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Td[(stance.)-618(Hence,)-378(the)-353(content)-353(of)-353(this)-352(instance)-353(must)-353(be)-352(such)-353(that)-353(all)]TJ 0 -10.4608 Td[(and)-383(only)-383(answers)-383(that)-383(are)-383(certain)-383(\050as)-383(e)15(xplained)-384(a)1(bo)14(v)15(e\051)-383(and)-383(con-)]TJ 0 -10.4607 Td[(sistent)-342(with)-341(the)-342(local)-342(edit)-341(log)-342(are)-342(returned.)-585(As)-342(w)10(as)-341(sho)25(wn)-342(in)-342(the)]TJ 0 -10.4608 Td[(w)10(ork)-290(on)-290(data)-291(e)15(xcha)1(nge)-291([12,)-300(25],)-300(the)-290(certain)-290(answer)-291(semantics)-290(can)]TJ 0 -10.4608 Td[(in)-325(f)10(act)-325(be)-325(achie)25(v)15(ed)-325(through)-325(a)-326(form)-325(of)-325(\223data)-325(translation\224,)-344(b)20(uilding)]TJ 0 -10.4608 Td[(peer)-368(instances)-367(called)]TJ/F72 8.9664 Tf 79.6032 0 Td[(canonical)-368(univer)10(sal)-367(solutions)]TJ/F67 8.9664 Tf 107.6257 0 Td[(.)-663(In)-368(our)-367(case,)]TJ -187.2289 -10.4607 Td[(the)-241(source)-241(data)-241(consists)-241(of)-241(edit)-241(logs)-241(so)-241(we)-242(generali)1(ze)-242(this)-241(to)]TJ/F72 8.9664 Tf 214.697 0 Td[(update)]TJ -214.697 -10.4608 Td[(tr)15(anslation)]TJ/F67 8.9664 Tf 39.2274 0 Td[(.)-665(A)-368(k)10(e)15(y)-368(aspect)-368(of)-368(the)-368(canonical)-368(uni)25(v)15(ersal)-369(solutions)-368(is)]TJ -39.2274 -10.4608 Td[(the)]TJ/F72 8.9664 Tf 12.9983 0 Td[(placeholder)-228(values)]TJ/F67 8.9664 Tf 69.8239 0 Td[(or)]TJ/F72 8.9664 Tf 9.5105 0 Td[(labeled)-228(nulls)]TJ/F67 8.9664 Tf 47.9192 0 Td[(for)-228(unkno)25(wn)-227(v)25(alues)-228(that)-228(are)]TJ -140.2519 -10.4608 Td[(nonetheless)-276(needed)-276(in)-277(order)-276(to)-276(v)25(alidate)-276(mappings)-277(wit)1(h)-277(e)15(xistentials)]TJ 0 -10.4607 Td[(\050such)-298(as)]TJ/F44 8.9664 Tf 32.232 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9963 Td[(3)]TJ/F67 8.9664 Tf 6.8222 0.9963 Td[(in)-298(Example)-298(2\051.)-907(The)-298(labeled)-298(nulls)-298(are)-298(internal)-298(book-)]TJ -47.173 -10.4608 Td[(k)10(eeping)-354(\050e.g.,)-379(queries)-354(can)-354(join)-353(on)-354(their)-354(equality\051,)-379(b)20(ut)-354(tuples)-354(with)]TJ 0 -10.4608 Td[(labeled)-317(nulls)-318(are)-317(discarded)-317(in)-318(order)-317(to)-318(produce)-317(certain)-317(answers)-318(to)]TJ 0 -10.4608 Td[(queries.)-390(\050W)80(e)-277(can)-277(additionally)-277(return)-277(tuples)-277(with)-276(labeled)-277(nulls,)-284(i.e.,)]TJ 0 -10.4607 Td[(a)-246(superset)-245(of)-246(the)-245(certain)-246(answers,)-246(which)-246(may)-245(be)-246(desirable)-245(for)-246(some)]TJ 0 -10.4608 Td[(applications.\051)]TJ 10.1867 -16.9704 Td[(E)]TJ/F67 7.1731 Tf 5.9268 0 Td[(X)-62(A)-61(M)-63(P)-61(L)-62(E)]TJ/F67 8.9664 Tf 36.9011 0 Td[(3.)]TJ/F72 8.9664 Tf 11.208 0 Td[(Continuing)-359(our)-360(e)20(xample)10(,)-387(assume)-359(that)-360(the)-359(peer)10(s)]TJ -64.2227 -10.4608 Td[(have)-250(the)-250(following)-250(local)-250(edit)-250(lo)10(gs)-250(\050wher)37(e)-250(`)]TJ/F31 8.9664 Tf 151.2521 0 Td[(+)]TJ/F72 8.9664 Tf 7.1675 0 Td[(')-250(signi\002es)-250(insertion\051:)]TJ/F31 8.9664 Tf -100.8055 -25.3055 Td[(\001)]TJ/F44 8.9664 Tf 7.6795 0 Td[(G)]TJ ET 1 0 0 1 353.4456 337.2209 cm q []0 d 0 J 0.3985 w 0 0.1992 m 56.8561 0.1992 l S Q 1 0 0 1 -0.1992 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -353.2464 -326.7601 cm BT /F31 8.9664 Tf 357.9288 329.8984 Td[(+)]TJ ET 1 0 0 1 369.3803 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -369.3803 -326.7601 cm BT /F31 8.9664 Tf 374.0627 329.8984 Td[(1)-1000(2)-1000(3)]TJ ET 1 0 0 1 410.1025 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -56.8561 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -353.2464 -316.2994 cm BT /F31 8.9664 Tf 357.9288 319.4376 Td[(+)]TJ ET 1 0 0 1 369.3803 316.2994 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -369.3803 -316.2994 cm BT /F31 8.9664 Tf 374.0627 319.4376 Td[(3)-1000(5)-1000(2)]TJ ET 1 0 0 1 410.1025 316.2994 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -56.6569 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 56.8561 0.1992 l S Q 1 0 0 1 -353.4456 -315.9009 cm BT /F31 8.9664 Tf 435.6011 340.7577 Td[(\001)]TJ/F44 8.9664 Tf 7.6795 0 Td[(B)]TJ ET 1 0 0 1 421.5097 337.2209 cm q []0 d 0 J 0.3985 w 0 0.1992 m 43.282 0.1992 l S Q 1 0 0 1 -0.1993 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -421.3104 -326.7601 cm BT /F31 8.9664 Tf 425.9929 329.8984 Td[(+)]TJ ET 1 0 0 1 437.4443 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -437.4443 -326.7601 cm BT /F31 8.9664 Tf 442.1268 329.8984 Td[(3)-1000(5)]TJ ET 1 0 0 1 464.5924 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -43.0827 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 43.282 0.1992 l S Q 1 0 0 1 -421.5097 -326.3616 cm BT /F31 8.9664 Tf 490.1827 340.7577 Td[(\001)]TJ/F44 8.9664 Tf 7.6795 0 Td[(U)]TJ ET 1 0 0 1 475.9997 337.2209 cm q []0 d 0 J 0.3985 w 0 0.1992 m 43.282 0.1992 l S Q 1 0 0 1 -0.1993 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -475.8004 -326.7601 cm BT /F31 8.9664 Tf 480.4829 329.8984 Td[(+)]TJ ET 1 0 0 1 491.9343 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -491.9343 -326.7601 cm BT /F31 8.9664 Tf 496.6168 329.8984 Td[(2)-1000(5)]TJ ET 1 0 0 1 519.0824 326.7601 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -43.0827 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 43.282 0.1992 l S Q 1 0 0 1 -475.9997 -326.3616 cm BT /F72 8.9664 Tf 316.812 304.3832 Td[(The)-250(update)-250(tr)15(anslation)-250(constructs)-250(local)-250(instances)-250(that)-250(contain:)]TJ/F44 8.9664 Tf 75.8212 -15.9071 Td[(G)]TJ ET 1 0 0 1 375.88 284.9393 cm q []0 d 0 J 0.3985 w 0 0.1992 m 40.7222 0.1992 l S Q 1 0 0 1 -0.1992 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -375.6808 -274.4786 cm BT /F31 8.9664 