An Efficient Characterization of Fuzzy Temporal Interval Relations

Transcription

1 A Effcet Characterzato of Fuzz Teporal Iterval Relatos Steve Schockaert, Marte De Cock, ad Etee E. Kerre Abstract Fuzz teporal terval relatos have bee defed to port teporal kowledge represetato ad reasog the presece of vagueess. The ost portat pedet to use these fuzz relatos real world applcatos s the lack of a characterzato that s both eas to pleet ad coputatoall effcet. I ths paper, we provde such a characterzato for the portat class of pecewse lear fuzz te tervals, whch covers all tpes of fuzz te tervals that we are lkel to ecouter applcatos. Furtherore, we dscuss a ore elegat characterzato for the specal case of trapezodall shaped fuzz tervals. I. INTRODUCTION Ver ofte the teporal forato that s at our dsposal s of a qualtatve ature. A tpcal eaple s teporal forato that s etracted fro atural laguage (e.g. [1]). Alle [2] has proposed a set of 13 jotl ehaustve ad utual eclusve qualtatve relatos betwee tervals of the real le to reaso wth ths kd of forato. For eaple, f we kow that A happeed before B, ad C happeed durg B, we a deduce that A happeed before C. The uderlg asto s that the te spa of evets ca be odelled as a terval. I practce however, ths s ot alwas true, as teporal forato s soetes pervaded wth ucertat ad vagueess. I [3], Dubos ad Prade dscuss how Alle s deftos of the qualtatve terval relatos ca be geeralzed whe the edpots of the tervals are ucerta. For eaple, assue that we kow that evet A lasted fro just before a to a lttle over 1.00 p, ad B lasted fro aroud a tll the earl eveg. I the approach take [3], the edpots of the te spas of A ad B are represeted as possblt dstrbutos, whch allows to epress for eaple the possblt that evet A happeed durg B. I [4] Badalo ad Gaco propose a fraework for qualtatve reasog wth ucerta teporal relatos, whch allows to deduce, for eaple, the possblt that A happeed before C whe we ol kow that the possblt that A happeed before B s 0.6 ad the possblt that B happeed durg C s 0.8. O the other had, teporal forato ca also be affected b vagueess, at least two dfferet was. Frst of all, the qualtatve relatos a be vague, epressg for eaple that A ad B bega at approatel the sae te, or that A happeed log before B. As show [5], ths ca be odelled b akg use of a graded equalt relato. Secodl, the te spa of evets a be vague. For eaple hstorcal evets ted to have a gradual begg The authors are wth the Departet of Appled Matheatcs ad Coputer Scece, Ghet Uverst, Krjgslaa 281 (S9), B-9000 Get, Belgu (eal: Steve.Schockaert,Marte.DeCock,Etee.Kerre@UGet.be). ad/or edg (e.g. the Cold War, the Great Depresso, the Russa Revoluto,...). I [6] t was suggested to represet the te spa of such vague evets as fuzz sets, whch we call fuzz tervals ths cotet, ad geeralzed deftos of the qualtatve terval relatos were gve. Alteratve deftos of the terval relatos whch cope to soe etet wth vague teporal relatos as well, were gve [7]. As poted out [5], approaches for rakg fuzz ubers a also be sutable for ths purpose (e.g. [8]). It sees however that all of these approaches, portat propertes of the orgal terval