Ladkin, P.B., Models of Axioms for Time Intervals, Proceedings of AAAI-87, the Sixth National Conference on Artificial Intelligence, Morgan Kaufmann 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a generalized network which constraints are preconvex. Key words : temporal reasoning, interval algebra, constraint satisfaction problems. 1 Introduction Within the context of the formalization of temporal reasoning, the interval model is the one that has been considered in the majority of cases [1, 3, 6]. Its objects are the rational intervals and the relations between these objects are relations such as meet , during , etc. Allen defined the interval network as a set of constraints between intervals and presented an algorithm for solving the problem of the consistency of such a network. This ....

P. Ladkin. Models of axioms for time intervals. AAAI-87. Proceedings of the Sixth National Conference on Artificial Intelligence, Volume 1, 234--239, Maurgan Kaufmann 1987.

....d informatique de Paris Nord, Institut Galil ee, Universit e Paris Nord, Avenue Jean Baptiste Cl ement, F 93430 Villetaneuse. y condotta irit.fr, farinas irit.fr, Institut de recherche en informatique de Toulouse, Universit e Paul Sabatier, 118 route de Narbonne, F 31062 Toulouse Cedex. kin [8] introduced the first order theory of intervals and proved that this theory is decidable and possesses the quantifier elimination property. Allen [1] introduced the interval network as a set of relational constraints between a finite number of interval variables and proposed a path consistency ....

....is decidable by means of the path consistency method in time polynomial in the length of the network. 2 Models This section is devoted to the semantical analysis of the relationship between rational intervals and rational rectangles. 2. 1 Interval models According to Allen, Hayes [2] and Ladkin [8], an interval model is a relational structure of the form M = I; m) where I is a nonempty set of intervals and m is a binary relation on I such that : ffl m ffi m Gamma1 ffi m m. ffl m ffi m = m ffi m ffi m. ffl m m Gamma1 = ffl m and m Gamma1 are serial. ffl m [ m ffi m [ ....

P. Ladkin. Models of axioms for time intervals. AAAI-87. Proceedings of the 6th National Conference on Artificial Intelligence, Volume 1, 234--239, Morgan Kaufmann, 1987.

....to it by each relationship 2. the relationships are mutually exclusive 3. the relationships have a transitive behaviour We shall below refer to this theory as IA . Allen s theory is formally re defined in terms of the single period relation Meets in [3] and formally analyzed by P. Ladkin in [9] from where we reproduce the axioms Meets is noted by k and Phi denotes the or exclusive logical connective: M 1 8p; q; r; s: pkq pks rkq = rks M 2 8p; q; r; s: pkq rks = pks Phi 9t: pktks Phi 9t: rktkq M 3 8p: 9q; r: qkpkr M 4 8p; q; r; s: pkqks pkrks = q = r 1 Let s call it IAH ....

....(y; z) def = x = y and (ii) second(x; y; z) def = x = z. Notice that over S seen as the tuples that satisfy the relation and pairs(S) are by definition exactly the same. We just introduce pairs for the sake of following the signature of our language. We show similarly to Ladkin [9] that the elements and the pairs of an unbounded linear order S form a model for IP . Theorem 1 (a model) Given an infinite set S and an unbounded strict linear order on it then the IP structure hS; pairs(S) first; secondi forms a model of IP. Proof: sketch) It is easy to see that every ....

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P. Ladkin. Models of axioms for time intervals. In Proc. AAAI'87, pages 234--239, 1987.

....to instants) qualitative relations between these intervals, and an algebra for reasoning about relations between intervals. The appeal of Allen s approach has triggered a variety of research enterprises within and beyond temporal reasoning. For example, Allen and Hayes [2, 10] and Ladkin [14] develop axiomatic frameworks for the theory; Vilain, Kautz, van Beek [20, 21] and Nkel [19] study the computational complexity of Allen s reasoning scheme and of some variants; Gsgen [9] Mukerjee and Joe [18] Freksa [7] and Hernndez [12] transfer the approach to the spatial domain; Ligozat ....