Tf 380.3632 277.6168 Td[(1)-1000(2)-1000(3)]TJ ET 1 0 0 1 416.403 274.4786 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -40.7222 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -375.6808 -264.0178 cm BT /F31 8.9664 Tf 380.3632 267.1561 Td[(3)-1000(5)-1000(2)]TJ ET 1 0 0 1 416.403 264.0178 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -40.523 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 40.7222 0.1992 l S Q 1 0 0 1 -375.88 -263.6193 cm BT /F44 8.9664 Tf 437.6744 288.4761 Td[(B)]TJ ET 1 0 0 1 427.8102 284.9393 cm q []0 d 0 J 0.3985 w 0 0.1992 m 27.1481 0.1992 l S Q 1 0 0 1 -0.1992 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -427.611 -274.4786 cm BT /F31 8.9664 Tf 432.2934 277.6168 Td[(3)-1000(5)]TJ ET 1 0 0 1 454.7591 274.4786 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -27.1481 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -427.611 -264.0178 cm BT /F31 8.9664 Tf 432.2934 267.1561 Td[(3)-1000(2)]TJ ET 1 0 0 1 454.7591 264.0178 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -27.1481 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -427.611 -253.5571 cm BT /F31 8.9664 Tf 432.2934 256.6953 Td[(1)-1000(3)]TJ ET 1 0 0 1 454.7591 253.5571 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -27.1481 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -427.611 -243.0963 cm BT /F31 8.9664 Tf 432.2934 246.2346 Td[(3)-1000(3)]TJ ET 1 0 0 1 454.7591 243.0963 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -26.9489 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 27.1481 0.1992 l S Q 1 0 0 1 -427.8102 -242.6978 cm BT /F44 8.9664 Tf 477.8885 288.4761 Td[(U)]TJ ET 1 0 0 1 466.1663 284.9393 cm q []0 d 0 J 0.3985 w 0 0.1992 m 30.681 0.1992 l S Q 1 0 0 1 -0.1992 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -465.9671 -274.4786 cm BT /F31 8.9664 Tf 470.6495 277.6168 Td[(2)-1197(5)]TJ ET 1 0 0 1 496.648 274.4786 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -30.681 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -465.967 -264.0178 cm BT /F31 8.9664 Tf 470.6495 267.1561 Td[(3)-1197(2)]TJ ET 1 0 0 1 496.648 264.0178 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -30.681 -10.4607 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -465.967 -253.5571 cm BT /F31 8.9664 Tf 470.6495 256.6953 Td[(5)]TJ/F44 8.9664 Tf 13.5741 0 Td[(c)]TJ/F43 5.9776 Tf 3.9895 -0.9962 Td[(1)]TJ ET 1 0 0 1 496.648 253.5571 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -30.681 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -465.967 -243.0963 cm BT /F31 8.9664 Tf 470.6495 246.2346 Td[(2)]TJ/F44 8.9664 Tf 13.5741 0 Td[(c)]TJ/F43 5.9776 Tf 3.9895 -0.9963 Td[(2)]TJ ET 1 0 0 1 496.648 243.0963 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -30.681 -10.4608 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -465.967 -232.6355 cm BT /F31 8.9664 Tf 470.6495 235.7738 Td[(3)]TJ/F44 8.9664 Tf 13.5741 0 Td[(c)]TJ/F43 5.9776 Tf 3.9895 -0.9962 Td[(3)]TJ ET 1 0 0 1 496.648 232.6355 cm q []0 d 0 J 0.3985 w 0.1992 0 m 0.1992 10.4608 l S Q 1 0 0 1 -30.4817 -0.3985 cm q []0 d 0 J 0.3985 w 0 0.1992 m 30.681 0.1992 l S Q 1 0 0 1 -466.1663 -232.237 cm BT /F72 8.9664 Tf 316.812 219.657 Td[(Examples)-250(of)-250(certain)-250(answer)10(s)-250(query)-250(semantics)-250(at)]TJ/F66 8.9664 Tf 174.9857 0 Td[(P)]TJ/F9 4.9813 Tf 4.2561 -0.9962 Td[(B)-47(i)1(oS)-51(QL)]TJ/F72 8.9664 Tf 26.7238 0.9962 Td[(:)]TJ/F46 8.9664 Tf -192.6405 -20.3006 Td[(\017)]TJ/F64 7.9701 Tf 9.0908 0 Td[(quer)-30(y)]TJ/F44 8.9664 Tf 23.6752 0 Td[(ans)]TJ/F31 8.9664 Tf 14.7339 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)-171(y)]TJ/F31 8.9664 Tf 14.1793 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(U)]TJ/F31 8.9664 Tf 7.2366 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(x;)-171(z)]TJ/F31 8.9664 Tf 13.9926 0 Td[(\051)]TJ/F44 8.9664 Tf 3.5838 0 Td[(;)-172(U)]TJ/F31 8.9664 Tf 11.3323 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(y)-36(;)-172(z)]TJ/F31 8.9664 Tf 13.6293 0 Td[(\051)]TJ/F64 7.9701 Tf 9.3284 0 Td[(retur)-25(ns)]TJ/F46 8.9664 Tf 28.479 0 Td[(f)]TJ/F31 8.9664 Tf 4.6077 0 Td[(\0502)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(2\051)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0958 0 Td[(\0503)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(3\051)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf -214.1276 -10.4608 Td[(\0505)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5\051)]TJ/F46 8.9664 Tf 8.1914 0 Td[(g)]TJ/F72 8.9664 Tf 4.6077 0 Td[(;)]TJ/F46 8.9664 Tf -34.1772 -15.0105 Td[(\017)]TJ/F64 7.9701 Tf 9.0909 0 Td[(quer)-30(y)]TJ/F44 8.9664 Tf 22.4137 0 Td[(ans)]TJ/F31 8.9664 Tf 14.7339 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)-171(y)]TJ/F31 8.9664 Tf 14.1794 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8253 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(U)]TJ/F31 8.9664 Tf 7.2366 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)-171(y)]TJ/F31 8.9664 Tf 14.1793 0 Td[(\051)]TJ/F64 7.9701 Tf 8.067 0 Td[(retur)-25(ns)]TJ/F46 8.9664 Tf 27.2435 0 Td[(f)]TJ/F31 8.9664 Tf 4.6077 0 Td[(\0502)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5\051)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(\0503)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0958 0 Td[(2\051)]TJ/F46 8.9664 Tf 8.1914 0 Td[(g)]TJ/F72 8.9664 Tf 4.6077 0 Td[(.)]TJ -205.4505 -15.5091 Td[(Mor)37(eo)10(ver)111(,)-287(if)-280(the)-280(edit)-279(lo)10(g)]TJ/F31 8.9664 Tf 89.5488 0 Td[(\001)]TJ/F44 8.9664 Tf 7.6795 0 Td[(B)]TJ/F72 8.9664 Tf 9.9282 0 Td[(would)-280(have)-280(al)1(so)-280(contained)-280(the)-280(cur)15(a-)]TJ -107.1565 -10.4608 Td[(tion)-289(deletion)]TJ/F31 8.9664 Tf 48.018 0 Td[(\050)]TJ/F46 8.9664 Tf 5.1197 0 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Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(o)]TJ/F67 