A generalized Fellegi-Holt paradigm for automatic error localization 5. Implied edits for general edit operations

In this section, a result will be derived that establishes whether a given path of edit operations of the form (3.1) can be used to make a given record consistent with a given system of edit rules (i.e., is a feasible solution to the error localization problem). This result uses the FM elimination technique discussed in Section 2.

Let x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhaaaa@38CA@ be a given record and let y t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaaaaa@3A1E@ be any record that can be obtained by applying, in sequence, the edit operations g 1 , , g t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab gAci8kaacYcacaWGNbWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3F11@ to x : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhacaGG6aaaaa@3988@

y t = g t g t 1 g 1 ( x ) . ( 5.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja am4za8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIyiYBcaWGNb WdamaaBaaaleaapeGaamiDaiabgkHiTiaaigdaa8aabeaak8qacqWI yiYBcqWIVlctcqWIyiYBcaWGNbWdamaaBaaaleaapeGaaGymaaWdae qaaOWdbmaabmaapaqaa8qacaWH4baacaGLOaGaayzkaaGaaiOlaiaa ywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiwdacaGGUaGaaG ymaiaacMcaaaa@57F4@

Write g n ( x ) = T n x + S n α n + c n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeWaaeWaa8aa baWdbiaahIhaaiaawIcacaGLPaaacqGH9aqpcaWHubWdamaaBaaale aapeGaamOBaaWdaeqaaOWdbiaahIhacqGHRaWkcaWHtbWdamaaBaaa leaapeGaamOBaaWdaeqaaOWdbiaahg7apaWaaSbaaSqaa8qacaWGUb aapaqabaGcpeGaey4kaSIaaC4ya8aadaWgaaWcbaWdbiaad6gaa8aa beaakiaacYcaaaa@4AAE@ for n { 1 , , t } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaad6gacqGHiiIZdaGadaWdaeaapeGaaGymaiaacYcacqGHMacV caGGSaGaamiDaaGaay5Eaiaaw2haaiaac6caaaa@41E4@ From (5.1) it follows by induction that

y 1 = T 1 x + S 1 α 1 + c 1 , y 2 = T 2 T 1 x + S 2 α 2 + c 2 + T 2 ( S 1 α 1 + c 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbuaabaqaciaaaeaacaWH5bWdamaaBaaaleaapeGaaGymaaWdaeqa aaGcpeqaaiabg2da9iaahsfapaWaaSbaaSqaa8qacaaIXaaapaqaba GcpeGaaCiEaiabgUcaRiaahofapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeGaaCySd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRa WkcaWHJbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcaaeaa caWH5bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpeqaaiabg2da9i aahsfapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaaCiva8aadaWg aaWcbaWdbiaaigdaa8aabeaak8qacaWH4bGaey4kaSIaaC4ua8aada WgaaWcbaWdbiaaikdaa8aabeaak8qacaWHXoWdamaaBaaaleaapeGa aGOmaaWdaeqaaOWdbiabgUcaRiaahogapaWaaSbaaSqaa8qacaaIYa aapaqabaGcpeGaey4kaSIaaCiva8aadaWgaaWcbaWdbiaaikdaa8aa beaak8qadaqadaWdaeaapeGaaC4ua8aadaWgaaWcbaWdbiaaigdaa8 aabeaak8qacaWHXoWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiab gUcaRiaahogapaWaaSbaaSqaa8qacaaIXaaapaqabaaak8qacaGLOa GaayzkaaGaaiilaaaaaaa@6487@

