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SkillPilot Curriculum Graph Specification

This document defines the SkillPilot curriculum graph as a mathematical structure, including its entities, relations, derived semantics (inheritance), and validity constraints.

The intent is that independent implementations interpret and validate graphs in the same way.

1. Notation and conventions

  • \(G\) is a finite set of goals (also called skills or nodes).
  • A binary relation \(X \subseteq G \times G\) is a set of ordered pairs \((a,b)\).
  • For any relation \(X\), \(X^+\) denotes the transitive closure of \(X\).
    Informally, \((a,b)\in X^+\) means there exists a directed path from \(a\) to \(b\) following edges in \(X\).
  • A directed graph \((G,X)\) is acyclic iff there is no \(g \in G\) such that \((g,g)\in X^+\).

2. Goals and attributes

Each goal \(g \in G\) is a distinct entity.

2.1 Attribute domains

  • \(\text{UUID}\): the set of UUID values.
  • \(\Sigma^*\): the set of finite strings over an alphabet \(\Sigma\).
  • \(\mathbb{R}_{>0}\): strictly positive real numbers.
  • \(P\): a set of phases. If phase ordering is used, phases form a totally ordered set \((P,\le)\).

2.2 Attribute mappings

Each goal \(g\in G\) has the following attributes:

  • \(Id: G \to \text{UUID}\)
  • \(Title: G \to \Sigma^*\)
  • \(Phase: G \to P\)
  • \(Weight: G \to \mathbb{R}_{>0}\)

2.3 Identifier uniqueness

Identifiers MUST be unique:

\[ \forall g,h\in G:\ g\neq h \Rightarrow Id(g)\neq Id(h) \]

3. Relations

The curriculum graph is defined using two primary relations on \(G\):

  • a hierarchy relation called Contains
  • a dependency relation called Direct Requires

4. Contains relation

4.1 Definition

The Contains relation is a binary relation:

\[ C \subseteq G \times G \]

\((p,c)\in C\) means parent \(p\) contains child \(c\).

Note: \(C\) is the direct containment relation (“direct contains”).
Indirect containment (ancestor/descendant) is derived via the transitive closure \(C^+\).

Edges in \(C\) are interpreted as hierarchical grouping (e.g., Module contains Topic).

4.2 Containment constraint (polyhierarchy)

\((G,C)\) MUST be acyclic (containment cannot contain cycles):

\[ \neg \exists g\in G:\ (g,g)\in C^+ \]

This allows multiple parents per node (a polyhierarchy).
If you want a strict tree/forest, see the recommended rule in §8.3.

4.3 Ancestors and descendants

Define:

\[ Ancestors(g) = \{\, a\in G \mid (a,g)\in C^+ \,\} \]
\[ Descendants(g) = \{\, d\in G \mid (g,d)\in C^+ \,\} \]

5. Direct Requires relation

5.1 Definition

The Direct Requires relation is a binary relation:

\[ R_d \subseteq G \times G \]

\((u,v)\in R_d\) means \(u\) is a direct prerequisite of \(v\).
Equivalently: to learn/attempt \(v\), \(u\) must be satisfied first.

5.2 DAG constraint

\((G,R_d)\) MUST be acyclic:

\[ \neg \exists g\in G:\ (g,g)\in R_d^+ \]

6. Effective Requires semantics

Direct prerequisites can be specified at higher-level nodes and inherited by their descendants in the hierarchy. This yields the Effective Requires relation.

6.1 Effective Requires relation

Define \(R_{eff}\subseteq G\times G\) by:

\[ (u,v)\in R_{eff} \iff \big((u,v)\in R_d\big) \ \lor\ \big(\exists a\in Ancestors(v): (u,a)\in R_d\big) \]

Interpretation:

  • A goal inherits all direct prerequisites declared on its ancestors.
  • Only direct prerequisites declared in \(R_d\) are inherited from ancestors.

Note (with multiple parents): If a node has multiple parents, it inherits the union of prerequisites from all ancestor paths.