8.9664 Tf 6.5752 -3.809 Td[(and)-269(each)]TJ -184.898 -10.4608 Td[(relation)]TJ/F44 8.9664 Tf 29.6334 0 Td[(R)]TJ/F67 8.9664 Tf 9.2773 0 Td[(on)-250(the)-250(RHS)-250(by)]TJ/F44 8.9664 Tf 55.2953 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(i)]TJ/F67 8.9664 Tf 3.1618 -3.809 Td[(;)]TJ/F46 8.9664 Tf -113.4944 -17.85 Td[(\017)]TJ/F67 8.9664 Tf 9.0908 0 Td[(F)15(or)-229(each)]TJ/F44 8.9664 Tf 32.8605 0 Td[(R)]TJ/F67 8.9664 Tf 9.0929 0 Td[(in)]TJ/F31 8.9664 Tf 9.033 0 Td[(\006)]TJ/F67 8.9664 Tf 6.6556 0 Td[(,)]TJ/F46 8.9664 Tf 4.3357 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F67 8.9664 Tf 4.7416 -3.809 Td[(is)-229(e)15(xtended)-230(with)-229(rules)-230(to)-229(remo)15(v)15(e)-230(t)1(uples)]TJ -77.7403 -10.4608 Td[(in)]TJ/F44 8.9664 Tf 9.2173 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(r)]TJ/F67 8.9664 Tf 6.3996 -3.809 Td[(and)-250(add)-250(those)-250(in)]TJ/F44 8.9664 Tf 60.7647 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(`)]TJ/F67 8.9664 Tf 3.5976 -3.809 Td[(:)]TJ/F31 8.9664 Tf -51.9537 -16.5464 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(i)]TJ/F9 4.9813 Tf 1.9311 -0.9962 Td[(R)]TJ/F31 8.9664 Tf 5.6369 0.9962 Td[(\051)]TJ/F44 8.9664 Tf 21.5165 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 4.3071 Td[(i)]TJ/F31 8.9664 Tf 3.1618 -4.3071 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5837 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051)]TJ/F46 8.9664 Tf 5.6316 0 Td[(^)-229(:)]TJ/F44 8.9664 Tf 14.335 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 4.3071 Td[(r)]TJ/F31 8.9664 Tf 4.158 -4.3071 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051)]TJ/F46 8.9664 Tf 6.1435 0 Td[(!)]TJ/F44 8.9664 Tf 11.7752 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 4.3071 Td[(o)]TJ/F31 8.9664 Tf 4.1594 -4.3071 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051)]TJ -130.2135 -14.6849 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(`)]TJ/F9 4.9813 Tf 2.5829 -0.9962 Td[(R)]TJ/F31 8.9664 Tf 5.6369 0.9962 Td[(\051)]TJ/F44 8.9664 Tf 21.5165 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 4.3071 Td[(`)]TJ/F31 8.9664 Tf 3.5976 -4.3071 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051)]TJ/F46 8.9664 Tf 6.1435 0 Td[(!)]TJ/F44 8.9664 Tf 11.7752 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 4.3071 Td[(o)]TJ/F31 8.9664 Tf 4.1594 -4.3071 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051)]TJ/F44 8.9664 Tf 3.5837 0 Td[(:)]TJ/F67 8.9664 Tf -149.1602 -20.241 Td[(Note)-292(that)-291(the)-292(pre)25(vious)-292(description)-291(and)-292(Figure)-292(2)-291(do)-292(not)-292(incorpo-)]TJ -8.9664 -10.4608 Td[(rate)-263(the)-263(treatment)-263(of)-264(trust)-263(conditions.)-349(W)80(e)-263(sho)25(w)-263(ho)25(w)-264(to)-263(incorporate)]TJ 0 -10.4608 Td[(them)-250(at)-250(the)-250(end)-250(of)-250(Section)-250(3.3.)]TJ 8.9664 -10.4607 Td[(Some)-374(of)-373(the)-374(ne)25(w)-374(mappings)-374(in)]TJ/F46 8.9664 Tf 115.5167 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F67 8.9664 Tf 6.0354 -3.809 Td[(contain)-374(ne)15(g)5(ation,)-404(and)-374(thus)]TJ -141.5396 -10.4608 Td[(one)-326(might)-325(w)10(onder)-326(if)-326(this)-325(af)25(fects)-326(the)-326(decidability)-325(of)-326(query)-326(e)25(v)25(alua-)]TJ 0 -10.4608 Td[(tion.)-1037(Note)-319(that)-320(the)-319(ne)15(g)5(ated)-320(atoms)-319(occur)-319(only)-320(on)-319(the)-320(LHS)-319(of)-320(the)]TJ 0 -10.4608 Td[(implication)-257(and)-258(e)25(v)15(ery)-257(v)25(ariable)-257(in)-258(the)-257(LHS)-257(also)-258(occurs)-257(in)-257(a)-258(positi)25(v)15(e)]TJ 0 -10.4607 Td[(atom)-250(there:)-311(we)-251(call)-250(such)-251(an)-250(assertion)-251(a)]TJ/F72 8.9664 Tf 141.014 0 Td[(tgd)-250(with)-251(safe)-250(ne)40(gation)]TJ/F67 8.9664 Tf 79.1131 0 Td[(.)-311(The)]TJ -220.1271 -10.4608 Td[(results)-250(of)-250([12,)-250(25])-250(generalize)-250(to)-250(tgds)-250(with)-250(safe)-250(ne)15(g)5(ation.)]TJ 10.1868 -17.7419 Td[(T)]TJ/F67 7.1731 Tf 5.9268 0 Td[(H)-62(E)-61(O)-62(R)-62(E)-61(M)]TJ/F67 8.9664 Tf 37.6991 0 Td[(3.1.)]TJ/F72 8.9664 Tf 17.9328 0 Td[(Let)]TJ/F46 8.9664 Tf 13.7507 0 Td[(M)]TJ/F72 8.9664 Tf 13.313 0 Td[(be)-256(weakly)-255(acyclic)-256(and)]TJ/F44 8.9664 Tf 81.3731 0 Td[(I)]TJ/F72 8.9664 Tf 7.0221 0 Td[(be)-256(an)-255(instance)]TJ -187.2044 -10.4608 Td[(of)-401(all)-401(local)-401(contrib)20(utions)-402(and)-401(r)37(ejections)-401(tables.Then)]TJ/F46 8.9664 Tf 197.2769 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F72 8.9664 Tf 6.2813 -3.809 Td[(is)-401(also)]TJ -214.5793 -10.4608 Td[(weakly)-301(acyclic,)-313(hence)-301(c)15(hase)]TJ/F47 5.9776 Tf 101.9261 -1.3948 Td[(M)]TJ/F12 4.9813 Tf 8.5314 1.7158 Td[(0)]TJ/F31 8.9664 Tf 3.1929 -0.321 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(I)]TJ/F31 8.9664 Tf 4.7301 0 Td[(\051)]TJ/F72 8.9664 Tf 6.2798 0 Td[(terminates)-301(in)-300(polynomial)-301(time)10(,)]TJ -128.2441 -10.4608 Td[(and)-240(mor)37(eo)10(ver)-240(it)-240(yields)-240(an)-240(instance)-240(of)-240(the)-240(of)-241(i)1(nput)-241(and)-240(output)-240(tables)]TJ 0 -10.4608 Td[(that)-250(is)-250(a)-250(canonical)-250(univer)10(sal)-250(solution)-250(for)]TJ/F44 8.9664 Tf 148.1138 0 Td[(I)]TJ/F72 8.9664 Tf 6.9718 0 Td[(with)-250(r)37(espect)-250(to)]TJ/F46 8.9664 Tf 54.712 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F72 8.9664 Tf 2.6844 -3.809 Td[(.)]TJ/F67 8.9664 Tf -214.5367 -17.7419 Td[(T)80(o)-349(recompute)-348(the)-349(relation)-349(input)-349(and)-348(output)-349(instances)-349(based)-349(on)]TJ -8.9664 -10.4608 Td[(the)-279(e)15(xtensional)-279(data)-279(in)-279(the)-279(local)-279(contrib)20(utions)-279(and)-279(rejection)-279(tables,)]TJ 0 -10.4608 Td[(we)-250(use)-250(a)-250(procedure)-250(based)-250(on)-250(the)-250(chase)-250(of)-250([12].)]TJ 10.1868 -19.7884 Td[(D)]TJ/F67 7.1731 Tf 6.922 0 Td[(E)-61(FI)-61(N)-62(I)-62(T)-62(I)-62(O)-61(N)]TJ/F67 8.9664 Tf 41.2357 0 Td[(3)-50(.)