Periodicity-based Temporal Constraints Paolo Terenziani*,
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1 Periodicity-based Temporal Constraints Paolo Terenziani*, Luca Anselma, Stefania Montani* *{terenz DI, Univ.. del Piemonte Orientale A. Avogadro, Italy DI, Università di Torino, Italy
2 Outline Introduction Representation formalism Reasoning mechanisms Conclusions 2
3 Outline Introduction Representation formalism Reasoning mechanisms Conclusions 3
4 Introduction Periodic events are widely studied in many research areas, such as Artificial Intelligence (AI) and Temporal Databases (TDB). In AI, approaches on periodicity-based temporal constraints: qualitative constraints [Terenziani,, 97] or durations [Longanantharaj & Gimbrone,, 95] between periodic events depend on the specific periodicity in which such events occur. In TDB, several approaches to model user-defined periodicities [Tuzhilin&Clifford Tuzhilin&Clifford,, 95].
5 Introduction Goals Comprehensive framework: - user-defined periodicities; - periodicity-based temporal constraints; - qualitative [Allen, 83] + quantitative [Dechter et al., 91] temporal constraints. 5
6 Outline Introduction Representation formalism Reasoning mechanisms Conclusions 6
7 Representation Formalism Periodicity-based constraint: <Ev, Per, Constr> Ev: : pair of events Per: : user-defined periodicity Constr: : temporal constraints 7
8 Representation Formalism Intuitive semantics <<ev1, ev2>, P, C> For each occurrence p i of P : 1. exactly one instance e1 i of ev1 in p i ; 2. exactly one instance e2 i of ev2 in p i ; 3. C(e1 i, e2 i ) holds. p 1 C p 2 C p 3 C e1 1 e2 1 e1 2 e2 2 e1 3 e2 3 time 8
9 Representation Formalism Regarding periodicities and constraints, we rely on contributions in literature: - periodicities: Leban et al. s collection formalism [86]; - temporal constraints: Dechter et al. s STP [91]. 9
10 Representation Formalism Examples Leban et al. collection formalism: Monday is the first day of the week Mondays := [1] \ Days:during:Weeks 10
11 Representation Formalism Examples Leban et al. collection formalism: Monday is the first day of the week Mondays := [1] \ Days:during:Weeks Dechter et al. s STP: Tom finishes to work between 10 and 30 min after Mary 10m E TW E MW 30m Tom starts to work after Mary 0 < S TW S MW < + 11
12 Representation Formalism Examples Periodicity-based constraint: On Monday, Tom starts to work after Mary, and finishes between 10 and 30 min after her <<TW, MW>, Mondays, <(0,+ ),(10m,+ ), (-,+ ),[10m,30m]>> (<<ev1,ev2>, P, <S ev1,s ev2 >,<S ev1,e ev2 >, <E ev1,s ev2 >,<E ev1,e ev2 >>) 12
13 Outline Introduction Representation formalism Reasoning mechanisms Conclusions 13
14 Constraint-propagation propagation-based temporal reasoning based on intersection <<ev1, ev2>, P 1, C 1 > <<ev1, ev2>, P 2, C 2 > <<ev1, ev2>,?,?>; and composition and composition <<ev1, ev2>, P 1, C 1 <<ev2, ev3>, P 2, C 2 > <<ev1, ev3>,?,?>. 14
15 Extensional calculus: exploding periodicities and generating all the instances of events (repetitions). 15
16 Extensional calculus: exploding periodicities and generating all the instances of events (repetitions). Two main drawbacks: - Too many instances; - Output of constraint propagation not user- friendly and perspicuous. 16
17 Extensional calculus: exploding periodicities and generating all the instances of events (repetitions). Two main drawbacks: - Too many instances; - Output of constraint propagation not user- friendly and perspicuous. We describe an intensional calculus: - perspicuous output; - correct but not complete; - can be combined with a complete extensional calculus. 17
18 Intensional calculus Modular intensional calculus: separately computed on periodicities ( P ) and constraints ( C ) C C are the standard ones in STP (i.e., [a,b] C [c,d] = [a,b] [c,d] and C [c,d] = [a+c,b+d]) P? 18