6.2 Effective prerequisite set

For convenience, define the set of effective prerequisites of a node:

\[ Pre_{eff}(v) = \{\, u\in G \mid (u,v)\in R_{eff} \,\} \]

7. Validity constraints

A SkillPilot curriculum graph is valid iff all constraints in this section hold.

7.1 Effective Requires must be acyclic

The dependency graph induced by effective prerequisites MUST be acyclic:

\[ \neg \exists g\in G:\ (g,g)\in R_{eff}^+ \]

This constraint is stricter than acyclicity of \(R_d\) alone because inheritance via \(C\) can introduce cycles.

Non-normative example (illustrative):
Let \((A,B)\in C\) (i.e., \(A\) contains \(B\)). Suppose \((X,A)\in R_d\) and \((B,X)\in R_d\).
Then \(B \to X \to A\) exists in \(R_d\), but inheritance adds \((X,B)\in R_{eff}\) (since \(A\) is an ancestor of \(B\)), creating a cycle \(B \to X \to B\) in \(R_{eff}\).

7.2 Local minimality

A direct prerequisite MUST NOT be redundantly stated on a node if it is already inherited from an ancestor.

\[ \forall g\in G,\ \forall u\in G: (u,g)\in R_d \Rightarrow \neg \exists a\in Ancestors(g): (u,a)\in R_d \]

7.3 Transitive minimality

A direct prerequisite edge MUST NOT be present if the prerequisite relationship already follows from other effective prerequisite paths.

Formally, for each \((u,g)\in R_d\), remove that single direct edge and recompute effective requirements; the prerequisite must no longer be implied transitively.

Let:

\[ R_d' = R_d \setminus \{(u,g)\} \]

and let \(R_{eff}'\) be the effective relation computed from \(R_d'\) using the definition in §6.1.

Then the constraint is:

\[ \forall (u,g)\in R_d:\ (u,g)\notin (R_{eff}')^+ \]

Interpretation: every edge in \(R_d\) is necessary to preserve prerequisite reachability under the inheritance rules.

The following are common modeling rules that typically improve graph quality. They may be treated as warnings or enforced as hard constraints depending on the product needs.

8.1 Avoid requiring descendants

A goal SHOULD NOT require its own descendant:

\[ (u,v)\in R_d \Rightarrow u \notin Descendants(v) \]

This prevents “inside-out” prerequisite definitions that often indicate a modeling error (e.g., a parent depending on one of its parts).

8.2 Avoid prerequisites along containment edges

Often, prerequisites SHOULD be modeled between peer concepts rather than between ancestors/descendants in the hierarchy. Common guidance:

  • For \((u,v)\in R_d\): \(u \notin Ancestors(v)\) and \(u \notin Descendants(v)\)

If your product needs exceptions, treat this as a heuristic.

8.3 Optional: At most one parent per node (tree/forest mode)

If you want a strict tree/forest hierarchy, enforce:

\[ \forall c\in G:\ \left|\{\,p\in G \mid (p,c)\in C\,\}\right| \le 1 \]

9. Learning availability and progression

Let \(M\subseteq G\) be the set of goals the learner has mastered.

9.1 Available next goals

The learner’s frontier (available next goals) is:

\[ Frontier(M) = \left\{ g \in G\setminus M \ \middle|\ \forall u\in G:\ (u,g)\in R_{eff}^+ \Rightarrow u\in M \right\} \]

Interpretation: a goal is available if all of its prerequisite goals (including transitive prerequisites) are mastered.

10. Phase ordering

Assume phases are totally ordered by \(\le\).

10.1 Monotone prerequisite flow

In typical curricula, prerequisites SHOULD point backward in time or remain within the same phase:

\[ (u,v)\in R_{eff} \Rightarrow Phase(u)\le Phase(v) \]

If the system supports remedial or non-linear paths, violations of this rule may be allowed as explicit exceptions.

