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BCDM Temporal Domains - Time is linear and totally ordered - Chronons are the basic time unit - Time domains are isomorphic to subsets of the domain of Natural numbers D VT = {t 1,t 2, …, t k }(valid time) D TT = {t’ 1,t’ 2, …, t’ h } {UC} (transaction time) D TT D VT (bitemporal chronons)

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BCDM Data Attribute names: D A ={A 1, A 2, …, A n } Attribute domains D D ={D 1, D 2, …, D n } Schema of a bitemporal relation: R = A i1, A i2, …, A ij T Domain of a bitemporal relation: D i1 D i2 … D ij D TT D VT Tuple of a relation r(R): x = (a 1, a 2, …, a j | t B )

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BCDM Example. Relation Employee with Schema: (name,salary,T) “Andrea was earning 60K at valid times 10, 11, 12 Such a tuple has been inserted into Employee at time 12, and is current now (say now=13)” (Andrea, 60k | {(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), ……}) VT TT 10 12 13 11

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BCDM Example. Relation Employee with Schema: (name,salary,T) “Andrea was earning 60K at valid times 10, 11, 12 Such a tuple has been inserted into Employee at time 12, and is current now (say now=13)” (Andrea, 60k | {(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), (UC,10), (UC,11), (UC,12)}) VT TT 10 12 13 11 UC

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BCDM Bitemporal relation: set of bitemporal tuples. Constraint: Value equivalent tuples are not allowed. (Bitemporal) DB: set of (bitemporal) relations

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BCDM Semantics (another viewpoint) NameSalaryT Andrea60K{(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)} (12,10) {Employee(Andrea,60K)} (12,11) {Employee(Andrea, 60K)} (12,12) {Employee(Andrea, 60K), Employee(John,50K)} (12,13) {Employee(John,50K)} (13,10) {Employee(Andrea,60K)} (13,11) {Employee(Andrea, 60K)} …….. (UC,12) {Employee(Andrea, 60K)}

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BCDM PROPERTIES Consistent extension (of “classical” SQL DB) A temporal DB is a set of “classical” DBs, one for each bitemporal chronon Uniqueness of representation (from the constraint about value equivalent tuples)

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BCDM Semantics of UC e.g., the DB’s clock thicks time 14 NameSalaryT Andrea60K{(12,10), (12,11), (12,12), (13,10), (13,11),(13,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)} NameSalaryT Andrea60K{(12,10), (12,11), (12,12), (13,10), (13,11), (13,12), (14,10), (14,11), (14,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)}

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UC semantics

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Deletion

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BCDM deletion (e.g., at time 15) delete(Employee, (Andrea,60K)) NameSalaryT Andrea60K{(12,10), (12,11), (12,12), (13,10), (13,11), (13,12), (14,10), (14,11), (14,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)} NameSalaryT Andrea60K{(12,10), (12,11), (12,12), (13,10), (13,11), (13,12), (14,10), (14,11), (14,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)}

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Insertion

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BCDM insertion (e.g., at time 16) insert(Employee, (Andrea,60K|{12,13})) insert(Employee, (Mary,70K|{16})) NameSalaryT Andrea60K{(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), (14,10), (14,11), (14,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)} NameSalaryT Andrea60K{(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), (14,10), (14,11), (14,12), (16,12),(16,13),(UC,12),(UC,13) } John50K{(12,12),(12, 13)} Mary70K{(16,16),(UC,16)}

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BCDM π D (r)={z | ∃x r (z[D]=x[D]) ∧ ∀ y r (y[D]=z[D] ⇒ y[T] z[T]) ∧ ∀ t z[T] ∃y r (y[D]=z[D] ∧ t y[T])} Algebraic Operators (Ex. Projection) - No value-equivalent tuple generated (uniqueness of representation!) - Coalescing!

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Example

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BCDM BCDM algebraic operators are a consistent extension of SQL’s ones (reducibility and equivalence) Algebraic Operators Properties

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BCDM Reducibility rTrT ρ t T (r T ) ρtTρtT op T (r T ) op T op op(ρ t T (r T )) ρtTρtT ρ t T (op T (r T )) =

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BCDM Equivalence r τtτt τ t (r) op T op op(r) τtτt τ t (op(r))=op T (τ t (r))

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BCDM PROBLEM Semantically clear but ….. inefficient (not suitable for a “direct” implementation) (1) Not 1-NF (2) UC (at each thick of the clock, all current tuples should be updated!)

