Competence-Based Skill Functions and Minimal Sets of Skills

Abstract

As we know, there is some relationship, such as precedence relation, among skills. Each precedence relation induces a competence structure. Thus, we study competence-based skill functions, which rely on competence structures and go from somethings observable to somethings invisible. Conversely, competence-based problem functions go from somethings invisible to somethings observable. In fact, these two dual types of functions based on competence structures are symmetry. Remarkably, there are two kinds of special competence-based skill functions: one is disjunctive, while the other is conjunctive. The former delineates knowledge spaces, which are symmetrical to simple closure spaces delineated by the latter. Based on these facts, we shows some theoretical results on competence-based skill functions, then design the corresponding algorithms for delineating knowledge structures. Sometimes for competence-based skill functions, some skills are maybe reducible. Thus, we discuss what kind of skills are reducible and obtain sufficient and some necessary conditions for skills being reducible for competence-based skill functions. Based on this, we design algorithms to reduce reducible skills and get minimal sets of skills. By comparison, for competence-based skill functions, we can find minimal sets of skills with the smallest cardinality whenever sets of skills are finite. For each algorithm, we take a corresponding example to illustrate the detailed procedure.

1. Introduction and Preliminaries

Knowledge space theory (KST) was a new idea recognized by behaviourist from the 1980s. It mainly concerned with the solution behavior to a given set of problems or items in a specific knowledge domain. The framework of KST can be integrated into psychological theory by bringing into the picture something invisible, such as skills, competencies, … responsible for something observable. This kind of development may be seen somewhat analogous to traditional mental testing [1]. This kind of stimulus-response were proved successful in applications. KST is arresting and has been applied so well in educational environment by the assumptions that there is a curriculum scheduled the covered content, and that there is a distinctive and complete definition of the knowledge domain. Based on KST, ALEKS Corporation built in 1990s has been exploiting and promoting educational services. Note that its initial products were assessment on Internet-delivered and tutoring services. Following Marshall [2] and others (Falmagne et al. [3], Albert et al. [4], Lukas and Albert [5]), we assume sets of ‘skills’ exist. Skills consists of approaches, algorithms, or strategies. A competence is a subset of skills required to solve a given item. A competence state refers to a subset of skills that an individual maters. The performance level refers to subsets of problems that individuals master, which is observable.

Note that answers to items in the specified knowledge domain reflect latent cognitive abilities of individuals, which can be represented by skills and thereby operating on the competence level. Or, more accurately, skills are understood as a theoretical construction accounting for the performance, and are applied to explain individuals’ knowledge states. Falmagne et al. [3] was the firstly to propose a method to linking somethings observable to somethings invisible in the process of learning by mappings from sets of items to subsets of skills relevant for mastering them. Afterwards, fruitful independent investigations come out to generalize this framework, refer to [6,7,8,9,10].

Doignon [6] introduced the concept of skill map into KST to generate knowledge structures from the view of underlying ability. Afterwards, Duntsch and Gediga [7,8] proposed the notion of skill function, which links observable solution behaviors to some invisible cognitive constructions [7,8,11,12] by assigning skills to items.

Symmetrically, problem functions also play important roles in linking something invisible to something observable by mappings from sets of items to collections of competence states. The way to delineate knowledge structure via a skill function focuses on identifying underlying skills and operates on the competence level [9,10,13,14]. For info more about the problem function, refer to [9,13,14].

Korossy [9,10] developed an extension of KST by taking competence-performance into consideration. In recent years, the generalization of KST based on competence was substantially developed for assessing individuals’ knowledge and guiding for further learning [12,15,16,17,18,19]. Competence structure is one of the central concepts in competence-based extension of KST. Each element in a competence structure is a subset of skills that may occur. In many cases, there exists some precedence relation among skills from the point of logical or pedagogical dependencies, which provide reasons for rational analysis. A precedence relation can induce a competence structure.

We recall that every skill function relying on competence structure establish a relationship between items and competence states, where each competency assigned is sufficient to solve the item. Such skill functions can also be regarded as the extension of general skill functions. Heller [11] introduced two kinds of special skill functions: disjunctive skill functions and conjunctive skill functions. The former delineates knowledge space, while the latter delineate simple closure spaces. They are dual to each other and also beautiful in a sense of symmetry.

Notice that Duntsch [7] illustrated a method for produce knowledge structures by skill functions through listing all subsets of skills. Analogous methods can be found in [1,8,11]. However, for skill functions, there are difficulties in delineating knowledge structures by listing all competence states when the number of competencies is too large. We discuss some theoretical results on competence-based skill functions and the two kinds of special cases: the disjunctive one and the conjunctive one, and design the corresponding algorithms for delineating the knowledge structures by avoiding listing all competence states.

It is worth noting that our algorithm for disjunctive competence-based skill functions is only concerned with atoms of competence spaces. That is to say, we skip over competence states that are not atoms, which is an improvement compared to the methods in [6,7,8]. Dually, based on the work of Doignon [6,7,8,11,12], as well as the concept of the intersection generation group introduced by Sun et al. [20], we propose the algorithm for conjunctive competence-based skill functions to be related to elements in certain minimal intersection generation groups of simple closure spaces. Throughout the procedure of this algorithm, to delineate knowledge structures, it is unnecessary to list competence states that are not in intersection generation groups. Besides, our algorithm can delineate knowledge structures via competence-based skill functions by finding competencies and weak competencies, whose cardinality is smaller than that of all competence states in general. In short, our methods are an improvement compared to the corresponding existing ones.

Given that for any skill function, maybe we can re-write knowledge states induced by some competence states by ones delineated by other competence states. If reducible skills are moved, the relation between items and competence states can be presented without the knowledge structure being changed. This idea can also be found out in rough set theory [21,22]. Analogously, for disjunctive competence-based skill functions, we develop an algorithm for finding its minimal sets of skills. The most important point is to remove union-reducible elements. This algorithm relies on corresponding competencies and weak competencies.

The rest parts of this article are constructed as follows. Section 2 outlines fundamental concepts in KST and competence-based KST. In Section 3 and Section 4, we mainly discuss some theoretical results and the corresponding algorithms based on two kinds of special competence-based skill functions. Section 5 discusses some theoretical results for common competence-based skill functions and illustrates an approach for delineating knowledge structures via competence-based skill functions. In Section 6, we investigate reducible skills and design algorithms for finding reducible skills. For our algorithms, we take the corresponding examples to illustrate the detailed procedures. Finally, the main results of the research are summarized in the last section.

2. An Overview of KST and Competence-Based KST

Let $Q,S$ be two sets, where Q consists of items and S consists of skills relevant to solving these items. A knowledge state refers to a subset of Q that an individual can solve. A pair $(Q,\mathcal{K})$ is said to be a knowledge structure if $\mathcal{K}$ is a collection of knowledge states with $⌀,Q\in \mathcal{K}$. Similarly, the competence state refers to a subset of S that an individual has. A pair $(S,\mathcal{C})$ is said to be a competence structure if $\mathcal{C}$ is a collection of competence states with $⌀,S\in \mathcal{C}$. If there is no confusion, we sometimes write $\mathcal{K}$ instead of $(Q,\mathcal{K})$ and $\mathcal{C}$ instead of $(S,\mathcal{C})$.

