# Aligned Induction

## Optimisation Summary

Haskell commentary on the implementation of Tractable and Practicable Inducers/Optimisation Summary

To summarise, the fud inducer and the fud decomposition inducer are defined explicitly.

### Sections

Practicable highest-layer shuffle content alignment valency-density fud inducer definition

Practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer definition

### Practicable highest-layer shuffle content alignment valency-density fud inducer definition

Let $A$ be a substrate histogram, $A \in \mathcal{A}_{z}$. The scaled shuffle histogram, $A_R$, is defined $A_R = \mathrm{scalar}(1/|R|) * \sum_{r \in R} L_r$ where $X \in \mathrm{enums}(\mathrm{shuffles}(\mathrm{history}(A)))$, $L = \mathrm{map}(\mathrm{his},\mathrm{flip}(X))$, $R \subseteq {1 \ldots z!^n}$ and $n = |V_A|$.

The practicable highest-layer shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘}$, is defined, (Haskell) $\begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,d}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D,d}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray}$ where (i) the shuffle content alignment valency-density computer is (Haskell) $\begin{eqnarray} I_{\mathrm{csd}}^{ * }((A,A_R,F)) = (I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) \end{eqnarray}$ (ii) the valency capacity computer is $\begin{eqnarray} I_{\mathrm{cvl}}^{ * }(F) := (I_{\approx\mathrm{pow}}^{ * }((w,1/m)) : W = \mathrm{der}(F),~w = |W^{\mathrm{C}}|,~m = |W|) \end{eqnarray}$ (iii) the highest-layer limited-layer limited-underlying limited-breadth fud tree searcher is (Haskell) $\begin{eqnarray} Z_{P,A,A_R,\mathrm{L,d}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,d}},\{\emptyset\}) \end{eqnarray}$ (iv) the highest-layer limited-layer limited-underlying limited-breadth fud tree searcher neighbourhood function is $\begin{eqnarray} &&P_{P,A,A_R,\mathrm{L,d}}(F) = \{G : \\ &&\hspace{2em}G \in P_{P,A,A_R,\mathrm{L}}(F),\\ &&\hspace{2em}(F \neq \emptyset \implies ~\mathrm{maxr}(\mathrm{el}(Z_{P,A,A_R,F,\mathrm{D,d}})) < \mathrm{maxr}(\mathrm{el}(Z_{P,A,A_R,G,\mathrm{D,d}})))\} \end{eqnarray}$ (v) the limited-layer limited-underlying limited-breadth fud tree searcher neighbourhood function is (Haskell) $\begin{eqnarray} &&P_{P,A,A_R,\mathrm{L}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \{T : K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})),\\ &&\hspace{5em}H \in \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})),\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),~T = I^{\mathrm{TPT}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray}$ (vi) the limited-underlying tuple set list maximiser is (Haskell) $\begin{eqnarray} Z_{P,A,A_R,F,\mathrm{B}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{B}},P_{P,A,A_R,F,\mathrm{B}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{B}}) \end{eqnarray}$ (vii) the limited-underlying tuple set list maximiser optimiser function is $\begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{B}} = \\ &&\hspace{2em}\{(K,I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F),A))-I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R))) :\\ &&\hspace{20em}K \in \mathrm{tuples}(V_A,F)\} \end{eqnarray}$ (viii) the limited-underlying tuple set list maximiser neighbourhood function is (Haskell) $\begin{eqnarray} &&P_{P,A,A_R,F,\mathrm{B}}(B) = \{(J,X_{P,A,A_R,F,\mathrm{B}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in B,~w \in \mathrm{vars}(F) \cup V_A \setminus K,~J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray}$ (ix) the limited-underlying tuple set list maximiser initial subset is (Haskell) $\begin{eqnarray} R_{P,A,A_R,\emptyset,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,\emptyset,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w,u \in V_A,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \\ R_{P,A,A_R,F,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,F,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \cup V_A ,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray}$ (x) the contracted decrementing linear non-overlapping fuds list maximiser is (Haskell) $\begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{n},-,K} =\\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{n},-,K},N_{P,A,A_R,F,\mathrm{n},-,K},\mathrm{top}(\mathrm{pmax}),R_{P,A,A_R,F,\mathrm{n},-,K}) \end{eqnarray}$ (xi) the contracted decrementing linear non-overlapping fuds list maximiser optimiser function is $\begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{n},-,K} = \{(H,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}H \in \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}},~G = \mathrm{depends}(F \cup H,\mathrm{der}(H))\} \end{eqnarray}$ (xii) the contracted decrementing linear non-overlapping fuds list maximiser initial subset is (Haskell) $\begin{eqnarray} &&R_{P,A,A_R,F,\mathrm{n},-,K} = \{(\{M^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(\{M^{\mathrm{T}}\})) : \\ &&\hspace{5em}Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},~M = \{J^{\mathrm{CS}\{\}} : J \in Y\}\} \end{eqnarray}$ (xiii) the contracted decrementing linear non-overlapping fuds list maximiser neighbourhood function is (Haskell) $\begin{eqnarray} &&N_{P,A,A_R,F,\mathrm{n},-,K}(C) = \{(H \cup \{N^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(H \cup \{N^{\mathrm{T}}\})) :\\ &&\hspace{10em}(H,\cdot) \in C,~M = \mathrm{der}(H),\\ &&\hspace{10em}w \in M,~|\{w\}^{\mathrm{C}}| > 2,~Q \in \mathrm{decs}(\{w\}^{\mathrm{CS}\{\}}),\\ &&\hspace{10em}N = \{Q\} \cup \{\{u\}^{\mathrm{CS}\{\}} : u \in M,~u \neq w\}\} \end{eqnarray}$ (xiv) the highest-layer limited-derived derived variables set list maximiser is (Haskell) $\begin{eqnarray} Z_{P,A,A_R,F,\mathrm{D,d}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{D,d}}) \end{eqnarray}$ (xv) the highest-layer limited-derived derived variables set list maximiser initial subset is (Haskell) $\begin{eqnarray} R_{P,A,A_R,F,\mathrm{D,d}} &=& \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \setminus V_A \setminus \mathrm{vars}(\mathrm{depends}(F,\{w\})),\\ &&\hspace{2em}J = \{w,u\},~|J^{\mathrm{C}}| \leq \mathrm{wmax}\} \end{eqnarray}$ (xvi) the limited-derived derived variables set list maximiser optimiser function is $\begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{D}} = \{(K,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{10em}K \subseteq \mathrm{vars}(F),~K \neq \emptyset,~G = \mathrm{depends}(F,K)\} \end{eqnarray}$ (xvi) the limited-derived derived variables set list maximiser neighbourhood function is (Haskell) $\begin{eqnarray} &&P_{P,A,A_R,F,\mathrm{D}}(D) = \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in D,~w \in \mathrm{vars}(F) \setminus V_A \setminus K,\\ &&\hspace{2em}J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{wmax},~\mathrm{der}(\mathrm{depends}(F,J)) = J\} \end{eqnarray}$ where the alignmenter is such that $I_{\mathrm{a}}^{ * }(A) \approx \mathrm{algn}(A)$, the partition decrements are $\begin{eqnarray} \mathrm{decs}(Q) := \{P : P \in \mathrm{parents}(Q),~|P|=|Q|-1\} \end{eqnarray}$ the tuples are defined (Haskell) $\begin{eqnarray} \mathrm{tuples}(V,F) := \{K : K \subseteq \mathrm{vars}(F) \cup V,~(\mathrm{der}(F) \neq \emptyset \implies K \cap \mathrm{der}(F) \neq \emptyset)\} \end{eqnarray}$ $\mathrm{el} = \mathrm{elements}$, $\mathrm{his} = \mathrm{histograms} \in \mathcal{F} \to \mathrm{P}(\mathcal{A})$, and $\mathrm{apply} \in \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{A}) \times \mathcal{A} \to \mathcal{A}$.

### Practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer definition

The practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer is implemented (Haskell) $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{d}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,d}})),~\{D\} = Q \end{eqnarray}$ where (i) $D_{\emptyset} = \{((\emptyset,\emptyset),\emptyset)\}$, (ii) the summed shuffle content alignment valency-density computer is $\begin{eqnarray} &&I_{\mathrm{Scsd}}^{ * }((A,D)) = \\ &&\hspace{2em}\sum (I_{\mathrm{a}}^{ * }(A * C * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }((A * C)_{R(A * C)} * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) : (C,F) \in \mathrm{cont}(D) \end{eqnarray}$ (iii) the highest-layer limited-models infinite-layer substrate fud decompositions tree searcher is $\begin{eqnarray} Z_{P,A,\mathrm{D,F,d}} = \mathrm{searchTreer}(\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}}),P_{P,A,\mathrm{D,F,d}},R_{P,A,\mathrm{D,F,d}}) \end{eqnarray}$ (iv) the highest-layer limited-models infinite-layer substrate fud decompositions tree searcher neighbourhood function is $\begin{eqnarray} &&P_{P,A,\mathrm{D,F,d}}(D) = \{E : \\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2},\{(\mathrm{size}(B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),~\mathrm{size}(B)>0,\\ &&\hspace{4em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,B,B_{R(B)},\mathrm{L,d}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,B,B_{R(B)},F_{\mathrm{L}},\mathrm{D,d}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ and (v) the highest-layer limited-models infinite-layer substrate fud decompositions tree searcher initial subset is $\begin{eqnarray} &&R_{P,A,\mathrm{D,F,d}} = \{\{((\emptyset,G),\emptyset)\} : \\ &&\hspace{2em}G \in \mathrm{maxd}(\mathrm{order}(D_{\mathrm{F}},\{G : \\ &&\hspace{4em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_{R(A)},\mathrm{L,d}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,A,A_{R(A)},F_{\mathrm{L}},\mathrm{D,d}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\}))\} \end{eqnarray}$ Now, given a set of search parameters $P$, the fud decomposition is $\begin{eqnarray} D_{\mathrm{Scsd},P} \in \mathrm{maxd}(I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{d}}^{‘ * }(A)) \end{eqnarray}$

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