Diagnosing cognitive errors possessed by examinees can be considered as a pattern classification problem that is designed to classify a sequential input of stimuli into one of several predetermined groups. The sequential inputs in this paper's context are item responses, and the predetermined groups are various states of knowledge resulting from misconceptions or different degrees of incomplete knowledge in a domain. In this study, the foundations of a combinatorial algorithm that will provide the universal set of states of knowledge will be introduced. Each state of knowledge is represented by a list of "can/cannot" cognitive tasks and processes (cognitively relevant attributes or latent variables) that are usually unobservable. A Boolean descriptive function is introduced as a mapping between the attribute space spanned by latent attribute variables and the item response space spanned by the item score variables. This function uncovers the unobservable content of a "black box." Once all possible classes are retrieved explicitly and expressed by a set of ideal item response patterns described by a "can/cannot" list of latent attributes, the notion of bug distributions and statistical pattern classification techniques will enable the accurate diagnosis of students' states of knowledge. Moreover, investigations on algebraic properties of these logically-derived-ideal-response patterns will provide insight into the structures of the test and dataset. There are 11 references and three illustrative tables. (Author/SLD)