-50(1)-1106(\050)-50(C)]TJ/F67 7.1731 Tf 31.8803 0 Td[(O)-62(N)-61(S)-62(I)-62(S)-61(T)-62(E)-61(N)-62(T)-374(S)-62(Y)-61(S)-62(T)-61(E)-62(M)-375(S)-61(T)31(A)49(T)-61(E)]TJ/F67 8.9664 Tf 101.5246 0 Td[(\051)-50(.)]TJ/F72 8.9664 Tf 10.383 0 Td[(An)-190(instance)]TJ/F31 8.9664 Tf -202.1324 -10.4608 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(I)-79(;)-171(J)]TJ/F31 8.9664 Tf 14.7578 0 Td[(\051)]TJ/F72 8.9664 Tf 5.6736 0 Td[(of)]TJ/F31 8.9664 Tf 9.0657 0 Td[(\006)]TJ/F47 5.9776 Tf 6.6555 3.809 Td[(0)]TJ/F72 8.9664 Tf 2.6844 -3.809 Td[(,)-236(wher)37(e)]TJ/F44 8.9664 Tf 28.0335 0 Td[(I)]TJ/F72 8.9664 Tf 6.82 0 Td[(is)-233(an)-233(instance)-233(of)-233(the)-233(local)-233(r)37(ejections)-233(and)-233(con-)]TJ -77.2742 -10.4608 Td[(trib)20(utions)-237(tables)-236(and)]TJ/F44 8.9664 Tf 75.9363 0 Td[(J)]TJ/F72 8.9664 Tf 8.0543 0 Td[(of)-237(the)-236(input)-237(and)-237(output)-236(tables,)-240(is)]TJ/F66 8.9664 Tf 117.2573 0 Td[(consistent)]TJ/F72 8.9664 Tf -201.2479 -10.4608 Td[(if)]TJ/F44 8.9664 Tf 7.2267 0 Td[(J)]TJ/F72 8.9664 Tf 8.1735 0 Td[(is)-250(a)-250(canonical)-250(univer)10(sal)-250(solution)-250(for)]TJ/F44 8.9664 Tf 131.9207 0 Td[(I)]TJ/F72 8.9664 Tf 6.9718 0 Td[(with)-250(r)37(espect)-250(to)]TJ/F46 8.9664 Tf 54.712 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F72 8.9664 Tf 2.6844 -3.809 Td[(.)]TJ/F67 8.9664 Tf -213.7438 -19.7884 Td[(T)80(o)-280(r)1(ecap,)]TJ/F72 8.9664 Tf 35.9763 0 Td[(publishing)]TJ/F67 8.9664 Tf 40.3707 0 Td[(a)-279(peer)]TJ/F66 8.9664 Tf 24.4242 0 Td[(P)]TJ/F67 8.9664 Tf 5.4784 0 Td[(')55(s)-279(edit)-280(log)-279(means)-280(producing)-279(a)-280(ne)25(w)]TJ -115.216 -10.4608 Td[(instance)-202(of)]TJ/F66 8.9664 Tf 40.4766 0 Td[(P)]TJ/F67 8.9664 Tf 5.4784 0 Td[(')55(s)-202(local)-202(rejections)-202(and)-203(cont)1(rib)20(utions)-203(tables,)-211(while)-202(hold-)]TJ -45.955 -10.4608 Td[(ing)-398(the)-398(other)-397(peers')-398(local)-398(rejections)-397(and)-398(contrib)20(utions)-398(tables)-398(the)]TJ 263.0137 653.5492 Td[(same)]TJ/F67 5.9776 Tf 18.4256 3.809 Td[(2)]TJ/F67 8.9664 Tf 3.4869 -3.809 Td[(.)]TJ/F72 8.9664 Tf 4.8906 0 Td[(Recomputing)]TJ/F66 8.9664 Tf 49.1652 0 Td[(P)]TJ/F67 8.9664 Tf 5.4785 0 Td[(')55(s)-206(instance)-413(means)-206(computing)-207(a)-206(ne)25(w)-206(instance)]TJ -81.4468 -10.4608 Td[(of)]TJ/F66 8.9664 Tf 9.8124 0 Td[(P)]TJ/F67 8.9664 Tf 5.4784 0 Td[(')55(s)-261(input)-262(and)-261(output)-261(tables)-262(that)-261(is)-261(a)-262(canonical)-261(uni)25(v)15(ersal)-261(solution)]TJ -15.2908 -10.4608 Td[(with)-235(respect)-234(to)-235(the)-234(internal)-235(mappings)]TJ/F46 8.9664 Tf 134.1501 0 Td[(M)]TJ/F47 5.9776 Tf 11.0211 3.809 Td[(0)]TJ/F67 8.9664 Tf 2.6844 -3.809 Td[(.)-305(By)-234(de\002nition,)-238(after)-235(each)]TJ -147.8556 -10.4608 Td[(update)-383(e)15(xchange)-383(the)-383(system)-384(must)-383(be)-383(in)-383(a)-383(consistent)-383(state.)-710(\050The)]TJ 0 -10.4607 Td[(initial)-250(state,)-250(with)-250(empty)-250(instances,)-250(is)-250(tri)25(vially)-250(consistent.\051)]TJ/F66 11.9552 Tf 0 -18.3631 Td[(3.2)-1000(Pr)18(o)10(v)10(enance)-250(Expr)18(essions)-250(and)-250(Graphs)]TJ/F67 8.9664 Tf 8.9664 -12.9515 Td[(As)-215(e)15(xplained)-216(in)-215(Section)-215(2.2,)-222(CDSS)-216(operation)-215(needs)-215(a)-215(data)-216(pro)15(v)15(e-)]TJ -8.9664 -10.4607 Td[(nance)-319(model)-320(that)-319(is)-319(more)-319(e)15(xpressi)25(v)15(e)-320(than)-319(wh)5(y-pro)15(v)15(enance)-319(or)-320(lin-)]TJ 0 -10.4608 Td[(eage.)-583(W)80(e)-195(describe)-194(the)-195(essential)-195(aspects)-194(of)-195(the)-195(model)-194(in)-195(this)-195(section,)]TJ 0 -10.4608 Td[(relying)-220(on)-219(some)-220(properties)-219(that)-220(were)-219(sho)25(wn)-220(in)-220(our)-219(theoretical)-220(study)]TJ 0 -10.4608 Td[(of)-220(the)-219(model)-220([16],)-225(which)-220(also)-219(justi\002es)-220(the)-219(choice)-220(of)-219(\223semirings\224)-220(for)]TJ 0 -10.4607 Td[(the)-250(formalism.)]TJ 8.9664 -10.4608 Td[(In)-347(our)-347(model)-347(the)-347(pro)15(v)15(enance)-346(of)-347(a)-347(source)-347(\050base\051)-347(tuple)-347(is)-347(repre-)]TJ -8.9664 -10.4608 Td[(sented)-337(by)-337(its)-337(o)25(wn)-337(tuple)-337(id)-337(\050we)-338(call)-337(this)-337(a)]TJ/F72 8.9664 Tf 154.0223 0 Td[(pr)45(o)10(venance)-337(tok)10(en)]TJ/F67 8.9664 Tf 63.6886 0 Td[(\051,)-359(and)]TJ -217.7109 -10.4608 Td[(that)-341(of)-341(a)-340(tuple)-341(computed)-341(through)-341(update)-340(translation)-341(is)-341(an)-341(e)15(xpres-)]TJ 0 -10.4607 Td[(sion)-241(o)15(v)15(er)-242(a)]TJ/F72 8.9664 Tf 41.085 0 Td[(semiring)]TJ/F67 8.9664 Tf 31.3819 0 Td[(,)-243(with)-241(domain)-242(the)-241(set)-242(of)-241(all)-241(pro)15(v)15(enance)-242(tok)10(ens;)]TJ -72.4669 -10.4608 Td[(tw)10(o)-213(operations,)]TJ/F31 8.9664 Tf 56.8351 0 Td[(+)]TJ/F67 8.9664 Tf 9.0741 0 Td[(and)]TJ/F46 8.9664 Tf 14.854 0 Td[(\001)]TJ/F67 8.9664 Tf 2.5598 0 Td[(;)-225(and)-213(one)-212(unary)-213(function)-213(for)-212(e)25(v)15(ery)-213(mapping.)]TJ -83.323 -10.4608 Td[(Intuiti)25(v)15(ely)65(,)]TJ/F46 8.9664 Tf 41.4041 0 Td[(\001)]TJ/F67 8.9664 Tf 5.2014 0 Td[(is)-295(used)-294(to)-295(combine)-294(the)-295(pro)15(v)15(enance)-295(of)-294(tuples)-295(in)-294(a)-295(join)]TJ -46.6055 -10.4608 Td[(operation,)-358(mappi)1(ng)-337(functions)-336(re\003ect)-336(the)-336(f)10(act)-336(that)-336(the)-336(correspond-)]TJ 0 -10.4607 Td[(ing)-253(mapping)-252(w)10(as)-253(in)40(v)20(olv)15(ed)-253(in)-252(the)-253(deri)25(v)25(ation,)-253(and)]TJ/F31 8.9664 Tf 174.5642 0 Td[(+)]TJ/F67 8.9664 Tf 9.4329 0 Td[(is)-253(used)-252(in)-253(cases)]TJ -183.9971 -10.4608 Td[(when)-281(a)-281(tuple)-281(can)-281(be)-281(deri)26(v)14(e)1(d)-281(in)-281(multiple)-281(w)10(ays)-281(\050e.g.)