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Task An efficient implementation must be devised The implementation must be proven to respect the semantics. Core issue here: efficient (1-NF) implementations hardly grant uniqueness of representation.

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An example of implementation: TSQL2 (Snodgrass et al., 1995) Temporal attribute T four temporal attributes (TT S, TT E, VT S, VT E ) Attribute value: a timestamp or UC Bitemporal tuple: A 1,….A n | TT S, TT E, VT S, VT E Bitemporal relation: set of bitemporal tuples Notice: value-equivalent tuples are allowed!

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An example of implementation: TSQL2 (Snodgrass et al., 1995) NameSalaryT Andrea60K{(12,10), (12,11), (12,12),(13,10), (13,11), (13,12), (UC,10), (UC,11), (UC,12)} John50K{(12,12),(12, 13)} NameSalaryTT S TT E VT S VT E Andrea60K12UC1012 John50K12 13 SEMANTICS BCDM TSQL2

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Semantics of TSQL2 representation

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From BCDM to TSQL2

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Property

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Insertion and Deletion in TSQL2

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An example of implementation: TSQL2 (Snodgrass et al., 1995) Efficient implementation (data model): - 1-NF - UC managed efficiently - clear semantics (mapping onto BCDM) BUT to get efficiency, we loose the uniqueness of representation property

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Problem: no uniqueness of representation NameSalaryT Andrea60K{(10,2), (10,3), (11, 2),(11,3), (12,1), (12,2),(12,3),(12,4),(13,1),(13,2),(13,3),(13,4)} NameSalaryTT S TT E VT S VT E Andrea60K101123 Andrea60K12UC14 NameSalaryTT S TT E VT S VT E Andrea60K12UC11 Andrea60K10UC23 Andrea60K12UC44 BCDM SEMANTICS TSQL2 (a) TSQL2 (b) Example. At time 10, the fact that Andrea earned 60K from 2 to 3 inserted in Employee. At time 12, such a tuple is updated: Andrea earned 60K from 1 to 4. At time 13, the tuple is (logically) deleted.

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Problem: no uniqueness of representation VT TT 1 2 1213 3 4 1011 TSQL2 implementation: “covering” rectangles

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Problem: no uniqueness of representation VT TT 1 2 1213 3 4 1011 TSQL2 Representation (a) NameSalaryTT S TT E VT S VT E Andrea60K101123 Andrea60K12UC14

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Problem: no uniqueness of representation VT TT 1 2 1213 3 4 1011 TSQL2 Representation (b) NameSalaryTT S TT E VT S VT E Andrea60K12UC11 Andrea60K10UC23 Andrea60K12UC44

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Problem: no uniqueness of representation VT TT 1 2 1213 3 4 1011 Other TSQL2 Representations!!

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Problem: no uniqueness of representation NameSalaryTT S TT E VT S VT E Andrea60K101123 Andrea60K12UC14 NameSalaryTT S TT E VT S VT E Andrea60K12UC11 Andrea60K10UC23 Andrea60K12UC44 Potentially, an enormous problem! e.g., Return all employees earning more than 50K for at most 3 consecutive time chronons NameTT S TT E VT S VT E Andrea12UC14 ?

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Problem: no uniqueness of representation One must grant that the temporal DB implementation respects its underlying semantics, independently of the representation DB1 DB2 op 1, …, op k DB1’ DB2’ op 1, …, op k Given two “semantically equivalent” temporal DBs, and given any sequence of operations, the results are always “semantic equivalent” Otherwise …..We cannot trust DB’s results!

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Problem: no uniqueness of representation Solution. Step 1. Formal definition of “semantic equivalence” Snapshot equivalence: Informally: two relations (Databases) are snapshot equivalent if they are identical at each bitemporal chronon

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Problem: no uniqueness of representation Solution. Step 2. Definition of manipulation and algebraic operators that preserve snapshot equivalence e.g., proofs given about TSQL2 (bitemporal) operators rB1rB1 rB2rB2 op B i op B i (r B 1 ) op B i (r B 2 ) snapshot equivalent snapshot equivalent

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Snapshot Equivalence Valid-timeslice operator σ B t1 (r) = {z (n+1) | x r (z[A]=x[A] z[T v ] = {t2 | (t1, t2) x[T]} z[T v ] } Transaction-timeslice operator

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Snapshot Equivalence

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Uniqueness of representation (BCDM)

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Timeslice operators in TSQL2

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TSQL2 property

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