John Goldsmith (1995) defines phonological rules as mappings between two different levels of sounds representation[1]—in this case, the abstract or underlying level and the surface level—and Bruce Hayes (2009) describes them as "generalizations" about the different ways a sound can be pronounced in different environments.
[2] That is to say, phonological rules describe how a speaker goes from the abstract representation stored in their brain, to the actual sound they articulate when they speak.
For example, the English plural written -s may be pronounced as [s] (in "cats"), [z] (in "cabs", "peas"), or as [əz] (in "buses"); these forms are all theorized to be stored mentally as the same -s, but the surface pronunciations are derived through a series of phonological rules.
Phonological rules are often written using distinctive features, which are (supposedly[note 3]) natural characteristics that describe the acoustic and articulatory makeup of a sound; by selecting a particular bundle, or "matrix," of features, it is possible to represent a group of sounds that form a natural class and pattern together in phonological rules.
For a formal analysis, it is often required to implement notation conventions in addition to those previously introduced to account for the variety of changes that occur as simply as possible.