Knowledge space theory (KST) provides a useful and efficient mathematical psychological framework for assessing individuals’ knowledge and facilitating further learning (Doignon & Falmagne, 1985). In KST, the knowledge of an individual is represented by his/her so-called knowledge state K, that is the set of all the items q in a knowledge domain Q he or she proved to master. The collection K of the states that can be observed in the population, which contains at least ∅ and the knowledge domain Q, is the knowledge structure (Doignon and Falmagne, 1985, Falmagne et al., 2006). Knowledge states and knowledge structures are the fundamental concepts in KST, and the construction of these states and structures constitutes a critically important problem. Several methods exist for constructing knowledge structures, such as query-routine (Koppen, 1994, Koppen and Doignon, 1990) and the ps-query routine (Cosyn & Thiery, 2000), which are designed to build knowledge structures and knowledge states through interviews with experienced experts. Schrepp (2003) describes a method of explorative data analysis, which allows for the identification of logical implications between items in a dichotomous questionnaire or test. There exist also methodologies to extract skill maps from data collections (Spoto et al., 2016). Additionally, an effective method to construct knowledge structures involves the analysis of item-skill relationships. Skill s represents a method, a strategy or an ability for a subject to solve a set of items. Given a domain S of skills, a particular subset T of S represents an individual’s mastery of skills (Anselmi et al., 2017, Doignon, 1994, Falmagne et al., 1990, Heller et al., 2017, Heller et al., 2013). The skill assignment is represented by the so-called skill map, which depicts the relationship between items and skills. Skills in KST enable the analysis of behavior’s interpretation. However, for a given skill s, different items may require varying proficiency levels of s for successful resolution, and individuals may exhibit different proficiency in the same skill. All these factors can significantly influence the solution of items within the domain Q. Thus, it is natural to introduce the concept of fuzzy skill pairs (s,l), where l∈[0,1] represents the proficiency level of skill s. Specifically, when skill proficiency is limited to only 0 or 1, fuzzy skills become classical skills. This implies that skills are special cases of fuzzy skills. Similar to skill maps, fuzzy skill assignments are represented by fuzzy skill maps (Sun et al., 2021). Additionally, some knowledge structures can be delineated via conjunctive and disjunctive models by fuzzy skill maps.

Forward-graded (FG) structures and backward-graded (BG) structures have garnered significant attention in recent years (Spoto and Stefanutti, 2020, Spoto et al., 2012). Moreover, any learning space (Falmagne & Doignon, 2011) is FG in at least one item, while the complement of any learning space is a BG structure (Spoto et al., 2013). Furthermore, Basic Local Independence Models (BLIM) applied to FG or BG structures are unidentifiable (Heller, 2017, Spoto et al., 2012). Therefore, Forward-graded (FG) and backward-graded (BG) structures hold particular importance in KST. For a skill map (Q,S,τ), if a skill s∈S satisfies s∈τ(q) and s∉τ(p) for all p∈Q∖, s is called a specific skill for an item q∈Q. This type of skill is crucial in defining FG and BG knowledge structures (Spoto et al., 2012). Spoto and Stefanutti (2020) established necessary and sufficient conditions for delineating FG and BG knowledge structures from skill maps.

Building upon some of the theoretical results in previous articles, we aim to establish the connection between FG (or BG) knowledge structures and fuzzy skill maps. Spoto and Stefanutti (2020) found that the closure space (respectively, knowledge space) delineated by the skill map is backward (respectively, forward) graded in an item q∈Q if and only if S contains a specific skill for q. This implies that the presence of certain specific skills indicates the forward-gradedness or backward-gradedness of the delineated knowledge structure in item q. However, the situation is different in fuzzy skill maps. Specific skills cannot be used in fuzzy skill maps. In place of specific skills, we incorporate specific upper and lower fuzzy skills, which are essential for delineating BG and FG knowledge structures through the use of fuzzy skill maps. We find that a fuzzy skill map with specific upper fuzzy skills in item q is a necessary and sufficient condition for having the BG simple closure space in q. At the same time, a fuzzy skill map with specific lower fuzzy skills in item q is a necessary and sufficient condition for having the FG knowledge space in q. In particular, when the proficiency of skills takes value only in 0 or 1, specific skills in skill maps are both specific upper and lower fuzzy skills in fuzzy skill maps. In this way, it can be seen that Theorem 1 in Spoto and Stefanutti (2020) is a special case of Theorem 1 and Theorem 2 in our paper. On the other hand, Spoto and Stefanutti (2020) established the necessary and sufficient conditions for forward-graded closure spaces and backward-graded knowledge spaces. However, in fuzzy skill maps, for the similar conditions of Theorem 2 in Spoto and Stefanutti (2020), we find only conjunctive models can delineate forward-graded closure spaces, while disjunctive models do not necessarily delineate backward-graded knowledge spaces (See Example 4). The BLIM model applied to FG or BG structures presents unidentifiability issues, and the characteristics of the skill maps delineating FG and BG structures was given in Spoto and Stefanutti (2020). Since fuzzy skill maps generalize classical skill maps, as an application, the corresponding probabilistic framework for fuzzy skills is introduced and the unidentifiability of CBLIM with fuzzy skills is discussed at last.

The paper is organized as follows. Section 2 provides a brief overview of knowledge space theory and fuzzy set theory. Basic concepts about fuzzy skill maps are introduced. In Section 3, we explore the necessary and sufficient conditions for delineating FG and BG knowledge structures from fuzzy skill maps and we also discuss their relation to the theory presented in Spoto and Stefanutti (2020). Section 4 introduces the probabilistic framework to fuzzy skill maps and discusses the unidentifiability of CBLIM with fuzzy skills. Finally, in Section 5, we provide a summary of the main results of our study.