Inducer preliminaries
Haskell commentary on the implementation of Tractable and Practicable Inducers/Inducer preliminaries
Sections
Sized cardinal substrate histograms
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.