19 Intensional calculus on periodicities Intensional level taking into account only the repetition in a typical common period (i.e., P1 P P2 or P P2). Issues: 1. a one-to to-one one correspondence between the instances of periodicities; 2. collection formalism must be extended to intensionally express the common period; 3. an algorithm must be devised to compute the common period on the basis of two periodicities. 19
20 1. One-to to-one one correspondence From the semantics of <<ev1,ev2>,p1,c1> p1 1 p1 2 p1 3 C1 C1 C1 e1 1 e2 1 e1 2 e2 2 e1 3 e2 3 time 20
21 1. One-to to-one one correspondence From the semantics of <<ev1,ev2>,p1,c1> and of <<ev1,ev2>,p2,c2> p2 1 p2 2 p2 3 C2 e1 1 e2 1 C2 e1 2 e2 2 C2 e1 3 e2 3 time 21
22 1. One-to to-one one correspondence From the semantics, <<ev1,ev2>,p1,c1> and <<ev1,ev2>,p2,c2> are consistent only if: p1 1 p1 2 p1 3 p2 1 p2 2 p2 3 C1 C1 C1 C2 e1 1 e2 1 C2 e1 2 e2 2 C2 e1 3 e2 3 time 22
23 1. One-to to-one one correspondence From the semantics, <<ev1,ev2>,p1,c1> and <<ev1,ev2>,p2,c2> are consistent only if: p1 1 p1 2 p1 3 p2 1 p2 2 p2 3 C1 C2 e1 1 e2 1 C1 C2 e1 2 e2 2 C1 C2 e1 3 e there is a 1 to 1 correspondence between occurrences of P1 and occurrences of P2; time 23
24 1. One-to to-one one correspondence From the semantics, <<ev1,ev2>,p1,c1> and <<ev1,ev2>,p2,c2> are consistent only if: p1 1 p1 2 p1 3 p2 1 p2 2 p2 3 C1 C2 e1 1 e2 1 C1 C2 e1 2 e2 2 C1 C2 e1 3 e there is a 1 to 1 correspondence between occurrences of P1 and occurrences of P2; 2. the corresponding occurrences intersect. time 24
25 1. One-to to-one one correspondence From the semantics, <<ev1,ev2>,p1,c1> and <<ev1,ev2>,p2,c2> are consistent only if: p1 1 p1 2 p1 3 p2 1 p2 2 p2 3 C1 C2 e1 1 e2 1 C1 C2 e1 2 e2 2 C1 C2 e1 3 e there is a 1 to 1 correspondence between occurrences of P1 and occurrences of P2; 2. the corresponding occurrences intersect. E.g., Tue&Wed and Wed&Thu satisfy both the constraints; Tue&Wed and Days do not satisfy 1; Tue&Wed and Thu&Fry do not satisfy 2. time 25
26 2. Extending collection language Collection language extended with - pairwise intersection P Mon&Tue); (e.g., Sun&Mon P - pairwise restricted union P (e.g., Sun&Mon Mon&Tue). (e.g., Sun&Mon P Operators are defined only in case there is a one-to to-one one correspondence between events and take into account corresponding pairs of events. Additionally, union is restricted to provide an empty result if the corresponding pairs do not intersect in time (so that only convex time intervals are coped with) (e.g., no Sun&Mon P Thur&Fri). 26
27 3. Algorithm: simplification rules. Intensional operations perform two types of simplifications: redundancy elimination (intersection of Working-Days and Mondays should be just Mondays and not Working-Days P Mondays); empty periodicity detection (intersection of Mondays and Wednesdays should be empty and not Mondays P Wednesdays). 27
28 3. Algorithm: simplification rules. Periodicities are user-defined defining a priori all the intersections and compositions unfeasible. Relations between user-defined periodicities: P, P, i P, ni P, # P (exhaustive and mutually exclusive) E.g., P1 P P2 iff (there is a one-to to-one one correspondence between the occurrences of P1 and P2 and) temporal inclusion ( )) holds between each corresponding pair of occurrences. 28
29 3. Algorithm: simplification rules. Sets of rules to compute the relation holding between two basic periodicities. E.g., if P1 = n \ P2:during:P3, then P1 P P3, P1 # P P2, P2 # P P3. (if Mon = [1] \ Days:during:Weeks, then Mon P Weeks, Mon # P Days, Days # P Weeks). 29
30 3. Algorithm: simplification rules. Basic periodicities. P1 P P2 P1 P P2 P1 i P P2 P1 P P2 P1 P2 P1 P P2 P P2 P2 P1 P1 P P2 P1 ni P P2 P1 # P P2 E.g., Mon P Weeks Mon P Weeks=Mon and P Weeks=Weeks Weeks. 30
31 3. Algorithm: simplification rules. Composite periodicities. Difficult to find rules to determine which one of the five relations holds between two composite periodicities. Operations between composite periodicities are decomposed by considering pairwise the basic periodicities composing them. E.g., (P1 P P2) P (P 1 P P 2) is empty if Pi ni P P j (i,j {1,2}). Certain simplifications can only be captured considering composite periodicities as a whole not simplification-complete. complete. 31