and, in general,

y t = T t T 1 x + S t α t + c t + n = 2 t T t T n ( S n 1 α n 1 + c n 1 ) , ( 5.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja aCiva8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIVlctcaWHub WdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaahIhacqGHRaWkcaWH tbWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiaahg7apaWaaSbaaS qaa8qacaWG0baapaqabaGcpeGaey4kaSIaaC4ya8aadaWgaaWcbaWd biaadshaa8aabeaak8qacqGHRaWkdaaeWbqaaiaahsfapaWaaSbaaS qaa8qacaWG0baapaqabaGcpeGaeS47IWKaaCiva8aadaWgaaWcbaWd biaad6gaa8aabeaaa8qabaGaamOBaiabg2da9iaaikdaaeaacaWG0b aaniabggHiLdGcdaqadaWdaeaapeGaaC4ua8aadaWgaaWcbaWdbiaa d6gacqGHsislcaaIXaaapaqabaGcpeGaaCySd8aadaWgaaWcbaWdbi aad6gacqGHsislcaaIXaaapaqabaGcpeGaey4kaSIaaC4ya8aadaWg aaWcbaWdbiaad6gacqGHsislcaaIXaaapaqabaaak8qacaGLOaGaay zkaaGaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa iwdacaGGUaGaaGOmaiaacMcaaaa@738D@

where the sum over n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaad6gaaaa@38BC@ is defined to be zero when t = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadshacqGH9aqpcaaIXaGaaiOlaaaa@3B35@ Moreover, all terms involving S n α n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahofapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaaCySd8aa daWgaaWcbaWdbiaad6gaa8aabeaaaaa@3C96@ vanish in these expressions when g n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgapaWaaSbaaSqaa8qacaWGUbaapaqabaaaaa@3A02@ does not contain any free parameters.

The path of edit operations P = [ g 1 , , g t ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadcfacqGH9aqpdaWadaWdaeaapeGaam4za8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacaGGSaGaeyOjGWRaaiilaiaadEgapaWaaS baaSqaa8qacaWG0baapaqabaaak8qacaGLBbGaayzxaaaaaa@4317@ can be applied to x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhaaaa@38CA@ to obtain a record that is consistent with the edits (2.1) if, and only if, there exists a y t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaaaaa@3A1E@ of the form (5.2) that satisfies A y t + b 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahgeacaWH5bWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiab gUcaRiaahkgacqWIzkszcaWHWaaaaa@3F39@ and all relevant additional restrictions of the form (3.2) on α 1 , , α t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaOGaai Olaaaa@406F@ Using (5.2), A y t + b 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahgeacaWH5bWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiab gUcaRiaahkgacqWIzkszcaWHWaaaaa@3F39@ can be written as:

( A T t T 1 ) x + ( A S t ) α t + n = 2 t ( A T t T n S n 1 ) α n 1 + b t 0 , ( 5.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbmaabmaapaqaa8qacaWHbbGaaCiva8aadaWgaaWcbaWdbiaadsha a8aabeaak8qacqWIVlctcaWHubWdamaaBaaaleaapeGaaGymaaWdae qaaaGcpeGaayjkaiaawMcaaiaahIhacqGHRaWkdaqadaWdaeaapeGa aCyqaiaahofapaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOa GaayzkaaGaaCySd8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH RaWkdaGfWbqabSWdaeaapeGaamOBaiabg2da9iaaikdaa8aabaWdbi aadshaa0WdaeaapeGaeyyeIuoaaOWaaeWaa8aabaWdbiaahgeacaWH ubWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabl+Uimjaahsfapa WaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaaC4ua8aadaWgaaWcbaWd biaad6gacqGHsislcaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaaC ySd8aadaWgaaWcbaWdbiaad6gacqGHsislcaaIXaaapaqabaGcpeGa ey4kaSIaaCOya8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIzk szcaWHWaGaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik aiaaiwdacaGGUaGaaG4maiaacMcaaaa@73CD@

with b t = b + A c t + n = 2 t A T t T n c n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahkgapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja aCOyaiabgUcaRiaahgeacaWHJbWdamaaBaaaleaapeGaamiDaaWdae qaaOWdbiabgUcaRmaaqadabaGaaCyqaiaahsfapaWaaSbaaSqaa8qa caWG0baapaqabaGcpeGaeS47IWKaaCiva8aadaWgaaWcbaWdbiaad6 gaa8aabeaak8qacaWHJbWdamaaBaaaleaapeGaamOBaiabgkHiTiaa igdaa8aabeaaa8qabaGaamOBaiabg2da9iaaikdaaeaacaWG0baani abggHiLdaaaa@51D5@ a vector of constants.

Interestingly, (5.3) and the possible additional restrictions of the form (3.2) constitute a linear system of the form (2.1) on the extended record ( x , α 1 , , α t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbmaabmaapaqaa8qaceWH4bWdayaafaWdbiaacYcaceWHXoGbauaa paWaaSbaaSqaaiaaigdaaeqaaOWdbiaacYcacqGHMacVcaGGSaGabC ySdyaafaWdamaaBaaaleaacaWG0baabeaaaOWdbiaawIcacaGLPaaa daahaaWcbeqaaOGamai2gkdiIcaaiiaacqWFUaGlaaa@472D@ Therefore, FM elimination may be used to remove all free parameters from this system. This yields a system of implied restrictions for x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhacaGGUaaaaa@397C@ Moreover, a repeated application of the fundamental property of FM elimination establishes that x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhaaaa@38CA@ satisfies this system of implied edits if, and only if, there exist parameter values for α 1 , , α t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3FB3@ that, together with x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhacaGGSaaaaa@397A@ satisfy (5.3) and (3.2). Hence, it follows that the path of edit operations P = [ g 1 , , g t ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadcfacqGH9aqpdaWadaWdaeaapeGaam4za8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacaGGSaGaeyOjGWRaaiilaiaadEgapaWaaS baaSqaa8qacaWG0baapaqabaaak8qacaGLBbGaayzxaaaaaa@4317@ can lead to a consistent record for x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhaaaa@38CA@ if, and only if, x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaahIhaaaa@38CA@ satisfies the system of implied edits obtained by eliminating α 1 , , α t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3FB3@ from (5.3) and (if relevant) additional restrictions of the form (3.2).

Example. Consider the following edits in x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@ and x 2 : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGG6aaaaa@3AA4@

x 1 0 , ( 5.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyzImRa aGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcaca aI1aGaaiOlaiaaisdacaGGPaaaaa@4872@

x 2 0 , ( 5.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyzImRa aGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcaca aI1aGaaiOlaiaaiwdacaGGPaaaaa@4874@

x 1 + x 2 5. ( 5.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda caGGUaGaaGOnaiaacMcaaaa@4B79@

Let g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgaaaa@38B5@ be the edit operation that transfers an amount of at most four units between x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@ and x 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGSaaaaa@3A96@ in either direction: g ( ( x 1 , x 2 ) ) = ( x 1 + α , x 2 α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgadaqadaWdaeaapeWaaeWaa8aabaWdbiaadIhapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8 qacaaIYaaapaqabaaak8qacaGLOaGaayzkaaaccaGae8NmGikacaGL OaGaayzkaaGaeyypa0ZaaeWaa8aabaWdbiaadIhapaWaaSbaaSqaa8 qacaaIXaaapaqabaGcpeGaey4kaSIaeqySdeMaaiilaiaadIhapaWa aSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyOeI0IaeqySdegacaGLOa GaayzkaaGae8NmGikaaa@50CE@ with 4 α 4. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiabgkHiTiaaisdacqGHKjYOcqaHXoqycqGHKjYOcaaI0aGaaiOl aaaa@3FED@ For this single edit operation, the system of transformed edits (5.3) is:

x 1 + α 0 , ( 5.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa eqySdeMaeyyzImRaaGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaI1aGaaiOlaiaaiEdacaGGPaaaaa@4AF6@

x 2 α 0 , ( 5.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyOeI0Ia eqySdeMaeyyzImRaaGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaI1aGaaiOlaiaaiIdacaGGPaaaaa@4B03@

x 1 + x 2 5. ( 5.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda caGGUaGaaGyoaiaacMcaaaa@4B7C@