-403(through)-281(dif)25(fer)20(-)]TJ 0 -10.4608 Td[(ent)-349(mappings\051.)-606(Using)-348(the)-349(terminology)-348(of)-349(proof-theoretic)-349(datalog)]TJ 0 -10.4608 Td[(e)25(v)25(aluation)-315(semantics,)-332(e)25(v)15(ery)-315(summand)-316(in)-315(a)-315(pro)15(v)15(enance)-316(e)15(xpression)]TJ 0 -10.4607 Td[(corresponds)-337(to)-336(a)]TJ/F72 8.9664 Tf 63.3401 0 Td[(derivation)-337(tr)37(ee)]TJ/F67 8.9664 Tf 56.5097 0 Td[(for)-337(that)-336(tuple)-337(in)-337(the)-337(result)-336(of)-337(the)]TJ -119.8498 -10.4608 Td[(\223program\224)-250(formed)-250(by)-250(the)-250(mappings.)]TJ/F66 8.9664 Tf 0 -13.2954 Td[(Notation.)]TJ/F67 8.9664 Tf 46.748 0 Td[(W)80(e)-301(write)]TJ/F64 8.9664 Tf 35.5501 0 Td[(Pv)]TJ/F31 8.9664 Tf 10.4637 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(R)]TJ/F31 8.9664 Tf 7.0357 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(t)]TJ/F31 8.9664 Tf 3.3278 0 Td[(\051\051)]TJ/F67 8.9664 Tf 9.8658 0 Td[(for)-301(the)-301(pro)15(v)15(enance)-301(e)15(xpression)-301(of)]TJ -120.1586 -10.4608 Td[(the)-191(tuple)]TJ/F44 8.9664 Tf 32.3144 0 Td[(t)]TJ/F67 8.9664 Tf 5.0404 0 Td[(in)-191(relation)]TJ/F44 8.9664 Tf 37.7927 0 Td[(R)]TJ/F67 8.9664 Tf 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Td[(we)-337(introduce)-337(a)-337(ne)25(w)-337(relation)]TJ/F44 8.9664 Tf 105.5127 0 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F31 8.9664 Tf 3.1618 1.4943 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)]TJ/F31 8.9664 Tf 9.9574 0 Td[(\026)]TJ/F44 8.9664 Tf -0.6382 0 Td[(y)]TJ/F31 8.9664 Tf 4.8601 0 Td[(\051)]TJ/F67 8.9664 Tf 6.6049 0 Td[(and)-337(we)-337(replace)]TJ/F31 8.9664 Tf 58.3512 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(m)]TJ/F45 5.9776 Tf 8.1188 -0.9962 Td[(i)]TJ/F31 8.9664 Tf 3.1618 0.9962 Td[(\051)]TJ/F67 8.9664 Tf 6.6049 0 Td[(with)]TJ -223.1613 -10.4607 Td[(the)-250(mapping)-250(rules)]TJ/F31 8.9664 Tf 59.1505 -16.446 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(m)]TJ/F47 5.9776 Tf 8.1188 4.3071 Td[(0)]TJ/F45 5.9776 Tf 0 -5.8846 Td[(i)]TJ/F31 8.9664 Tf 3.1618 1.5775 Td[(\051)]TJ/F44 8.9664 Tf 21.5165 0 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9963 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F31 8.9664 Tf 3.1618 1.4944 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5837 0 Td[(x;)]TJ/F31 8.9664 Tf 9.9574 0 Td[(\026)]TJ/F44 8.9664 Tf -0.6382 0 Td[(y)]TJ/F31 8.9664 Tf 4.8602 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(\036)]TJ/F31 8.9664 Tf 5.4849 0 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)]TJ/F31 8.9664 Tf 9.9574 0 Td[(\026)]TJ/F44 8.9664 Tf -0.6382 0 Td[(y)]TJ/F31 8.9664 Tf 4.8602 0 Td[(\051)]TJ -106.1064 -13.4496 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(m)]TJ/F47 5.9776 Tf 8.1188 4.3071 Td[(00)]TJ/F45 5.9776 Tf 0 -5.8846 Td[(i)]TJ/F31 8.9664 Tf 4.8707 1.5775 Td[(\051)]TJ/F44 8.9664 Tf 21.5165 0 Td[(R)]TJ/F31 8.9664 Tf 7.0357 0 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)]TJ/F31 8.9664 Tf 11.2754 2.3661 Td[(\026)]TJ/F44 8.9664 Tf -1.9562 -2.3661 Td[(f)]TJ/F31 8.9664 Tf 5.4483 0 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5837 0 Td[(x)]TJ/F31 8.9664 Tf 5.2235 0 Td[(\051\051)]TJ/F67 8.9664 Tf 9.4091 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4982 Td[(i)]TJ/F31 8.9664 Tf 3.1617 1.4944 Td[(\050)-63(\026)]TJ/F44 8.9664 Tf 3.5838 0 Td[(x;)]TJ/F31 8.9664 Tf 9.9574 0 Td[(\026)]TJ/F44 8.9664 Tf -0.6382 0 Td[(y)]TJ/F31 8.9664 Tf 4.8602 0 Td[(\051)]TJ/F67 8.9664 Tf -178.078 -16.446 Td[(Note)-206(that)]TJ/F31 8.9664 Tf 34.5723 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(m)]TJ/F47 5.9776 Tf 8.1188 3.809 Td[(0)]TJ/F45 5.9776 Tf 0 -5.4747 Td[(i)]TJ/F31 8.9664 Tf 3.1618 1.6657 Td[(\051)]TJ/F67 8.9664 Tf 5.43 0 Td[(mirrors)]TJ/F44 8.9664 Tf 28.2428 0 Td[(m)]TJ/F45 5.9776 Tf 8.1188 -0.9962 Td[(i)]TJ/F67 8.9664 Tf 5.008 0.9962 Td[(b)20(ut)]TJ/F72 8.9664 Tf 13.1259 0 Td[(does)-206(not)-206(pr)45(oject)-206(any)-206(attrib)20(ute)1(s)]TJ/F67 8.9664 Tf 107.4215 0 Td[(.)-295(Note)]TJ -216.7836 -10.4608 Td[(also)-216(that)]TJ/F31 8.9664 Tf 31.7708 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(m)]TJ/F47 5.9776 Tf 8.1188 3.809 Td[(00)]TJ/F45 5.9776 Tf 0 -5.4747 Td[(i)]TJ/F31 8.9664 Tf 4.8707 1.6657 Td[(\051)]TJ/F67 8.9664 Tf 5.5221 0 Td[(deri)25(v)15(es)-216(the)-216(actual)-216(data)-217(instance)-216(from)-216(the)-216(pro)15(v)15(enance)]TJ -53.8661 -10.4608 Td[(encoding.)]TJ 10.1868 -19.6379 Td[(E)]TJ/F67 7.1731 Tf 5.9268 0 Td[(X)-62(A)-61(M)-63(P)-61(L)-62(E)]TJ/F67 8.9664 Tf 36.9011 0 Td[(9.)]TJ/F72 8.9664 Tf 11.208 0 Td[(Since)-303(tuples)-302(in)]TJ/F44 8.9664 Tf 55.9559 0 Td[(B)]TJ/F72 8.9664 Tf 10.1327 0 Td[(\050as)-303(shown)-302(in)-303(the)-302(gr)15(aph)-303(of)-302(Ex-)]TJ -130.3113 -10.4608 Td[(ample)-192(5\051)-193(can)-192(be)-193(derived)-192(thr)45(ough)-193(mappings)]TJ/F44 8.9664 Tf 152.6263 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9962 Td[(1)]TJ/F72 8.9664 Tf 5.8767 0.9962 Td[(and)]TJ/F44 8.9664 Tf 15.1752 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9962 Td[(4)]TJ/F72 8.9664 Tf 4.151 0.9962 Td[(,)-204(we)-192(can)-193(r)37(ep-)]TJ -194.0667 -10.4608 Td[(r)37(esent)-317(their)-316(pr)45(o)10(venance)-317(using)-317(two)-316(r)37(elations)]TJ/F44 8.9664 Tf 160.8278 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(1)]TJ/F72 8.9664 Tf 6.9904 1.4943 Td[(and)]TJ/F44 8.9664 Tf 16.2889 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(4)]TJ/F72 8.9664 Tf 6.9904 1.4943 Td[(\050named)]TJ -212.2133 -10.4607 Td[(based)-378(on)-379(their)-378(origins)-378(of)]TJ/F44 8.9664 Tf 96.1705 0 Td[(B)]TJ/F72 8.9664 Tf 7.4197 0 Td[(,)-410(and)]TJ/F31 8.9664 Tf 22.7638 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9963 Td[(1)]TJ/F31 8.9664 Tf 4.1511 0.9963 