32 3. Algorithm Adaptation of Floyd-Warshall Warshall s algorithm: 1. for each ev k in periodic events 2. for each ev i, ev j in periodic events 3. let PBC i,j the constraint between ev i and ev j 4. PBC i,j PBC i,j (PBC PBC k,j ) PBC i,j PBC i,j PBC i,k PBC k,j It applies the intersection and composition operations a cubic number of times. 32
33 Extensional constraint propagation Algorithm not complete in consistency checking. E.g., (i) every Mondays ev1 starts 20 hours before ev2,, and (ii) every Mondays ev2 starts 10 hours before ev3. Must consider jointly periodicities and constraints extensional algorithm. 33
34 Extensional constraint propagation Complete extensional algorithm: 1. generate the extensions of the periodicities relative to the time span LCM, least common multiple of the periodicities; 2. generate the instances of the events and add the temporal constraints in the KB; 3. add the constraints that the instances are contained in the time intervals of the proper periodicity; 4. propagate the constraints via an all-pairs shortest path algorithm. 34
35 Outline Introduction Representation formalism Reasoning mechanisms Conclusions 35
36 Conclusions The approach deals with: (i) qualitative and (ii) quantitative periodicity-based temporal constraints, and (iii) considers also user-defined periodicities. It integrates and extends the STP framework and the Leban s formalism. 36
37 Conclusions We have: - singled out five relations between periodicities and used them to - defined the operations of intersection and composition between both basic and composite periodicities; - described an intensional approach, which is correct and provides perspicuous (intensional) output, but is not complete, and - an extensional approach, which can be used to check consistency, and which is correct and complete. 37
38 Conclusions Future work Apply the approach in the domain of clinical therapies. In clinical therapies often events are periodic. Benchmarks of examples from clinical guidelines from Azienda Ospedaliera San Giovanni Battista, Torino, Italy and Cancer Research, London, UK. 38
39 The end. Thanks for your attention
40 Additional slides
41 Intensional calculus on periodicities - P1 P P2 iff NOT P1 P P2 and ( ) temporal inclusion ( )) holds between each corresponding pair of occurrences; - P1 i P P2 iff NOT P1 P P2 and NOT P1 P P2 and ( )) the temporal intersection between each corresponding pair of occurrences is not empty; - P1 ni P P2 iff ( )( ) the temporal intersection between each corresponding pair of occurrences is empty; - P1 # P P2 iff there is not any one-to to-one one correspondence between the occurrences of P1 and P2. 41
42 Intensional calculus on periodicities Example: On Monday, Tom starts to work after Mary, and finishes between 10m and 30m after her (α) <<TW, MW>, Mondays, <(0,+ ), (10m,+ ),(-,+ ),[10m,30m]>>; On working days, Tom works for 8 hours (β) <<TW>,Working-Days, <[480m,480m]>>; On working days, Tom starts to work 10m to 20m after Alice, and finishes between 10m and 30m after her (γ) <<TW,AW>,Working-Days, <[10m,20m],(10m,+ ), (-,+ ),[10m,30m]>>. 42
43 Intensional calculus on periodicities Gives: On Monday, Alice works for 7h50m to 8h20m (δ) <<AW>,Mondays,<[470m,500m]>>; On working days, Mary starts to work 8h20m before to 20m after Alice, and finishes to work between 20m before to 20m after her (ε) <<MW, AW>, Working-Days, < (-500m,20m),( (-20m,510m),( (-500m, 500m,-470m), (-20m,20m)>>.( 43
44 Calculus on composite periodicities In the following, we hypothesize that composite periodicities are put in normal form, as below: (P11 P P12 P P P1s1) P P (Ph1 P Ph2 P P Phsh) where each Pij is a basic periodicity. We also hypothesize that all input composite periodicities (if any) have already been simplified (this can be done in a pre-compilation step). Temporal reasoning on composite periodicities requires to calculate: the intersection P of normal forms; the restricted union P of normal forms (to compute respectively). 44
