Query Answering over Fact Bases for Fuzzy Interval Ontologies

Transcription

1 Query Answering over Fac Bases for Fuzzy Inerval Onologies Gerald S. Plesniewicz, Vu Nguen Thi Min, Dmiry Masherov Naional Research Universiy MPEI 1: Absrac. Temporal onologies conain evens ha are conceps and roles wih references o emporal inervals. Therefore, a emporal onology induces he inerval onology. We consider fuzzy inerval onologies wrien in a fuzzy Boolean exension of Allen s inerval logic. Synacically, he exended logic ELA is he se of all Boolean combinaions of proposiional variables and senences of Allen s inerval logic. Semanics of ELA is defined using fuzzy inerpreaions of proposiional variables and aomic senences of Allen s logic. An inerval onology in ELA is a finie se ELA senences (formulas). A fac is an esimae of a formula i.e. an expression of he form r φ s where φ is a ELA formula and 0 r s 1. A fac base for an inerval onology is a finie se of facs wih formulas from he onology. We presen a mehod of finding answers o queries addressed o fac bases for fuzzy inerval onologies. The mehod uses analyical ableaux. Keywords: knowledge represenaion, onologies, fuzzy onologies, emporal logics, Allen s inerval logic, query answering 1 Inroducion Temporal onologies conain evens ha are conceps and roles wih references o emporal inervals. Therefore, a emporal onology induces he inerval onology. Consider an example. Example 1. Suppose, we should define he srucure of he concep Agen in some onology for a muli-agen sysem. Then we may wrie declaraions such as Agen[Name: Sring, Carry_ou: Acion(*)], Acion[Name: Sring, Inerval: (Ineger,Ineger), Procedure: Program]. The erms Agen(Name=rob07) and Agen(Name=rob07).Carry_ou.Inerval denoe he robo rob07 and he emporal inervals of he acions carrying ou by rob07. Le he robo rob07 is able o carry ou he acions a, b and c, i.e. Agen(Name=rob07).Carry_ou = {a, b, c}. These acions spend cerain ime. Thus, emporal inervals A, B and C are associaed wih he acions. Suppose, here is he following knowledge abou he inervals: (1) If p is rue hen here is no ime poin a which boh acions a and b are carried ou;

2 94 Gerald S. Plesniewicz e al. (2) If q is rue hen he acion b is carried ou only when he acion c is carried ou. Consider he quesion: (3) Wha Allen s relaions are impossible beween he C and A if boh condiions p and q ake place? In Allen s inerval logic (see [1, 2]) wih implicaion, he saemens (1) and (2) can be wrien as he inerval onology O ={p A bb*b, q B edfs C}(see furher). The query (3) is wrien as?x p q C x A. (End of Example 1.) In Allen s inerval logic LA, here are 7 basic relaions beween inervals: e (equals), b (before), m (mees), o (overlaps), f (finishes), s (sars), d (during). (See Table 1 for inerpreaion of hese relaions, where A and A + denoe he lef and he righ ends of he inerval A). Le r(a θ B) be he se of inequaliies characerized of he basic Allen s relaion θ (see he hird column of Table 1). For example, r(a f B) = {A > B, А + В +, B + A + }. Table 1. Basic relaions of Allen s inerval logic Inerval relaion Illusraion Inequaliies for he ends of inervals A b B ===A=== ===B=== В > А + A m B ===A=== А + В, В А + A o B A d B A s B A f B A e B ===A=== ===A=== ===A=== ===A=== =====A===== B >A, A + > В, B + >A + A >B, B + >A + A B, B A, B + > A + A > B, А + В +, B + A + A B, B A, А + В +, А + В + The invered relaions are marked by aserisks: b* (afer), m* (me-by), o* (overlapped-by), f* (finished-by), s* (sared-by), d* (conains); so, A α*b ó B α A. Le Ω 0 = {e, b, m, o, f, s, d} and Ω = Ω 0 U { b*, m*, o*, f*, s*, d*} = {e, b, m, o, f, s, d, b*, m*, o*, f*, s*, d*}. A senence (formula) of LA is an expression of he form A ω B where ω is any subse of he se Ω and A, B are inerval variables. If ω = {α}, hen insead of A{α}B we wrie simply A α B. If ω ={α 1, α 2,, α k } hen we wrie A α 1 α 2 α k B insead of A{α 1, α 2,, α k }B. By definiion, he formula A α 1 α 2 α k B is rue if i is rue a leas

