We present a general notion of properties that
are characterized by local conditions that are invariant
under a sufficiently rich class of symmetries.
Our framework generalizes two popular models of
testing graph properties as well as the algebraic
invariances studied by Kaufman and Sudan (STOC'08).

We show that, in the aforementioned models of
testing graph properties, characterization by
such invaraint local conditions is closely
related to proximity oblivious testing
(as defined by Goldreich and Ron, STOC'09).
In contrast to this relation, we show that, in general,
characterization by invaraint local conditions
is neither necessary nor sufficient
for proximity oblivious testing.
Furthermore, we show that easy testability is {\em not}\/ guaranteed
even when the property is characterized by local conditions
that are invariant under a 1-transitive group of permutations.

Merely correcting numerous typos (spelling mistakes). Oded

We present a general notion of properties that
are characterized by local conditions that are invariant
under a sufficiently rich class of symmetries.
Our framework generalizes two popular models of
testing graph properties as well as the algebraic
invariances studied by Kaufman and Sudan (STOC'08).

We show that, in the aforementioned models of
testing graph properties, characterization by
such invaraint local conditions is closely
related to proximity oblivious testing
(as defined by Goldreich and Ron, STOC'09).
In contrast to this relation, we show that, in general,
characterization by invaraint local conditions
is neither necessary nor sufficient
for proximity oblivious testing.
Furthermore, we show that easy testability is {\em not}\/ guaranteed
even when the property is characterized by local conditions
that are invariant under a 1-transitive group of permutations.