45 Calculus on composite periodicities Intersection of normal forms. (α) ((P11 P P P1s1) P P (Pg1 P P Pgsg)) P ((P 11 P P P P 1t1) P P P (P h1 P P P hth)). Emptiness detection.. If there are two basic periodicities Pil and P jk such that Pil #P P jk,, then the intersection of the two normal forms is undefined. If there exist two unions of basic periodicities (Pi1 P P Pisi) ) and (P j1 P P P jtj) ) such that Pil in (Pi1 P P P PisiP isi) ) and P jk in (P j1 P P P P jtjp jtj) Pil nip P jk, the intersection of the two normal forms is empty. If no emptiness is detected, the resulting periodicity can be obtained by applying the redundancy elimination rules to the formula obtained by concatenating by P the two input periodicities (i.e., to a formula like (α))( )). 45
46 Calculus on composite periodicities Intersection of normal forms. (α) ((P11 P P P1s1) P P (Pg1 P P Pgsg)) P ((P 11 P P P P 1t1) P P P (P h1 P P P hth)). Redundancy elimination.. If there are two unions of basic periodicities (Pi1 P P Pisi) ) and (P j1 P P P jtj) ) such that P jk in (P j1 P P P P jtjp jtj) s.t. Pil in (Pi1 P P P PisiP isi) Pil P P jk,, then all the other P jp with p k p k can be removed from (P j1 P P P jtj) ) because they surely are unnecessary for computing the result of the intersection (and vice versa, by exchanging the role of the two union sets). Moreover, if P jk in (P j1 P P P P jtjp jtj) ) s.t. Pil in (Pi1 P P P PisiP isi) Pil nip P jk, P jk can be removed (and vice versa). 46
47 Calculus on composite periodicities Union of normal forms. (β)) ((P11 P P P1s1) P P (Pg1 P P Pgsg)) P ((P 11 P P P P 1t1) P P P (P h1 P P P hth)). By applying distributive property, the composite periodicity above can be rewritten as follows: ((P11 P P P1s1) P (P 11 P P P P 1t1)) P P P ((Pg1 P P Pgsg) ) ) P (P h1 P P P hth)) which reduces to the intersection of unions in the form (Pi1 P P Pisi) P P (P j1 P P P jtj). 47
48 Calculus on composite periodicities Union of normal forms. (Pi1 P P Pisi) P P (P j1 P P P jtj). Emptiness detection.. If there exist two basic periodicities Pil and P jk such that Pil #P P jk,, then the union of the two normal forms is undefined. Moreover, let us consider the formula obtained after the application of the distributive property. If there is a union of unions (Pi1 P P Pisi) P P (P j1 P P P jtj) ) which is empty, the overall intersection is empty (from our definition, restricted union may introduce emptiness). Working on the corresponding basic periodicities, this translates to the following rule: if Pil in (Pi1 P P Pisi) ) and P jk in (P j1 P P P jtj) Pil nip P jk, then (Pi1 P P Pisi) P P (P j1 P P P jtj) ) (and the whole union of normal forms) is empty. If no emptiness is detected, the resulting periodicity can be obtained by applying the redundancy elimination rules (to a formula like (β)).( 48
49 Calculus on composite periodicities Union of normal forms. (Pi1 P P Pisi) P P (P j1 P P P jtj). Redundancy elimination.. Every union in the form (Pi1 P P Pisi) P P (P j1 P P P jtj) ) can be simplified as follows: if Pil in (Pi1 P P P PisiP isi) ) and P jk (P j1 P P P jtj) ) s.t. Pil P P jk,, then Pil can be removed from (Pi1 P P Pisi) (and vice versa). Moreover, the rules concerning intersection of normal forms can also be applied at this point. 49
50 Conclusions Related works Several approaches dealing with constraints between periodic events: [Ligozat,, 91; Morris et al., 96]: only periodicity-independent independent (always( always) ) qualitative constraints; [Bettini et al., 02] multiple user-defined granularities between non-repeated events; [Longanantharaj&Gimbrone,, 96] only qualitative constraints; [Terenziani,, 97] periodicity-based duration constraints. 50
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