I also add the following restrictions of the form (3.2) on α : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHjaacQdaaaa@3A26@

α 4 , ( 5.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHjabgwMiZkabgkHiTiaaisdacaGGSaGaaGzbVlaaywW7 caaMf8UaaGzbVlaaywW7caGGOaGaaGynaiaac6cacaaIXaGaaGimai aacMcaaaa@498D@

α 4. ( 5.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHjabgsMiJkaaisdacaGGUaGaaGzbVlaaywW7caaMf8Ua aGzbVlaaywW7caGGOaGaaGynaiaac6cacaaIXaGaaGymaiaacMcaaa a@4892@

This yields five linear constraints (5.7) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa a@3896@ (5.11) on x 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGccaGGSaaaaa@3A95@ x 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGSaaaaa@3A96@ and α , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHjaacYcaaaa@3A18@ from which α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHbaa@3968@ may be removed by FM elimination to obtain:

x 1 4 , ( 5.12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyzImRa eyOeI0IaaGinaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl aacIcacaaI1aGaaiOlaiaaigdacaaIYaGaaiykaaaa@4A1C@

x 2 4 , ( 5.13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyzImRa eyOeI0IaaGinaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl aacIcacaaI1aGaaiOlaiaaigdacaaIZaGaaiykaaaa@4A1E@

x 1 + x 2 0 , ( 5.14 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHLjYScaaIWa GaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda caGGUaGaaGymaiaaisdacaGGPaaaaa@4C3C@

x 1 + x 2 5. ( 5.15 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda caGGUaGaaGymaiaaiwdacaGGPaaaaa@4C33@

According to the theory, any record ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbmaabmaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiaacYcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpe GaayjkaiaawMcaaGGaaiab=jdiIcaa@3FFD@ that satisfies (5.12) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa a@3896@ (5.15) can be made consistent with the original edits (5.4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa a@3896@ (5.6) by transferring a certain amount 4 α 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiabgkHiTiaaisdacqGHKjYOcqaHXoqycqGHKjYOcaaI0aaaaa@3F3B@ between x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@ and x 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGUaaaaa@3A98@ The example record ( x 1 , x 2 ) = ( 2 , 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbmaabmaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiaacYcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpe GaayjkaiaawMcaaGGaaiab=jdiIkabg2da9maabmaapaqaa8qacqGH sislcaaIYaGaaiilaiaaiodaaiaawIcacaGLPaaacqWFYaIOaaa@473A@ is inconsistent with the original edit rules (5.4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa a@3896@ (5.6) but satisfies (5.12) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa a@3896@ (5.15). This implies that the record can be made consistent with the original edits by applying g . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadEgacaGGUaaaaa@3967@ It is easy to see that this is true; any choice 2 α 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaaikdacqGHKjYOcqaHXoqycqGHKjYOcaaIZaaaaa@3E4B@ will do.

It is interesting to note that, for the special case that P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaadcfaaaa@389E@ consists of the single FH operation that imputes x j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaGccaGGSaaaaa@3AC9@ the transformed system of edits (5.3) is obtained by replacing every occurrence of x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa@3A0F@ in the original edits by an unrestricted parameter α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHjaac6caaaa@3A1A@ Eliminating α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeg7aHbaa@3968@ from (5.3) is equivalent in this case to eliminating x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa@3A0F@ directly from the original edits. In this sense, the above result generalizes the fundamental property of FM elimination for FH operations to all edit operations of the form (3.1).

In general, the set of records defined by expression (5.2) depends on the way the edit operations are ordered. Thus, two paths consisting of the same set of edit operations in a different order need not yield the same solution to the error localization problem. In this respect, general edit operations differ from FH operations (Scholtus 2014).

Report a problem on this page

Is something not working? Is there information outdated? Can't find what you're looking for?

Please contact us and let us know how we can help you.

Privacy notice

Date modified: