# Aligned Induction

## Inducer preliminaries

Haskell commentary on the implementation of Tractable and Practicable Inducers/Inducer preliminaries

### Sections

Computers

Sized cardinal substrate histograms

Inducers

### Computers

The set of computers is discussed in section ‘Computation time and representation space’ P402 and ‘Computation of alignment’ P563. The set of computers, $\mathrm{computers}$, is a type class that formalises computation time and representation space. Define the application of a computer, $\mathrm{apply} \in \mathrm{computers} \to (\mathcal{X} \to \mathcal{Y})$. Define the shorthand $I^{ * } := \mathrm{apply}(I)$. Define the domain of the application, $\mathrm{domain} \in \mathrm{computers} \to \mathrm{P}(\mathcal{X})$, and the range of the application, $\mathrm{range} \in \mathrm{computers} \to \mathrm{P}(\mathcal{Y})$, such that $\forall I \in \mathrm{computers}~(I^{ * } \in \mathrm{domain}(I) \to \mathrm{range}(I))$ and $\forall I \in \mathrm{computers}~(\mathrm{dom}(I^{ * }) = \mathrm{domain}(I))$. The computation or application time is defined as $\mathrm{time} \in \mathrm{computers} \to (\mathcal{X} \to \mathbf{N}_{>0})$. Define the shorthand $I^{\mathrm{t}} := \mathrm{time}(I)$. The representation space is defined as $\mathrm{space} \in \mathrm{computers} \to (\mathcal{X} \to \ln \mathbf{N}_{>0})$. Define the shorthand $I^{\mathrm{s}} := \mathrm{space}(I)$. See appendix `Computers’ P1134 for a more formal definition.

### Sized cardinal substrate histograms

The set of integral substrate histograms, $\mathcal{A}_{U,\mathrm{i},V,z}$, discussed in the ‘Overview’ section ‘States, histories and histograms’ P13, is generalised for the purposes of tractable and practicable induction. The sized cardinal substrate histograms, $\mathcal{A}_z$, are defined in section ‘Distinct geometry sized cardinal substrate histograms’ P280. Let the set of sized cardinal substrate histograms $\mathcal{A}_z$ be the set of complete integral cardinal substrate histograms of size $z$ and dimension less than or equal to the size such that the independent is completely effective $\begin{eqnarray} \mathcal{A}_z &=& \{A : A \in \mathcal{A}_{\mathrm{c}} \cap \mathcal{A}_{\mathrm{i}},~\mathrm{size}(A) = z,~|V_A| \leq z,~A^{\mathrm{U}} = A^{\mathrm{XF}} = A^{\mathrm{C}}\} \end{eqnarray}$ where $A^{\mathrm{CS}} = \mathrm{cartesian}(U_A)(V_A)$ and $U_A = \mathrm{implied}(\mathrm{implied}(A))$ and $V_A = \mathrm{vars}(A)$. There is no single system that contains all the substrate histograms. The infinite implied system, $U_A$ where $A \in \mathcal{A}_z$, contains the substrate variables, $V_A \subset \mathrm{vars}(U_A)$, and all the partition variables in the power functional definition set on $V_A$, $\forall F \in \mathcal{F}_{U_A,V_A}~(\mathrm{vars}(F) \subset \mathrm{vars}(U_A))$. Define the subset of the sized cardinal substrate histograms, $\mathcal{A}_z$, for which the independent, $A^{\mathrm{X}}$, is integral, and therefore also a substrate histogram, as the integral-independent substrate histograms, $\begin{eqnarray} \mathcal{A}_{z,\mathrm{xi}} &=& \{A : A \in \mathcal{A}_z,~A^{\mathrm{X}} \in \mathcal{A}_{\mathrm{i}}\}~=~\{A : A, A^{\mathrm{X}} \in \mathcal{A}_z\} \subset \mathcal{A}_z \end{eqnarray}$

### Inducers

The set of inducers is discussed in section ‘Tractable alignment-bounding’ P568. The inducers are computers $I_z \in \mathrm{inducers}(z) \subset \mathrm{computers}$ such that (i) the domain is a set of substrate histograms which are at least a superset of the integral-independent substrate histograms, $\mathcal{A}_{z,\mathrm{xi}} \subseteq \mathrm{domain}(I_z) \subseteq \mathcal{A}_{z}$, (ii) the finite time and space application returns a rational-valued function of the substrate models set, $I_z^{ * }(A) \in \mathcal{M}_{U_A,V_A} \to \mathbf{Q}$, and (iii) the maximum of the inducer application, $\mathrm{maxr} \circ I_z^{ * }$, is positively correlated with the finite alignment-bounded iso-transform space ideal transform maximum function, $\mathrm{maxr} \circ X_{z,\mathrm{xi,T,y,fa,j}}$ (defined in section ‘Iso-transform-independent conditional’ P555). That is, the induction correlation of inducer $I_z$ is positive.

The literal derived alignment integral-independent substrate ideal formal-abstract transform inducer $I_{z,\mathrm{a,l}}^{‘} \in \mathrm{inducers}(z)$ is a literal finite approximation of the maximisation of the derived alignment, $\begin{eqnarray} I_{z,\mathrm{a,l}}^{‘ * }(A) &=& \{(T,I_{\approx\mathrm{ln}\mathbf{Q}}^{ * }(\mathrm{algn}(A * T))) : \\ &&\hspace{2em}T \in \mathcal{T}_{U_A,V_A},~A^{\mathrm{X}} * T = (A * T)^{\mathrm{X}},~A = A * T * T^{\dagger A}\} \end{eqnarray}$ Now consider the definition of the class of tractable inducers.

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