3 Query Answering over Fac Bases for Fuzzy Inerval Onologies 95 one formula A α i B (1 i k). The senences of he form A α B wih α Ω 0 are called aomic. The fuzzy Boolean exension ELA Allen s inerval logic is defined as follows. SYNTAX of ELA: proposiional variables are ELA formulas; every LA formula belongs o ELA, i.e. LA ELA; if φ and ψ are ELA formulas hen ~φ, φ /\ ψ and φ \/ ψ are also ELA formulas, and φ ψ is ELA formula considered as shorhand for ~φ \/ ψ. An (inerval) onology is a finie se of ELA formulas. Le P(O) be he se of all proposiional variables enering he formulas from O, and A(O) be he se of all aomic senences enering he formulas from O. Le В(О) = U{r(β) β А(О)}. For example, if О = {A o B (B mf C) /\ p, q A s C, C o*a} hen Р(О) = {p, q} and А(О) = {B m C, B f C, A s C, A o C}, and B(O) = {B + C, C B +, B > C, B + C +, C + B +, A C, C A, C + > A +, C >A, A + > C, C + >A + }. SEMANTICS of ELA is defined using fuzzy inerpreaions. A fuzzy inerpreaion of an onology O is any funcion... from Р(О) B(О) o [0,1] = {x 0 x 1} wih he following consrains: (a) If X <Y and Y X belong o В(О) hen X <Y + Y X = 1; (b) If X =Y, X <Y and Y <X belong o В(О) hen X=Y + max{ X <Y, Y <X }=1; (c) If X <Y, Y <Z and X <Z belong o В(О) hen X <Z min{ X <Y, Y <Z }, and he similar consrains which are obained by replacing signs < by signs or =. We expand he funcion... o А(О) by A θ B = min{ V V r(a θ B)}. Furher, we expand... o formulas by he usual rules of Zadeh s fuzzy logic: ~ φ = 1 φ, φ /\ ψ = min{ φ, ψ }, φ \/ ψ = max{ φ, ψ } [3]. Le r and s be numbers from [0,1] and φ be a ELA formula. Expressions of he forms φ > r, φ r, φ < r and φ r are called esimaes of he formula φ, and expressions of he form r φ s (where 0 r s 1) are called bilaeral esimaes of φ. The esimaes are inerpreed naurally. Le... be any inerpreaion of he onology {φ}. Then φ > r ó df φ > r, φ r ó df φ r, φ < r ó df φ < r, φ r ó df φ r, r φ s ó df r φ s. The se EST of all esimaes for ELA formulas can be considered as a crisp logic wih fuzzy inerpreaions. As every logic, EST has he relaion = of logical consequence. Le E EST and σ EST. We sae E = σ when here is no fuzzy inerpreaion... : E [0,1] such ha all esimaes from E are rue bu he esimae σ is false. We consider esimaes wih he relaion as facs. For any inerval onology O = {φ 1, φ 2,, φ n } (φ i ELA), any se Fb = {r 1 φ 1 s 1, r 2 φ 2 s 2,, r n φ n s n } (0 r i, s i 1) of bilaeral esimaes is called a fac base for he onology O. We can query a fac base and ge he appropriae answers. Le ψ = ψ[x 1, x 2,, x n ] be an ELA formula in which some of is Allen s connecives are replaced wih variables x 1, x 2,, x n whose values are in Ω. A query is an expression of he form? (x 1, x 2,, x m ) ψ[ x 1, x 2,, x m ], (1.1)