Note that we say a collection $\mathcal{A}$ is union-closed if for any ${\mathcal{A}}^{\prime }\subseteq \mathcal{A}$, the union $\bigcup {\mathcal{A}}^{\prime }\in \mathcal{A}$. Analogously, we say a collection $\mathcal{A}$ is intersection-closed if for any ${\mathcal{A}}^{\prime }\subseteq \mathcal{A}$, the intersection $\bigcap {\mathcal{A}}^{\prime }\in \mathcal{A}$.

Let $\mathcal{K}$ be a collection of knowledge states on Q. We say $(Q,\mathcal{K})$ is a knowledge space if $(Q,\mathcal{K})$ is a union-closed knowledge structure. Moreover, We say $(Q,\mathcal{K})$ is a closure space if $(Q,\mathcal{K})$ is intersection-closed and $Q\in \mathcal{K}$. A simple closure space refers to a closure space with $⌀\in \mathcal{K}$. Alternatively, simple closure space is an intersection-closed knowledge structure.

For a collection $\mathcal{G}$, the collection $\{\bigcup \mathcal{H}\mid \mathcal{H}\subseteq \mathcal{G}\}$ is said to be the span of $\mathcal{G}$, denoted by $\mathbb{S}\left(\mathcal{G}\right)$. Alternatively, we say $\mathcal{G}$ spans $\mathbb{S}\left(\mathcal{G}\right)$. Let $(Q,\mathcal{K})$ be a knowledge structure. An element $A\in \mathcal{K}$ is called an atom if there exists $q\in Q$ such that A is a minimal element containing q. This is equivalent to there exists $q\in Q$ such that $q\in A$ and for any $K\in \mathcal{K}$, $q\in K$ implies $A\subseteq K$.

Symmetrically, Sun et al. [20] introduced the concept of an intersection generation group. For a collection $\mathcal{G}$, a subcollection $\mathcal{H}\subseteq \mathcal{G}$ is said to be an intersection generation group of $\mathcal{G}$ if every $G\in \mathcal{G}$, there exists a subcollection ${\mathcal{H}}^{\prime }\subseteq \mathcal{H}$ satisfying $K=\bigcap {\mathcal{H}}^{\prime }$. In that case, write $\mathbb{I}\left(\mathcal{H}\right)=\mathcal{G}$.

The concepts of competence structure and competence space are analogous to the concepts of knowledge structure and knowledge space, respectively. Moreover, other concepts in competence structures or competence spaces are also analogous to those in knowledge structures or knowledge spaces.

For any set X, we write $\mathcal{P}\left(X\right)=\{A:A\subseteq X\}$ and ${\mathcal{P}}^{*}\left(X\right)=\mathcal{P}\left(X\right)\backslash \{⌀\}$.

A skill function refers a triple $(Q,S,\mu )$, in which $\mu$ is a mapping from Q to ${\mathcal{P}}^{*}\left({\mathcal{P}}^{*}\left(S\right)\right)$ such that for any $q\in Q$, elements in $\mu \left(q\right)$ are pairwise incomparable. For any $q\in Q$, each element in $\mu \left(q\right)\in {\mathcal{P}}^{*}\left(S\right)$ and can be considered as an approach to solve q, called a competency for solving q or a competency in $(Q,S,\mu )$.

For a skill function $(Q,S,\mu )$ and $T\subseteq S$. We say a subset $K\subseteq Q$ is the knowledge state delineated byT via $(Q,S,\mu )$ if $K=\{q\in Q\mid C\subseteq T\mathrm{for}\mathrm{some}C\in \mu (q\left)\right\}$. In particular, we call a skill function $(Q,S,\mu )$ is disjunctive if $C\in \mu \left(q\right)$ are singletons for all $q\in Q$. Dually, we call a skill function $(Q,S,\mu )$ is conjunctive if for any $q\in Q$, there exists $C\subseteq S$ such that $\mu \left(q\right)=\left\{C\right\}$.

Symmetrically, a problem function refers to a triple $(Q,S,p_{\mu })$, in which $p_{\mu }$ is a mapping from $\mathcal{P}\left(S\right)$ to $\mathcal{P}\left(Q\right)$. Thereby, knowledge states $K\subseteq Q$ can be specified by $p_{\mu }\left(T\right)=\{q\in Q\mid C\subseteq T\mathrm{for}\mathrm{some}C\in \mu \left(q\right)\}$ for a certain $T\subseteq S$.

For more about KST and competence-based KST, refer to [1,3,6,9,10,13,14,23,24,25].

If not emphasized, $Q,S$ are not confined to finite sets. Note that in this article, all theoretical results can be applied for arbitrary sets $Q,S$. However, given the operability of computers, all algorithms can be realized for finite $Q,S$.

3. Knowledge Structures Delineated by Disjunctive Competence-Based Skill Functions

Korossy [9,10] introduced and investigated the competence-performance approach as a skill-based extension of KST. Korossy assumed that there are two distinguished levels in solving items: the performance one and the competence one. The former refers to observable behaviors and can be presented by knowledge structures, while the latter refers to theoretical abilities used to explain observable behaviors and is characterized by competence structures.

Note that skill functions go from the something observable to something invisable and problem functions go the other way around. In other words, the two kinds of functions are symmetrical. It is worth noting that skill-based extensions of KST provide extra explanatory power if the solution behavior on a specific knowledge domain can be forecasted by a finite set of skills.

Definition 1.

A competence-based skill function on a competence structure $(S,\mathcal{C})$ is a triple $(Q,S,{\mu }_{\mathcal{C}})$, in which ${\mu }_{\mathcal{C}}$ is a mapping from Q to ${\mathcal{P}}^{*}\left(\mathcal{C}\right)$ such that elements in each ${\mu }_{\mathcal{C}}\left(q\right)$ are pairwise incomparable. Write

$${\mathcal{A}}_{{\mu }_{\mathcal{C}}}=\{C\in \mathcal{C}\mid C\in p_{{\mu }_{\mathcal{C}}}\left(q\right)forsomeq\in Q\}.$$

Each element in ${\mathcal{A}}_{{\mu }_{\mathcal{C}}}$ is said to be a competence in $(Q,S,{\mu }_{\mathcal{C}})$.

Definition 2.

Let $(Q,S,{\mu }_{\mathcal{C}})$ a competence-based skill function. For any $T\in \mathcal{C}$, the set

$$p_{{\mu }_{\mathcal{C}}}\left(T\right)=\{q\in Q\mid C\subseteq T\mathrm{for}\mathrm{some}C\in {\mu }_{\mathcal{C}}\left(q\right)\}$$

is said to be the knowledge state delineated by $T\in \mathcal{C}$ via $(Q,S,{\mu }_{\mathcal{C}})$.

For a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$, we write

$${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{C}\}.$$

Sometimes, a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ is briefly denoted by ${\mu }_{\mathcal{C}}$.

It is an easy observation that the empty subset ⌀ of Q is delineated by the element $⌀\in \mathcal{C}$, and Q is delineated by the element $S\in \mathcal{C}$. Thus, both ⌀ and Q in ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, implying $(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})$ is a knowledge structure.

Definition 3.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a competence-based skill function. A competence-based problem function induced by $(Q,S,{\mu }_{\mathcal{C}})$ is a triple $(Q,S,p_{{\mu }_{\mathcal{C}}})$, where $p_{{\mu }_{\mathcal{C}}}$ is a mapping from $\mathcal{C}$ to $\mathcal{P}\left(Q\right)$ such that

$$p_{{\mu }_{\mathcal{C}}}\left(T\right)=\{q\in Q\mid C\subseteq T\mathrm{for}\mathrm{some}C\in {\mu }_{\mathcal{C}}\left(q\right)\}$$

for any $T\in \mathcal{C}$.

Sometimes, the competence-based problem function $(Q,S,p_{{\mu }_{\mathcal{C}}})$ is briefly denoted by $p_{{\mu }_{\mathcal{C}}}$. It is an easy observation that $p_{{\mu }_{\mathcal{C}}}$ is monotonic, moreover, $p_{{\mu }_{\mathcal{C}}}(⌀)=⌀$ and $p_{{\mu }_{\mathcal{C}}}\left(S\right)=Q$. A competence-based problem function $(Q,S,p_{{\mu }_{\mathcal{C}}})$ can be considered as an extension of the problem function $(Q,S,p_{\mu })$ by setting the domain of $p_{\mu }$ to $\mathcal{C}$.

We come to investigate the approach to produce knowledge structures by disjunctive competence-based skill functions.

Definition 4.

For any competence space $(S,\mathcal{C})$, a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$ is said to be a disjunctive if for any $q\in Q$, each element in $\mu \left(q\right)$ is an atom of $\mathcal{C}$.

A disjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ can be conceived as the generalization of a disjunctive skill function $(Q,S,\mu )$ by setting the range to ${\mathcal{P}}^{*}\left(\mathcal{C}\right)$.

Lemma 1.

Let $(S,\mathcal{C})$ be a competence space. Then, a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$ is disjunctive if and only if the competence-based problem function $p_{{\mu }_{\mathcal{C}}}$ induced by $(Q,S,{\mu }_{\mathcal{C}})$ is union-preserving, i.e.,

$$p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{A})={\bigcup }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right)$$

for any subcollection $\mathcal{A}\subseteq \mathcal{C}$.

Proof. 

Assume that $(Q,S,{\mu }_{\mathcal{C}})$ is disjunctive. For any $\mathcal{A}\subseteq \mathcal{C}$, we have

$$p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{A})=\{q\in Q\mid C\subseteq \bigcup \mathcal{A}\mathrm{for}\mathrm{some}C\in {\mu }_{\mathcal{C}}\left(q\right)\}.$$

Note that $p_{{\mu }_{\mathcal{C}}}$ is monotone, then $p_{{\mu }_{\mathcal{C}}}\left(A\right)\subseteq p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{A})$ for every $A\in \mathcal{A}$. It follows that

$${\bigcup }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right)\subseteq p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{A}).$$

Conversely, if $q\in p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{A})$, then $C\subseteq \bigcup \mathcal{A}$ for some $C\in {\mu }_{\mathcal{C}}\left(q\right)$. Note that C is an atom in $\mathcal{C}$, then there exists $s\in C$ such that $C\subseteq B$ for every $B\in \mathcal{C}$ containing s. Also note that $s\in A_s$ for some $A_s\in \mathcal{A}$. Thus, $C\subseteq A_s$, implying $q\in p_{{\mu }_{\mathcal{C}}}\left(A_s\right)$. Hence, $q\in {\bigcup }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right)$.

Now, we assume that $p_{{\mu }_{\mathcal{C}}}$ is union-preserving. If $(Q,S,{\mu }_{\mathcal{C}})$ is not disjunctive, then there exists $q\in Q$ and $C\in {\mu }_{\mathcal{C}}\left(q\right)$ such that C is not an atom in $\mathcal{C}$. Then, for any $s\in C$, there exists $A_s\in \mathcal{C}$ containing s such that $A_s⫋C$. Note that ${\bigcup }_{s\in C}{A}_s=C$. By the union-preserving of $p_{{\mu }_{\mathcal{C}}}$, we have $q\in p_{{\mu }_{\mathcal{C}}}\left(C\right)={\bigcup }_{s\in C}{p}_{{\mu }_{\mathcal{C}}}\left(A_s\right)$. Thus, $q\in p_{{\mu }_{\mathcal{C}}}\left(A_{s_0}\right)$ for some $s_0\in C$. Thus, we infer that there exists $C^{\prime }\in {\mu }_{\mathcal{C}}\left(q\right)$ such that $C^{\prime }\subseteq A_{s_0}$. It follows that $C^{\prime }⫋C$, which contradict the incomparability of elements in ${\mu }_{\mathcal{C}}$. □

Theorem 1.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a disjunctive competence-based skill function on the competence space $(S,\mathcal{C})$ and $(Q,S,p_{{\mu }_{\mathcal{C}}})$ the competence-based problem function induced by $(Q,S,{\mu }_{\mathcal{C}})$. Then

$$(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})=\mathbb{S}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}\right).$$

Proof. 

For any $K\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, consider the corresponding $T\in \mathcal{C}$ such that $K=p_{{\mu }_{\mathcal{C}}}\left(T\right)$. Let

$$\mathcal{G}=\{C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\mid C\subseteq T\}\mathrm{and}{T}^{\prime }=\bigcup \mathcal{G}.$$

Obviously, $\mathcal{G}\subseteq {\mathcal{A}}_{{\mu }_{\mathcal{C}}}$ and $T^{\prime }\subseteq T$. By the monotonicity of $p_{{\mu }_{\mathcal{C}}}$, we have $p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)\subseteq K$. For any $q\in K$, by the choice of T, there exists $C_q\in {\mu }_{\mathcal{C}}\left(q\right)\subseteq {\mathcal{A}}_{{\mu }_{\mathcal{C}}}$ such that $C_q\subseteq T$. Note that $C_q\subseteq T^{\prime }$, then $q\in p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)$. So $K=p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)$. By Lemma 1, the mapping $p_{{\mu }_{\mathcal{C}}}$ is union-preserving, hence

$$K=p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{G})=\bigcup \{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{G}\}.$$

Note that

$$\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{G}\}\subseteq \{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}.$$

Thus, $K\in \mathbb{S}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}\right)$. Conversely, for any $\mathcal{D}\subseteq {\mathcal{A}}_{{\mu }_{\mathcal{C}}}$, observe that $\mathcal{C}$ is union-closed, then $\bigcup \mathcal{D}\in \mathcal{C}$. It follows that $p_{{\mu }_{\mathcal{C}}}(\bigcup \mathcal{D})\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Therefore,

$$(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})=\mathbb{S}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}\right).$$

□

Theorem 1 implies that for any competence space $(S,\mathcal{C})$ and a disjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$, the pair $(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})$ is a knowledge space. Based on this fact, we give an algorithm for yielding knowledge structures by disjunctive competence-based skill functions $(Q,S,{\mu }_{\mathcal{C}})$, which is only concerned with the atoms of $(S,\mathcal{C})$.

The algorithm for delineating knowledge structures by disjunctive competence-based skill functions is summarized as follows.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a disjunctive competence-based skill function on the competence space $(S,\mathcal{C})$ and $(Q,S,p_{{\mu }_{\mathcal{C}}})$ the competence-based problem function induced by $(Q,S,{\mu }_{\mathcal{C}})$.

Step 1. 

List the set ${\mathcal{A}}_{{\mu }_{\mathcal{C}}}$ of all competencies in $(Q,S,{\mu }_{\mathcal{C}})$;

Step 2. 

Find out $p_{{\mu }_{\mathcal{C}}}\left(C\right)$ for each $C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}$;

Step 3. 

Compute all unions of subcollections of $\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}$;

Step 4. 

Set ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ be the set of all unions obtained in step 2, i.e., ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\mathbb{S}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}\right)$.

The following example illustrates the detailed procedure of the above algorithm.

Example 1.

Let $Q=\{q_1,q_2,q_3,q_4\}$ and $S=\{s_1,s_2,s_3,s_4\}$. Consider the competence structure $(S,\mathcal{C})$, where

$$\mathcal{C}=\{⌀,\left\{s_1\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_4\},\{s_3,s_4\},$$

$$\{s_1,s_2,s_3\},\{s_1,s_2,s_4\},\{s_1,s_3,s_4\},\{s_2,s_3,s_4\},S\}.$$

Consider the given disjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ defined as

$${\mu }_{\mathcal{C}}\left(q\right)=\left\{\begin{array}{cc}\{\left\{s_1\right\},\{s_2,s_4\}\},\hfill & q=q_1;\hfill \\ \left\{\{s_1,s_2\}\right\},\hfill & q=q_2;\hfill \\ \{\left\{s_3\right\},\{s_1,s_2\}\},\hfill & q=q_3;\hfill \\ \left\{\left\{s_3\right\}\right\},\hfill & q=q_4.\hfill \end{array}\right.$$

Note that

$${\mathcal{A}}_{{\mu }_{\mathcal{C}}}=\{\left\{s_1\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_2,s_4\}\}.$$

Then all knowledge states delineated by elements in ${\mathcal{A}}_{{\mu }_{\mathcal{C}}}$ are

$$p_{{\mu }_{\mathcal{C}}}\left(\left\{s_1\right\}\right)=\left\{q_1\right\},p_{{\mu }_{\mathcal{C}}}\left(\left\{s_3\right\}\right)=\{q_3,q_4\},p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_2\}\right)=\{q_1,q_2,q_3\},p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_4\}\right)=\left\{q_1\right\}.$$

By Theorem 1, ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ is the span of $\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\}.$ Eventually,

$${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{⌀,\left\{q_1\right\},\{q_3,q_4\},\{q_1,q_2,q_3\},\{q_1,q_3,q_4\},Q\}.$$

Remark 1.

Doignon [6] provided a method to delineate knowledge structures via disjunctive skill functions though listing all subsets of the skills. Similar methods can be found in [1,11]. Throughout the procedure of our algorithm to delineate knowledge spaces, we focus on atoms of competence spaces instead of all competence states. That is to say, we skip over competence states that are not atoms, which is some improvement compared to existing methods.

4. Knowledge Structures Delineated by Conjunctive Competence-Based Skill Functions

Note that in KST, the conjunctive model and disjunctive model are dual to each other. Thus, it makes some sense to investigate the competence-based conjunctive skill functions. Moreover, knowledge structures delineated by competence-based conjunctive skill functions are simple closure space, which are symmetrical to ones delineated by competence-based disjunctive skill functions.

Definition 5.

For any intersection-closed competence structure $(S,\mathcal{C})$, a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$ is said to be a conjunctive if for any $q\in Q$, there exists $C\in \mathcal{C}$ such that ${\mu }_{\mathcal{C}}\left(q\right)=\left\{C\right\}$.

A conjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ can be conceived as the generalization of a disjunctive skill function $(Q,S,\mu )$ on $(S,\mathcal{C})$.

Lemma 2.

Let $(S,\mathcal{C})$ be an intersection-closed competence structure. Then, a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$ is conjunctive if and only if the competence-based problem function $p_{{\mu }_{\mathcal{C}}}$ induced by $(Q,S,{\mu }_{\mathcal{C}})$ is intersection-preserving, i.e.,

$$p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{A})={\bigcap }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right).$$

for any subcollection $\mathcal{A}\subseteq \mathcal{C}$.

Proof. 

Assume that $(Q,S,{\mu }_{\mathcal{C}})$ is conjunctive. For any $\mathcal{A}\subseteq \mathcal{C}$, we have

$$p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{A})=\{q\in Q\mid C\subseteq \bigcap \mathcal{A}\mathrm{for}\mathrm{some}C\in {\mu }_{\mathcal{C}}\left(q\right)\}.$$

By the monotonicity of $p_{{\mu }_{\mathcal{C}}}$, we have $p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{A})\subseteq {\bigcap }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right)$. Conversely, let $q\in p_{{\mu }_{\mathcal{C}}}\left(A\right)$ for every $A\in \mathcal{A}$. Since $(Q,S,{\mu }_{\mathcal{C}})$ is conjunctive, there exists unique $C_q\in \mathcal{C}$ such that ${\mu }_{\mathcal{C}}\left(q\right)=\left\{C_q\right\}$. Then, infer $C_q\subseteq A$ for every $\bigcap \mathcal{A}$. It follows that $C_q\subseteq \bigcap \mathcal{A}$, implying $q\in p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{A})$. Hence, ${\bigcap }_{A\in \mathcal{A}}{p}_{{\mu }_{\mathcal{C}}}\left(A\right)\subseteq p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{A})$.

Conversely, we assume that $p_{{\mu }_{\mathcal{C}}}$ is intersection-preserving. If $(Q,S,{\mu }_{\mathcal{C}})$ is not conjunctive, then there exists $q\in Q$ such that ${\mu }_{\mathcal{C}}\left(q\right)$ containing at least two distinct nonempty sets in $\mathcal{C}$. Let $B_1,B_2\in \mathcal{C}\backslash \{⌀\}$ such that $B_1,B_2\in {\mu }_{\mathcal{C}}\left(q\right).$ Note that $q\in p_{{\mu }_{\mathcal{C}}}\left(B_1\right)$ and $q\in p_{{\mu }_{\mathcal{C}}}\left(B_2\right)$. By the hypothesis, we have $q\in p_{{\mu }_{\mathcal{C}}}\left(B_1\right)\cap p_{{\mu }_{\mathcal{C}}}\left(B_2\right)=p_{{\mu }_{\mathcal{C}}}(B_1\cap B_2)$. Thus, we infer that there exists $C\in {\mu }_{\mathcal{C}}\left(q\right)$ such that $C\subseteq B_1\cap B_2$. Since $B_1\ne B_2$, then either $B_1⫋B_2$ or $B_2⫋B_1$. If $B_1⫋B_2$, then there exists $s\in B_1$ such that $s\notin B_2$. Then $s\in B_1$ and $s\in C$. Otherwise, we infer $s\in B_2$ and $s\in C$. Both cases contradict the incomparability of ${\mu }_{\mathcal{C}}\left(q\right).$ □

Theorem 2.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a conjunctive competence-based skill function on the competence structure $(S,\mathcal{C})$. If $(S,\mathcal{C})$ has an intersection generation group ${\mathcal{I}}_{\mathcal{C}}$, then

$$(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})=\mathbb{I}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}\right).$$

Proof. 

For any $K\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, consider the corresponding $T\in \mathcal{C}$ such that $K=p_{{\mu }_{\mathcal{C}}}\left(T\right)$. By the hypothesis, there exists $\mathcal{J}\subseteq {\mathcal{I}}_{\mathcal{C}}$ such that $T=\bigcap \mathcal{J}$. By Lemma 2, the mapping $p_{{\mu }_{\mathcal{C}}}$ is intersection-preserving, hence

$$K=p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{J})=\bigcap \{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{J}\}.$$

Note that

$$\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{J}\}\subseteq \{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}.$$

Thus, $K\in \mathbb{I}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}\right)$. Conversely, for any $\mathcal{D}\subseteq {\mathcal{I}}_{\mathcal{C}}$, observe that $\bigcap \mathcal{D}\in \mathcal{C}$. It follows that $p_{{\mu }_{\mathcal{C}}}(\bigcap \mathcal{D})\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Therefore, $(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})=\mathbb{I}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}\right).$ □

Theorem 2 implies that for any intersection-closed competence structure $(S,\mathcal{C})$ and a conjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$, the pair $(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})$ is a simple closure space. Based on this fact, we provide an algorithm for delineating knowledge structures by conjunctive competence-based skill functions $(Q,S,{\mu }_{\mathcal{C}})$, which is related to the intersection generation group of $(S,\mathcal{C})$.

The procedure to delineate the knowledge structures by conjunctive competence-based skill functions is summarized as the following.

Let $(S,\mathcal{C})$ be a competence structure with an intersection generation group ${\mathcal{I}}_{\mathcal{C}}$ of $\mathcal{C}$, $(Q,S,{\mu }_{\mathcal{C}})$ a conjunctive competence-based skill function on $(S,\mathcal{C})$ and $(Q,S,p_{{\mu }_{\mathcal{C}}})$ the competence-based problem function induced by $(Q,S,{\mu }_{\mathcal{C}})$.

Step 1. 

List all elements in ${\mathcal{I}}_{\mathcal{C}}$;

Step 2. 

Find out $p_{{\mu }_{\mathcal{C}}}\left(C\right)$ each $C\in {\mathcal{I}}_{\mathcal{C}}$;

Step 3. 

Compute all intersections of subcollections of $\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}$;

Step 4. 

Set ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ be the set of all intersections obtained in step 2, i.e., ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\mathbb{I}\left(\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}\right)$.

The following example illustrates the detailed procedure of the above algorithm.

Example 2.

Let $Q=\{q_1,q_2,q_3,q_4\}$ and $S=\{s_1,s_2,s_3,s_4\}$. Consider the intersection-closed competence structure $(S,\mathcal{C})$, where

$$\mathcal{C}=\{⌀,\left\{s_1\right\},\left\{s_3\right\},\left\{s_4\right\},\{s_1,s_3\},\{s_1,s_4\},\{s_2,s_4\},\{s_3,s_4\},$$

$$\{s_1,s_2,s_4\},\{s_1,s_3,s_4\},\{s_2,s_3,s_4\},S\}.$$

Consider the conjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ defined as

$${\mu }_{\mathcal{C}}\left(q\right)=\left\{\begin{array}{cc}\left\{\{s_1,s_4\}\right\},\hfill & q=q_1;\hfill \\ \left\{\left\{s_3\right\}\right\},\hfill & q=q_2;\hfill \\ \left\{\left\{s_1\right\}\right\},\hfill & q=q_3;\hfill \\ \left\{\{s_2,s_3,s_4\}\right\}\hfill & q=q_4.\hfill \end{array}\right.$$

It is easy to check that the collection

$${\mathcal{I}}_{\mathcal{C}}=\{\{s_1,s_3\},\{s_1,s_2,s_4\},\{s_1,s_3,s_4\},\{s_2,s_3,s_4\},S\}$$

is the minimal intersection generation group of $\mathcal{C}$. All values of the corresponding competence-based problem function $p_{{\mu }_{\mathcal{C}}}$ at elements in ${\mathcal{I}}_{\mathcal{C}}$ are

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_3\}\right)=\{q_2,q_3\},p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_2,s_4\}\right)=\{q_1,q_3\},p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_3,s_4\}\right)=\{q_1,q_2,q_3\},$$

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_3,s_4\}\right)=\{q_2,q_4\},p_{{\mu }_{\mathcal{C}}}\left(S\right)=Q.$$

According to Theorem 2, $(Q,{\mathcal{K}}_{{\mu }_{\mathcal{C}}})$ is the set of all intersections of subcollections of $\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{I}}_{\mathcal{C}}\}$. Eventually,

$${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{⌀,\left\{q_2\right\},\left\{q_3\right\},\{q_1,q_3\},\{q_2,q_3\},\{q_2,q_4\},\{q_1,q_2,q_3\},Q\}.$$

Remark 2.

Doignon [6] provided a method to delineateg knowledge structures via conjunctive skill functions through all subsets of the skills. However, throughout the procedure of our algorithm to delineate knowledge structures, it is unnecessary to list competence states not in the intersection generation groups, which is some improvement compared to Doignon’s method.

5. Knowledge Structures Delineated by Competence-Based Skill Functions

Definition 6.

Let $(S,\mathcal{C})$ be a competence structure and $(Q,S,{\mu }_{\mathcal{C}})$ a competence-based skill function. An element $C\in \mathcal{C}$ is said to be a weak competency in $(Q,S,{\mu }_{\mathcal{C}})$ if $C\notin {\mu }_{\mathcal{C}}\left(Q\right)$ and $C=\bigcup \mathcal{G}$ for some subcollection $\mathcal{G}\subseteq {\mu }_{\mathcal{C}}\left(Q\right)$.

For any competence structure $(S,\mathcal{C})$ and any competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$, the empty subset ⌀ of $\mathcal{C}$ is a weak competency in $(Q,S,{\mu }_{\mathcal{C}})$ since it is the union of the empty subfamily of ${\mu }_{\mathcal{C}}\left(Q\right)$. It is trivial that ${\mu }_{\mathcal{C}}\left(Q\right)$ is also a weak competency in $(Q,S,{\mu }_{\mathcal{C}})$.

For any competence structure $(S,\mathcal{C})$ and any competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on $(S,\mathcal{C})$, the set of all competencies and all weak competencies in $(Q,S,{\mu }_{\mathcal{C}})$ is denoted by ${\mathcal{H}}_{{\mu }_{\mathcal{C}}}$.

Theorem 3.

Let $(S,\mathcal{C})$ be a competence structure and $(Q,S,{\mu }_{\mathcal{C}})$ a competence-based skill function. Then

$${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{p_{{\mu }_{\mathcal{C}}}\left(H\right)\mid H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\}.$$

Proof. 

For any $K\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ and the corresponding $T\in \mathcal{C}$ such that $K=p_{{\mu }_{\mathcal{C}}}\left(T\right)$. For any $q\in K$, there exists $C_q\in {\mu }_{\mathcal{C}}\left(q\right)$ such that $C_q\subseteq T$. Take $T^{\prime }={\bigcup }_{q\in K}{C}_q$. Note that $T^{\prime }\subseteq T$, then $p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)\subseteq p_{{\mu }_{\mathcal{C}}}\left(T\right)$ by the monotonicity of $p_{{\mu }_{\mathcal{C}}}$. For any $q\in K$, by the choice of $T^{\prime }$, we have $q\in p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)$, hence $K\subseteq p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)$. It follows that $K=p_{{\mu }_{\mathcal{C}}}\left(T\right)=p_{{\mu }_{\mathcal{C}}}\left(T^{\prime }\right)$. Note that if $T^{\prime }\in {\mu }_{\mathcal{C}}$, then $T^{\prime }$ is a competency; otherwise, $T^{\prime }$ is a weak competency. Thus, ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}\subseteq \{p_{{\mu }_{\mathcal{C}}}\left(H\right)\mid H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\}$. Note that ${\mathcal{H}}_{{\mu }_{\mathcal{C}}}\subseteq \mathcal{C}$ and $\mathcal{K}=\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in \mathcal{C}\}$, then $\{p_{{\mu }_{\mathcal{C}}}\left(H\right)\mid H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\}\subseteq {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Hence, ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{p_{{\mu }_{\mathcal{C}}}\left(H\right)\mid H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\}$. □

Based on Theorem 3, we provide an algorithm to delineate knowledge structures by competence-based skill functions summarized as follows, the procedure of which is only related to competencies and weak competencies.

Let $(S,\mathcal{C})$ be a competence structure, $(Q,S,{\mu }_{\mathcal{C}})$ a competence-based skill function on $(S,\mathcal{C})$, and $(Q,S,p_{{\mu }_{\mathcal{C}}})$ the competence-based problem function induced by $(Q,S,{\mu }_{\mathcal{C}})$.

Step 1. 

Find the set ${\mathcal{H}}_{{\mu }_{\mathcal{C}}}$;

Step 2. 

Compute $p_{{\mu }_{\mathcal{C}}}\left(H\right)$ for each $H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}$;

Step 3. 

Set ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{p_{{\mu }_{\mathcal{C}}}\left(H\right)\mid H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\}$.

The following example represents the details of this algorithm.

Example 3.

Let $Q=\{q_1,q_2,q_3,q_4\}$ and $S=\{s_1,s_2,s_3,s_4\}$. Consider the competence structure $(S,\mathcal{C})$, where

$$\mathcal{C}=\{⌀,\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},\{s_3,s_4\},\{s_1,s_2,s_3\},\{s_1,s_2,s_4\},\{s_2,s_3,s_4\},S\}.$$

Define a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ as follows:

$${\mu }_{\mathcal{C}}\left(q\right)=\left\{\begin{array}{cc}\{\{s_2,s_3\},\{s_3,s_4\}\},\hfill & q=q_1,\hfill \\ \{\{s_1,s_2\},\{s_1,s_3\}\},\hfill & q=q_2,\hfill \\ \{\left\{s_2\right\},\{s_3,s_4\}\},\hfill & q=q_3,\hfill \\ \left\{\left\{s_3\right\}\right\},\hfill & q=q_4.\hfill \end{array}\right.$$

All competencies in $(Q,S,{\mu }_{\mathcal{C}})$ are

$$\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},\{s_3,s_4\},$$

and all weak competencies in $(Q,S,{\mu }_{\mathcal{C}})$ are

$$⌀,\{s_1,s_2,s_3\},\{s_2,s_3,s_4\},S.$$

Thus,

$${\mathcal{H}}_{{\mu }_{\mathcal{C}}}=\{⌀,\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},\{s_3,s_4\},\{s_1,s_2,s_3\},\{s_2,s_3,s_4\},S\}.$$

All knowledge states delineated by elements in ${\mathcal{H}}_{{\mu }_{\mathcal{C}}}$ are

$$p_{{\mu }_{\mathcal{C}}}(⌀)=⌀,p_{{\mu }_{\mathcal{C}}}\left(\left\{s_2\right\}\right)=\left\{q_3\right\},p_{{\mu }_{\mathcal{C}}}\left(\left\{s_3\right\}\right)=\left\{q_4\right\},p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_2\}\right)=\{q_2,q_3\},$$

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_3\}\right)=\{q_2,q_4\},p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_3\}\right)=\{q_1,q_3,q_4\},p_{{\mu }_{\mathcal{C}}}\left(\{s_3,s_4\}\right)=\{q_1,q_3,q_4\},$$

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_2,s_3\}\right)=Q,p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_3,s_4\}\right)=\{q_1,q_3,q_4\},p_{{\mu }_{\mathcal{C}}}\left(S\right)=Q.$$

According to Theorem 3,

$${\mathcal{K}}_{{\mu }_{\mathcal{C}}}=\{⌀,\left\{q_3\right\},\left\{q_4\right\},\{q_2,q_3\},\{q_2,q_4\},\{q_1,q_3,q_4\},Q\}.$$

Remark 3.

Heller [11] provide a method to delineate knowledge structures via skill functions through listing all competence states. Similar methods can be found in [1]. Our algorithm can delineate knowledge structures via competence-based skill functions by finding the set of competencies and weak competencies, whose cardinality is smaller than that of all competence states in general.

6. Reducible Skills in Competence-Based Skill Functions

Competence-based skill functions are bridges between items and competence states. For a knowledge domain, the more skills an individual has, the more items he can solve. Thus, individuals tend to master a minimal set of skills to solve items. Note that for any competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$, knowledge states delineated by some competence states in $(S,\mathcal{C})$ may be represented by knowledge states delineated by some other competence states. Therefore, it is natural to find minimal sets of skills of competence-based skill functions. For any finite sets Q, S and a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$, the minimal set of skills always exist and is unique.

Let us begin with an example to investigate minimal sets of skills of disjunctive competence-based skill functions.

Example 4.

Consider the disjunctive competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ defined in Example 1, let

$$M=\{s_1,s_2,s_3\}.$$

Then $M\subseteq S$ and

$${\mathcal{C}}_M=\{⌀,\left\{s_1\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},M\}.$$

Define

$${\mu }_{{\mathcal{C}}_M}=\left\{\begin{array}{cc}\left\{\left\{s_1\right\}\right\},\hfill & q=q_1;\hfill \\ \left\{\{s_1,s_2\}\right\},\hfill & q=q_2;\hfill \\ \{\left\{s_3\right\},\{s_1,s_2\}\},\hfill & q=q_3;\hfill \\ \left\{\left\{s_3\right\}\right\},\hfill & q=q_4.\hfill \end{array}\right.$$

Then

$${\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}=\{\left\{s_1\right\},\left\{s_3\right\},\{s_1,s_2\}\}.$$

The all knowledge states delineated by elements in ${\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ are

$$p_{{\mu }_{{\mathcal{C}}_M}}\left(\left\{s_1\right\}\right)=\left\{q_1\right\},p_{{\mu }_{{\mathcal{C}}_M}}\left(\left\{s_3\right\}\right)=\{q_3,q_4\},p_{{\mu }_{{\mathcal{C}}_M}}\left(\{s_1,s_2\}\right\})=\{q_1,q_2,q_3\}.$$

Eventually,

$${\mathcal{K}}_{{\mu }_{{\mathcal{C}}_M}}=\{⌀,\left\{q_1\right\},\{q_3,q_4\},\{q_1,q_2,q_3\},\{q_1,q_3,q_4\},Q\},$$

which is the same as ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Recall that

$${\mathcal{A}}_{{\mu }_{\mathcal{C}}}=\{\left\{s_1\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_2,s_4\}\}.$$

For $s_4$, there is unique element $\{s_2,s_4\}\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\backslash {\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$. Note that

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_4\}\right)=\left\{q_1\right\}=p_{{\mu }_{\mathcal{C}}}\left(\left\{s_1\right\}\right),$$

which implies that knowledge states delineated by $\{s_2,s_4\}\in {\mathcal{A}}_{{\mu }_{\mathcal{C}}}\backslash {\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ and $\left\{s_1\right\}\in {\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ are the same.

Example 4 inspires us to ask, for some competence-based skill functions, whether there exist subsets $M\subseteq S$ such that all knowledge states delineated by elements in ${\mathcal{A}}_{{\mu }_{\mathcal{C}}}\backslash {\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ can be replaced or represented by ones delineated by elements in ${\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a competence-based skill function on the competence structure $(S,\mathcal{C})$. For a nonempty subset $M\subseteq S$, let ${\mathcal{C}}_M$ be the restriction on $\mathcal{C}$, i.e.,

$${\mathcal{C}}_M=\{C\in \mathcal{C}\mid C\subseteq M\}$$

and define $(Q,M,{\mu }_{{\mathcal{C}}_M})$ by restricting the range of ${\mu }_{\mathcal{C}}$ on ${\mathcal{P}}^{*}\left({\mathcal{P}}^{*}\left(M\right)\right)$, i.e.,

$${\mu }_{{\mathcal{C}}_M}\left(q\right)={\mu }_{\mathcal{C}}\left(q\right)\bigcap {\mathcal{P}}^{*}\left({\mathcal{P}}^{*}\left(M\right)\right)=\{C\in {\mu }_{\mathcal{C}}\left(q\right)\mid C\subseteq M\}$$

for any $q\in Q$.

Definition 7.

For a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ on the competence structure $(S,\mathcal{C})$, a skill $s\in S$ is said to be a reducible if the competence-based skill function $(Q,S\backslash \left\{s\right\},{\mu }_{{\mathcal{C}}_{S\backslash \left\{s\right\}}})$ also delineates ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, i.e., ${\mathcal{K}}_{{\mu }_{{\mathcal{C}}_{S\backslash \left\{s\right\}}}}={\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. A subset $M\subseteq S$ is said to be a minimal set of skills for $(Q,S,{\mu }_{\mathcal{C}})$ if ${\mathcal{K}}_{{\mu }_{{\mathcal{C}}_M}}={\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ and for any $M^{\prime }\subseteq S$, ${\mathcal{K}}_{{\mu }_{{\mathcal{C}}_{M^{\prime }}}}={\mathcal{K}}_{{\mu }_{\mathcal{C}}}$ implies $M\subseteq M^{\prime }$.

Remark 4. 

(1)

In Definition 7, the ‘minimal cardinality’ means that for any $N\subseteq S$, if the knowledge structure delineated by $(Q,S,{\mu }_{{\mathcal{C}}_N})$ is also the same as ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, then the cardinality of N is not less than the cardinality of M;

(2)

For any competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$ and its minimal set of skills M, each competency in ${\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ is a minimal competence sufficient to solve at least one item in Q. In other words, for any $C\in {\mathcal{A}}_{{\mu }_{{\mathcal{C}}_M}}$ and any $s\in C$, the set $C\backslash \left\{s\right\}$ is not sufficient any longer to solve any item in Q.

Recall that a competence structure $(S,\mathcal{C})$ is said to be discriminative if for any $s,t\in S$, there exists $C\in \mathcal{C}$ such that $\{s,t\}\cap C=\left\{s\right\}$ or $\{s,t\}\cap C=\left\{t\right\}$. For any non-discriminative competence structure, we can construct a discriminative reduction, where the procedure is similar to the one in constructing discriminative knowledge structures. Thereafter, we always consider competence-based skill functions on discriminative competence structures.

Theorem 4.

Let $(S,\mathcal{C})$ be a competence space and $s\in S$ is a reducible skill in a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$. Then, $(S\backslash \left\{s\right\},{\mathcal{C}}_{S\backslash \left\{s\right\}})$ is also a competence space.

Proof. 

Note that

$${\mathcal{C}}_{S\backslash \left\{s\right\}}=\{C\in \mathcal{C}\mid s\notin C\}$$

and

$${\mu }_{{\mathcal{C}}_{S\backslash \left\{s\right\}}}\left(q\right)=\{C\in {\mu }_{\mathcal{C}}\left(q\right)\mid s\notin C\}.$$

For any subcollection $\mathcal{A}\subseteq {\mathcal{C}}_{S\backslash \left\{s\right\}}$, we have $\bigcup \mathcal{A}\in \mathcal{C}$, since $\mathcal{C}$ is union-closed. Obviously, $s\notin A$ for any $A\in \mathcal{A}$, whence $s\notin \bigcup \mathcal{A}$. It follows that $\bigcup \mathcal{A}\in {\mathcal{C}}_{S\backslash \left\{s\right\}}$. Trivially, $⌀\in {\mathcal{C}}_{S\backslash \left\{s\right\}}$ and $S\backslash \left\{s\right\}=\bigcup {\mathcal{C}}_{S\backslash \left\{s\right\}}\in {\mathcal{C}}_{S\backslash \left\{s\right\}}$. Thus, $(S\backslash \left\{s\right\},{\mathcal{C}}_{S\backslash \left\{s\right\}})$ is a competence space. □

According to Theorems 1 and 4, for any competence space $(S,\mathcal{C})$ and a competence-based skill function $(Q,S,{\mu }_{\mathcal{C}})$, if we do finite steps for the reduction and remove one reducible skill in S in each step, then we can also delineate ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Thus, for finite S, we summarize the steps to find sets of reducible skills for competence-based skill functions as follows.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a disjunctive competence-based skill function on a discriminative competence space $(S,\mathcal{C})$.

Step 1. 

Label all elements in S as $\{s_0,s_1,\cdots ,s_n\}$ and set $S_0=S$;

Step 2. 

List the set ${\mathcal{D}}_{s_0}=\{C\in \mathcal{C}\mid s_0\in C\}$ of all elements in $\mathcal{C}$ containing $s_0$;

Step 3. 

Compute the sets ${\mathcal{K}}_{s_0}=\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{D}}_{s_0}\}$ and ${\mathcal{K}}_{\overline{s_0}}={\mathcal{K}}_{{\mu }_{\mathcal{C}}}\backslash \{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{D}}_{s_0}\}$;

Step 4. 

If for any $K\in {\mathcal{K}}_{s_0}$, there exists a subcollection ${\mathcal{A}}_K\subseteq {\mathcal{K}}_{\overline{s_0}}$ such that $K=\bigcup {\mathcal{A}}_K$, then we remove $s_0$ from S and set $S_1=S\backslash \left\{s_0\right\}$. Otherwise, $s_0$ can not be removed, then we set $S_1=S_0$.

Step 5. 

For any $s_i$ ($1\le i\le n-1$), repeat Steps 2–4 by replacing $s_0$, $S_0$ with $s_i$, $S_i$, respectively. Then, we can decide whether $s_i$ can be removed and obtain $S_n$;

Step 6. 

Put $M=S_n$, which is a minimal set of skills for $(Q,S,{\mu }_{\mathcal{C}})$.

It is easy to verify that for any conjunctive competence-based skill function $(Q,S,{\mu }_C)$ on a discriminative intersection-closed competence structure $(S,\mathcal{C})$, each $s\in S$ is not a reducible skill.

Before we investigate the reducible skills of common competence-based skill functions, we look at the following example.

Example 5.

Consider the competence-based skill function defined in Example 3, let $M=\{s_1,s_2,s_3\}$ and

$${\mathcal{C}}_M=\{⌀,\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},M\}.$$

Then

$${\mathcal{C}}_{{\overline{s}}_4}=\{⌀,\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},\{s_1,s_2,s_3\}\},$$

and

$${\mathcal{H}}_{\overline{s_4}}=\{⌀,\left\{s_2\right\},\left\{s_3\right\},\{s_1,s_2\},\{s_1,s_3\},\{s_2,s_3\},M\}.$$

All knowledge states delineated by elements in ${\mathcal{H}}_{\overline{s_4}}$ are

$$p_{{\mu }_{\mathcal{C}}}(⌀)=⌀,p_{{\mu }_{\mathcal{C}}}\left(\left\{s_2\right\}\right)=\left\{q_3\right\},p_{{\mu }_{\mathcal{C}}}\left(\left\{s_3\right\}\right)=\left\{q_4\right\},p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_2\}\right)=\{q_2,q_3\},$$

$$p_{{\mu }_{\mathcal{C}}}\left(\{s_1,s_3\}\right)=\{q_2,q_4\},p_{{\mu }_{\mathcal{C}}}\left(\{s_2,s_3\}\right)=\{q_1,q_3,q_4\},p_{{\mu }_{\mathcal{C}}}\left(M\right)=Q.$$

Applying Theorem 3, we get

$${\mathcal{K}}_{{\mu }_{{\mathcal{C}}_M}}=\{⌀,\left\{q_3\right\},\left\{q_4\right\},\{q_2,q_3\},\{q_2,q_4\},\{q_1,q_3,q_4\},Q\},$$

which is the same as ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$.

We turn to investigate how to find reducible skills for competence-based skill functions. By applying Theorem 3, the following result can be easily derived.

Theorem 5.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a competence-based skill function on a competence structure $(S,\mathcal{C})$. Then s is a reducible skill of $(Q,S,{\mu }_{\mathcal{C}})$ if and only if for any knowledge state $K\in {\mathcal{K}}_{{\mu }_{\mathcal{C}}}$, there exists $H\in {\mathcal{H}}_{\overline{s}}$ such that $p_{{\mu }_{{\mathcal{C}}_S}}\left(H\right)=K$, where ${\mathcal{H}}_{\overline{s}}=\{H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\mid s\notin H\}$.

Remark 5.

Theorem 5 can be interpreted as: each competency in ${\mu }_{\mathcal{C}}\left(q\right)$ containing s is not sufficient to solve the item q when the skill s is removed.

Making use of Theorem 5, we design an algorithm for reducible skills for competence-based skill functions.

Let $(Q,S,{\mu }_{\mathcal{C}})$ be a competence-based skill function on a competence structure $(S,\mathcal{C})$.

Step 1. 

Label all elements in S as $\{s_0,s_1,\cdots ,s_n\}$ and set $S_0=S$;

Step 2. 

List the set ${\mathcal{D}}_{s_0}=\{C\in \mathcal{C}\mid s_0\in C\}$ of all elements in $\mathcal{C}$ containing $s_0$;

Step 3. 

Compute the sets ${\mathcal{K}}_{s_0}=\{p_{{\mu }_{\mathcal{C}}}\left(C\right)\mid C\in {\mathcal{D}}_{s_0}\}$ and ${\mathcal{H}}_{\overline{s_0}}=\{H\in {\mathcal{H}}_{{\mu }_{\mathcal{C}}}\mid s_0\notin H\}$;

Step 4. 

If for any $K\in {\mathcal{K}}_{s_0}$, there exists $H\in {\mathcal{H}}_{\overline{s_0}}$ such that $K=p_{{\mu }_{{\mathcal{C}}_{S_0}}}\left(H\right)$, then we remove $s_0$ from $S_0$ and set $S_1=S_0\backslash \left\{s_0\right\}$; otherwise, $s_0$ can not be removed, then we set $S_1=S_0$.

Step 5. 

For any $s_i$ ($1\le i\le n-1$), repeat Steps 2–4 by replacing $s_0$, $S_0$ with $s_i$, $S_i$ respectively. Then we can decide whether $s_i$ can be removed and obtain $S_n$;

Step 6. 

Put $M=S_n$, which is a minimal set of skills for $(Q,S,{\mu }_{\mathcal{C}})$.

Remark 6.

For any (disjunctive) competence-based skill function $(Q,M,{\mu }_{\mathcal{C}})$ on a competence structure (space) $(S,\mathcal{C})$, by applying the corresponding algorithm, we can remove at most one reducible skill in each procedure. Thus, for finite S with the cardinality n, let $R=\{s_0,\cdots ,s_{n-1}\}$ be an arrangement of all elements in S. After n procedures, we get a minimal set $M_R\subseteq S$ such that the competence-based skill function $(Q,M_R,{\mu }_{{\mathcal{C}}_{M_R}})$ delineates ${\mathcal{K}}_{{\mu }_{\mathcal{C}}}$. Note that the cardinality of such arrangements is $n!$. Thus, we can do at most $n\times n!$ procedures to produce all minimal sets of skills for $(Q,S,{\mu }_{\mathcal{C}})$. Finally, we pick the minimal set of skills with the smallest cardinality.

In the present article, we always assume that competence structures and competence-based skill functions are given, however, the process of constructing competence structures and competence-based skill functions are also time-consuming. Besides, the manuscript lacks practical examples with an educational psychology background. In the future, we will more focus on the corresponding empirical analysis.

7. Conclusions

In this paper, we mainly consider knowledge structures delineated by competence-based skill functions and minimal sets of skills for competence-based skill functions. Theorem 1 states that knowledge structures delineated by disjunctive competence-based skill functions are only concerned with atoms in competence spaces. Theorem 2 shows that knowledge structures delineated by conjunctive competence-based skill functions are only related to elements in some intersection generation groups. Besides, Theorem 3 reveals that knowledge structures delineated by competence-based skill functions are just in connection with competencies and weak competencies. Based on the theoretical results we obtained, we propose corresponding approaches for constructing knowledge structures from competence-based skill functions. These are different from the existing methods [26,27,28].

Given that for individuals, having more skills leads to solving more items. Thus, individuals tend to master the lesser or necessary skills for solving items. Note that in knowledge structures, some states delineated by competence states can be represented by ones delineated by some other competence states. This implies that maybe there are some reducible skills in the set of skills. So, we develop two theoretical results in Theorems 4 and 5 for this case, and then propose the corresponding algorithms to find the reducible skills and get minimal sets of skills for competence-based skill functions. When the reducible skills are moved, the relationship between items and skills can be represented without the knowledge structures being changed. For competence-based skill functions with finite skills, we can always find their minimal sets of skills with the smallest cardinality.