Definition: Information is available when at least one agent in the economy has it. (This is stronger than the ‘existence in principle’ of controlability.)
Definition: Sound information J(s) is a list of both what one does to maintain s and what one might do to change s into 1 - s, using true cause and effect relations. Thus J(s) = BI(s, s)
BI(s, 1-s) | truth. Denote an arbritrary belief as J’(s) - that however will not be sound since it would not be necessarily true.
Remark: True information is sound when the information concerning {1, 1} and {1, 0} is joined, or if the information on {0, 0} and {0, 1} is joined. One may e.g. know how to burn or not to burn a match, but not how to restore cinders into a match again (except for restarting the universe, but that is not likely controlable). Let 1 stand for match, and 0 for cinders. Then J(1) exists, but J(0) doesn’t (only partly, to maintain cinders as they are). Using sound information rather than basic information has analytical advantage. A Roman emperor may think that he maintains his good fortune by sacrificing to the gods. We rather discuss cases where governments deliberately abstain from wrong policies.
Remark: Consider the list {“If you happen to flip back to 0, use BI(0, 1) to go back to 1”}. Can we classify this as BI(1, 1) ? We could allow this if the cost of the temporary flip is low. For example, riding a bicycle requires continual readjustment of equilibrium. We can define BI(s, s) = {chance(s, 1-s)}
BI(1-s, s) | truth, as implied control information. But since this does not give BI(s,1-s), the implied control information does not give sound information. Stated differently, we are interested in durable states s, and not in flipping states. If we observe s then we want this to be caused by deliberate rejection of the use of BI(s, 1-s). We also regard cases in which implied control would be costly.
Definition: The tuple (J(1), J(0), s) is the state of a sound system. Note: Though the information is denoted as a function of s, information in a controlable state is the prime cause and s the prime effect.
Definition: Information is called compact iff J(0)