relatos are lost, whch akes the usutable as a bass for qualtatve reasog. Therefore [9], we have proposed alteratve deftos, based o the oto of relatedess easures for fuzz sets [10]. Oe of the ost portat ltatos of these deftos for practcal applcatos s that the fuzz teporal relatos are coputatoall rather epesve to evaluate. I ths paper we troduce a coputatoall effcet characterzato of these fuzz teporal relatos for a portat class of fuzz tervals. I Secto II, we frst recall the deftos of the fuzz teporal terval relatos that were troduced [9], ad geeralzed [11]. Net, Secto III we provde a characterzato of these fuzz teporal terval relatos for arbtrar pecewse lear fuzz tervals. Ths gves us a ethod to process fuzz teporal forato a effcet wa, as real world applcatos the te spas of vague evets wll lkel be represeted b pecewse lear fuzz tervals. Moreover, ever fuzz terval ca be approated b a pecewse lear fuzz terval wth a arbtrar hgh precso. I Secto IV we dscuss a specal class of pecewse lear fuzz tervals, ael trapezodall shaped fuzz tervals. Although the resultg characterzato s less geeral, t s ore elegat, ad thus ore apt to be used theoretcal arguets. Fall, Secto V cocludes the paper. A. Crsp te tervals II. FUZZY TEMPORAL RELATIONS Qualtatve relatos betwee te tervals are usuall defed as costrats o the boudar pots of these tervals [2]. For eaple, we sa that A = [a, a + ] s before B = [b, b + ] ff a + < b, ad that A happeed durg B ff b < a ad a + < b +. Vague teporal relatos lke A happeed log before B ca be represeted b usg a fuzz relato that epresses the degree to whch oe te pot s log before aother [5]. I the followg, we use the fuzz

2 For coveece, we wrte R S stead of R Tw S the reader of ths paper. We ca prove the followg characterzatos of the copostos of L ad L. Proposto 1 (Coposto): [11] Let α 1, α 2 1, 2 0; t holds that R ad L (α 1, 1) L (α 2, 2) = L (α 1+α 2+( 1, 2),a( 1, 2)) (a) L (a,.) L (α 1, 1) L (α 2, 2) = L (α 1+α 2,a( 1, 2)) L (α 1, 1) L (α 2, 2) = L (α 2 α 1+( 1, 2) 1,a( 1, 2)) L (α 1, 1) L (α 2, 2) = L (α 1 α 2+( 1, 2) 2,a( 1, 2)) Note that partcular whe α 1 = α 2 = 1 = 2 = 0, ths proposto correspods to the trastvt rules for the crsp relatos < ad. Fg. 1. (b) L (a,.) The fuzz relatos L ad L for a fed value a R relatos L ad L defed for a ad b R as 1 f b a > α + L (a, b) = 0 f b a α b a α otherwse ( > 0) epressg the degree to whch a s log before b, ad L (a, b) = 1 f b a α 0 f b a < α b a+α+ otherwse ( > 0) epressg the degree to whch a s before or at approatel the sae te as b. The role of α ad s llustrated Fgure 1. I both cases we assue 0 ad α R. Our terpretato of the cocepts log before ad before or at approatel the sae te as depeds largel o the values of the paraeters α ad. If α = = 0, the L degeerates to <, ad L degeerates to. O the other had, whe > 0 we have a gradual (lear) trasto betwee satsfg ad ot satsfg these cocepts. It s eas to see that L (a, b) = 1 L (α,)(b, a) (1) Recall that the ( T ) coposto R T S of a fuzz relato R fro R to R ad a fuzz relato S fro R to R s defed for a ad c R as (R T S)(a, c) = T (R(a, b), S(b, c)) b R where T s a arbtrar t or. I the followg, we wll