# Aligned Induction

## Tractable inducers

Although the literal derived alignment inducer, $I_{z,\mathrm{a,l}}^{‘}$, is finitely computable, it is intractable. Section ‘Tractable alignment-bounding’ P568 discusses the various intractabilities and the classes of limits and constraints on the structures of more tractable inducers.

### Sections

Intractable substrate volume

Intractable derived volume

Intractable search set elements

Intractable_literal_substrate_model_inclusion

### Intractable substrate volume

First, the substrate volume is intractable P588. The application of the transformer $I_{ * \mathrm{T}}$, in the literal derived alignment inducer, $I_{z,\mathrm{a,l}}^{‘}$, to a substrate histogram, $A$, and a substrate transform, $T \in \mathcal{T}_{U_A,V_A}$, is $I_{ * \mathrm{T}}^{ * }((T,A)) = A * T$. The volume, $|V_A^{\mathrm{C}}|$, grows exponentially with underlying dimension $n = |V_A|$, and so the space complexity of the transformer, $I_{ * \mathrm{T}}$, is exponential with respect to underlying dimension, $n$. To address this (i) the inducer models of the literal derived alignment inducer are expanded from substrate transforms, $\mathcal{T}_{U_A,V_A}$, to substrate fuds, $\mathcal{F}_{U_A,V_A}$, and (ii) the substrate fuds are then limited by intersecting with one of the class of limited-underlying subsets of the functional definition sets $\mathcal{F}_{\mathrm{u}} \subset \mathcal{F}$. A set of limited-underlying fuds, $\mathcal{F}_{\mathrm{u}}$, is defined such that a fud $F \in \mathcal{F}_{\mathrm{u}}$ is such that its transforms, $F \subset \mathcal{T}$, are each tractably computable. For example the underlying volume of the transforms may be limited by a maximum underlying volume limit $\mathrm{xmax} \in \mathbf{N}_{\geq 4}$. The set of inducer models is $\mathcal{F}_{U_A,V_A} \cap \mathcal{F}_{\mathrm{u}}$.

Note that, if the size, $z = \mathrm{size}(A)$, is less than the volume, $v = |A^{\mathrm{C}}|$, the independent is necessarily non-integral, $z < v \implies A^{\mathrm{X}} \notin \mathcal{A}_{\mathrm{i}}$. For this reason, any volume limits, for example, $\mathrm{xmax} \in \mathbf{N}_{\geq 4}$, should be chosen such that they are less than or equal to the size, $\mathrm{xmax} \leq z$.

### Intractable derived volume

Next, the derived volume is intractable P591. Both the computation time and computation space of the alignmenter applied to the transformed sample histogram, $I_{\mathrm{a}}^{ * }(A * T) \approx \mathrm{algn}(A * T)$, in the literal derived alignment inducer, $I_{z,\mathrm{a,l}}^{‘}$, vary with the derived volume, $w = |W^{\mathrm{C}}|$, where $W = \mathrm{der}(T)$. The derived volume, $w$, grows exponentially with derived dimension $m = |W|$ and so the time and space complexities are exponential, and therefore intractable, with respect to derived dimension, $m$. This is also the case where the implementation uses a fuder, $I_{\mathrm{ * F}}$, in a limited-underlying derived alignment fud inducer, because the application of a fud $F \in \mathcal{F}_{U_A,V_A} \cap \mathcal{F}_{\mathrm{u}}$ must still compute $(A * F)^{\mathrm{X}}$ in an independenter, $I_{\mathrm{X}}$, in order to compute derived alignment, $\mathrm{algn}(A * F)$. So a further compromise is made by intersecting the substrate fuds with one of the class of limited-derived subsets of the functional definition sets $\mathcal{F}_{\mathrm{d}} \subset \mathcal{F}$. A set of limited-derived fuds, $\mathcal{F}_{\mathrm{d}}$, is defined such that a fud $F \in \mathcal{F}_{\mathrm{d}}$ is such that the independent derived of the fud is tractable. For example the derived volume of the fud may be limited by a maximum derived volume limit of $\mathrm{wmax} \in \mathbf{N}_{\geq 4}$.

Although the limited-variables substrate fuds, $\mathcal{F}_{U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}}$, has coverage of the entire substrate even when the substrate volume, $v$, is greater than the underlying volume limit, for example $v > \mathrm{xmax}$, the derived volume is still strictly limited, $w \leq \mathrm{wmax}$. In section ‘Summation aligned decomposition inducers’ P582 it is conjectured that a summation aligned decomposition $D \in \mathcal{D}_{\Sigma}(A)$ is such that the content alignment equals the summation alignment, \begin{eqnarray} \mathrm{algn}(A * D^{\mathrm{T}}) - \mathrm{algn}(A^{\mathrm{X}} * D^{\mathrm{T}}) &=& \mathrm{alignmentSum}(A,D) \end{eqnarray} where \begin{eqnarray} \mathrm{alignmentSum}(A,D) &=& \sum \mathrm{algn}(A * C * T) : (C,T) \in \mathrm{cont}(D) \end{eqnarray} and $\mathrm{cont} = \mathrm{elements} \circ \mathrm{contingents}$. Thus, insofar as the content alignment approximates to the derived alignment, summing the derived alignments of the contingent fuds avoids the computation of the nullable transform, $D^{\mathrm{T}}$, which may have intractable derived volume, for example $w > \mathrm{wmax}$, where $w =|W^{\mathrm{C}}|$ and $W = \mathrm{der}(D^{\mathrm{T}})$. Just as above where the set of inducer models is increased from substrate transforms, $\mathcal{T}_{U_A,V_A}$, to substrate fuds, $\mathcal{F}_{U_A,V_A}$, the inducer models is again expanded to the substrate fud decompositions $\mathcal{D}_{\mathrm{F},U_A,V_A}$. The set of inducer models is then the limited-variables substrate fud decompositions, $\mathcal{D}_{\mathrm{F},U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}}))$. Given a limited-variables substrate fud decomposition $D \in \mathcal{D}_{\mathrm{F},U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}}))$, the inducer computes the tractable sum of the contingent derived alignments, $\sum \mathrm{algn}(A * C * F^{\mathrm{T}}) : (C,F) \in \mathrm{cont}(D)$.

### Intractable search set elements

Third, the computation of the search set models is intractable for two reasons, (i) fud flattening, and (ii) layer variables cardinality P599. The computation of the finite substrate fud set, $\mathcal{F}_{U_A,V_A}$, requires the exclusion of duplicate nested partitions. This is done by checking for the uniqueness of the flattened partitions. This check is intractable so the substrate fud set, $\mathcal{F}_{U_A,V_A}$, is replaced by the intersection of (i) the infinite-layer substrate fud set $\mathcal{F}_{\infty,U_A,V_A}$, which dispenses with the check, and (ii) one of the class of the sets of limited-layer fuds, $\mathcal{F}_{\mathrm{h}}$. The limited-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{h}}$, places a limit on the number of layers, for example, a maximum layer limit of $\mathrm{lmax} \in \mathbf{N}_{>0}$.

The layer variables cardinality intractability is because of exponential time complexity of the computation of the layers of the fuds. This is addressed by defining one of the class of the sets of limited-breadth fuds, $\mathcal{F}_{\mathrm{b}}$. For example a maximum layer breadth limit of $\mathrm{bmax} \in \mathbf{N}_{>0}$. Together the classes of limits are intersected together to form the class of limited-models $\mathcal{F}_{\mathrm{q}} = \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}} \cap \mathcal{F}_{\mathrm{h}} \cap \mathcal{F}_{\mathrm{b}}$. The set of inducer models is then the limited-models infinite-layer fud decompositions, $\mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}})$.

### Intractable literal substrate model inclusion

Last, the computation of the literal substrate model inclusion is intractable P606. Consider the remaining two constraints, (i) formal-abstract equality, $A^{\mathrm{X}} * T = (A * T)^{\mathrm{X}}$, and (ii) ideal transform, $A = A * T * T^{\dagger A}$. As described in section ‘Intractable literal substrate model inclusion’ P606, both of these inclusion tests of the model are intractable with respect to substrate volume. Now consider how inducers can be made tractable while adhering to these constraints as closely as possible.

First, the formal-abstract equality is weakened to the independent-formal constraint, $A^{\mathrm{X}} * T = (A^{\mathrm{X}} * T)^{\mathrm{X}}$, in the derived alignment substrate ideal independent-formal transform inducer, $I_{z,\mathrm{a,fx,j}}^{‘}$. This constraint is still intractable, so it is replaced by constraining the transforms to be non-overlapping, $\begin{eqnarray} \neg \mathrm{overlap}(T) &\implies& A^{\mathrm{X}} * T = (A^{\mathrm{X}} * T)^{\mathrm{X}} \end{eqnarray}$ in the derived alignment substrate ideal non-overlapping transform inducer, $I_{z,\mathrm{a,n,j}}^{‘}$. If the ideality inclusion test is dropped and the inducer model set of substrate transforms, $\mathcal{T}_{U_A,V_A}$, is replaced by the limited-models infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$, then the tractable limited-models derived alignment substrate non-overlapping infinite-layer fud inducer $I_{z,\mathrm{a,F,\infty,n,q}}^{‘} \in \mathrm{inducers}(z)$, given substrate histogram $A \in \mathcal{A}_{z}$, can be defined $\begin{eqnarray} &&I_{z,\mathrm{a,F,\infty,n,q}}^{‘ * }(A) = \\ &&\hspace{5em}\{(F,I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A * F^{\mathrm{T}}))) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}\} \end{eqnarray}$ where $\mathcal{F}_{\mathrm{n}}$ is the set of non-overlapping fuds. Although this inducer is tractable, the non-overlapping constraint is weaker than the formal-abstract equality constraint and the ideality constraint has been dropped altogether. The entropy of the doubly-independent formal independent histogram, $\mathrm{entropy}((A^{\mathrm{X}} * T)^{\mathrm{X}})$, is expected to be greater than the entropy of the abstract histogram, $\mathrm{entropy}((A * T)^{\mathrm{X}})$, whereas if the formal-abstract equality constraint holds then the entropies would be equal. The abstract-non-formal entropy substrate ideal independent-formal transform inducer, $I_{z,\mathrm{e,fx,j}}^{‘} \in \mathrm{inducers}(z)$, maximises the entropy difference between the abstract and the formal independent. The inducer is defined $\begin{eqnarray} &&I_{z,\mathrm{e,fx,j}}^{‘ * }(A) = \\ &&\hspace{2em}\{(T,I_{\approx\ln\mathbf{Q}}^{ * }(\mathrm{entropy}((A * T)^{\mathrm{X}}) - \mathrm{entropy}((A^{\mathrm{X}} * T)^{\mathrm{X}}))) : \\ &&\hspace{5em}T \in \mathcal{T}_{U_A,V_A},~A^{\mathrm{X}} * T = (A^{\mathrm{X}} * T)^{\mathrm{X}},~A = A * T * T^{\dagger A}\} \end{eqnarray}$ Derived alignment approximates to the sized entropy difference between the abstract histogram and the derived histogram, $\begin{eqnarray} \mathrm{algn}(A * T) & \approx & z \times (\mathrm{entropy}((A * T)^{\mathrm{X}}) - \mathrm{entropy}(A * T)) \end{eqnarray}$ so the abstract-non-formal entropy inducer, $I_{z,\mathrm{e,fx,j}}^{‘}$, weakly maximises the derived alignment.

The abstract-non-formal entropy inducer is defined in terms of the entropies of histograms in the derived variables, $\mathrm{entropy}((A * T)^{\mathrm{X}})$ and $\mathrm{entropy}((A^{\mathrm{X}} * T)^{\mathrm{X}})$, and so ignores the entropies of the underlying components, $\{(C,\mathrm{entropy}(A * C)) : (\cdot,C) \in T^{-1}\}$. The discussion considers the actualisations, which alter the relative independence of the derived and underlying, and then proposes an inducer that maximises the midisation pseudo-alignment, $\mathrm{algn}(A) - \mathrm{algn}(A * T * T^{\dagger A}) - \mathrm{algn}((A * T)^{\mathrm{X}} * T^{\odot A})$. However, the ideality constraint restricts the midisation pseudo-alignment to be equal to the negative surrealisation alignment, so the ideality constraint is dropped. The midisation pseudo-alignment substrate independent-formal transform inducer $I_{z,\mathrm{m,fx}} \in \mathrm{inducers}(z)$, given substrate histogram $A \in \mathcal{A}_{z}$, is defined $\begin{eqnarray} &&I_{z,\mathrm{m,fx}}^{ * }(A) = \\ &&\hspace{2em}\{(T,I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A) - \mathrm{algn}(A * T * T^{\dagger A}) - \mathrm{algn}((A * T)^{\mathrm{X}} * T^{\odot A}))) : \\ &&\hspace{5em}T \in \mathcal{T}_{U_A,V_A},~A^{\mathrm{X}} * T = (A^{\mathrm{X}} * T)^{\mathrm{X}}\} \end{eqnarray}$ The computation of midisation is intractable, so a further approximation is required. Maximisation of midisation tends to move component alignments from off-diagonal states to on-diagonal states, balancing the high derived alignment of longer diagonals with the high on-diagonal component alignments of shorter diagonals. Thus the midisation pseudo-alignment varies with the derived alignment valency density. The derived alignment valency-density substrate independent-formal transform inducer $I_{z,\mathrm{ad,fx}}^{‘} \in \mathrm{inducers}(z)$, given substrate histogram $A \in \mathcal{A}_{z}$, is defined $\begin{eqnarray} &&I_{z,\mathrm{ad,fx}}^{‘ * }(A) = \\ &&\hspace{2em}\{(T,I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A * T)/w_T^{1/m_T})) : T \in \mathcal{T}_{U_A,V_A},~A^{\mathrm{X}} * T = (A^{\mathrm{X}} * T)^{\mathrm{X}}\} \end{eqnarray}$ where derived variables $W_T = \mathrm{der}(T)$, derived volume $w_T = |W_T^{\mathrm{C}}|$, derived dimension $m_T = |W_T|$ and $I_{\approx\mathbf{R}}^{ * }$ computes an approximation to a real number. The geometric average of the derived valencies is $w_T^{1/m_T}$.

If the independent formal constraint is replaced by constraining the transforms to be non-overlapping, and the inducer model set of substrate transforms, $\mathcal{T}_{U_A,V_A}$, is replaced by the limited-models infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$, then the tractable limited-models derived alignment valency-density substrate non-overlapping infinite-layer fud inducer $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘} \in \mathrm{inducers}(z)$, given substrate histogram $A \in \mathcal{A}_{z}$, can be defined as $\begin{eqnarray} &&I_{z,\mathrm{ad,F,\infty,n,q}}^{‘ * }(A) = \\ &&\hspace{5em}\{(F,I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A * F^{\mathrm{T}})/w_F^{1/m_F})) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}\} \end{eqnarray}$ This derived alignment valency-density fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$, addresses the formal-abstract equality constraint, $A^{\mathrm{X}} * T = (A * T)^{\mathrm{X}}$, but ignores the ideal transform constraint, $A = A * T * T^{\dagger A}$. As described above, intractable derived volume can be addressed by expanding the inducer models to the substrate fud decompositions, $\mathcal{D}_{\mathrm{F},U_A,V_A}$, and summing the derived alignments of the contingent fuds, $\begin{eqnarray} \sum \mathrm{algn}(A * C * F^{\mathrm{T}}) : (C,F) \in \mathrm{cont}(D) \end{eqnarray}$ Using a similar method, sections ‘Decomposition alignment’ P491 and ‘Tractable decomposition inducers’ P621 show how maximising the sum of the contingent alignment valency-densities, $\begin{eqnarray} \sum \mathrm{algn}(A * C * F^{\mathrm{T}})/w_F^{1/m_F} : (C,F) \in \mathrm{cont}(D) \end{eqnarray}$ of limited-models non-overlapping infinite-layer fud decompositions, $\mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}))$, removes alignments along the decomposition path and tends to independent leaf components. When fully decomposed the nullable transform of the decomposition is ideal, $A * D^{\mathrm{T}} * D^{\mathrm{T}\dagger A} = A$. The tractable limited-models summed alignment valency-density substrate aligned non-overlapping infinite-layer fud decomposition inducer, given non-independent substrate histogram $A \in \mathcal{A}_{z} \setminus \{A^{\mathrm{X}}\}$, is defined $\begin{eqnarray} &&I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘ * }(A) = \\ &&\hspace{2em}\{(D,I_{\approx\mathbf{R}}^{ * }(\sum \mathrm{algn}(A * C * F^{\mathrm{T}})/w_F^{1/m_F} : (C,F) \in \mathrm{cont}(D))) : \\ &&\hspace{5em}D \in \mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}})),\\ &&\hspace{10em}\forall (C,F) \in \mathrm{cont}(D)~(\mathrm{algn}(A * C * F^{\mathrm{T}})>0)\} \end{eqnarray}$

Here the model has been extended from transforms, $M \in \mathcal{T}_{U,V}$, to functional definition set decompositions, $M \in \mathcal{D}_{\mathrm{F},\infty,U,V}$. At the same time the set of fud decompositions has been restricted to those having (a) fuds that are non-overlapping, $\mathcal{F}_{\mathrm{n}}$, (b) fuds with a limited-underlying, limited-derived, limited-layer and limited-breadth structure, $\mathcal{F}_{\mathrm{q}} = \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}} \cap \mathcal{F}_{\mathrm{h}} \cap \mathcal{F}_{\mathrm{b}}$, and (c) fuds with derived alignment, $\mathrm{algn}(A * C * F^{\mathrm{T}})>0$. The tractable optimal model is $\begin{eqnarray} D_{\mathrm{Sd}}~\in~\mathrm{maxd}(I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘ * }(A)) \end{eqnarray}$ The limited-models summed alignment valency-density substrate aligned non-overlapping infinite-layer fud decomposition inducer, $I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘}$, limits the optimisation to make aligned induction tractable. By additionally imposing a sequence on the search and other constraints, tractable induction can be made practicable in the highest-layer summed shuffle content alignment valency-density fud decomposition inducer, $I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{d}}^{‘}$.

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