Ladkin, P.B., Models of axioms for time intervals, Proceedings AAAI-87, Morgan Kaufmann 1987, 234-239.

....(p : t : s) Phi 9t (r : t : q) M3) 8p 9q; r (q : p : r) M4) 8p; q; r; s (p : q : s p : r : s q = r) M5) 8p; q (p : q 9r; s; t (r : p q : s r : t : s) Here p : q : r is the same as p : q q : r; Phi stands for exclusive OR, i.e. exactly one of the alternatives must hold. Ladkin [4] claimed that the last axiom is dependent on the first three axioms. Galton [3] disproved this by (manually) constructing a structure which satisfies the first three axioms but falsifies the fifth. Galton s 3 element structure may be produced automatically by finite model searching tools. But ....

P. Ladkin, "Models of axioms for time intervals," Proc. AAAI-87, 234--239.

.... Hayes (1989) This logic uses one primitive relation, meets, and one primitive object, the time period. Allen Hayes present a brief axiomatization of interval logic (5 axioms) and show how the logic can be extended to distinguish between time points, moments of time, and time intervals. Ladkin (1987) shows that all the semantic models of this axiomatization are isomorphic to an interval structure based on an unbounded linear order. Ladkin (1987) and van Benthem (1983) present similar axiomatizations that are characterized by interval structures on an unbounded, dense, linear order. Thus the ....

....of interval logic (5 axioms) and show how the logic can be extended to distinguish between time points, moments of time, and time intervals. Ladkin (1987) shows that all the semantic models of this axiomatization are isomorphic to an interval structure based on an unbounded linear order. Ladkin (1987) and van Benthem (1983) present similar axiomatizations that are characterized by interval structures on an unbounded, dense, linear order. Thus the Allen Hayes model is weaker in that it does not require the time line to be dense (i.e. if t1 t3, the dense time requires that there exists a t2 ....

Ladkin, P. Models of Axioms for Time Intervals, (1987) Proc. of the National Conference of the American Association for Artificial Intelligence (AAAI-87), Morgan-Kaufman.

....; also denoted by 1 The programs we used and an enumeration of the ORD Horn subclass can be obtained from the authors or by anonymous ftp from duck.dfki.uni sb.de as pub papers RR 93 11.programs.tar.Z. 2 Other underlying models of the time line are also possible, e.g. the rationals [5, 21]. For our purposes these distinctions are not significant, however. 3 Basic Interval Sym Pictorial Endpoint Relation bol Example Relations X before Y OE xxx X Gamma Y Gamma , X Gamma Y , Y after X yyy X Y Gamma , X Y X meets Y m xxxx X Gamma Y ....

P. B. Ladkin. Models of axioms for time intervals. In AAAI-87 [1], pages 234--239.

....The theory of temporal incidence allows to infer that the time periods during which these tasks utilize the common resource do not overlap. In turn it enables some temporal constraint propagation. A lot of research in artificial intelligence focussed on formalizing time and temporal incidence [49, 3, 32, 46, 47, 4, 16, 33, 51], however it turns out not to be a simple task. First because a theory of time must naturally reflect commonsense intuitions about time. Second because it must be adequate to describe events that happen and change the values of fluents without contradicting those intuitions. For instance, ....

.... is defined, we note by pairs(S) the set of ordered pairs of distinct elements of S: pairs(S) f(x; y) j x; y 2 S x yg. Over a set of pairs we define the following relations: i) first(x; y; z) def = x = y and (ii) second(x; y; z) def = x = z. Now we show similarly to Ladkin [32] that the elements and the pairs of an unbounded linear order S form a model for IP . Theorem 1 (a model) Given an infinite set S and an unbounded strict linear order on it, the IP structure hS; pairs(S) first; secondi forms a model of IP. Proof: sketch) It is easy to prove that every ....