11. Summary of required validity conditions

A curriculum graph \((G,C,R_d)\) is valid iff:

  1. \(Id\) is injective on \(G\)
  2. \((G,C)\) is acyclic (containment DAG / polyhierarchy; multiple parents allowed)
  3. \((G,R_d)\) is a DAG
  4. \(R_{eff}\) (computed from \(C\) and \(R_d\)) is acyclic
  5. \(R_d\) satisfies local minimality
  6. \(R_d\) satisfies transitive minimality

Everything else in this specification is either derived (definitions) or recommended modeling guidance.

12. Filters and scoped evaluation (Optimistic vs. Pessimistic)

A filter restricts the global curriculum graph to a subset of nodes (e.g., Grade 12 AND Subject: Mathematics AND Track: Advanced).

12.1 Filter predicate and induced subgraph

A filter is modeled as a predicate:

\[ F: G \to \{0,1\}. \]

It selects the filtered node set:

\[ G_F = \{\, g \in G \mid F(g)=1 \,\}. \]

The induced (restricted) relations are:

\[ C_F = C \cap (G_F \times G_F), \qquad R_{d,F} = R_d \cap (G_F \times G_F). \]

The effective relation inside the filtered graph, \(R_{eff,F}\), is computed from \(C_F\) and \(R_{d,F}\) using the same definition as in §6 (i.e., by inheriting requirements along \(C_F\)).

12.2 Optimistic mode

In optimistic mode, we first apply the filter and then compute availability inside the filtered graph only.
Intuition: when a learner is in the filtered scope (e.g., Grade 12), we temporarily assume that missing prerequisites from outside the scope do not block progress.

Let \(M \subseteq G\) be the learner’s mastered set (global). Define the filtered mastery:

\[ M_F = M \cap G_F. \]

Then the optimistic frontier is:

\[ Frontier_{opt}(M,F) = \left\{ g \in G_F \setminus M_F \ \middle|\ \forall u\in G_F:\ (u,g)\in R_{eff,F}^+ \Rightarrow u\in M_F \right\}. \]

12.3 Pessimistic mode or strict mode

In pessimistic mode or strict mode, candidate goals are still restricted to the filtered set, but prerequisites are enforced globally (including nodes outside the filter).

Let \(R_{eff}\) be computed on the full graph \((G,C,R_d)\). Then:

\[ Frontier_{pess}(M,F) = \left\{ g \in G_F \setminus M \ \middle|\ \forall u\in G:\ (u,g)\in R_{eff}^+ \Rightarrow u\in M \right\}. \]

12.4 Diagnostic: missing prerequisites

For diagnosis, define the set of missing prerequisites of a goal \(g\):

\[ Missing(g,M) = \{\, u \in G \mid (u,g)\in R_{eff}^+ \land u\notin M \,\}. \]

To distinguish gaps inside vs. outside the filter:

\[ \begin{aligned} Missing_{in}(g,M,F) &= Missing(g,M)\cap G_F,\\ Missing_{out}(g,M,F) &= Missing(g,M)\setminus G_F. \end{aligned} \]

Operationally, one can start with optimistic mode for efficiency and exploration; if a learner struggles with a goal, switch to pessimistic mode (or compute \(Missing_{out}\)) to identify prerequisite gaps outside the current filter.

12.5 Optional: relaxed pessimism via a prerequisite scope

A “weakened” pessimistic approach can be modeled by choosing a scope set \(S \subseteq G\) of prerequisites that must be enforced (e.g., only prerequisites from the last one or two phases, or only prerequisites up to a bounded depth).

Define:

\[ Frontier_{scope}(M,F,S) = \left\{ g \in G_F \setminus M \ \middle|\ \forall u\in S:\ (u,g)\in R_{eff}^+ \Rightarrow u\in M \right\}. \]

Special cases:

  • \(S = G\) gives the fully pessimistic mode.
  • Choosing \(S\) smaller than \(G\) yields a relaxed pessimistic check that can be widened iteratively if needed.