowledge Representation Systems Conclusion

Representation Based on Dating Schemes Absolute dating – When events are instantaneous and their complete linear order is known. – A time stamp is associated with each event eg. sec:min:hour – Time comparisons are reduced to numerical calculations. Duration between events can be calculated. Domains –Database transactions using a central clock –Appointment scheduling

Pseudo Dates –Assign a number to each event such that numerical ordering reflects the linear ordering. –Used when absolute time is unknown, but the linear order is known. (e1, T1)=> event e1 occurs at time T1 (e2, T2) => event e2 occurs at time T2 If e3 occurs b/w e1 and e2 then (e3, (T1+T2)/2) –Duration information is lost.

More complex representations –Events that occur over a range but still remain instantaneous. –An event is associated with a pair (e1,l1) –This representation does not work with pseudo-times

Constraint Propagation Approach A graph-based representation. Arcs => relations Nodes => time points (events) Can represent binary relations (e1<e2) Cannot represent disjunctions involving more than two time points (e1<e2 OR e2<e3) When new information is added, it may constraint existing information. Constraints based on transitivity can be used to update graph incrementally.

Constraint Propagation Approach (cont.) Representation becomes complex when events take place over intervals. (non-instantaneous) Some relationships cannot be captured by point-based representations. E1(e1,l1) and E2(e2, l2) E1 :before E2 => (l1<e2) E1 :overlaps E2 => (e1 < e2) & (e2 < l1) & (l1 < l2)

Constraint Propagation Approach (cont.) Allen’s constraint propagation algorithm –Based on intervals, rather than points.

Constraint Propagation Approach (cont.) Ability to capture 2^13 constraints compared to 181 expressible by point-based representation. A NP-complete algorithm exists that can handle such a representation. Allen developed a heuristic O(n 3 ) algorithm. This algorithm is complete with respect to any situation that can be expressed as point-based constraints. The algorithm also handles more complex situations but without the guarantee of completeness.

Duration based representations This representation makes use of duration information. Makes use of an acyclic directed graph that has a well defined start and end node. Each node represents an event and the arcs indicate the associated duration. Each node indicates the earliest and latest starting time. This technique is very useful for scheduling applications, for which it was initially developed.

Duration based representations (cont.)

This technique is useful for domains where duration is known. One unknown duration can render this technique useless. Partial ordering of endpoints cannot be represented by this technique.

Duration based representations (cont.) Dean and McDermott developed a similar representation. Rather than using durations of events as a base, they represent all information as durations between time points. More qualitative information can be described by using infinity as a value for unknown duration values.

Temporal Logic For a temporal representation to be successful, it must be embedded within a more general representation that can encode general assertions about the world. Green(frog1,t1) frog1 is green at time t1 frog1 is green over the time interval t1 This technique does not work with all predicates sun rose over the interval t1 is not equivalent to saying sun rose at every interval in the interval t1. Such predicates take intervals as arguments----Rise(t1,t2)

Temporal Knowledge Representation Systems The RHET system developed at Rochester integrates a temporal reasoner to a general purpose reasoning system. It is a horn based AI representation language that has as a subcomponent the TIMELOGIC temporal reasoning system developed by Koomen and based on Allen's interval logic. RHET is a hybrid system, rather than using a single uniform proof technique, each predicate defined in RHET could potentially use its own specialized techniques for computing its truthhood. [A(x,y)] [B(x)] => [P(x,y)]

Temporal Knowledge Representation Systems (cont.) All temporal relations are represented as [TimeRel t1 r t2] If all variables are bounded then the temporal database is used directly. TIMELOGIC determines set of all possible bindings and this is passed to RHET. Once all the variables are bounded then this information is passed to TIMELOGIC which evaluates the consequences of this binding using constraint propagation.

Conclusion Persistence problem –If P is true over some time interval T1, what can be said about P being true after T1. –Techniques designed to tackle this problem are crude and depend on assumptions. Current techniques on time representation work well in select domains. Not effective in domains where agent has limited knowledge of the world.