We initiate a study of learning and testing dynamic environments,
focusing on environment that evolve according to a fixed local rule.
The (proper) learning task consists of obtaining the initial configuration
of the environment, whereas for non-proper learning it suffices to predict
its future values. The testing task consists of checking whether
the environment has indeed evolved from some initial configuration
according to the known evolution rule.
We focus on the temporal aspect of these computational problems,
which is reflected in the requirement that only a small portion
of the environment is inspected in each time slot
(i.e., the time period between two consecutive applications
of the evolution rule).

We present some general observations, an extensive study of two special cases,
two separation results, and a host of open problems.
The two special cases that we study refer to linear rules of evolution
and to rules of evolution that represent simple movement of objects.
Specifically, we show that evolution according to any linear rule
can be tested within a total number of queries that is sublinear
in the size of the environment, and that evolution according to
a simple one-dimensional movement can be tested within a total number
of queries that is independent of the size of the environment.

correcting a few typos etc

We initiate a study of learning and testing dynamic environments,
focusing on environment that evolve according to a fixed local rule.
The (proper) learning task consists of obtaining the initial configuration
of the environment, whereas for non-proper learning it suffices to predict
its future values. The testing task consists of checking whether
the environment has indeed evolved from some initial configuration
according to the known evolution rule.
We focus on the temporal aspect of these computational problems,
which is reflected in the requirement that only a small portion
of the environment is inspected in each time slot
(i.e., the time period between two consecutive applications
of the evolution rule).

We present some general observations, an extensive study of two special cases,
two separation results, and a host of open problems.
The two special cases that we study refer to linear rules of evolution
and to rules of evolution that represent simple movement of objects.
Specifically, we show that evolution according to any linear rule
can be tested within a total number of queries that is sublinear
in the size of the environment, and that evolution according to
a simple one-dimensional movement can be tested within a total number
of queries that is independent of the size of the environment.

We added a new section (i.e., Sec 1.3) of proof overviews,
corrected the proof of Thm 2.2 and added details to the proof of Thm 7.3.

We initiate a study of learning and testing dynamic environments,
focusing on environment that evolve according to a fixed local rule.
The (proper) learning task consists of obtaining the initial configuration
of the environment, whereas for non-proper learning it suffices to predict
its future values. The testing task consists of checking whether
the environment has indeed evolved from some initial configuration
according to the known evolution rule.
We focus on the temporal aspect of these computational problems,
which is reflected in the requirement that only a small portion
of the environment is inspected in each time slot
(i.e., the time period between two consecutive applications
of the evolution rule).

We present some general observations, an extensive study of two special cases,
two separation results, and a host of open problems.
The two special cases that we study refer to linear rules of evolution
and to rules of evolution that represent simple movement of objects.
Specifically, we show that evolution according to any linear rule
can be tested within a total number of queries that is sublinear
in the size of the environment, and that evolution according to
a simple one-dimensional movement can be tested within a total number
of queries that is independent of the size of the environment.

This version contains an additional section (current Section 7)
that studies the testing of evironments of moving objects when
the number of objects s relatively very small (i.e., the ``sparse case'').

We initiate a study of learning and testing dynamic environments,
focusing on environment that evolve according to a fixed local rule.
The (proper) learning task consists of obtaining the initial configuration
of the environment, whereas for non-proper learning it suffices to predict
its future values. The testing task consists of checking whether
the environment has indeed evolved from some initial configuration
according to the known evolution rule.
We focus on the temporal aspect of these computational problems,
which is reflected in the requirement that only a small portion
of the environment is inspected in each time slot
(i.e., the time period between two consecutive applications
of the evolution rule).

We present some general observations, an extensive study of two special cases,
two separation results, and a host of open problems.
The two special cases that we study refer to linear rules of evolution
and to rules of evolution that represent simple movement of objects.
Specifically, we show that evolution according to any linear rule
can be tested within a total number of queries that is sublinear
in the size of the environment, and that evolution according to
a simple one-dimensional movement can be tested within a total number
of queries that is independent of the size of the environment.