. Function composition operator: 
.
. A projection (p. 
).
. The inverse of , i.e. the set of states that map
to a given state under (p. ).
. (For a simple projection 
).
A generalized inverse. Since only maps certain components of
S' to S, 
is s' with those components replaced by
their corresponding components in s (p. ).
. For a simple reduction from an
environment E' to E, the function mapping an action a from E
to the action that implements it in E'
(p. ).
. The chain-environment of n states (p. ).
E. An environment.
. G a goal of E and E' uniformly
reducible to E. The existential goal of G in E': the set of
all E'-states that map to a goal state under some binding (p. ).
. The serial product. The Cartesian product
of 
and 
in which actions from the two environments must be
taken separately (p. ).
. The parallel product. The Cartesian product
of and in which actions from the two environments must be
taken simultaneously (p. ).
. (E' an environment uniformly reducible to E) The
leftmost-ready binding map from E' to E (p. ).
p. A policy.
. The standard policy for single-material environment
E and goal G (p. ).
. The singleton environment (the environment with
exactly one state). Used to represent a self-resetting tool (p. ).