4 96 Gerald S. Plesniewicz e al. where ψ = ψ[x 1, x 2,, x n ] is an ELA formula in which some of is Allen s connecives are replaced wih variables x 1, x 2,, x n whose values are in Ω. (For example, he expression?(x 1, x 2 ) (p \/ A bs B) B x 1 od C /\ ~A x 2 D is a query.) The answer o query (1.1), addressed o he fac base Kb, is he se of all uples (g, h; α 1, α 2,, α m ) wih α i Ω and g, h [0,1] such ha Kb = g ψ[α 1, α 2,, α m ] h wih maximal g and minimal h. So, we have g = max{r Kb = r ψ[α 1, α 2,, α m ]}and g = min{s Kb = ψ[α 1, α 2,, α m ] s}. Remarks. 1) I is easy o prove ha he maximum and he minimum exis. 2) Since φ r ó φ r ó 1 φ 1 r ó ~ φ r ó ~ φ 1 r and r φ s ó r φ s ó r φ, φ s ó φ r, ~ φ 1 s, hen any fac base wih bilaeral esimaes is equivalen o a fac base wih lower esimaes i.e. of he form φ i r i. We will consider furher only fac bases wih lower esimaes. Example 3. Consider he onology O from Example 1 as a fuzzy onology wih he fac base Fb = {р А bb*в 0.6, q В edfs С 0.9}. In he nex secion we show ha he se {(0.6, d), (0.6, e), (0.6, f), {(0.6, s)} is he answer o he query?x p q ~ C x A. (End of Example 3.) Generally, we can associae wih any fuzzy logic he crisp logic of esimaes whose senences are expressions of he form r φ s where φ are formulas of he fuzzy logic and 0 r s 1. Umbero Sraccia have sudied a fuzzy descripion logic which are he logics of esimaes for descripion logics [4, 5]. The logic of esimaes for proposiional logic was considered in [6] where he mehod of query answering over fac bases was described. In he paper, we presen he mehod (based on analyical ableaux [6]) for finding he answers o queries addressed a fac base for an inerval onology. 2 Finding Answers o Queries Addressed o a Fac Base The mehod of analyical ableaux can be applied o he problem of finding answers o queries addressed o fac bases for fuzzy inerval onologies. We show, by example, how o do his. Example 3. Consider again he inerval onology O and he is fac base from Example 2: Fb = {р А bb*в 0.6, q В edfs С 0.9}. In Fig.1, i is shown he deducion ree consruced sep by sep from Fb and he esimae p q ~ C x A < g which is corresponded o he body of he query?x p q ~ C x A. Consrucing he deducion ree, we sar wih he iniial branch conaining he formulas р А bb*в 0.6, q В edfs С 0.9. A he firs sep we apply he rule from Table 2 in he fourh row and second column (denoe by T2(4,2) his rule) o he formula р А bb*в 0.6 and we pu he label [1] on he righ of he formula. As a resul of he applicaion of he rule T2(4,2), he fork wih he esimaes p 0.4 and А bb*в 0.6 are added o he iniial branch and he label 1: is pu on he lef of each of he esimaes. A he sep 2, he rule T2(4,2) is applied o q В edfs С 0.9. As a resul, he fork wih he esimaes q 0.1 and В edfs С 0.9 are added o each of wo curren branches. A he sep 8, he rule T8(1,2) is applied o he esimaes q

5 Query Answering over Fac Bases for Fuzzy Inerval Onologies and q >1 g. As a resul, we ge he inequaliy g 0.9 ha means he esimaes q 0.1 and q 1 g are inconsisen (and herefore, he firs branch is inconsisen) if and if g 0.9. A sep 9, he rule T8(1,2) is applied o he esimaes p 0.4 and p >1 g. As a resul, we obain ha he second branch is inconsisen if and only if g 0.6. A sep 10, he rule T8(1,2) is applied o he esimaes q 0.1 and q 1 g. As a resul, we obain ha he hird branch is inconsisen if and only if g 0.9. Thus, he firs, second and hird branch are inconsisen if and only if g min{0.9, 0.6, 0.9} = 0.6. A sep 12, he rule T4(1,1) is applied o he esimaes В edfs С 0.9 and В edfs С 0.9, and as resul, he esimae А bb*dfmm*oo*s C 0.6 is obained. Indeed, using Table 4 which is a fragmen of he Allen s able of composiions (see [2]), we have bb* ᵒ edfs = b ᵒ e U b ᵒ d U b ᵒ f U b ᵒ s U b* ᵒ e U b* ᵒd U b* ᵒ f U b* ᵒ s = b U bdmos U bdmos U b U b* U b*dfm*o* U b* U b*dfm*o* = bb*dfmm*oo*s. A sep 13, he rule T4(1,3) is applied o he esimae А bb*dfmm*oo*s C 0.6, and we have C b*bd*f*m*mo*os*a 0.6. A sep 14, he subsiuion {x := defs, g := 0.6} is applied o he esimae C x A < g, and we have C b*bd*f*m*mo*os* A < 0.6. Finally, a sep15 we obain he conradicion: C b*bd*f*m*mo*os* A 0.6 and C b*bd*f*m*mo*os* A < 0.6. From he subsiuion we obain he following answer o he query?x p q ~ C x A: {(0.6, d), (0.6, e), (0.6, f), {(0.6, s)}. (End of Example 3.) p q ~ C x A < g [3].. р А bb*в 0.6 [1] q В edfs С 0.9 [2] 1: p 0.4 [9] 1: А bb*в 0.6 [12] 2: q 0.1 [8] 2: В edfs С 0.9 2: q 0.1 [10] 2: В edfs С 0.9 [12] : p q >1 g [4] 3: p q >1 g [5] 3: p q >1 g [6] 3: p q >1 g [7] 3: ~ C x A < g 3: ~ C x A < g 3: ~ C x A < g 3: ~ C x A < g [11] 4: p >1 g 5: p >1 g [9] 6: p >1 g 7: p >1 g 4: q >1 g [8] 5: q >1 g 6: q >1 g [10] 7: q < 1 g 8: g 0.9 X 9: g 0.6 X 10: g 0.9 X 11: C x A < g {x := defs, g := 0.6} [14] 12: А bb*dfmm*oo*s C 0.6 [13] 13: C b*bd*f*m*mo*os* A 0.6 [15] 14: C b*bd*f*m*mo*os* A < 0.6 [15] 15: X Fig. 1. Deducion ree for Example 2

6 98 Gerald S. Plesniewicz e al. Remark. In Example 2, he Tables 2, 3 and 4 were used o consruc he deducion ree in Fig. 1. Generally, he Tables 5, 6, 7 may be needed. The inference rules enering all hese ables are formed a complee sysem for query answering over fac bases for onologies wrien in he language ELA. Table 2. Inference rules for proposiional connecives ~ φ > φ < 1 ~ φ φ 1 ~ φ < φ > 1 ~ φ φ 1 φ /\ ψ > φ > ψ > φ /\ ψ φ > ψ > φ /\ ψ < φ < ψ < φ /\ ψ φ ψ φ \/ ψ > φ > ψ > φ \/ ψ φ ψ φ \/ ψ < φ < ψ < φ \/ ψ φ ψ φ ψ > φ <1 ψ > φ ψ φ 1 ψ φ ψ < φ >1 ψ < φ ψ φ 1 ψ Fragmen of Allen s able of composiions Table 3. b d f s b b bdmos b b b* Ω b*dfm*o* b* b*dfm*o* Table 4. Inference rules wih he composie relaions ω and ρ A ω B r B ρ C s A ω ᵒ ρ C min{r, s} A ω B r B ρ C s A ω ᵒ ρ C max{r, s} A ω B r B ω*a r A ω B r B ω*a r A ω B r A ρ B s A ω B min{r, s} A ω B r A ρ B s A ω B max{r, s} A αω B r A α B r A ω B r A αω B r A α B r A ω B r Table 5. Inference rules for modificaion of esimaes (X A + ) (X A + ) > (X A + ) (X A + ) < (X >A ) (X > A ) > (X >A ) (X >A ) <

7 Query Answering over Fac Bases for Fuzzy Inerval Onologies 99 (A X) (A X) (A X) < (A X) < (A X) (A X) (A X) < (A X) < X {B +, B } Table 6. Inference rules for Allen s connecives А b В > (В A + ) > А b В (В A + ) А b В < (В A + ) < А b В < (В A + ) < А m В > А m В А m В < А m В (A + В ) > (В A + ) > (A + В ) (В A + ) > (A + В ) < (В A + ) < (A + В ) (В A + ) < А o В > А o В А o В < _ А o В _ (B > A ) > (A + > B ) > (A + < B + ) > (B > A ) (A + > B ) (A + < B + ) (B > A ) < (A + > B ) < (A + < B + ) < (B > A ) (A + > B ) (A + < B + ) А f В > А f В А f В < А f В (A > B ) > (A + B + ) > (A + B + ) > (A > B ) (A + B + ) (B + A + ) > (A > B ) < (A + B + ) < (A + B + ) < (A > B ) (A + B + ) (A + B + ) < А s В > А s В А s В < А s В (A > B ) > (B + A + ) > (B + > A + ) > (A > B ) (B + A + ) (B + > A + ) > (A B ) < (B + A + ) < (B + > A + ) < (A B ) (B + A + ) (B + > A + ) А d В > А d В А d В < _ А d В _ (A > B ) > (B + < A + ) > (A > B ) (B + < A + ) (A > B ) < (B + < A + ) < (A > B ) (B + < A + ) А e В > А e В > А e В < А e В _ (B A ) > (A B ) > (B + A + ) > (A + B + ) > (B A ) (A B ) (B + A + ) (A + B + ) (B A ) < (A B ) < (B + A + ) < (A + B + ) < (B A ) (A B ) (B + A + ) (A + B + )

8 100 Gerald S. Plesniewicz e al. Table 7. Inference rules for conrary pairs (where V {X Y, X > Y}) p < x p x p > x p _ x (V) < x (V) x (V) > x (V) x V {X Y, X > Y}) 3 Conclusion We have defined he fuzzy Boolean exension of Allen s inerval logic and considered onologies wrien in he exension. Fac bases for such onologies consis of bilaeral esimaes for formulas from he onologies. We have considered he problem of query answering over fac bases. For decision of his problem he analyical ableaux mehod was applied. Acknowledgemen This work was suppored by Russian Foundaion for Basic Research (projec ) and Minisry of Educaion and Science of Kazakhsan (projec 0115 RK 00532). References 1. Allen J.A.: Mainaining Knowledge abou Temporal Inervals. Communicaions of he ACM, 20 (11), (1983). 2. Allen J.W., Ferguson G.: Acions and Evens in Inerval Temporal Logic. Journal of Logic and Compuaion, 4 (5), (1994). 3. Perfilieva I., Mockor I.: Mahemaical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrech, Sraccia U.: A fuzzy Descripion Logic: Proceedings of he Naional Conference on Arificial Inelligence, (1988). 5. Sraccia U.: Reasoning wihin Fuzzy Descripion Logics. Journal of Arificial Inelligence Research Proceedings of he Naional Conference on Arificial Inelligence, 14 (1), (2001). 6. D Aggosino, M., Gabbay D., Hahnle R., Possega J. (eds.): Handbook of Tableaux Mehods, Kluwer Academic Publishers, Dordrech, Plesniewicz G.S.: Query Answering over Fac Bases in Fuzzy Proposiional Logic IFSA Congress and NAFIPS Annual Meeing, Edmonon, Canada, June 24-28, (2013).