[Article contains additional citation context not shown here]

P. Ladkin. Models of axioms for time intervals. In Proc. AAAI'87, pages 234--239, 1987.

....a weak representation is a set of elements, interpreted as intervals, together with an assignment of one of the 13 atomic relations to each pair of intervals, in such a way that the structural constraints are met. The problem of determining all representations of Allen s algebra was considered by Ladkin (Ladkin 1987) (under a different name) Ladkin proved, by using quantifier elimination, that the exists only one countable representation of Allen s algebra A 2 , namely, the set of intervals in the rationals. In the next subsection, we indicate why Ladkin s result is a special case of a general result which ....

Ladkin, P. 1987. Models of axioms for time intervals.

....Gal90] For example, it is hard to imagine that the time spent by a ball, thrown vertically into the air, at the top of its trajectory can be represented by anything but a time point. If in some other cases intervals are more natural, they can be easily defined as ordered pairs of points [Sho87a, Lad87b] In this way, it is irrelevant whether an interval includes its endpoints or not and the truth at the endpoints problem presented by Allen simply disappears. If intervals are introduced in this way, Allen s thirteen temporal relations can be defined in terms of the natural point relations ; ....

.... consistency of PA networks (with or without the 6= relation) computing the minimal PA network and finding a solution of a PA network or its corresponding interval network [vB89a, vBC90, vB90b, vB92] These (and other) problems have also been discussed in by Ladkin and Maddux in [LM88a, LM88b, Lad87b, Lad87c, Lad87d, Lad87a, Lad86a, Lad86b] Finally, Len Schubert and his colleagues have implemented really practical algorithms for reasoning in PA networks as part of their natural language understanding system [MS88a, MS88b, MS90, GS93, GSS93] The work of [GA89] has also addressed similar ....

Peter Ladkin. Models of Axioms for Time Intervals. In Proceedings of AAAI-87, pages 234--239, 1987.

....an explicit decision procedure) On the other hand, 7, 51] reformulated Allen s Interval Calculus as a formal theory in firstorder logic in terms of the single relation meets. Allen and Hayes s axioms define precisely the theory of intervals over an unbounded linear order, not necessarily dense [66]. This weaker theory is still decidable, but does not admit elimination of quantifiers (I refer to Ladkin s several reports from Krestel Institute for further details) Allen s arguments against time points [4] are: i) they are not necessary since instants can be represented as very short periods, ....

P. Ladkin. Models of axioms for time intervals. In Proc. AAAI'87, pages 234--239, 1987.

....For example, it is hard to imagine that the time spent by a ball, thrown vertically into the air, at the top of its trajectory can be represented by anything but a time point. If in some other cases intervals are more natural, they can be easily defined as ordered pairs of points [Sho87a] Lad87b] In this way, it is irrelevant whether an interval includes its endpoints or not and the truth at the endpoints problem presented by Allen simply disappears. If intervals are introduced in this way, Allen s thirteen temporal relations can be defined in terms of the natural point relations ; ....

....Further experimental work is needed to determine the advantages or drawbacks of the algorithms in the TMM in other applications. 3. 3 Deciding First Order Statements About Intervals Ladkin has studied decision procedures for arbitrary first order statements about intervals [Lad87a] Lad88] In [Lad87b] he shows how to complete the Allen and Hayes axioms [AH85] in order to obtain a countably categorical, complete and decidable theory LM which is equivalent to the theory of the structure INT (Q) i.e. the set of intervals over the rationals along with thirteen binary relations similar to ....

Peter Ladkin. Models of Axioms for Time Intervals. In Proceedings of AAAI-87, pages 234--239, 1987.

....the or exclusive logical connective) AH 1 8p; q; r; s (pkq pks rkq ) rks) AH 2 8p; q; r; s (pkq rks ) 9t (pktks) Phi pks Phi 9t (rktkq) AH 3 8p 9q; r qkpkr AH 4 8p; q; r; s (pkqks pkrks ) q = r) AH 5 8p; q (pkq ) 9r; s; t (rkpkqks rktks) We call it AH. It has been also by Ladkin [13] who (i) relates AH to other period based theories of time, ii) completely characterizes its models, and (iii) proposes a completion to obtain an axiomatization of the theory of rational intervals which is proved to be countably categorical [24] and, therefore, complete. The completion is ....

P. Ladkin. Models of axioms for time intervals. In Proc. AAAI'87, pages 234--239, 1987.

....is an axiomatization of Th(hINT (Q) Qi) Let us see how our theory relates to previous ones. The original Allen and Hayes s theory (let us call it IAH ) 2] is exclusively based on intervals. For the sake of comparison, we can introduce instants using the following technique used by Ladkin [16]: we take pairs of intervals one meeting the other, apply the equivalence relation having the same meeting point and associate a instant to each class Let us call this extension IAH I . We obtain a theory whose class of models is the same as our instant period axiomatization, i.e. the ....

P. Ladkin. Models of axioms for time intervals. In Proc. AAAI'87, pages 234--239, 1987.

....programs we used and an enumeration of the ORD Horn subclass can be obtained from the authors or by anonymous ftp from duck.dfki.uni sb.de as pub papers DFKI others RR 93 11.programs.tar.Z. 2 Other underlying models of the time line are also possible, e.g. the rationals (Allen Hayes 1985; Ladkin 1987). For our purposes these distinctions are not significant, however. Basic Interval Sym Endpoint Relation bol Relations X before Y OE X Gamma Y Gamma , X Gamma Y , Y after X X Y Gamma , X Y X meets Y m X Gamma Y Gamma , X Gamma Y , Y ....

Ladkin, P. B. 1987. Models of axioms for time intervals.

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Ladkin, P.B., Models of Axioms for Time Intervals, Proceedings of AAAI-87, the Sixth National Conference on Artificial Intelligence, Morgan Kaufmann 1987.

....validity of propositions which are true between these two points. Two points define a convex interval, as in [All83] We would say that his representation uses points, but his ontology is that of convex intervals. It doesn t matter much in this case since the two are logically interdefinable [Lad87c] and a well known view of Quine [Qui61, Qui69, Qui70] shows that we can consider ourselves equally ontologically committed to two classes of things when they are logically interdefinable (although of course it matters what sort of logical interdefinability one considers, and what sorts of ....

....union of convex intervals, we must choose a representation for the union of convex intervals we care about. Standard logical languages use predicate or function symbols with a fixed number of places. An Allen Hayes type convex interval theory corresponds to a theory using pairs of points (see [Lad87c] Thus, one can use either two point arguments, like Shoham, or one interval argument, like Allen [All84] to construct theories of truth over convex intervals. However, union of convex intervals correspond to no fixed number of points one unionof convex interval may have four maximal convex ....

Peter B. Ladkin. Models of axioms for time intervals. In Proceedings of the 6th National Conference on Artificial Intelligence (AAAI-87), pages 234--239. Morgan Kaufmann, 1987.

....from the work of Allen [1] in temporal reasoning. In these problems, the underlying domain of values is infinite, and each constraint is an infinite relation. Allen adapted a constraint propagation algorithm to an infinite domain to help in the analysis of such problems. See [2] 6] 13] 20] [22], 24] 25] 26] 27] 28] 29] 30] 31] 32] 45] 46] 50] 55] 62] 63] 64] and [66] We formulate CSP concepts and methods using relation algebras. We believe this clarifies the mathematics of binary constraint satisfaction methods, and allows problems with finite or ....

Ladkin, P.B., Models of Axioms for Time Intervals, Proceedings of AAAI-87, the Sixth National Conference on Artificial Intelligence, Morgan Kaufmann 1987, pp234--239; Kestrel Institute Technical Report KES.U.87.4. (long version), Palo Alto CA, 1987.

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P.B. Ladkin, Models of axioms for time intervals, Proceedings AAAI'87 (1987) 234-239.

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