3. Transaction theory

Transcription

1 3. Transaction theory 6 Serializability Thomas Leich Transaction Management Last updated:

2 3. Transaction theory 6 Serializability 7 View Serializability Thomas Leich Transaction Management Last updated:

3 3. Transaction theory 6 Serializability 7 View Serializability 8 Conflict Serializability Thomas Leich Transaction Management Last updated:

4 3. Transaction theory 6 Serializability 7 View Serializability 8 Conflict Serializability 9 Conflict Graph Thomas Leich Transaction Management Last updated:

5 3. Transaction theory 6 Serializability 7 View Serializability 8 Conflict Serializability 9 Conflict Graph 10 Closure Properties Thomas Leich Transaction Management Last updated:

6 3. Transaction theory 6 Serializability 7 View Serializability 8 Conflict Serializability 9 Conflict Graph 10 Closure Properties 11 Transaction Abort and Fault Tolerance Thomas Leich Transaction Management Last updated:

7 Serializability Serializability Introduction to the topic Formalization of schedules Serializability concepts Comparison of serializability concepts Thomas Leich Transaction Management Last updated:

8 Serializability Introduction to Serializability T 1 : read(a); A := A 10; write(a); read(b); B := B + 10; write(b); T 2 : read(b); B := B 20; write(b); read(c); C := C + 20; write(c); Execution alternatives for two transactions: Serial, e.g. T1 before T 2 Interleaved, e.g. alternating steps of T 1 and T 2 Thomas Leich Transaction Management Last updated:

9 Serializability Interleaved execution: Example Execution 1 Execution 2 Execution 3 T 1 T 2 T 1 T 2 T 1 T 2 read(a) read(a) read(a) A 10 read(b) A 10 write(a) A 10 read(b) read(b) B 20 write(a) B + 10 write(a) B 20 write(b) write(b) read(b) read(b) read(b) write(b) B 20 read(c) B + 10 write(b) B + 10 read(c) read(c) C + 20 write(b) C + 20 write(b) C + 20 write(c) write(c) write(c) Thomas Leich Transaction Management Last updated:

10 Serializability Results of different executions A B C A + B + C Initial Value After execution After execution After execution Thomas Leich Transaction Management Last updated:

11 Serializability Simplified model Lock-Unlock-Model T 1 : lock A; unlock A; lock B; unlock B; T 2 : lock B; unlock B; lock C; unlock C; Thomas Leich Transaction Management Last updated:

12 Serializability Interleaved Executions with Lock/Unlock: Example Execution 1 Execution 2 Illegal execution T 1 T 2 T 1 T 2 T 1 T 2 lock A lock A lock A unlock A lock B lock B lock B unlock A unlock A unlock B unlock B lock B lock B lock B unlock B unlock B lock C lock C read C unlock B unlock B unlock C unlock C unlock C Thomas Leich Transaction Management Last updated:

13 Serializability Serializability An interleaved execution of a set of transactions is called serializable, if the result of the interleaved execution is identical with the result of a (randomly chosen) serial execution of the same set of transactions transaction. Thomas Leich Transaction Management Last updated:

14 Serializability The concept of schedules u 4 u 3 u 2 u 1 v 4 v 3 v 2 v 1 Scheduler... w 2 v 3 w 1 u 1 v 2 v 1 w 4 w 3 w 2 w 1 Thomas Leich Transaction Management Last updated:

15 Serializability The Read-Write Model Transaction T is a finite sequence of operations (steps) p i of the form r(x k ) or w(x k ): T = p 1 p 2 p 3 p n with p i {r(x k ), w(x k )} A complete transaction T has as last step either an Abort a or a Commit c: T = p 1 p n a or T = p 1 p n c. Thomas Leich Transaction Management Last updated:

