Definition. Compositionality, computability, and complexity › Complex indirect constant $\mu$-systems (§ 8.3.4) [pagin2021-CIC]

A complex indirect constant $\mu$-system $\mathrm{\Phi }$ must satisfy the following.

1. $R_{\mathrm{\Phi }}$ is finite.
2. For any rule, the rewrite variables on the rhs must be a subset of those on the left.
3. Rewrite variables $v_1,v_2,...$ take all and only object language grammatical terms as instances.
4. Rewrite variables $y_1,y_2,...$ take all and only expressions in $C_m$ as instances.
5. Every atomic rule $r\in R_{\mathrm{\Phi }}$ is of the form $${\mu }_i(t)\to e$$ where ${\mu }_i\in S$, $t\in A_O$ and $e$ is a simple or complex expression in $C_m$.
6. For every pair $⟨{\mu }_i\in S,t\in A_o⟩$ there is an atomic rule.
7. Every direct complex rule $r$ has the form $${\mu }_i(\alpha (v_1,...,v_n))\to K({\mu }_{k_1}(v_1),...,{\mu }_{k_n}(v_n))\Leftarrow \mathrm{Gr}(\alpha (v_1,...,v_n))$$ where Gr corresponds to grammaticality.
8. Every indirect complex rule has the form $${\mu }_i(\alpha (v_1,...,v_n))\to f({\mu }_{k_1}(v_1),...,{\mu }_{k_n}(v_n),v_1,...,v_n)\Leftarrow \mathrm{Gr}(\alpha (v_1,...,v_n))$$ where $v_1,...,v_n$ are optional and $f\in F$.
9. For every pair $⟨{\mu }_i\in S,\alpha \in {\mathrm{\Sigma }}_o^{\prime }$, there is a unique complex (indirect or direct) rule.
10. Every indirect ground rule is of the $F$-form $$f(y_1,...,y_n,...,t_n)\to K^{\prime }(f^{\prime }(y_1,...,y_n))$$ or the $G$-form $$g(e_1,...,e_n)\to K(e_1,...,e_n)$$ where $f,f^{\prime }\in F$, $g\in G$, $K,K^{\prime }$ are operators over ${\mathrm{\Sigma }}_m^{\prime }$, $e_1,...,e_n\in A_m$ and $t_1,...,t_j\in A_o$, and at most one of the primed operators is null.
11. Every indirect recurisve rule in $R_{\mathrm{\Phi }}$ has an $F$-form $$f(y_1,...,y_n,\sigma (v_1,...,v_n))\to K(f(y_1,...,y_n,v_1)...,f(y_1,...,y_n,v_n))$$ or $G$-form $$g(K(y_1,...,y_n))\to K^{\prime }(g^{\prime }(y_1,...,y_n))$$ where $f\in F$, $g,g^{\prime }\in G$, $K,K^{\prime }$ are operators over ${\mathrm{\Sigma }}_m^{\prime }$, $\sigma \in {\mathrm{\Sigma }}_O^{\prime }$, and at most one of the primed operators is null.