use the Łukasewcz t or T w, ad ts dual t coor S w, defed for all a ad b [0, 1] as T w (a, b) = a(0, a + b 1) S w (a, b) = (1, a + b) B. Fuzz te tervals We represet the te spa of vague evets as cove ad upper secotuous oralsed fuzz sets R, whch we call fuzz (te) tervals. Recall that a fuzz set A R wth a bouded port s cove ad upper secotuous ff for each α ]0, 1] the α level set { A() α} s a closed terval. If A ad B are fuzz te tervals, we caot ake use of boudar pots to defe (vague) teporal relatos, sce the boudares of fuzz te tervals a be precse. Istead we propose the deftos that are show the secod colu of Table I, where I w deotes the Łukasewcz plcator, defed for a ad b [0, 1] b I w (a, b) = (1, 1 a + b) ad T w s the Łukasewcz t or as defed above. For eaple bb (A, B) epresses the degree to whch the begg of A s log before the begg of B, whle eb (A, B) epresses the degree to whch the ed of A s before or at approatel the sae te as the begg of B. If there s o cause for cofuso, we soetes ot the subscrpt (α, ). It ca be show that f A ad B are crsp tervals ad α = = 0, these deftos cocde wth the correspodg deftos for crsp tervals, whch are show the frst colu of Table I. Furtherore, (geeralzatos of) the trastvt propertes of the crsp terval relatos hold for these geeralzed deftos [11]. Cosder for eaple the fuzz te tervals A ad B that are depcted Fgure 2. Table II shows the degree to whch the eght fuzz teporal relatos fro Table I are satsfed for dfferet values of the paraeters α ad. I geeral, we have that creasg the values of α ad causes a crease (resp. a decrease) the degree to whch the fuzz relatos volvg L (resp. L ) are satsfed. The a proble wth the deftos of the fuzz teporal relatos s that t s uclear how to evaluate the practce as the volve rea ad fa ragg over the real le. The ost obvous soluto s to appl dscretzato techques, but these are coputatoall epesve. Assue

3 TABLE I DEFINITION OF THE QUALITATIVE TEMPORAL RELATIONS BETWEEN FUZZY TIME INTERVALS A AND B, AND THEIR CORRESPONDENCE WITH THE CLASSICAL DEFINITIONS WHEN A = [a, a + ] AND B = [b, b + ] ARE CRISP INTERVALS. Crsp tervals Fuzz te tervals a < b ( )( A ( )( B < )) bb (A, B) = Tw(A(), f Iw(B(), L (, ))) a b ( )( B ( )( A )) bb (A, B) = f Iw(B(), Tw(A(), L (, ))) a + < b + ( )( B ( )( A < )) ee (A, B) = Tw(B(), f Iw(A(), L (, ))) a + b + ( )( A ( )( B )) ee (A, B) = f Iw(A(), Tw(B(), L (, ))) a + < b ( )( )( A B < ) eb (A, B) = f Iw(A(), f Iw(B(), L (, ))) a + b ( )( )( A B ) eb (A, B) = f Iw(A(), f Iw(B(), L (, ))) a < b + ( )( )( A B < ) be (A, B) = Tw(A(), Tw(B(), L (, ))) a b + ( )( )( A B ) be (A, B) = Tw(A(), Tw(B(), L (, ))) Fg. 2. Two fuzz te tervals A ad B III. A CHARACTERIZATION FOR PIECEWISE LINEAR FUZZY INTERVALS A. Pecewse lear fuzz tervals I the followg, we use A = ((a 0 /λ 0 ), (a 1 /λ 1 )), where a 0 a 1 ad λ 0, λ 1 [0, 1], to deote a lear fuzz set R,.e. for each R, we have λ 0 + ( a 0 ) λ1 λ0 a 1 a 0 f a 0 < a 1 A() = λ 1 f = a 1 0 otherwse for eaple that we choose soehow pots ad pots j (1, j ) such that a T w(a( ), a T w(b( j ), L (, j ))) (2) =1 j=1 s a reasoable approato of be (A, B). Evaluatg (2) stll requres Θ( 