16 Serializability Interleaved transactions SHUFFLE(T ): set of all possible interleaved executions of the steps of all transactions T i in the set T All steps of the transaction T i are included only once The relative order of the steps of a transaction is preserved T 1 := r 1 (x)w 1 (x) T 2 := r 2 (x)r 2 (y)w 2 (y) SHUFFLE(T ) = {r 1 (x)w 1 (x)r 2 (x)r 2 (y)w 2 (y), r 2 (x)r 1 (x)w 1 (x)r 2 (y)w 2 (y),... } Thomas Leich Transaction Management Last updated:

17 Serializability Schedule A complete schedule is a SHUFFLE-Element of a set of complete transactions. A schedule is a prefix of a complete schedule. r 1 (x)r 2 (x)w 1 (x) r }{{} 2 (y)a 1 w 2 (y)c 2 a schedule }{{} a complete schedule Thomas Leich Transaction Management Last updated:

18 Serializability Serial schedule A serial schedule s for T is a complete schedule of the following form: s := T ρ(1) T ρ(n) for a permutation ρ of {1,..., n} Resulting serial schedules for two transactions: T 1 := r 1 (x)w 1 (x)c 1 and T 2 := r 2 (x)w 2 (x)c 2 : s 1 := r 1 (x)w 1 (x)c }{{ 1 r } 2 (x)w 2 (x)c }{{ 2 } T 1 T 2 s 2 := r 2 (x)w 2 (x)c }{{ 2 r } 1 (x)w 1 (x)c }{{ 1 } T 2 T 1 Thomas Leich Transaction Management Last updated:

19 Serializability Correctness criteria A schedule s is correct, if its effect (the result of the execution of the schedule) is equivalent to the effect of a (randomly chosen) serial schedule s regarding the same set of transactions (s s ). If a schedule s is equivalent to a serial schedule s, then s is serializable (to s ). Open: How to define "being equivalent"? Thomas Leich Transaction Management Last updated:

20 View Serializability View serializability Idea: Effect of a transaction only depends on what values were seen by the transaction. It reads the value that was written last Special treatment necessary for the initialization and for the final result of a schedule Thomas Leich Transaction Management Last updated:

21 View Serializability View serializability: Preparations Artificial additional transactions: 1 Initial transaction T 0 : initially writes all involved objects 2 Terminal transaction T : reads all involved objects in the end Thomas Leich Transaction Management Last updated:

22 View Serializability Reads-from-relation If s is the relation temporally before in the schedule s, then r j (x) reads x from T i if and only if: wi (x) s r j (x) and k(wi (x) s w k (x) w k (x) s r j (x)). Reads-from-relation RF(s) for a schedule s: RF (s) := { (T i, x, T j ) r j (x) reads x from T i } Thomas Leich Transaction Management Last updated:

23 View Serializability View equivalence Two schedules s and s are view equivalent, if: 1 op(s) = op(s ) op(s): the set of all steps occurring in s including a and c (Sets must be identical for both schedules) 2 RF(s) = RF (s ) Schedules s and s have the same Reads-from-relation Thomas Leich Transaction Management Last updated:

24 View Serializability View equivalence: Example I Given two complete schedules s 1 and s 2 consisting of two transactions T 1 and T 2 with: s 1 := r 1 (x)r 2 (y)w 1 (y)w 2 (y)c 1 c 2 s 2 := r 1 (x)w 1 (y)r 2 (y)w 2 (y)c 2 c 1 Thomas Leich Transaction Management Last updated:

25 View Serializability View equivalence: Example II For calculating the reads-from-relations RF(s 1 ) and RF (s 2 ), s 1 and s 2 are expanded by the initial transaction T 0 and the terminal transaction T in the following way: s 1 := w 0 (x)w 0 (y)c 0 }{{} T 0 s 2 := w 0 (x)w 0 (y)c 0 } {{ } T 0 r 1 (x)r 2 (y)w 1 (y)w 2 (y)c 1 c 2 r (x), r (y)c }{{} T r 1 (x)w 1 (y)r 2 (y)w 2 (y)c 2 c 1 r (x), r (y)c }{{} T Thomas Leich Transaction Management Last updated:

26 View Serializability View equivalence: Example III Resulting reads-from-relations: RF (s 1 ) := {(T 0, x, T 1 ), (T 0, y, T 2 ), (T 0, x, T ), (T 2, y, T )} RF (s 2 ) := {(T 0, x, T 1 ), (T 1, y, T 2 ), (T 0, x, T ), (T 2, y, T )} View equivalence of s 1 and s 2 : Comparison of reads-from-relations Since RF (s 1 ) RF(s 2 ), s 1 and s 2 are not view equivalent Thomas Leich Transaction Management Last updated:

27 View Serializability View Serializability A schedule s is view serializable, only if s is view equivalent to a serial schedule. Set of all view serializable schedules: VSR (view serializability) For n transactions, there exist n! serial schedules Problems: Exponential complexity for testing Testing requires complete schedules (blind writes and transaction aborts) Thomas Leich Transaction Management Last updated:

28 View Serializability View Serializability: Example Obviously, schedule s 2 is view serializable, because s 2 is serial. Possible serial schedules s and s for s 1 : s = T 1 T 2 = r 1 (x)w 1 (y)c 1 r 2 (y)w 2 (y)c 2 RF(s ) := {(T 0, x, T 1 ), (T 1, y, T 2 ), (T 0, x, T ), (T 2, y, T )} RF(s ) RF(s 1 ) s = T 2 T 1 = r 2 (y)w 2 (y)c 2 r 1 (x)w 1 (y)c 1 RF(s ) := {(T 0, y, T 2 ), (T 0, x, T 1 ), (T 0, x, T ), (T 1, y, T )} RF(s ) RF(s 1 ) RF (s ) RF(s 1 ) and RF(s ) RF(s 1 ): Schedule s 1 is not serializable! Thomas Leich Transaction Management Last updated:

29 Conflict Serializability Conflict Serializability Idea: Only the relative order of operations and not the actually read value is of importance in conflict situations It is not necessary to know which transaction has most recently written a value, only whether the value has been written before or after a transaction Thomas Leich Transaction Management Last updated:

30 Conflict Serializability Conflicts T 1 T 2 read A read A order independent T 1 T 2 read A write A order dependent T 1 T 2 write A read A order dependent T 1 T 2 write A write A order dependent Thomas Leich Transaction Management Last updated:

31 Conflict Serializability Conflict Serializability Conflict relation C of s: C(s) := { (p, q) p, q are in conflict and p s q} Conflict matrix: r i (x) w i (x) r j (x) w j (x) Thomas Leich Transaction Management Last updated:

32 Conflict Serializability Adjusted conflict relation conf(s) is an adjusted conflict relation, which includes no aborted transactions conf(s) := C(s) { (p, q) (p t q t ) t aborted(s)} aborted(s): Set of aborted transactions in schedule s Thomas Leich Transaction Management Last updated:

33 Conflict Serializability Adjusted conflict relation: Example Given schedule s: s = r 1 (x)w 1 (x)r 2 (x)r 3 (y)w 2 (y)c 2 a 1 c 3 Conflict relation for s: C(s) := {(w 1 (x), r 2 (x)), (r 3 (y), w 2 (y))} Removing aborted transaction T 1 from s: conf(s) := {(r 3 (y), w 2 (y))} Thomas Leich Transaction Management Last updated:

34 Conflict Serializability Conflict equivalence Two schedules s and s are conflict equivalent (s c s ) if: 1 op(s) = op(s ) 2 conf(s) = conf(s ) Thomas Leich Transaction Management Last updated:

35 Conflict Serializability Conflict serializability A schedule s is conflict serializable, if and only if s is conflict equivalent to a serial schedule. Class of all conflict serializable schedules: CSR (conflict serializability) Thomas Leich Transaction Management Last updated:

36 Conflict Serializability Conflict serializability: Example I Given two schedules s and s : s = r 1 (x)r 1 (y)w 2 (x)w 1 (y)r 2 (z)w 1 (x)w 2 (y) s = r 1 (y)r 1 (x)w 1 (y)w 2 (x)w 1 (x)r 2 (z)w 2 (y) Question: Are schedules s and s conflict equivalent? Step 1: op(s) = op(s ) applies, since all database operations occurring in s occur in s as well; also applies vice versa Thomas Leich Transaction Management Last updated:

37 Conflict Serializability Conflict serializability: Example II Step 2: Adjusted conflict relations conf(s) = {(r 1 (x), w 2 (x)), (w 2 (x), w 1 (x)), (r 1 (y), w 2 (y)), (w 1 (y), w 2 (y))} conf(s ) = {(r 1 (x), w 2 (x)), (w 2 (x), w 1 (x)), (r 1 (y), w 2 (y)), (w 1 (y), w 2 (y))} conf(s) = conf(s ) applies; so the conflict relations are equal and therefore s and s are conflict equivalent Thomas Leich Transaction Management Last updated:

38 Conflict Serializability Conflict serializability: Example III Test for conflict serializability by comparison with serial schedules Given schedule s: s = r 1 (x)r 1 (y)w 2 (x)w 1 (y)r 2 (z)w 1 (x)w 2 (y) Thomas Leich Transaction Management Last updated:

39 Conflict Serializability Conflict serializability: Example IV Adjusted conflict relation for s: conf(s) = {(r 1 (x), w 2 (x)), (w 2 (x), w 1 (x)), (r 1 (y), Possible serial schedule s 1 : w 2 (y)), (w 1 (y), w 2 (y))} s 1 = T 1 T 2 = r 1 (x)r 1 (y)w 1 (y)w 1 (x)c 1 w 2 (x)r 2 (z)w 2 (y)c 2 Conflict relation of s1 is not equal to the conflict relation of s: conf(s 1 ) = {(r 1 (x), w 2 (x)), (w 1 (x), w 2 (x)), (r 1 (y), w 2 (y)), (w 1 (y), w 2 (y))} Thomas Leich Transaction Management Last updated:

40 Conflict Serializability Conflict serializability: Example V Possible serial schedule s 2 as a candidate: s 2 = T 2 T 1 = w 2 (x)r 2 (z)w 2 (y)c 2 r 1 (x)r 1 (y)w 1 (y)w 1 (x)c 1 Conflict relation of s 2 is not equal to the conflict relation of s: conf(s 2 ) = {(w 2 (x), r 1 (x)), (w 2 (y), r 1 (y)), (w 2 (y), w 1 (y)), (w 2 (x), w 1 (x))} Therefore: s / CSR, which means that schedule s is not conflict serializable Thomas Leich Transaction Management Last updated:

41 Conflict Serializability Conflict serializability: Example VI Schedule: s 3 = r 1 (x)r 2 (x)w 2 (y)c 2 w 1 (x)c 1 is obviously conflict serializable, since only one conflict occurs Thomas Leich Transaction Management Last updated:

42 Conflict Serializability Graph-based Test Conflict graph G(s) = (V, E) for schedule s: 1 Set of vertices V includes all transactions occurring in s 2 Set of edges E includes all directed edges between two conflicting transactions: (t, t ) E t t ( p t)( q t ) with (p, q) conf(s) Thomas Leich Transaction Management Last updated:

43 Conflict Serializability Chronological sequence of three transactions T 1 T 2 T 3 r(y) w(y) w(x) r(y) w(x) w(z) r(u) w(x) s = r 1 (y)r 3 (u)r 2 (y)w 1 (y)w 1 (x)w 2 (x)w 2 (z)w 3 (x) Thomas Leich Transaction Management Last updated:

