(c) Oded Shmueli Transactions Lecture 2 (BHG, Chap. 2) The formal foundation

(c) Oded Shmueli Partial order L=(Σ, <), Σ is the domain, < is a binary relation on Σ that is:  irreflexive, for all a  Σ, a  a (i.e., a < a is false).  transitive, for all a, b, c in Σ, a < b and b < c implies a < c. If a < b then a is a predecessor of b and b follows a. If neither a < b nor b < a then a and b are incomparable. L’=(Σ’, <‘) is a restriction of L=(Σ, <) on domain Σ’ if Σ’  Σ and for all a, b  Σ’, a <‘ b iff a < b. L’ is a prefix of L, L’ ≤ L, if L’ is a restriction of L and for each a  L’, all predecessors of a in L are in Σ’.

(c) Oded Shmueli Partial order and DAGs A partial order L=(Σ, <) can be viewed as a directed graph G=(N, E):  N = Σ.  (a, b)  E iff a < b. G is acyclic as, by transitivity, cyclic would imply a < a for some a  Σ. G is also transitively closed. Conversely, given a DAG G=(N,E), we can construct a partial order (Σ, <) by transitively closing G to produce (N, E+) and setting Σ = N and a < b iff (a, b)  E+.

(c) Oded Shmueli Transactions In the system context a transaction is a particular program execution that manipulates the database using read and write operations. In the theory context a transaction is a modeling of such an execution where the operations against the database are modeled as well as their order. Since a transaction may be generated by concurrent programs, a transaction is best modeled as a partial order. We will not model all aspects of transactions:  No initial values.  Values read or written.  Analysis will apply to any situation (view each write as an arbitrary function of all read values).  Can model input and output statements via unique data items.

(c) Oded Shmueli Transactions, informally T = (S,<), partial order:  S is the collection of read operations and write operations (once).  a or c, not both are in S.  all operations precede a or c.  a < b indicates a happened before b.  for all x, if Wi[x] and Ri[x] are in S, they are not incomparable. r2[x] r2[y] w2[z] c2

(c) Oded Shmueli Transactions, formally Ti is a partial order with ordering relation <i:  Ti  {ri[x], wi[x] | x is a data item}  {ai, ci}  ai  T iff ci  T.  if t  T is either ai or ci then for all other p  T, p <i t.  If ri[x], wi[x]  T then either ri[x] < wi[x] or wi[x] < ri[x].

(c) Oded Shmueli Complete History Two operations conflict if they operate on the same data item and one is a write. A complete history over transaction set T={T1,…,Tn} is a partial order (H,<H):  H is the union of the Ti’s, H =  i Ti.  <H contains the union of the <i, <H   i <i.  for any two conflicting p, q  H: p <H q or q <H p.

(c) Oded Shmueli History Histories model system-wide, not necessarily complete, executions. A History is a prefix of a complete history. We usually represent histories as DAGs. In DAG representation, usually not all transitive edges are drawn.

(c) Oded Shmueli Committed Projection of a History Ti committed (aborted) if ci (ai) present. C(H): restriction of H to the set of operations of transactions committed in H. C(H) is a complete history. C(H) defines the semantics of a history H, that is the kind of database state transformation performed. For this interpretation to be sound, the system need achieve this effect.

(c) Oded Shmueli History example T1=r1[x]  w1[x]  c1 T3=r3[x]  w3[y]  w3[x]  c3 T4=r4[y]  w4[x]  w4[y]  c4 w4[x] w4[y] r1[x]c1 r4[y] w1[x] r3[x]w3[y]w3[x] c4 c3 H1 – complete history w4[x] w4[y] c1 r4[y] w1[x] r3[x]w3[y]w3[x] H1’ –history, prefix of H1 All transactions committed T3, T4 active r1[x]

(c) Oded Shmueli C(H) T1=r1[x]  w1[x]  c1 T3=r3[x]  w3[y]  w3[x]  c3 T4=r4[y]  w4[x]  w4[y]  c4 w4[x] w4[y] c1 r4[y] w1[x] r3[x]w3[y]w3[x] H1’ –history, prefix of H1 Committed Projection of H1’, restriction to the domain of committed transactions c1w1[x]r1[x]

(c) Oded Shmueli Serializable Histories Define equivalence of histories. Define serial histories. Define serializable histories.

(c) Oded Shmueli (Conflict) Equivalence of Histories Histories H and H’ are equivalent:  H and H’ have the same set of transactions and operations.  H and H’ have the same order on conflicting operations of transactions that are not aborted in H. Formally, for conflicting pi and pj such that ai, aj  H, if pi <H pj then pi <H’ pj (implying pi <H pj iff pi <H’ pj) Informally, in ordering conflicting operations we determine what’s computed, so equivalent histories perform the same database state transformation. Formally CSR ==> VSR.