2 ) basc arthetc operatos. I the et secto we provde a effcet characterzato of the fuzz teporal relatos that s eact for pecewse lear fuzz tervals. Moreover, ths characterzato ca be used to approate the fuzz teporal relatos for arbtrar fuzz te tervals a wa that s uch ore effcet (.e. kwadratc the uber of le segets that are used to approate the fuzz te tervals). TABLE II EVALUATION OF THE FUZZY TEMPORAL RELATIONS FOR THE FUZZY INTERVALS A AND B FROM FIGURE 2 FOR DIFFERENT VALUES OF THE PARAMETERS (α, ) (α, ) (0,0) (0,20) (20,0) (20,20) (40,0) (40,20) bb (A, B) bb (A, B) ee (A, B) ee (A, B) eb (A, B) eb (A, B) be (A, B) be (A, B) Note that whe a 0 = a 1 ad λ 1 > 0, A degeerates to a fuzz sgleto ad the value λ 0 s gored. Ever pecewse lear fuzz terval A s equal to the uo of a fte set of lear fuzz sets, where the uo of two fuzz sets A ad B R s defed as (A B)() = a(a(), B()) for all R. Due to the followg two propostos, we ol eed to provde a characterzato for lear fuzz sets. Proposto 2: Let A = =1 A, B = j=1 B j, A = ((a 0/λ 0), (a 1/λ 1)) ad B j = ((b j 0 /δj 0 ), (bj 1 /δj 1 )) for 1 ad 1 j. It holds that eb (A, B) = =1 j=1 eb (A, B j ) eb (A, B) = =1 j=1 eb (A, B j ) be (A, B) = a a =1 j=1 be (A, B j ) a =1 be (A, B) = a j=1 be (A, B j ) Proof: As a eaple we prove the frst equalt. We have that eb (A, B) = f = f = f = I w(a(), f I w(b(), L (, ))) I w( a A (), f =1 I w(a (), f =1 f =1 I w(a (), I w( a j=1 B j(), L (, ))) I w(b j (), L (, ))) j=1 f j=1 I w(b j (), L (, )))

4 = f =1 j=1 = =1 I w(a (), f I w(b j (), L (, ))) j=1 eb (A, B j ) Note that the proof of Proposto 2 s depedet of the fact that the relatos L ad L are used, e.g. for a arbtrar fuzz relato S R we also have that f I w(a(), f I w(b(), S(, ))) = f =1 j=1 I w(a (), f I w(b j (), S(, ))) Ufortuatel, a slar characterzato for bb (A, B), bb (A, B), ee (A, B), ad ee (A, B) s ot vald. Note that a proof aalogous to the proof of Proposto 2 s ot possble, sce geeral f a(, ) a(f (, ) (, f ) ), To obta characterzatos of the reag fuzz teporal relatos, we wll use the followg lea. Lea 1: Let A = ((a 0 /λ 0 ), (a 1 /λ 1 )), ad a 0 < a 1. Let C() = T w (A(), L (, )) for all R. The C s a pecewse lear fuzz set. Furtherore, f λ 0 < λ 1, for all R t holds that T w (A(), L (, )) 0 f + α < a 0 λ 0 a 0+α+λ 0 f a 0 λ 0 + α < a 0 = A( + α) f a 0 + α < a 1 f a 1 + α λ 1 provded a1 a0 λ 1 λ 0, ad T w (A(), L (, )) 0 f + α < a 1 λ 1 a = 1+α+λ 1 f a 1 λ 1 + α < a 1 f a 1 + α λ 1 provded a1 a0 λ 1 λ 0. Moreover, f λ 0 λ 1, for all R t holds that T w (A(), L (, )) 0 f + α < a 0 λ 0 a = 0+α+λ 0 f a 0 λ 0 + α < a 0 f a 0 + α λ 0 Eaple 1: Cosder the lear fuzz set A 1 Fgure 3(a). The correspodg pecewse lear fuzz set C 1 defed for all R b C 1 () = T w (A(), L (20,20)(, )) s show Fgure 3(b). Proposto 3: Let A = =1 A, B = j=1 B j, A = ((a 0/λ 0), (a 1/λ 1)) ad B j = ((b j 0 /δj 0 ), (bj 1 /δj 1 )) for 1 ad 1 j. We ca alwas fd lear fuzz sets A k ad B l (1 k, 1 l, ad ) such that A = k=1 A k, B = B l ad bb (A, B) = a j=1 bb (A k, B j ) k=1 bb (A, B) = a =1 bb (A, B l) ee (A, B) = a =1 ee (A, B l) ee (A, B) = a k=1 j=1 ee (A k, B j ) Proof: As a eaple, we prove the secod