44 Conflict Graph Conflict graph G(s): T 1 T 2 T 3 Thomas Leich Transaction Management Last updated:

45 Conflict Graph Properties of conflict graph G(s) 1 If s is a serial schedule, the given conflict graph is an acyclic graph 2 For every acyclic graph G(s), a serial schedule s can be constructed in order to make s conflict serializable to s (Test e.g. by topological sorting) 3 If a graph contains cycles, the respective schedule is not conflict serializable Thomas Leich Transaction Management Last updated:

46 Conflict Graph Topological Sorting (Recursive) iteration through the graph can detect the absence of cycles Each node is visited once processing status per node and time stamp for entering and leaving is noted repeated reentry on arbitrary unprocessed nodes Topological sorting can be done in the same run Times of leaving provides the order sorted sequence defines a conflict-equivalent serial schedule see example of Saake/Sattler: Algorithmen und Datenstrukturen (the absent-minded professor...) Thomas Leich Transaction Management Last updated:

47 Conflict Graph Topological Sorting: Input socks panties watch shoes trousers shirt belt tie jacket Thomas Leich Transaction Management Last updated:

48 Conflict Graph Topological Sorting: Result 11/16 panties socks 17/18 shoes watch 9/10 trousers 12/15 shirt 1/8 13/14 6/7 belt tie 2/5 jacket 3/4 Thomas Leich Transaction Management Last updated:

49 Conflict Graph Topological Sorting: Alternative result 17/18 panties socks 15/16 shoes watch 13/14 9/12 trousers shirt 5/8 10/11 1/4 belt tie 6/7 jacket 2/3 Thomas Leich Transaction Management Last updated:

50 Conflict Graph Conflict Graphs and Serializability For every schedule s applies: G(s) acyclical s CSR Thomas Leich Transaction Management Last updated:

51 Closure Properties Problems during run-time Verification of serializability properties during run-time Only incomplete schedules available during run-time Observation of incomplete schedules is necessary Transactions that have not performed a commit yet can still be aborted anytime How does this affect the accuracy of the verification? Thomas Leich Transaction Management Last updated:

52 Closure Properties Closure Properties 1 Prefix Closure If property E applies to a schedule s, E also applies to any prefix of s. If E is fulfilled at the end of a schedule, E must not have been violated before. 2 Commit Closure If E applies to s, E also applies to CP(s) ("committed projection"). If E applies to a set of transactions, it still applies if some of them are aborted. 3 Prefix-commit Closure (PCC) Prefix-commit closure is the conjunction. Thomas Leich Transaction Management Last updated:

53 Closure Properties Closure Properties VSR-Schedules are not prefix-commit closed not prefix closed: blind writes can repair not commit closed: last writes can be changed by Abort CSR-Schedules are prefix-commit closed prefix closed: cycles in the graph remain cycles commit closed: Aborts can not generate any cycles Thomas Leich Transaction Management Last updated:

54 Closure Properties CSR vs VSR: Example I Does CSR VSR or VSR CSR? Given: schedule s: s = r 1 (y)r 3 (w) r 2 (y)w 1 (y) w }{{} 1 (x)w 2 (x) w }{{} 2 (z)w 3 (x)c 2 c 1 c 3 T 2 T 1 T 1 T 2 Thomas Leich Transaction Management Last updated:

55 Closure Properties CSR vs VSR: Example II Schedule s is not conflict serializable, since conflict graph G(s) contains a cycle. T 1 T 3 T 2 Thomas Leich Transaction Management Last updated:

56 Closure Properties CSR vs VSR: Example III Is s view serializable? Determining of the reads-from-relation RF RF(s) = {(T 0, y, T 1 ), (T 0, w, T 3 ), (T 0, y, T 2 ), (T 3, x, T ), (T 1, y, T ), (T 2, z, T ), (T 0, w, T )} Serial schedule s = T 2 T 1 T 3 : s = r 2 (y)w 2 (x)w 2 (z)c 2 r 1 (y)w 1 (y)w 1 (x)c 1 r 3 (w)w 3 (x)c 3 Reads-from-relation for schedule s : RF (s ) = {(T 0, y, T 1 ), (T 0, w, T 3 ), (T 0, y, T 2 ), (T 3, x, T ), (T 1, y, T ), (T 2, z, T ), (T 0, w, T )} Thomas Leich Transaction Management Last updated:

57 Closure Properties CSR vs VSR: Example IV RF (s) = RF(s ) applies, therefore, schedule s is view serializable Therefore: Conflict serializability is more restrictive than view serializability (CSR VSR) Generally: CSR VSR Cause: blind writes can repair violations caused by conflicts Thomas Leich Transaction Management Last updated:

58 Transaction Abort and Fault Tolerance Fault Tolerance In terms of fault tolerance, the following schedule s is not acceptable: s = r 1 (x)w 1 (x)r 2 (x)a 1 w 2 (x)c 2... though it is serializable in VSR and CSR! Thomas Leich Transaction Management Last updated:

59 Transaction Abort and Fault Tolerance Recoverability RC s is recoverable, if the following condition is fulfilled: (T i reads from T j in s) (c i s) (c j s c i ) Thomas Leich Transaction Management Last updated:

60 Transaction Abort and Fault Tolerance Recoverability: Example s 1 = w 1 (x)w 1 (y)r 2 (u)w 2 (x)r 2 (y)w 2 (y)c 2 w 1 (z)c 1 In s 1, T 2 reads data object y from T 1 but c 2 comes before c 1 s 1 is not recoverable s 2 is recoverable s 2 = w 1 (x)w 1 (y)r 2 (u)w 2 (x)r 2 (y)w 2 (y)w 1 (z)c 1 c 2 But: Problems when aborting T 1 instead of c 1 (dirty read)! Thomas Leich Transaction Management Last updated:

61 Transaction Abort and Fault Tolerance Avoiding cascading aborts Schedule s avoids cascading aborts ACA, if the following condition is fulfilled: (T i reads x from T j in s ) (c j s r i (x)) A transaction may only read data that has last been written by an already committed transaction. Thomas Leich Transaction Management Last updated:

62 Transaction Abort and Fault Tolerance Avoiding cascading aborts: Example Schedule s 2 from the last example does not belong into the class ACA However, s 3 avoids cascading aborts: s 3 = w 1 (x)w 1 (y)r 2 (u)w 2 (x)w 1 (z)c 1 r 2 (y)w 2 (y)c 2 Therefore: s 3 ACA Thomas Leich Transaction Management Last updated:

63 Transaction Abort and Fault Tolerance Problems with Before-Images DB Content Operation x = 1 (initial value) x = 2 w 1 (x 2) [ BF x,t1 = 1 ] x = 3 w 2 (x 3) [ BF x,t2 = 2 ] a 1 rollback of w 1 (x 2) with BF x := 1. Overwriting of T 2 has to be maintained x = 3 a 2 rollback of w 2 (x 3) with BF x :=?? Thomas Leich Transaction Management Last updated:

64 Transaction Abort and Fault Tolerance Strictness ST Schedule s is strict, if the following applies: (w j (x) s p i (x) j i) (a j s p i (x) c j s p i (x), (p {r, w})) No written object of an incomplete transaction may be read or overwritten Thomas Leich Transaction Management Last updated:

65 Transaction Abort and Fault Tolerance Strictness: Example s 3 / ST s 4 is strict, therefore s 4 ST: s 4 = w 1 (x)w 1 (y)r 2 (u)w 1 (z)c 1 w 2 (x)r 2 (y)w 2 (y)c 2 Thomas Leich Transaction Management Last updated:

66 Transaction Abort and Fault Tolerance Relation between the concepts VSR CSR RC ACA ST serial often using strict locking protocols acceptable using cascading repair Thomas Leich Transaction Management Last updated: