Subject: more defs: snapshot equivalent

From: "Rick Snodgrass" <rts>
Date: Sat, 21 Nov 1992 15:33:28 MST

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\subsection{Snapshot Equivalent}

\entry{Definition}         
Informally, two tuples are {\em snapshot equivalent\/} if the snapshots of the
tuples at all times are identical.

Let temporal relation schema $R$ have $n$ time dimensions, $D_i$, $i =
1, \ldots, n$, and let $\tau^i$, $i = 1, \ldots, n$ be corresponding
timeslice operators, e.g., the valid timeslice and transaction
timeslice operators. Then, formally, tuples $x$ and $y$
are snapshot equivalent if 

\[t_i \in D_i,~ i = 1, ~\ldots,~ n,~
\tau^n_{t_n}( \ldots (\tau^1_{t_1}(x)) \ldots ) =
\tau^n_{t_n}( \ldots (\tau^1_{t_1}(y)) \ldots )\]

\noindent
Similarly, two relations are {\em snapshot equivalent\/} if at every time their
snapshots are equal.
{\em Snapshot equivalence\/} is a binary relation that can be applied to
tuples and to relations.

\entry{Alternative Names}   
Weakly equal, temporally weakly equal, weak equivalence.

\entry{Discussion}
Weak equivalence has been used by Ullman to relate two algebraic
expressions (Ullman, Principles of Database Systems, Second Edition,
page 309). Hence, ``temporally weakly equal'' is preferable to
``weakly equal.'' (E7)

In comparing ``temporally weakly equal'' with ``snapshot equivalent'',
the former term is longer and more wordy, and is
somewhat awkward, in that it contains two adverbs (-E2). 
``Temporally weak'' is not intuitive---in what way is it weak?
Snapshot equivalent explicitly identifies the source of the
equivalence (+E8).

\subsection{Snapshot-Equivalence Preserving Operator}

\entry{Definition}         

A unary operator $F$ is {\em snapshot-equivalence preserving\/} if
relation $r$ is snapshot equivalent to $r'$ implies $F(r)$ is snapshot
equivalent to $F(r')$. This definition may be extended to operators
that accept two or more argument relation instances.

\entry{Alternative Names}   
Weakly invariant operator, is invariant under weak binding of belongs to.

\entry{Discussion}
This definition does not rely on the term ``weak binding'' (+E7).

\subsection{Snapshot Equivalence Class}

\entry{Definition}         
A {\em snapshot equivalence class\/} is a set of relation instances that
are all snapshot equivalent to each other.

\entry{Alternative Names}   
Weak relation.

\entry{Discussion}
``Weak relation'' is not intuitive, as the concept identifies a set of relation
instances, not a single instance (-E8).

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