Td[(\051)]TJ/F72 8.9664 Tf 6.9762 0 Td[(and)]TJ/F31 8.9664 Tf 16.8421 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9963 Td[(4)]TJ/F31 8.9664 Tf 4.151 0.9963 Td[(\051)]TJ/F72 8.9664 Tf 3.5838 0 Td[(,)-410(r)37(espectively\051,)]TJ -185.4634 -10.4608 Td[(using)-250(the)-250(mapping)-250(rules:)]TJ/F44 8.9664 Tf 73.1675 -16.446 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9963 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(1)]TJ/F31 8.9664 Tf 4.1511 1.4944 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(i;)-171(c;)-172(n)]TJ/F31 8.9664 Tf 20.8933 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(G)]TJ/F31 8.9664 Tf 7.2158 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(i;)-171(c;)-172(n)]TJ/F31 8.9664 Tf 20.8933 0 Td[(\051)]TJ/F44 8.9664 Tf -84.4242 -13.4496 Td[(B)]TJ/F31 8.9664 Tf 7.4197 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(i;)-171(n)]TJ/F31 8.9664 Tf 12.808 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4982 Td[(1)]TJ/F31 8.9664 Tf 4.151 1.4944 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(i;)-171(c;)-172(n)]TJ/F31 8.9664 Tf 20.8933 0 Td[(\051)]TJ/F44 8.9664 Tf -76.5428 -13.4496 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(4)]TJ/F31 8.9664 Tf 4.151 1.4943 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(i;)-171(n;)-172(c)]TJ/F31 8.9664 Tf 20.8933 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(B)]TJ/F31 8.9664 Tf 7.4197 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(i;)-171(c)]TJ/F31 8.9664 Tf 11.2386 0 Td[(\051)]TJ/F44 8.9664 Tf 3.5838 0 Td[(;)-172(U)]TJ/F31 8.9664 Tf 11.3323 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(n;)-172(c)]TJ/F31 8.9664 Tf 13.6442 0 Td[(\051)]TJ/F44 8.9664 Tf -107.1174 -13.4495 Td[(B)]TJ/F31 8.9664 Tf 7.4197 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(i;)-171(n)]TJ/F31 8.9664 Tf 12.8081 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8253 0 Td[(:-)]TJ/F44 8.9664 Tf 7.7199 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9963 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(4)]TJ/F31 8.9664 Tf 4.1511 1.4944 Td[(\050)]TJ/F44 8.9664 Tf 3.5837 0 Td[(i;)-171(n;)-172(c)]TJ/F31 8.9664 Tf 20.8933 0 Td[(\051)]TJ/F72 8.9664 Tf -149.7103 -29.8956 Td[(Recall)-250(fr)45(om)-250(e)20(xample)-250(6)-250(that)]TJ/F64 8.9664 Tf 98.7816 0 Td[(Pv)]TJ/F31 8.9664 Tf 10.4637 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(B)]TJ/F31 8.9664 Tf 7.4197 0 Td[(\0503)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(2\051\051)-285(=)]TJ/F44 8.9664 Tf 24.0624 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9962 Td[(1)]TJ/F31 8.9664 Tf 4.1511 0.9962 Td[(\050)]TJ/F64 8.9664 Tf 3.5837 0 Td[(Pv)]TJ/F31 8.9664 Tf 10.4637 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(G)]TJ/F31 8.9664 Tf 7.2158 0 Td[(\0503)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0958 0 Td[(5)]TJ/F44 8.9664 Tf 4.6077 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(2\051\051)1(\051)]TJ -214.7058 -10.4608 Td[(+)]TJ/F44 8.9664 Tf 9.1409 0 Td[(m)]TJ/F43 5.9776 Tf 8.1188 -0.9962 Td[(4)]TJ/F31 8.9664 Tf 4.1511 0.9962 Td[(\050)]TJ/F64 8.9664 Tf 3.5837 0 Td[(Pv)]TJ/F31 8.9664 Tf 10.4637 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(B)]TJ/F31 8.9664 Tf 7.4196 0 Td[(\0503)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5\051\051)]TJ/F64 8.9664 Tf 11.7752 0 Td[(Pv)]TJ/F31 8.9664 Tf 10.4637 0 Td[(\050)]TJ/F44 8.9664 Tf 3.5838 0 Td[(U)]TJ/F31 8.9664 Tf 7.2366 0 Td[(\0502)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5\051\051)1(\051)]TJ/F72 8.9664 Tf 15.359 0 Td[(.)-309(The)-248(\002r)10(st)-248(part)-248(of)-247(this)-248(pr)45(o)10(venance)]TJ -119.4542 -10.4608 Td[(e)20(xpr)37(ession)-286(is)-286(r)37(epr)37(esented)-285(by)-286(the)-286(tuple)]TJ/F44 8.9664 Tf 138.217 0 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9962 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(1)]TJ/F31 8.9664 Tf 4.1511 1.4943 Td[(\0503)]TJ/F44 8.9664 Tf 8.1915 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5)]TJ/F44 8.9664 Tf 4.6077 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(2\051)]TJ/F72 8.9664 Tf 10.7547 0 Td[(and)-286(the)-286(second)]TJ -184.6712 -10.4607 Td[(by)-284(the)-284(tuple)]TJ/F44 8.9664 Tf 44.9899 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9963 Td[(B)]TJ/F43 5.9776 Tf 5.8963 -0.4981 Td[(4)]TJ/F31 8.9664 Tf 4.1511 1.4944 Td[(\0503)]TJ/F44 8.9664 Tf 8.1914 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(2)]TJ/F44 8.9664 Tf 4.6077 0 Td[(;)]TJ/F31 8.9664 Tf 4.0957 0 Td[(5\051)]TJ/F72 8.9664 Tf 8.1915 0 Td[(.)-412(\050Both)-284(can)-283(be)-284(obtained)-284(by)-284(assigning)]TJ/F44 8.9664 Tf 136.7634 0 Td[(i)]TJ/F31 8.9664 Tf 6.2916 0 Td[(=)]TJ -231.9359 -10.4608 Td[(3)]TJ/F44 8.9664 Tf 4.6077 0 Td[(;)-172(n)]TJ/F31 8.9664 Tf 12.2145 0 Td[(=)-286(2)]TJ/F44 8.9664 Tf 14.3351 0 Td[(;)-172(c)]TJ/F31 8.9664 Tf 10.645 0 Td[(=)-286(5)]TJ/F72 8.9664 Tf 16.5767 0 Td[(in)-250(eac)15(h)-250(of)-250(the)-250(rules)-250(abo)10(ve)15(.\051)]TJ/F67 8.9664 Tf -49.4126 -19.6379 Td[(As)-245(we)-246(detail)-245(in)-245(Section)-246(5,)-246(we)-245(can)-246(further)-245(optimize)-245(this)-246(represen-)]TJ -8.9664 -10.4608 Td[(tation)-250(in)-250(order)-250(to)-250(ef)25(\002ciently)-250(implement)-250(it)-250(in)-250(an)-250(RDBMS.)]TJ/F72 10.9589 Tf 2.7397 -18.4385 Td[(4.1.3)-1000(Derivation)-250(T)92(esting)]TJ/F67 8.9664 Tf 6.2267 -12.9515 Td[(In)-241(addition)-242(to)-241(computing)-241(instances)-242(and)-241(their)-241(pro)15(v)15(enance,)-243(we)-242(can)]TJ -8.9664 -10.4607 Td[(use)-361(datalog)-361(rules)-361(to)-361(determine)]TJ/F72 8.9664 Tf 114.7916 0 Td[(how)-361(a)-361(tuple)-361(was)-361(derived.)]TJ/F67 8.9664 Tf 99.1532 0 Td[(This)-361(is)]TJ -213.9448 -10.4608 Td[(useful)-352(in)-351(tw)10(o)-352(conte)15(xts:)-513(\002rst,)-377(to)-352(assess)-351(whether)-352(a)-351(tuple)-352(should)-352(be)]TJ 0 -10.4608 Td[(trusted,)-202(which)-191(requires)-190(that)-190(it)-191(be)-190(deri)25(v)25(able)-191(only)-190(from)-190(trusted)-191(sources)]TJ 0 -10.4608 Td[(and)-363(along)-364(trus)1(ted)-364(mappings;)-420(and)-363(second,)-391(to)-364(see)-363(if)-363(it)-363(is)-364(deri)25(v)25(able)]TJ 0 -10.4607 Td[(e)25(v)15(en)-204(if)-204(we)-204(remo)15(v)15(e)-204(certain)-204(tuples,)-214(w)1(hich)-205(is)-204(necessary)-204(for)-204(performing)]TJ 0 -10.4608 Td[(incremental)-250(maintenance)-250(of)-250(CDSS)-250(instances.)]TJ 8.9664 -10.4608 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Td[(encoding)-268(of)-268(pro)15(v)15(enance\051)-268(into)-268(a)-268(se)1(ries)-268(of)]TJ/F72 8.9664 Tf 147.0153 0 Td[(delta)-268(rules)]TJ/F67 8.9664 Tf 38.2671 0 Td[(.)-364(F)15(or)-268(the)-268(inser)20(-)]TJ -185.2824 -10.4607 Td[(tion)-334(delta)-334(rules)-335(we)-334(use)-334(ne)25(w)-334(relation)-335(names)-334(of)-334(the)-334(form)]TJ/F44 8.9664 Tf 205.0582 0 Td[(R)]TJ/F43 5.9776 Tf 7.0357 3.809 Td[(+)]TJ/F67 8.9664 Tf 6.0883 -3.809 Td[(,)]TJ/F44 8.9664 Tf 5.4276 0 Td[(P)]TJ/F43 5.9776 Tf 7.1637 3.9186 Td[(+)]TJ/F9 4.9813 Tf -2.5022 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1388 0 Td[(i)]TJ/F67 8.9664 Tf 3.4516 2.3863 Td[(,)]TJ -236.8618 -10.4608 Td[(etc.)-307(while)-240(for)-240(the)-240(deletion)-240(delta)-240(rules)-240(we)-240(use)-240(ne)25(w)-241(rela)1(tion)-241(names)-240(of)]TJ 0 -10.4608 Td[(the)-250(form)]TJ/F44 8.9664 Tf 32.8705 0 Td[(R)]TJ/F47 5.9776 Tf 7.0357 3.809 Td[(\000)]TJ/F67 8.9664 Tf 6.2543 -3.809 Td[(,)]TJ/F44 8.9664 Tf 4.4832 0 Td[(P)]TJ/F47 5.9776 Tf 7.1637 3.9186 Td[(\000)]TJ/F9 4.9813 Tf -2.5022 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1388 0 Td[(i)]TJ/F67 8.9664 Tf 3.6177 2.3863 Td[(,)-250(etc.)]TJ -55.0952 -10.4608 Td[(F)15(or)-361(the)-362(case)-361(of)]TJ/F66 8.9664 Tf 59.1415 0 Td[(incr)18(emental)-361(insertion)]TJ/F67 8.9664 Tf 86.0132 0 Td[(in)-361(the)-362(absence)-361(of)-362(peer)20(-)]TJ -154.1211 -10.4608 Td[(speci\002c)-394(trust)-394(conditions,)-429(the)-394(algorithm)-394(is)-394(simple,)-430(and)-394(analogous)]TJ 0 -10.4607 Td[(to)-399(the)]TJ/F66 8.9664 Tf 25.0943 0 Td[(counting)]TJ/F67 8.9664 Tf 36.9623 0 Td[(and)]TJ/F64 8.9664 Tf 16.5283 0 Td[(DRed)]TJ/F67 8.9664 Tf 26.4987 0 Td[(incremental)-399(vie)25(w)-400(maintenance)-399(algo-)]TJ -105.0836 -10.4608 Td[(rithms)-366(of)-365([17]:)-542(we)-366(can)-365(directly)-366(e)25(v)25(aluate)-366(the)-365(insertion)-366(delta)-366(rules)]TJ 0 -10.4608 Td[(until)-344(reaching)-345(a)-344(\002xpoint)-344(and)-345(then)-344(add)]TJ/F44 8.9664 Tf 142.1497 0 Td[(R)]TJ/F43 5.9776 Tf 7.0357 3.809 Td[(+)]TJ/F67 8.9664 Tf 9.1762 -3.809 Td[(to)]TJ/F44 8.9664 Tf 10.0637 0 Td[(R)]TJ/F67 8.9664 Tf 7.0357 0 Td[(,)]TJ/F44 8.9664 Tf 5.5412 0 Td[(P)]TJ/F43 5.9776 Tf 7.1637 3.9186 Td[(+)]TJ/F9 4.9813 Tf -2.5022 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1388 0 Td[(i)]TJ/F67 8.9664 Tf 6.5396 2.3863 Td[(to)]TJ/F44 8.9664 Tf 10.0637 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4982 Td[(i)]TJ/F67 8.9664 Tf 3.1617 1.4944 Td[(,)-368(etc.)]TJ -220.866 -10.4608 Td[(T)35(rust)-259(combines)-260(na)1(turally)-260(with)-259(the)-259(incremental)-259(insertion)-260(algorithm:)]TJ 0 -10.4608 Td[(the)-228(starting)-228(point)-229(for)-228(the)-228(algorithm)-228(is)-228(already-trusted)-228(data)-229(\050f)1(rom)-229(the)]TJ 0 -10.4607 Td[(prior)-349(instance\051,)-375(plus)-349(ne)25(w)-350(\223base\224)-349(insertions)-349(which)-350(can)-349(be)-350(directly)]TJ 0 -10.4608 Td[(tested)-190(for)-190(trust)-190(\050since)-190(their)-190(pro)15(v)15(enance)-190(is)-190(simply)-190(their)-190(source\051.)-290(Then,)]TJ 0 -10.4608 Td[(as)-264(we)-264(deri)25(v)15(e)-263(tuples)-264(via)-264(mapping)-264(rules)-263(from)-264(trusted)-264(tuples,)-267(we)-264(sim-)]TJ 0 -10.4608 Td[(ply)-253(apply)-253(the)-252(associated)-253(trust)-253(conditions)-253(to)-253(e)1(nsure)-253(that)-253(we)-253(only)-253(de-)]TJ 0 -10.4607 Td[(ri)25(v)15(e)-250(ne)25(w)-250(trusted)-250(tuples.)]TJ/F66 8.9664 Tf 8.9664 -10.4608 Td[(Incr)18(emental)-213(deletion)]TJ/F67 8.9664 Tf 80.8511 0 Td[(is)-213(signi\002cantly)-212(more)-213(comple)15(x.)-297(When)-213(a)-213(tu-)]TJ -89.8175 -10.4608 Td[(ple)-273(is)-272(deleted,)-279(it)-272(is)-273(possible)-273(to)-273(remo)15(v)15(e)-272(an)15(y)-273(pro)15(v)15(enance)-273(e)15(xpressions)]TJ 0 -10.4608 Td[(and)-384(tuples)-384(that)-383(are)-384(its)-384(immediate)-384(consequents)-383(and)-384(are)-384(no)-384(longer)]TJ 0 -10.4607 Td[(directly)-224(deri)25(v)25(able.)-301(Ho)25(we)25(v)15(er)40(,)-230(the)-224(pro)15(v)15(enance)-224(graph)-224(may)-224(include)-224(c)15(y-)]TJ 0 -10.4608 Td[(cles:)-440(it)-314(is)-315(possible)-315(to)-315(ha)20(v)15(e)-315(a)-315(\223loop\224)-314(in)-315(the)-315(pro)15(v)15(enance)-315(graph)-315(such)]TJ 0 -10.4608 Td[(that)-409(se)25(v)15(eral)-409(tuples)-408(are)-409(mutually)-409(deri)25(v)25(able)-409(from)-408(one)-409(another)40(,)-449(yet)]TJ 0 -10.4608 Td[(none)-337(are)-337(deri)25(v)25(able)-336(from)-337(edbs,)-359(i.e.,)-358(local)-337(contrib)20(utions)-337(from)-337(some)]TJ 0 -10.4607 Td[(peer)-328(in)-327(the)-328(CDSS.)-327(Hence,)-347(in)-328(order)-327(to)-328(\223g)5(arbage)-328(collect\224)-327(these)-328(no-)]TJ 0 -10.4608 Td[(longer)20(-deri)25(v)25(able)-392(tuples,)-427(we)-391(must)-392(test)-392(whether)-391(the)15(y)-392(are)-392(deri)25(v)25(able)]TJ 0 -10.4608 Td[(from)-378(trusted)-378(base)-378(data)-378(in)-378(local)-378(contrib)20(utions)-379(tabl)1(es;)-443(those)-378(tuples)]TJ 0 -10.4608 Td[(that)-279(are)-280(not)-279(must)-279(be)-279(recursi)25(v)15(ely)-280(deleted)-279(follo)25(wing)-279(the)-279(same)-280(proce-)]TJ 0 -10.4607 Td[(dure.)]TJ 8.9664 -10.4608 Td[(Suppose)-319(that)-318(we)-319(are)-319(gi)25(v)15(en)-318(each)-319(list)-319(of)-318(initial)-319(updates)]TJ/F31 8.9664 Tf 198.5409 2.2665 Td[(\026)]TJ/F44 8.9664 Tf -1.982 -2.2665 Td[(R)]TJ/F47 5.9776 Tf 7.0357 3.809 Td[(\000)]TJ/F67 8.9664 Tf 9.1119 -3.809 Td[(from)]TJ -221.6729 -10.4608 Td[(all)-235(of)-234(the)-235(peers.)