equalt. The defto of bb (A, B) cotas the forula T w (A(), L (, )) (3) whch we abbrevate as C() for all R. Fro Lea 1 we kow that C s a pecewse lear fuzz set. Furtherore, sce A = =1 A, (3) equals a =1 T w (A (), L (, )) (4) Accordg to Lea 1, for all {1,..., }, the fuzz sets C defed as C () = T w (A (), L (, )) are all pecewse lear fuzz sets. Fro (3) equals (4) we derve that C s the uo of all C s ( = 1,..., ). We ow decopose each of the orgal lear fuzz sets B j oe or ore lear fuzz sets B l of the for B l = ((b l 0/δ l 0), (b l 1/δ l 1)). The paraeters b l 0 ad b l 1 are chose such that there ests a l {1, 2,..., } wth C() = C l () for all [b l 0, b l 1]. Furtherore the are restrcted b the fact that the le seget defg B l s a subseget of the le seget that defes B j. Obvousl δ l 0 = B(b l 0) ad δ l 1 = B(b l 1). We obta bb (A, B) = f = f = f = I w(b(), T w (A(), L (, ))) I w( a B l(), T w (A(), L (, ))) I w(b l(), T w (A(), L (, ))) f I w(b l(), T w (A(), L (, ))) For / [b l 0, b l 1], we have that B l () = 0, ad as a cosequece I w (B l(), T w (A(), L (, ))) = 1

5 hece we obta for certa A l s bb (A, B) = f = I w(b l(), T w (A(), L (, ))) f I w (B l(), T w (A(), L (, ))) [b l 0,bl 1 ] = f I w (B l(), T w (A l (), L (, ))) [b l 0,bl 1 ] (a) Ital decoposto of A ad B to lear fuzz sets a =1 f I w (B l(), T w (A (), L (, ))) [b l 0,bl 1 ] Hece bb (A, B) a = f =1 a =1 bb (A, B l) I w(b l(), T w (A (), L (, ))) (b) T w(a (), L (, )) Coversel we fd a =1 bb (A, B l) = a =1 f I w (B l(), [b l 0,bl 1 ] a =1 = a =1 = a =1 = f I w (B l(), [b l 0,bl 1 ] T w (A (), L (, ))) f I w (B l(), [b l 0,bl 1 ] f I w (B l(), [b l 0,bl 1 ] T w ( a =1 A (), L (, ))) T w (A(), L (, ))) T w (A l (), L (, ))) f I w (B l(), T w (A l (), L (, ))) [b l 0,bl 1 ] = bb (A, B) Eaple 2: Cosder the pecewse lear fuzz tervals fgure 3(a), ad assue that we wat to evaluate bb (20,20)(A, B) usg Proposto 3. To fd out how the lear fuzz sets B 1, B 2 ad B 3 should be decoposed we appl Lea 1 to obta a represetato as a pecewse (c) New decoposto of B to lear fuzz sets Fg. 3. Obtag ew le segets for B to appl Proposto 3. lear fuzz set for C 1 () = T w (A 1 (), L (20,20)(, )) (5) C 2 () = T w (A 2 (), L (20,20)(, )) (6) C 3 () = T w (A 3 (), L (20,20)(, )) (7) C 4 () = T w (A 4 (), L (20,20)(, )) (8) C 1 ad C 2 are depcted Fgure 3(b). Furtherore, t holds that C 3 () = C 2 () ad C 4 () = C 3 ( 25) C 3 () for all R. We have that { T w (A(), L (20,20) (, )) = C 1 () for all C 2 () for all Fro ths we ca deduce that the le seget defg B 1 should ot be dvded to subsegets, as for all [10, 30], t holds that C() = C 1 (). Nether should the le seget defg B 3 be dvded to subsegets, as for all [70, 120] t holds that C() = C 2 (). However, B 2 should be decoposed to the lear fuzz sets B 2 = ((30/0), ( / 8 21 )) ad B 3 = (( / 8 21 ), (70/1)),

