Homomorphisms between problem spaces

In procedural knowledge space theory (PKST), a “problem space” is a formal representation of the knowledge that is needed for solving all of the problems of a certain type. The competence state of a real problem solver is a subset of the problem space which satisfies a specific condition, named the “sub-path assumption”. There could exist specific “symmetries” in a problem space that make certain parts of it “equivalent” up to those symmetries. Whenever an equivalence relation is introduced for elements in a problem space, the question almost naturally arises whether the collection of the induced equivalence classes forms, itself, a problem space. This is the main question addressed in the present article, which is restated as the problem of defining a homomorphism of one problem space into another problem space. Two types of homomorphisms are examined, which are named the “strong” and the “weak homomorphism”. The former corresponds to the usual notion of “operation preserving mapping”. The latter preserves operations in only one direction. Two algorithms are developed for testing the existence of homomorphisms between problem spaces. The notions and algorithms are illustrated in a series of three examples in which quite well-known neuro-psychological and cognitive tests are employed.