-304(Our)-235(goal)-235(is)-234(no)25(w)-235(to)-234(produce)-235(a)-234(set)-235(of)]TJ/F44 8.9664 Tf 184.3694 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(i)]TJ/F67 8.9664 Tf 5.2649 -3.809 Td[(update)-235(rela-)]TJ -196.67 -10.4608 Td[(tions)-265(for)-264(the)-265(peer)-264(relations)-265(and)-264(a)-265(corresponding)-264(set)]TJ/F44 8.9664 Tf 184.6949 0 Td[(P)]TJ/F47 5.9776 Tf 7.1637 3.9186 Td[(\000)]TJ/F9 4.9813 Tf -2.5021 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1387 0 Td[(i)]TJ/F67 8.9664 Tf 5.9897 2.3863 Td[(to)-265(apply)-264(to)]TJ -200.4849 -10.4608 Td[(each)-286(pro)15(v)15(enance)-285(relation.)-417(Figure)-285(3)-286(sho)25(ws)-285(pseudocode)-286(for)-285(such)-286(an)]TJ 0 -10.4608 Td[(algorithm.)-332(First,)-260(the)-257(algorithm)-258(deri)25(v)15(es)-257(the)-258(deletions)-257(to)-258(apply)-257(to)-258(the)]TJ 0 -10.4607 Td[(pro)15(v)15(enance)-307(mapping)-308(relations;)-336(based)-307(on)-308(these,)-321(it)-308(computes)-307(a)-308(ne)25(w)]TJ/F66 8.9664 Tf 263.0137 657.334 Td[(Algorithm)]TJ/F72 8.9664 Tf 42.0878 0 Td[(Pr)45(opa)10(gateDelete)]TJ/F67 8.9664 Tf -42.0878 -10.4608 Td[(1.)]TJ/F66 8.9664 Tf 17.9328 0 Td[(f)25(or)]TJ/F67 8.9664 Tf 13.4674 0 Td[(e)25(v)15(ery)]TJ/F44 8.9664 Tf 21.7971 0 Td[(P)]TJ/F9 4.9813 Tf 4.6616 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F67 8.9664 Tf 3.1617 1.4943 Td[(,)-250(let)]TJ/F44 8.9664 Tf 15.6909 0 Td[(R)]TJ/F43 5.9776 Tf 7.0357 3.809 Td[(0)]TJ/F46 8.9664 Tf 6.7109 -3.809 Td[(\040)]TJ/F44 8.9664 Tf 11.7752 0 Td[(R)]TJ/F67 8.9664 Tf -107.8702 -10.4608 Td[(2.)-1250(Initialize)]TJ/F44 8.9664 Tf 52.0491 0 Td[(c)]TJ/F46 8.9664 Tf 6.5493 0 Td[(\040)]TJ/F31 8.9664 Tf 11.7752 0 Td[(0)]TJ/F67 8.9664 Tf -70.3737 -10.4608 Td[(3.)]TJ/F66 8.9664 Tf 17.9328 0 Td[(r)18(epeat)]TJ/F67 8.9664 Tf -17.9328 -10.4607 Td[(4.)-2926(Compute)-250(all)]TJ/F44 8.9664 Tf 79.289 0 Td[(P)]TJ/F47 5.9776 Tf 7.1637 3.9186 Td[(\000)]TJ/F9 4.9813 Tf -2.5022 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1388 0 Td[(i)]TJ/F67 8.9664 Tf 5.8593 2.3863 Td[(based)-250(on)-250(their)-250(delta)-250(rules)]TJ -94.9486 -10.4608 Td[(5.)-2926(\050)]TJ/F46 8.9664 Tf 35.9461 0 Td[(\003)]TJ/F67 8.9664 Tf 6.8493 0 Td[(Propag)5(ate)-250(ef)25(fects)-250(of)-250(deletions)]TJ/F46 8.9664 Tf 108.2945 0 Td[(\003)]TJ/F67 8.9664 Tf 4.6077 0 Td[(\051)]TJ -155.6976 -10.4608 Td[(6.)]TJ/F66 8.9664 Tf 32.9603 0 Td[(f)25(or)]TJ/F67 8.9664 Tf 13.4674 0 Td[(each)-250(idb)]TJ/F44 8.9664 Tf 32.3683 0 Td[(R)]TJ/F67 8.9664 Tf -78.796 -10.4608 Td[(7.)]TJ/F66 8.9664 Tf 46.4277 0 Td[(do)]TJ/F67 8.9664 Tf 11.71 0 Td[(update)-250(each)-250(associated)]TJ/F44 8.9664 Tf 83.9064 0 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F67 8.9664 Tf 3.1618 1.4943 Td[(,)-250(by)-250(applying)]TJ/F44 8.9664 Tf 49.3148 0 Td[(P)]TJ/F47 5.9776 Tf 7.1637 3.9186 Td[(\000)]TJ/F9 4.9813 Tf -2.5022 -6.3049 Td[(R)]TJ/F45 5.9776 Tf 5.1388 0 Td[(i)]TJ/F67 8.9664 Tf 5.8593 2.3863 Td[(to)-250(it)]TJ -220.4787 -10.4608 Td[(8.)-5734(De\002ne)-391(ne)25(w)-392(relation)]TJ/F44 8.9664 Tf 134.674 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(c)]TJ/F43 5.9776 Tf 3.3423 0 Td[(+1)]TJ/F67 8.9664 Tf 13.25 -3.809 Td[(to)-391(be)-392(the)-391(union)-391(of)-392(all)]TJ/F44 8.9664 Tf -100.1643 -10.4608 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F67 8.9664 Tf 3.1618 1.4943 Td[(,)-250(projected)-250(to)-250(the)]TJ/F31 8.9664 Tf 63.0678 0 Td[(\026)]TJ/F44 8.9664 Tf -0.5638 0 Td[(x)]TJ/F67 8.9664 Tf 7.465 0 Td[(attrib)20(utes.)]TJ -141.5669 -10.4608 Td[(9.)-2926(\050)]TJ/F46 8.9664 Tf 35.946 0 Td[(\003)]TJ/F67 8.9664 Tf 6.8493 0 Td[(Check)-250(tuples)-250(whose)-250(pro)15(v)15(enance)-250(w)10(as)-250(af)25(fected)]TJ/F46 8.9664 Tf 164.2447 0 Td[(\003)]TJ/F67 8.9664 Tf 4.6077 0 Td[(\051)]TJ -211.6477 -10.4607 Td[(10.)]TJ/F66 8.9664 Tf 32.9603 0 Td[(f)25(or)]TJ/F67 8.9664 Tf 13.4674 0 Td[(each)-250(idb)]TJ/F44 8.9664 Tf 32.3683 0 Td[(R)]TJ/F67 8.9664 Tf -78.796 -10.4608 Td[(11.)]TJ/F66 8.9664 Tf 46.4277 0 Td[(do)]TJ/F67 8.9664 Tf 11.71 0 Td[(Let)]TJ/F44 8.9664 Tf 14.2319 0 Td[(R)]TJ/F100 5.9776 Tf 6.9725 -1.0627 Td[(ch)-1(k)]TJ/F67 8.9664 Tf 13.536 1.0627 Td[(be)-254(an)-255(empty)-254(temporary)-254(relation)-254(with)]TJ/F44 8.9664 Tf 133.209 0 Td[(R)]TJ/F67 8.9664 Tf 7.0357 0 Td[(')55(s)]TJ -174.9851 -10.4608 Td[(schema)]TJ -58.1377 -10.4608 Td[(12.)]TJ/F66 8.9664 Tf 58.1377 0 Td[(f)25(or)]TJ/F67 8.9664 Tf 13.4674 0 Td[(each)-250(tuple)]TJ/F44 8.9664 Tf 38.8419 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(c)]TJ/F31 8.9664 Tf 3.8405 -3.809 Td[(\050)-15(\026)]TJ/F44 8.9664 Tf 3.5837 0 Td[(a)]TJ/F31 8.9664 Tf 4.882 0 Td[(\051)]TJ/F67 8.9664 Tf 5.8254 0 Td[(not)-250(in)]TJ/F44 8.9664 Tf 22.9179 0 Td[(R)]TJ/F45 5.9776 Tf 7.0357 3.809 Td[(c)]TJ/F43 5.9776 Tf 3.3423 0 Td[(+1)]TJ/F31 8.9664 Tf 9.7412 -3.809 Td[(\050)-15(\026)]TJ/F44 8.9664 Tf 3.5837 0 Td[(a)]TJ/F31 8.9664 Tf 4.882 0 Td[(\051)]TJ/F67 8.9664 Tf -187.1171 -10.4608 Td[(13.)]TJ/F66 8.9664 Tf 71.6051 0 Td[(do)-250(if)]TJ/F67 8.9664 Tf 19.2755 0 Td[(in)-233(there)-233(e)15(xis)1(ts,)-237(in)-232(an)15(y)-233(pro)15(v)15(enance)-233(relation)]TJ/F44 8.9664 Tf -7.5655 -10.4608 Td[(P)]TJ/F9 4.9813 Tf 4.6615 -0.9962 Td[(R)]TJ/F45 5.9776 Tf 5.6369 -0.4981 Td[(i)]TJ/F67 8.9664 Tf 5.4034 1.4943 Td[(associated)-250(with)]TJ/F44 8.9664 Tf 57.2764 0 Td[(R)]TJ/F67 8.9664 Tf 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