Defining communication as the output of two complementary informative processes describes communication in an ideal noiseless environment where the second function is a perfect inverse of the other. Here, the second function perfectly ``undoes" what the first function ``does" and there is no noise introduced into the system.
In the real world, however, there is almost always noise present and communication is imperfect, except in the presence of error correcting codes that allow the recipient of a message other than the message that was sent to recover the original message [Ham86,Los90].
We refer to the output of a noisy communication channel as noise modified communication. Communications received that have been noise modified but can be understood accurately or used by a human, even given the noise induced changes in it, are acceptable noisy communications. These errors or noise may be induced by external processes and events, causing the informative processes to function differently than they would without the error producing process. Noise or errors in the information may be viewed as additive if the operations of each process are independent of the operations of the other processes and these processes are in series.
In many cases, the inversion provided by a process f or f-1 is imperfect, that is, for some x, it will be the case that For example, consider the case where the author who grew up in the northern United States refers to snow when talking with his daughter, who has spent her life in the southern half of the country. The author's hierarchy encodes a certain set of meanings as snow while his daughter decodes the term snow as a rather different phenomenon. Here the daughter's receiving function isn't the exact inverse of her father's encoding function.
The acceptability of noise-modified or imperfectly inverted communication may be best viewed as either binary or continuous. The exact nature of the acceptability function is an empirical question to be determined for any given communication channel. Binary acceptability occurs when a message is either acceptable or not acceptable. Acceptability may thus occur when the message is similar to at least a certain degree to an expected message. Acceptability may also be continuously valued, that is, the acceptability itself can take any of a range of values from, for example, 0 to 1, depending on how close the acceptability is to the best-case acceptability and the worst-case acceptability.