(c) Oded Shmueli Equivalence example r2[z]w2[y] w2[x] r1[x]r1[y] w1[y] c2 c1w1x] r2[z]w2[y] w2[x] r1[x]r1[y] w1[y] c2 c1 w1x] H2 H3  H2 r2[z] w2[y] w2[x] r1[x]r1[y] w1[x] c2 c1 w1y] H4 not equivalent to H2, H3, for example, w1[y], w2[y]

(c) Oded Shmueli Serializable Histories A complete history is serial if for all Ti, Tj all operations of Ti precede those of Tj or vice versa. We would like “correct” to mean “same as serial”. Technical problem: serial is complete by definition, history is not. “Solution”: allow serial histories over incomplete transactions. But, incomplete histories may be incorrect database transformation. A serial execution is a correct database state transformation. So, for a history H to be “correct” we require it to be “equivalent” to a complete history H’. H itself is not necessarily complete, C(H) is complete. Also, C(H) is the semantics of H. So, we define:  H is serializable (SR) if C(H) is equivalent to a serial history.

(c) Oded Shmueli The Serialization Graph Consider history H over T={T1,..,Tn} SG(H) has a node for each committed transaction in H. An edge from Ti to Tj if one of Ti’s operations conflicts with and precedes one of Tj’s operations.

(c) Oded Shmueli Serialization Graph r2[x] w2[y] r1[x]w1[x] w1[y] c2 c1 H5 r3[x]w3[x]c3 T2T1T3 SG(H5) Note: SG is not transitively closed in general, e.g., replace w3[x] with w3[z].

(c) Oded Shmueli Topological sort Consider a DAG G=(V,E). List the nodes of V as v1,…,vn so that for all edges (vi, vj), i<j. A directed graph is acyclic iff it has a topological sort. Finding a t.s.:  find a source v (no incoming edges).  delete edges outgoing from the source.  output v.

(c) Oded Shmueli The Serializability Theorem H is serializable iff SG(H) is acyclic (if) Equivalence of C(H) to a serial history Hs, in topological sort order of transactions in C(H). Conflicting operations appear in the same order in C(H) and Hs.

(c) Oded Shmueli The Serializability Theorem (if): detailed H over T={T1,…,Tn}. W.l.o.g., T1,…,Tm are committed in H. Consider SG(H). Sort it topologically Ti1,…,Tim. Let Hs= Ti1,…,Tim. Claim: H  Hs. Proof: Need to show: same operations, same order on conflicting operations.  H and Hs have the same set of operations.  Let pi (of Ti) and pj of (Tj) be conflicting operations.  All such operations are ordered in H.  There is an edge Ti  Tj in SG(H).  So, in the t.s., Ti must precede Tj.  So Ti precedes Tj in Hs. So pi precedes pj in Hs.

(c) Oded Shmueli The Serializability Theorem (Cont.) H is serializable iff SG(H) is acyclic (only if) Consider Hs equivalent to C(H). Ti  Tj in SG(H)  Ti precedes Tj in Hs. So, a cycle in SG(H) implies a transaction precedes itself in Hs, which is impossible.

(c) Oded Shmueli The Serializability Theorem (only if): detailed H is SR. Hs  C(H). Consider Ti  Tj in SG(H). This is due to conflicting pi (of Ti) and pj (of Tj) and pi precedes pj in C(H). Since Hs  C(H), pi precedes pj in Hs. Since Hs is serial, Ti precedes Tj in Hs. If there is a cycle T1  T2 …  Tk=T1 in SG(H):  Then, T1 precedes T2 in Hs, …precedes T1 in Hs. But T1 cannot precede itself  no cycle can exist.

(c) Oded Shmueli Example H6 = w1[x] w1[y] c1 r2[x] r3[y] w2[x] c2 w3[y] c3 SG(H6) = T1T3T2 There are two t.s.’s:  T1 T3 T2  T1 T2 T3 Both provide equivalent serial histories.

(c) Oded Shmueli Recoverable Histories Ti reads x from Tj if  Wj[x] < Ri[x]  aj  Ri[x]  Wj[x] < Wk[x] < Ri[X]  ak < Ri[x]  Note: i=j is possible. Ti reads from Tj if Ti reads some data item from Tj.

(c) Oded Shmueli Examples: Additional Requirements w1[x] r2[x] w2[y] c2  T1 may abort, not recoverable (RC) w1[x] r2[x] w2[y] is RC  if T1 aborts, so must T2 (not ACA) w1[x,2] w1[y,3] w2[y,1] c1 r2[x] a2  RC+ACA. We should put y=3. Seems ok. X=1 w1[x,2] w2[x,3] a1  should x be 1 (or 3)? If a2, should we put 2? Should be 1!

