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Fill this gap by an alternative characterization of WSR. We Let in__~t[L,S 3 be the set of all interpretations of (L,S). ~ In words, (L',S') (L,S) is weakly serializable if we can find a serial augmented log such that elem (L') c elem (L), S' c S and p[L,S] has the same input-output behavior as p[L',S'], for any interpretation p*. Lk[S(Lk) ] (a) (A,~) c S E N WSR (the empty augmented log is in SE and WSR) (b) RI[X]R2[Y]WI[ x] W2[ x] e WSR with weak serialization Rl[X]Wl[X]R2[Y]W2[x ] (c) Rl[X]R2[x]W2[x]Wi[x3e WSR with weak serialization Rl[X3Wl[X3 Note that there is no serialization containing both transactions.

Otherwise Yi 31 31 3r ' 3 By construction, HDB S depends only on V, n, S . Moreover, the following two laminas hold. ,xm}. (~(L,S) ESE) (P~Bs[L,S]("x") = y) be a database system with A m an m-ary predicate symbol of the language used to write database descriptions. Let HDB S = (HD,HF,H I) interpretation for DBS. 2: Let GPS be a type i general purpose scheduler Suppose that GPS is well-formed. t. WSR. Proof. t, consistency preservation. (i = 0,i). 1, GPS satisfies: (i) (~fDBS = (V,A,n,S,GPS) £dbs.

Thus, if any output log generated by a general purpose scheduler GPS is weakly serializable, any database system using GPS preserves consistency. 1: L (L,S) e SE iff = I V el em(L) cDOMAIN(S) ^ ~ i I ..... i n e [i,~)) (L =RilWil. R We say that 1 W n 1 ) n (L,S) e SE is a serial augmented log . 1 allows ik=l~, for some k,~. Hence, (L,S) e SE abstracts a serial computation of the transactions, possibly with repetitions. 2: (L,S) E W S R iff (L,S) e W F L ^ (9(L' ,S') ~SE) (elem(L') c elem(L) AS' c S ^ PHIL,S] ("x") [p - H L' ,S' ] ("x")) where H is the Herbrand interpretation of (L,S) and x .

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