6 Thus we obta that Fg. 4. If A = ((a 0 /λ 0 ), (a 1 /λ 1 )) ad λ 0 < λ 1 t holds that A() = L (0, a) (a, ), for [a 0, a 1 ] as C() = C 1 () for [30, ], ad C() = C 2() for [ , 70]. B. Lear fuzz tervals Because of Proposto 2 ad Proposto 3, the proble of fdg a characterzato for pecewse lear fuzz te tervals s reduced to fdg a characterzato for lear fuzz te tervals. I the reader of ths secto, we wll show how be (A, B) ca be evaluated whe A ad B are lear fuzz tervals. Slar results ca be obtaed for the other fuzz teporal relatos fro Table I. Due to lted space, we ot these results here. Proposto 4: Let A = ((a 0 /λ 0 ), (a 1 /λ 1 )), λ 0, λ 1 [0, 1], a 0 < a 1, B = ((b 0 /δ 0 ), (b 1 /δ 1 )), δ 0, δ 1 [0, 1], b 0 < b 1. Furtherore let a ad b be defed as follows: a = b = { a1 a 0 λ 1 λ 0 f λ 1 > λ 0 0 otherwse { b1 b 0 δ 0 δ 1 f δ 0 > δ 1 0 otherwse The characterzato of be (A, B) depeds o the relatve orderg of a, b ad. The results are suarzed Table III. Proof: (sketch) Frst assue that λ 0 < λ 1 ad δ 0 > δ 1. It the holds that A() = L (0, a) (a, ) for all [a 0, a 1 ], where a = a 1 + (1 λ 1)(a 1 a 0 ) λ 1 λ 0 Ths s llustrated Fgure 4. I the sae wa we have that B() = L (0, b ) (, b ) for all [b 0, b 1 ], where b = b 0 (1 δ 0)(b 1 b 0 ) δ 0 δ 1 be (A, B) = T w (A(), T w (B(), L (, ))) = T w (L (0, a) (a, ), [a 0,a 1] T w (L (0, b ) (, b ), L (, ))) [b 0,b 1] (9) It ca be show that geeral ( 0 1 ) T w (L (α (, ), 1, 1) L (α 2, 2) (, z)) [ 0, 1] provded 1 2, ad = T w (L (α 1, 1) (, ( 1, a( 0, z + α 2 ))), L (α 2, 2) ( 0, z)) (10) T w (L (α (, ), 1, 1) L (α 2, 2) (, z)) [ 0, 1] = T w (L (α 1, 1) (, 1), L (α 2, 2) (( 1, a( 0, α 1 )), z)) (11) provded 1 2. To coplete the proof, we ca appl (10) ad (11) to (9). Ths last part of the proof s ore or less straghtforward, but tedous as t volves a case aalss o the relatve orderg of a, b ad, as well as o the relatve orderg of a 0, a 1, b 0 + α, ad b 1. Now assue that λ 0 λ 1 ad δ 0 > δ 1. It holds that be (A, B) = T w (A(), T w (B(), L (, ))) = T w (λ 0, T w (B(), L (a 0, ))) = T w (λ 0, T w (L (0, b ) (, b ), L (a 0, ))) [b 0,b 1] ad aga we ca appl (10) or (11). The proof for δ 0 δ 1 s aalogous. Although the characterzato Table III s ore logwded tha the correspodg defto Table I, t akes pleetg the fuzz teporal relatos ver straghtforward. Moreover, the te coplet of evaluatg the fuzz teporal relatos for lear fuzz tervals s costat. Because of Proposto 2 ad Proposto 3, the te coplet of evaluatg the fuzz teporal relatos for two pecewse lear fuzz sets A ad B s kwadratc the uber of le segets that are used to defe A ad B. Note that practce t s ulkel that ore tha a few le segets are used to defe a fuzz te terval. IV. A CHARACTERIZATION FOR TRAPEZOIDALLY SHAPED FUZZY SETS I ths secto we dscuss the behavour of a specal class of pecewse lear fuzz tervals, ael trapezodall shaped fuzz sets. I partcular, we wll show that a ore

7 TABLE III CHARACTERIZATION OF THE FUZZY TEMPORAL RELATION be FOR LINEAR FUZZY TIME INTERVALS A AND B Suffcet codto Characterzato < b a be (A, B) = T w(λ 1, δ 0, L (0, (a a) 1, a(b 0 + α, a 0 )), L (0, b ) (a 0, a(b 0 + α, a 0 b 1 + b 0 )), L (a 0, b 1 )) < a b be (A, B) = T w(λ 1, δ 0, L (0, b ) ((a 1 α, b 1 ), b 0 ), L (0, ((a 1 α, b 1 + a 1 a 0 ), b 1 ), L a) (a 0, b 1 )) 0 < b a be (A, B) = T w(λ 1, δ 0, L (0, (a a) 1, a(b 0 + α, a 0 )), L (a 0, b 0 )) 0 < a b be (A, B) = T w(λ 1, δ 0, L (0, b ) ((a 1 α, b 1 ), b 0 ), L (a 1, b 1 )) 0 < a b be (A, B) = T w(λ 1, δ 0, L (a 1, b 0 )) 0 < b a be (A, B) = T w(λ 1, δ 0, L (a 1, b 0 )) 0 = a b be (A, B) = T w(λ 0, δ 0, L (0, b ) ((a 0 α, b 1 ), b 0 ), L (a 0, b 1 )) 0 = b a be (A, B) = T w(λ 1, δ 1, L (0, (a a) 1, a(b 1 + α, a 0 )), L (a 0, b 1 )) 0 = a < b be (A, B) = T w(λ 0, δ 0, L (a 0, b 0 )) 0 = b < a be (A, B) = T w(λ 1, δ 1, L (a 1, b 1 )) 0 = a = b be (A, B) = T w(λ 0, δ 1, L (a 0, b 1 )) Proposto 5: L (α 1, 1;λ 1) L (α = 2, 2;λ 2) L (α,a( 1, 2);T w(λ 1,λ 2)) L (α 1, 1;λ 1) L (α = 2, 2;λ 2) L (α,a( 1, 2);T w(λ 1,λ 2)) L (α 1, 1;λ 1) L (α = 2, 2;λ 2) L (α,a( 1, 2);T w(λ 1,λ 2)) L (α 1, 1;λ 1) L (α = 2, 2;λ 2) L (α,a( 1, 2);T w(λ 1,λ 2)) Fg. 5. The fuzz set A = [a 0, a 1, a 2, a 3 ; λ] elegat characterzato ca be obtaed for ths specal class of pecewse lear fuzz tervals. I the followg, we wll wrte A = [a 0, a 1, a 2, a 3 ; λ] to deote the fuzz set R that s defed for all R as a 0 a 1 a 0 f a 0 < < a 1 λ f a 1 a 2 A() = a 3 a 3 a 2 f a 2 < < a 3 0 otherwse where a 1 = a 0 + (a 1 a 0 )λ ad a 2 = a 3 (a 3 a 2 )λ. Furtherore, we pose a 0 a 1 a 2 a 3 ad λ [0, 1]. The fuzz set A s llustrated Fgure 5. I provg propertes of such trapezodall shaped fuzz sets, the followg lea s ofte ver useful. Lea 2: Let A = [a 0, a 1, a 2, a 3 ; λ]. For each R t holds that A() = (λ, L (0,a (a 1 a 0) 1, ), L (0,a (, a 3 a 2) 2)) I the reader of ths paper we wll soetes wrte L (α,;λ) (, ) as a shorthad for (λ, L (, )), ad L (α,;λ) (, ) as a shorthad for (λ, L (, )). To obta the characterzato for trapezodall shaped fuzz tervals, we use the followg geeralzato of Proposto 1. where α = α 1 + α λ λ 2 a( 1, 2 )T w (λ 1, λ 2 ) α = α 1 + α 2 + ( 1, 2 ) λ 1 1 λ a( 1, 2 )T w (λ 1, λ 2 ) α = α 1 + α λ λ 2 1 a( 1, 2 )T w (λ 1, λ 2 ) α = α 1 α λ λ 2 2 a( 1, 2 )T w (λ 1, λ 2 ) Note that for λ 1 = λ 2 = 1, Proposto 5 s equvalet to Proposto 1. The followg proposto provdes a characterzato for trapezodall shaped fuzz tervals. Proposto 6: Let A = [a 0, a 1, a 2, a 3 ; λ 1 ] ad B = [b 0, b 1, b 2, b 3 ; λ 2 ]. Furtherore, let s L a = a 1 a 0, s R a = a 3 a 2, s L b = b 1 b 0 ad s R b = b 3 b 2. It holds that bb (A, B) = a(t w (λ 1, 1 λ 2 ), bb (A, B) = (I w (λ 2, λ 1 ), ee (A, B) = a(t w (λ 2, 1 λ 1 ), ee (A, B) = (I w (λ 1, λ 2 ), (λ 1, L (α 1,a(,s L a,sl b ))(a 1, b 1 ))) a(1 λ 2, L (α 2,a(,s L a,sl b ))(a 1, b 1 ))) (λ 2, L (α 3,a(,s R a,sr b ))(a 2, b 2 ))) a(1 λ 1, L (α 4,a(,s R a,sr b ))(a 2, b 2 ))) eb (A, B) = a(s w (1 λ 1, 1 λ 2 ), L (α 5,a(,s R a,sl b )(a 2, b 1 ))

8 eb (A, B) = a(s w (1 λ 1, 1 λ 2 ), L (α 6,a(,s R a,sl b ))(a 2, b 1 )) be (A, B) = (T w (λ 1, λ 2 ), L (α 7,a(,s L a,sr b ))(a 1, b 2 )) be (A, B) = (T w (λ 1, λ 2 ), L (α 8,a(,s L a,sr b ))(a 1, b 2 )) where α 1 = α + (0, s L b )(1 λ 2 ) + a(, s L b )(1 λ 1 ) s L a + λ 1 (a(, s L b ), s L a ) α 2 = α + (0, s L a )(1 λ 1 ) + a(, s L a )(1 λ 2 ) s L b + λ 2 (a(, s L a ), s L b ) α 3 = α + (0, s R a )(1 λ 1 ) + a(, s R a )(1 λ 2 ) s R b + λ 2 (a(, s R a ), s R b ) α 4 = α + (0, s R b )(1 λ 2 ) + a(, s R b )(1 λ 1 ) s R a + λ 1 (a(, s R b ), s R a ) α 5 = α + (0, s L b )(1 λ 2 ) + (a(, s L b ), s R a ) λ 2 a(, s L b ) s R a λ 1 + a(, s L b, s R a )T w (λ 1, λ 2 ) α 6 = α + ( s L b, ( s L b )(1 λ 2 )) + a(, s L b )λ 2 + s R a λ 1 s R a a(, s R a, s L b )T w (λ 1, λ 2 ) α 7 = α + ( s R b, ( s R b )(1 λ 2 )) + s L a λ 1 + a(, s R b )λ 2 s L a a(, s L a, s R b )T w (λ 1, λ 2 ) α 8 = α + (0, s R b )(1 λ 2 ) + (a(, s R b ), s L a ) λ 1 s L a λ 2 a(, s R b ) + a(, s L a, s R b )T w (λ 1, λ 2 ) Proof: (sketch) As a eaple, we sketch how the characterzato for bb (A, B) ca be prove. We have bb (A, B) = Usg Lea 2 we ca show that T w (A(), f I w(b(), L (, ))) T w (A(), f I w(b(), L (, ))) = T w (L (0,a (a 1 a 0;λ 1) 1, ), f I w(l (0,b (b 1 b 0;λ 2) 1, ), L (, ))) Usg the fact that 1 T w (a, b) = I w (a, 1 b) ad (1) we fd fro Proposto 5 (usg the otatos of ths Proposto) that f I w(l (α (, ), 1, 1;λ 1) L (α 2, 2;λ 2) (z, )) = L (α,a( 1, 2);T w(λ 1,λ 2)) (z, ) (12) trapezodall shaped fuzz sets. For eaple, t s eas to see that be (A B, C D) = a(be (A, C), be (A, D), be (B, C), be (B, D)) where A, B, C, ad D are arbtrar fuzz sets R. Moreover, suffcet codtos ca be derved uder whch be (A B, C D) = (be (A, C), be (A, D), be (B, C), be (B, D)) However, a detaled dscusso of ths kd of teractos wth uo ad tersecto s outsde the scope of ths paper. V. CONCLUSION We have show how the evaluato of the fuzz teporal terval relatos for pecewse lear fuzz tervals bols dow to the evaluato for lear fuzz tervals. Furtherore, we have provded a characterzato for lear fuzz tervals that s both effcet to evaluate ad eas to pleet. Fall, we have provded ore elegat deftos for the specal case of trapezodall shaped fuzz sets. ACKNOWLEDGEMENT Steve Schockaert would lke to thak the Fud for Scetfc Research Fladers for fudg hs research. REFERENCES [1] M. Lapata ad A. Lascarde, Iferrg setece teral teporal relatos, Proceedgs of NAACL 04, pp , [2] J.F. Alle, Matag kowledge about teporal tervals, Coucatos of the ACM, vol. 26, pp , [3] D. Dubos ad H. Prade, Processg fuzz teporal kowledge, IEEE Trasactos o Sstes, Ma, ad Cberetcs, vol. 19, pp , [4] S. Badalo ad M. Gaco, A fuzz eteso of Alle s terval algebra, Lecture Notes Artfcal Itellgece, vol. 1792, pp , [5] D. Dubos, A. HadjAl, ad H. Prade, Fuzzess ad ucertat teporal reasog, Joural of Uversal Coputer Scece, vol. 9, pp , [6] G. Nagpál ad B. Motk, A fuzz odel for represetg ucerta, subjectve ad vague teporal kowledge otologes, Lecture Notes Coputer Scece, vol. 2888, pp , [7] H.J. Ohlbach, Relatos betwee fuzz te tervals, Proceedgs of the 11th It. Sp. o Teporal Represetato ad Reasog, pp , [8] D. Dubos ad H. Prade, Rakg fuzz ubers the settg of possblt theor, Iforato Sceces, vol. 30, pp , [9] S. Schockaert, M. De Cock, ad E.E. Kerrre, Iprecse teporal terval relatos, Lecture Notes Artfcal Itellgece, vol. 3849, pp , [10] S. Schockaert, M. De Cock, ad C. Corels, Relatedess of fuzz sets, Joural of Itellget ad Fuzz Sstes, vol. 6, pp , [11] S. Schockaert, M. De Cock, ad E.E. Kerre, Fuzzfg Alle s teporal terval relatos, preparato. Usg (12), ad oce ore Proposto 5 copletes the proof. Note that whle we have defed fuzz tervals as oralsed fuzz sets, ths secto we also cosder trapezodall fuzz sets A for whch hgta = A() = λ < 1. Ths s useful whe we cosder uos ad tersectos of