(c) Oded Shmueli Formally: Additional Requirements (i ≠ j) RC Ti reads from Tj and ci in H  cj < ci  Don’t commit if you read uncommitted data. ACA Ti reads, via ri[x], from Tj  cj < ri[x]  Only read data produced by committed transactions. Here i ≠ j. ST wj[x] < oi[x]  aj < oi[x] or cj < oi[x]  implement abort by restoring before-images. Each category is more restrictive.

(c) Oded Shmueli ST  ACA  RC Let H  ST.  Suppose Ti reads x from Tj in H.  Then, wj[x] < ri[x] and aj  ri[x].  By ST, cj < ri[x]. So, H  ACA and ST  ACA.  H9 = w1[x] w1[y] r2[u] w2[x] w1[z] c1 r2[y] w2[y] c2  ACA but  ST. So, ST  ACA. Let H  ACA.  Suppose Ti reads x from Tj in H and ci  H.  H  ACA  wj[x] < cj < ri[x].  ci  H  ri[x] < ci  cj < ci. So, H  RC and ACA  RC.  H8 = w1[x] w1[y] r2[u] w2[x] r2[y] w2[y] w1[z] c1 c2  RC but  ACA. So, ACA  RC.

(c) Oded Shmueli State of the world ST ACA RC Serial SR

(c) Oded Shmueli Prefix Commit Closed (PCC) Properties PCC property: if holds on history H then it holds for C(H’) for any prefix H’ of H. Any correctness criterion better be PCC. Otherwise, system fails after producing H’ s.t. the property does not hold on C(H’). ACA, ST, RC, SR are all PCC properties. SR: H is SR. Look at SG(H). Look at prefix H’. Look at C(H’). SG(C(H’)) is sub-graph of SG(H), hence acyclic. Hence C(H’) is SR.

(c) Oded Shmueli Operations other than read/write Two operations conflict if the order of their performance may matter. Computational effect: value returned, data items’ values. Need to extend definition of conflict. Theorems will apply. Same SG(H), theorem. Can create compatibility matrix. Important feature - ordering of conflicting operations.

(c) Oded Shmueli Operations other than read/write - example Consider increment (inc) that adds 1 and decrement (dec) that subtracts 1. No value is returned. Conflict table   n means conflict  y means no conflict readwriteincdec readynnn writennnn incnnyy decnnyy

(c) Oded Shmueli Operations other than read/write – example history r4[y] w4[x] w1[x] w1[y] dec4[y] c1 r3[x]inc3[y]c3 inc2[y]dec2[x] c2 c4 T2T3T4 T1 T1 T3 T2 T4 H11 SG(H11)

(c) Oded Shmueli View Equivalence Transactions are deterministic transformers. If a transaction reads the same values in two executions, it’ll produce the same values. So, if in two executions transactions read the same values, they’ll produce the same values. If, in addition, for all items x, the last transaction to write into x is the same one in the two executions, the final DB will be the same.

(c) Oded Shmueli View Equivalence, formally Final write: wi[x] in H, ai not in H, for all other wj[x], wj[x] < wi[x] or aj in H. H is view-equivalent to H’ if:  H, H’ are over the same set of transactions,  For all Ti, Tj s.t. ai, aj not in H (and H’), if Ti reads x from Tj in H, Ti also does so in H’.  Same final writes in H and H’.

(c) Oded Shmueli View Serializability We’d like a definition that captures “a history is view equivalent to a serial history”. And, use it as a correctness criterion. Let’s try “a history is v-serializable if it’s view equivalent to a serial history”. H12 = w1[x] w2[x] w2[y] c2 w1[y] c1 | w3[x] w3[y] c3. H12 is view equivalent to T1 T2 T3. Suppose the system crashes at |. Resulting execution, H12’ = w1[x] w2[x] w2[y] c2 w1[y] c1, is not view equivalent to either T1 T2 or T2 T1. So, “v-serializable” is not an appropriate correctness criterion. We need enforce PCC.

(c) Oded Shmueli View Serializability, formally H is VSR if if for each prefix H’ of H, C(H) is view equivalent to a serial history. “for each prefix” - so it’s a PCC property!

(c) Oded Shmueli View Serializability, properties CSR  VSR (next slide) VSR  CSR  W1[x] W2[x] W3[y] c2 W1[y] W3[y] c3 W1[z] c1 is VSR.  but bot CSR: T1  T2  T1 in SG(H). VSR more inclusive but not a practical notion (a scheduler that outputs exactly VSR histories will need to “solve” P=NP first).

(c) Oded Shmueli View Serializability, CSR  VSR CSR  VSR: Let H be SR. SG(H) is acyclic. Consider an arbitrary prefix H’ of H. SG(H’) is acyclic (subgraph of SG(H)). H’ is SR. H’  Hs where Hs is serial. In H’ and Hs:  Same read from: otherwise conflicting ops are in the wrong order.  Same final writes: similar reason. Conclusion: H’ is VSR. H’ chosen arbitrarily, so H is VSR.