Tuple Optimisation
Haskell commentary on the implementation of Tractable and Practicable Inducers/Tuple Optimisation
Sections
Contracted decrementing linear non-overlapping fuds list maximiser
Limited-underlying tuple set list maximiser
Contracted decrementing linear non-overlapping fuds list maximiser
Given the limitations of the limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘}$, and given that optimisation in a practicable inducer is necessary because of limited time or space, consider an alternative method of optimisation that maximises maximum function correlation by optimising within the tuple.
The cardinality of the second term of the upper bound on the cardinality of the neighbourhood function of the limited-path-models tuple partition inducer, $|\mathrm{B}(K^{\mathrm{CS}})| \leq \mathrm{bell}(\mathrm{xmax})$ where $K \in \mathrm{tuples}(V,F)$ and $|K^{\mathrm{C}}| \leq \mathrm{xmax}$, can be addressed by choosing only a subset $Q$ of the partitions of a tuple, $Q \subset \mathrm{B}(K^{\mathrm{CS}})$. The cardinality of the subset must be less than or equal to the maximum optimise step cardinality $|Q| \leq \mathrm{omax}$. If the subset, $Q$, is chosen arbitrarily then the practicable inducer correlation can be expected to be reduced. However, considered by itself a single partition of the subset $P \in Q$ cannot have an optimise function that depends on alignment because a fud $G$ constructed from it must be independent, $A * G^{\mathrm{T}} = (A * G^{\mathrm{T}})^{\mathrm{X}} \implies \mathrm{algn}(A * G^{\mathrm{T}}) = 0$, where $G = \mathrm{depends}(F \cup \{P^{\mathrm{T}}\},\{P\}) = \mathrm{depends}(F,K) \cup \{P^{\mathrm{T}}\}$. This is because the constructed fud, $G$, is mono-derived-variate, $|\mathrm{der}(G)|=1$.
Section ‘Substrate models computation’ P634 defines the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V) \in \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$, which constructs the layer-ordered fuds from sets of partition transforms of the tuple, $\{P^{\mathrm{T}} : P \in \mathrm{B}(K^{\mathrm{CS}})\}$. The discussion then goes on to define the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnmubh}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, which constructs the fuds with non-overlapping substrate transforms of the tuples, $\mathcal{T}_{U,K,\mathrm{n}}$, such that the regular substrate cardinalities may be computed. The latter fud tree constructs transforms of the limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}}$, which is defined as the limited-breadth non-overlapping substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{n},\mathrm{bmax}}$, applied to the tuple, \[ \begin{eqnarray} \mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}} &=& \{N : Y \in \mathrm{B}(K),~|Y| \leq \mathrm{mmax},~N \in \prod_{J \in Y} \mathrm{B}(J^{\mathrm{CS}})\} \end{eqnarray} \] Consider a stricter set of transforms of the partition-sets on the tuple such that the derived variables are pluri-variate and pluri-valent, which avoid necessarily independent derived histograms that have zero alignment. The pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$, is the intersection of the lower-limited-valency substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{umin}}$, where $\mathrm{umin}=2$, and the range-limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\mathrm{mran}}$, where $\mathrm{mran} = (2,\mathrm{mmax})$ and $\mathrm{mmax} \geq 2$, is defined \[ \begin{eqnarray} &&\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}} = \\ &&\hspace{2em}\{N : Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},~N \in \prod_{J \in Y} (\mathrm{B}(J^{\mathrm{CS}}) \setminus \{\{J^{\mathrm{CS}}\}\})\} \end{eqnarray} \] Therefore, given pluri-variate tuple, $|K|>1$, consider instead a subset of the transforms of the pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets set of the tuple $Q \subset \{N^{\mathrm{T}} : N \in \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}\}$ such that $|Q| \leq \mathrm{pmax}$, where the maximum tuple optimise limit is $\mathrm{pmax} = \lfloor \mathrm{omax}/\mathrm{bmax} \rfloor \in \mathbf{N}_{>0}$ and $\mathrm{omax} \geq \mathrm{bmax}$. The derived variables are pluri-variate and pluri-valent, $\forall T \in Q~(|\mathrm{der}(T)|>1)$, and $\forall T \in Q~\forall w \in \mathrm{der}(T)~(|U_w|>1)$. Each of these transforms has a corresponding partition, $\forall T \in Q~(T^{\mathrm{P}K} \in \mathrm{B}(K^{\mathrm{CS}}))$, but now are not restricted to zero derived alignment. That is, in some cases, $A * G^{\mathrm{T}} \neq (A * G^{\mathrm{T}})^{\mathrm{X}}$, where $T \in Q$ and $G = \mathrm{depends}(F \cup \{T\},\mathrm{der}(T))$.
The tuple transform, $T \in Q$, is non-overlapping, $\forall P_1, P_2 \in \mathrm{der}(T)~(P_1 \neq P_2 \implies \mathrm{vars}(P_1) \cap \mathrm{vars}(P_2) = \emptyset)$. That is, there exists a partition $Y$ of the tuple, $Y \in \mathrm{B}(K)$, which is such that $\exists M \in \mathrm{der}(T) :\leftrightarrow Y~\forall (P,J) \in M~(\mathrm{vars}(P)=J)$. However, the constructed exploded fud $G’ = \mathrm{explode}(G)$ is not necessarily non-overlapping if $\mathrm{overlap}(\mathrm{depends}(F,K))$, so the tuple transforms, $Q$, are chosen by maximising the shuffle content alignment valency-density rather than derived alignment valency-density.
Together the optimised subset is defined \[ \begin{eqnarray} &&Q = \mathrm{topd}(\mathrm{pmax})(\{(N^{\mathrm{T}},(\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}}))/\mathrm{cvl}(G)) : \\ &&\hspace{8em}N \in \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}},~G = \mathrm{depends}(F \cup \{N^{\mathrm{T}}\},N)\}) \end{eqnarray} \] Maximising the shuffle content alignment, $\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}})$, tends to minimise the formal alignment, $\mathrm{algn}(A^{\mathrm{X}} * G^{\mathrm{T}})$. The formal alignment is zero when non-overlapping, $\neg \mathrm{overlap}(G’) \implies \mathrm{algn}(A^{\mathrm{X}} * G^{\mathrm{T}}) = 0$.
As mentioned above, an alternative method to shuffle content alignment would be to simply exclude overlapping constructed exploded fuds, $\mathrm{overlap}(G’)$, from the optimisation, but note that each fud, $G’$, must be tested because it is insufficient to exclude it on the basis that the tuple, $K$, is overlapping, $\mathrm{overlap}(\mathrm{depends}(F,K)) \Longleftarrow \mathrm{overlap}(G’)$.
As shown in section ‘Substrate structures’ P181, the cardinality of the strong non-overlapping substrate transforms set is bounded \[ \mathrm{bell}(|K^{\mathrm{C}}|) \leq |\{N^{\mathrm{T}K} : N \in \mathcal{N}_{U,K,\mathrm{n}}\}| \leq \mathrm{bell}(|K|) \times \mathrm{bell}(|K^{\mathrm{C}}|) \] That is, to search for the optimised subset from the strong non-overlapping substrate transforms set, $Q \subset \{N^{\mathrm{T}} : N \in \mathcal{N}_{U,K,\mathrm{n}}\}$, would require computation time in some cases of $\mathrm{bell}(\mathrm{kmax}) \times \mathrm{bell}(\mathrm{xmax})$, if the tuple volume equals the maximum underlying volume limit, $|K^{\mathrm{C}}| = \mathrm{xmax}$. The lower bound is the cardinality of the partitions of the tuple, $\mathrm{bell}(\mathrm{xmax})$. So the search for non-overlapping substrate transforms of the tuple requires more computation time than the search for partition transforms of the tuple. As shown in section ‘Substrate models computation’ the cardinality of the searched list of the limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree $\mathrm{tfitnubh}(U)(V)$, is greater than or equal to the cardinality of the searched list of the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree $\mathrm{tfiubh}(U)(V)$, \[ |\mathrm{subpaths}(\mathrm{tfitnubh}(U)(V))| \geq |\mathrm{subpaths}(\mathrm{tfiubh}(U)(V))| \] However, only a subset of the derived histograms of the strong non-overlapping substrate transforms are non-independent, and hence aligned, so only this subset is searched. Even in the case where the maximum derived dimension equals the tuple dimension, $\mathrm{mmax}=k$, the exclusion of the partitions of the unary partition of the tuple, $\mathrm{B}(K^{\mathrm{CS}})$, because of the maximum derived dimension, $\mathrm{mmin}=2$, means that the cardinality of the pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$, is less than or equal to the cardinality of the partitions of the tuple, $|\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}| \leq |\mathrm{B}(K^{\mathrm{CS}})|$. This is because the Bell number is log-convex.
In the case where computational resources are still exceeded by this cardinality, $|\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}|$, consider searches restricted to subsets of the transforms of the pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$.
Consider the subset of the pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$, where the partition of the tuple is given, in the case where the tuple is at least bi-variate, $|K|>1$. Let the given partition be $Y \in \mathrm{B}(K) \setminus \{\{K\}\}$. Then \[ \prod_{J \in Y} (\mathrm{B}(J^{\mathrm{CS}}) \setminus \{\{J^{\mathrm{CS}}\}\}) \subset \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}} \] The cardinality of this set is $\prod_{J \in Y} (\mathrm{bell}(|J^{\mathrm{CS}}|)-1) < \mathrm{bell}(\mathrm{xmax})$. The computation time is comparable at least to $\mathrm{bell}(\mathrm{xmax}^{1/\mathrm{mmax}})^\mathrm{mmax} < \mathrm{bell}(\mathrm{xmax})$.
Consider an optimisation where the search is broken into two separate searches. The first search determines the partition of the tuple, $Y$, by searching the transforms of the intersection of the substrate self-cartesian partition-sets set, $\mathcal{N}_{U,K,\mathrm{c}}$, and the pluri-limited-tuple-derived-dimension non-overlapping substrate transforms set, $\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}}$, where $2 \leq \mathrm{mmax} \leq |K|$, which is \[ \mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}} = \{\{J^{\mathrm{CS}\{\}} : J \in Y\} : m \in \{2 \ldots \mathrm{mmax}\},~Y \in \mathrm{S}(K,m)\} \] The tuple partition search is optimised by maximising the shuffle content alignment valency-density, (Haskell) \[ \begin{eqnarray} &&Y \in \mathrm{maxd}(\{(Z,(\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}}))/\mathrm{cvl}(G)) : \\ &&\hspace{8em}m \in \{2 \ldots \mathrm{mmax}\},~Z \in \mathrm{S}(K,m),\\ &&\hspace{8em}N = \{J^{\mathrm{CS}\{\}} : J \in Z\},~G = \mathrm{depends}(F \cup \{N^{\mathrm{T}}\},N)\}) \end{eqnarray} \] Then, given the partition, $Y$, of the tuple, the second search, optimised by shuffle content alignment valency-density, is, (Haskell) \[ \begin{eqnarray} &&Q = \mathrm{topd}(\mathrm{pmax})(\{(N^{\mathrm{T}},(\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}}))/\mathrm{cvl}(G)) : \\ &&\hspace{5em}N \in \prod_{J \in Y} (\mathrm{B}(J^{\mathrm{CS}}) \setminus \{\{J^{\mathrm{CS}}\}\}),~G = \mathrm{depends}(F \cup \{N^{\mathrm{T}}\},N)\}) \end{eqnarray} \] The tuple partition search term is comparable to $\mathrm{bell}(\mathrm{kmax})$. The transforms search term is comparable to $\mathrm{bell}(\mathrm{xmax}^{1/\mathrm{mmax}})^\mathrm{mmax}$. The computation time of the transforms search, $Q$, dominates that of the tuple partition search $Y$.
Having discussed the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnmubh}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, which constructs the fuds with non-overlapping substrate transforms of the tuples, $\mathcal{T}_{U,K,\mathrm{n}}$, section ‘Substrate models computation’ goes on to consider the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted decrementing linear non-overlapping fuds infinite-layer fud tree $\mathrm{tfifdnmubh}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, which constructs the fuds with strong limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds, $\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}$, on the tuples, defined \[ \begin{eqnarray} &&\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}\\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : M \in \mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n,mmax}},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : Y \in \mathrm{B}(K),~|Y| \leq \mathrm{mmax},~M = \{J^{\mathrm{CS}\{\}} : J \in Y\},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \end{eqnarray} \] where the tree of self non-overlapping substrate decremented partition-sets is defined $\mathrm{tdec}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathrm{P}(\mathcal{R}_U))$ as \[ \mathrm{tdec}(U)(M) := \{(N,\mathrm{tdec}(U)(N)) : N \in \mathcal{N}_{U,M,-} \cap \mathcal{N}_{U,M,\mathrm{n,s}}\} \] and $\mathrm{tdec}(U)(\emptyset) :=\emptyset$. Explicitly this is \[ \begin{eqnarray} &&\mathrm{tdec}(U)(M) := \{(N,\mathrm{tdec}(U)(N)) :\\ &&\hspace{2em}w \in M,~Q \in \mathrm{decs}(\{w\}^{\mathrm{CS}\{\}}),~N = \{Q\} \cup \{\{u\}^{\mathrm{CS}\{\}} : u \in M,~u \neq w\}\} \end{eqnarray} \] where $\mathrm{decs} = \mathrm{decrements} \in \mathcal{R}_U \to \mathrm{P}(\mathcal{R}_U)$ and \[ \mathrm{decrements}(P) = \{P \setminus \{C,D\} \cup \{C \cup D\} : C,D \in P,~C \neq D\} \] Instead of a subset of the transforms of the pluri-valent pluri-limited-tuple-derived-dimension non-overlapping substrate partition-sets of the tuple, $Q \subset \{N^{\mathrm{T}} : N \in \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}\}$, consider a subset of the pluri-valent pluri-limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds of the tuple $Q \subset \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$ such that $|Q| \leq \mathrm{pmax}$. The pluri-valent pluri-limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds is defined \[ \begin{eqnarray} &&\mathcal{F}_{U,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}\\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : M \in \mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdecp}(U)(M))\})\} \\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},~M = \{J^{\mathrm{CS}\{\}} : J \in Y\},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdecp}(U)(M))\})\} \end{eqnarray} \] where $\mathrm{mmax} \geq 2$ and the tree of pluri-valent self non-overlapping substrate decremented partition-sets is defined $\mathrm{tdecp}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathrm{P}(\mathcal{R}_U))$ as \[ \mathrm{tdecp}(U)(M) := \{(N,\mathrm{tdecp}(U)(N)) : N \in \mathcal{N}_{U,M,\overline{2}} \cap \mathcal{N}_{U,M,-} \cap \mathcal{N}_{U,M,\mathrm{n,s}}\} \] and $\mathrm{tdecp}(U)(\emptyset) :=\emptyset$. Explicitly this is \[ \begin{eqnarray} &&\mathrm{tdecp}(U)(M) := \{(N,\mathrm{tdecp}(U)(N)) :\\ &&\hspace{10em}w \in M,~|\{w\}^{\mathrm{C}}| > 2,~Q \in \mathrm{decs}(\{w\}^{\mathrm{CS}\{\}}),\\ &&\hspace{10em}N = \{Q\} \cup \{\{u\}^{\mathrm{CS}\{\}} : u \in M,~u \neq w\}\} \end{eqnarray} \] If the tuple is pluri-variate and pluri-valent, $|K|>1$, and $\forall v \in K~(|U_v|>1)$, then the derived variables are pluri-variate and pluri-valent, $\forall H \in Q~(|\mathrm{der}(H)|>1)$, and $\forall H \in Q~\forall w \in \mathrm{der}(H)~(|U_w|>1)$. Each of these fuds has a corresponding partition, $\forall H \in Q~(H^{\mathrm{TP}K} \in \mathrm{B}(K^{\mathrm{CS}}))$, but now are not restricted to zero derived alignment. That is, in some cases, $A * G^{\mathrm{T}} \neq (A * G^{\mathrm{T}})^{\mathrm{X}}$, where $H \in Q$ and $G = \mathrm{depends}(F \cup H,\mathrm{der}(H))$. The tuple fud, $H$, is non-overlapping, $\neg \mathrm{overlap}(H)$. However, the constructed exploded fud $G’ = \mathrm{explode}(G)$ is not necessarily non-overlapping so the choice of tuple fuds, $Q \subset \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$, is made by maximising the shuffle content alignment valency-density, \[ \begin{eqnarray} &&Q = \mathrm{topd}(\mathrm{pmax})(\{(H,(\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}}))/\mathrm{cvl}(G)) : \\ &&\hspace{5em}H \in \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}},~G = \mathrm{depends}(F \cup H,\mathrm{der}(H))\}) \end{eqnarray} \] The cardinality of the contracted decrementing linear non-overlapping fuds is greater than or equal to the cardinality of the non-overlapping substrate transforms, $|\mathcal{F}_{U,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}| \geq |\mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}|$, so the computation time is increased. However, consider the optimisation of the tree of decrements. Define the contracted decrementing linear non-overlapping fuds list maximiser \[ \begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{n},-,K} =\\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{n},-,K},N_{P,A,A_R,F,\mathrm{n},-,K},\mathrm{top}(\mathrm{pmax}),R_{P,A,A_R,F,\mathrm{n},-,K}) \end{eqnarray} \] where (i) the optimiser function is \[ \begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{n},-,K} = \{(H,(I_{\mathrm{a}}^{ * }(A * G^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * G^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(G)) : \\ &&\hspace{10em}H \in \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}},~G = \mathrm{depends}(F \cup H,\mathrm{der}(H))\} \end{eqnarray} \] (ii) the initial subset is (Haskell) \[ \begin{eqnarray} &&R_{P,A,A_R,F,\mathrm{n},-,K} = \{(\{M^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(\{M^{\mathrm{T}}\})) : \\ &&\hspace{10em}Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},~M = \{J^{\mathrm{CS}\{\}} : J \in Y\}\} \end{eqnarray} \] and (iii) the neighbourhood function is (Haskell) \[ \begin{eqnarray} &&N_{P,A,A_R,F,\mathrm{n},-,K}(C) = \{(H \cup \{N^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(H \cup \{N^{\mathrm{T}}\})) :\\ &&\hspace{10em}(H,\cdot) \in C,~M = \mathrm{der}(H),\\ &&\hspace{10em}w \in M,~|\{w\}^{\mathrm{C}}| > 2,~Q \in \mathrm{decs}(\{w\}^{\mathrm{CS}\{\}}),\\ &&\hspace{10em}N = \{Q\} \cup \{\{u\}^{\mathrm{CS}\{\}} : u \in M,~u \neq w\}\} \end{eqnarray} \] Then the subset of the decrementing linear non-overlapping fuds is \[ \mathrm{dom}(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})) \subset \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}} \] So the choice of tuple fuds, $Q \subset \mathcal{F}_{U_A,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2}}$, can be made by maximising the shuffle content alignment valency-density of the elements of the tree maximiser, $Z_{P,A,A_R,F,\mathrm{n},-,K}$, (Haskell) \[ \begin{eqnarray} Q = \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})) \end{eqnarray} \] If it is the case that the $\mathrm{topd}(\mathrm{pmax})$ optimisation is such that each step of the list has no more than $\mathrm{pmax}$ fuds, $\forall (i,C) \in \mathrm{list}(Z_{P,A,A_R,F,\mathrm{n},-,K})~(|C| \leq \mathrm{pmax})$, then the cardinality of the elements must be less than (a) the maximum tuple optimise limit, $\mathrm{pmax}$, times (b) the length of the longest path of decrements, $|K^{\mathrm{C}}|/2$. Thus $|\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})| < \mathrm{pmax} \times \mathrm{xmax}/2$. The cardinality of the searched must be less than or equal to (i) the cardinality of the initial set, $|R_{P,A,A_R,F,\mathrm{n},-,K}| = |\mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}}| \leq \mathrm{bell}(|K|)-1$, plus (ii) (a) the cardinality of the elements, $|\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})|$, times (b) the cardinality of the decrements, which is less than $|\mathcal{N}_{U,M,\overline{2}} \cap \mathcal{N}_{U,M,-} \cap \mathcal{N}_{U,M,\mathrm{n,s}}|$. If the tuple is partitioned into a singleton component and a remainder component the worst case cardinality of decrements is less than $\mathrm{xmax}^2/2^3$. Thus cardinality of the searched set is constrained \[ |\mathrm{searched}(Z_{P,A,A_R,F,\mathrm{n},-,K})| \leq \mathrm{bell}(\mathrm{kmax}) + \mathrm{pmax} \times \mathrm{xmax}^3/2^4 \] If the maximum tuple optimise limit is one, $\mathrm{pmax} = 1$, this cardinality is smaller than that of the contracted non-overlapping substrate transforms, $|\mathcal{T}_{U,K,\mathrm{n}}| \leq \mathrm{bell}(\mathrm{kmax}) \times \mathrm{bell}(\mathrm{xmax})$.
A variation of the contracted decrementing linear non-overlapping fuds list maximiser is to restrict the neighbourhood optimisation to the maximum, $\mathrm{pmax}=1$, and apply $\mathrm{pmax}$ only to the initial set. To do this a tree tail maximiser is used with an inclusion function of $\mathrm{max}$ and the initial set is explicitly optimised beforehand. Define the maximum-roll contracted decrementing linear non-overlapping fuds tree maximiser (Haskell) \[ \begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mr}} =\\ &&\hspace{2em}\mathrm{maximiseTailTreer}(X_{P,A,A_R,F,\mathrm{n},-,K},N_{P,A,A_R,F,\mathrm{n},-,K},\mathrm{max},\\ &&\hspace{15em}\mathrm{top}(\mathrm{pmax})(R_{P,A,A_R,F,\mathrm{n},-,K})) \end{eqnarray} \] This restriction pushes the $\mathrm{pmax}$ path selection into the initial set rather than towards the end of the optimise path where sometimes different decrementing linear fuds roll to the same derived partition variables.
The decrementing maximiser initial set, $\mathrm{top}(\mathrm{pmax})(R_{P,A,A_R,F,\mathrm{n},-,K})$, is biased to larger tuple partition cardinalities, $|Y|$, where $Y \in \mathrm{B}(K)$. This is because the valency-capacity varies against the partition cardinality, $(\prod_{J \in Y} |J^{\mathrm{C}}|)^{1/|Y|} = |K^{\mathrm{C}}|^{1/|Y|}$. Therefore consider a variation of the maximum-roll maximiser that has an initial set of cardinality $\mathrm{pmax}$ for each of the possible tuple partition cardinalities, $m \in \{2 \ldots \mathrm{mmax}\}$. Define the maximum-roll-by-derived-dimension contracted decrementing linear non-overlapping fuds tree maximiser (Haskell) \[ \begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}} =\\ &&\hspace{2em}\mathrm{maximiseTailTreer}(X_{P,A,A_R,F,\mathrm{n},-,K},N_{P,A,A_R,F,\mathrm{n},-,K},\mathrm{max},R_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}}) \end{eqnarray} \] where the initial subset is \[ \begin{eqnarray} &&R_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}} = \\ &&\hspace{2em}\bigcup \{\mathrm{top}(\mathrm{pmax})(\{(\{M^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(\{M^{\mathrm{T}}\})) : \\ &&\hspace{10em}Y \in \mathrm{S}(K,m),~M = \{J^{\mathrm{CS}\{\}} : J \in Y\}\}) : m \in \{2 \ldots \mathrm{mmax}\}\} \end{eqnarray} \] Now the cardinality of the searched is constrained \[ |\mathrm{searched}(Z_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}})| \leq \mathrm{bell}(\mathrm{kmax}) + (\mathrm{mmax}-1) \times \mathrm{pmax} \times \mathrm{xmax}^3/2^4 \]
Another constraint that may be applied to reduce the cardinality of the searched set, $|\mathrm{searched}(Z_{P,A,A_R,F,\mathrm{n},-,K})|$, is to restrict the initial set such that the volume of each component of the tuple partition is limited. The maximum valency is $\mathrm{umax} \in \mathbf{N}_{>0}$. The initial partition-set forming the bottom layer of the decrementing linear fud is in the intersection of the substrate self-cartesian partition-sets set, the pluri-limited-tuple-derived-dimension non-overlapping substrate transforms set and the limited-valency substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}} \cap \mathcal{N}_{U,K,\mathrm{umax}}$. The pluri-limited-valency pluri-limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds is defined \[ \begin{eqnarray} &&\mathcal{F}_{U,\mathrm{n},-,K,\overline{\mathrm{b}},\mathrm{mmax},\overline{2},\mathrm{umax}}\\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : M \in \mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n},\overline{\mathrm{b}},\mathrm{mmax}} \cap \mathcal{N}_{U,K,\mathrm{umax}},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdecp}(U)(M))\})\} \\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},\\ &&\hspace{10em}(\forall J \in Y~(|J^{\mathrm{C}}| \leq \mathrm{umax})),~M = \{J^{\mathrm{CS}\{\}} : J \in Y\},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdecp}(U)(M))\})\} \end{eqnarray} \] Define the limited-valency contracted decrementing linear non-overlapping fuds list maximiser \[ \begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{n,w},-,K} =\\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{n},-,K},N_{P,A,A_R,F,\mathrm{n},-,K},\mathrm{top}(\mathrm{pmax}),R_{P,A,A_R,F,\mathrm{n,w},-,K}) \end{eqnarray} \] where the initial set is (Haskell) \[ \begin{eqnarray} &&R_{P,A,A_R,F,\mathrm{n,w},-,K} = \{(\{M^{\mathrm{T}}\},X_{P,A,A_R,F,\mathrm{n},-,K}(\{M^{\mathrm{T}}\})) : \\ &&\hspace{10em}Y \in \mathrm{B}(K),~2 \leq |Y| \leq \mathrm{mmax},\\ &&\hspace{10em}(\forall J \in Y~(|J^{\mathrm{C}}| \leq \mathrm{umax})),~M = \{J^{\mathrm{CS}\{\}} : J \in Y\}\} \end{eqnarray} \] In this case the choice of tuple fuds is (Haskell) \[ Q = \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n,w},-,K})) \] The effect of the valency limit is to make the partitions of the tuple larger and more regular. For example, a binary partition such that all but one of the tuple variables, having common valency $d$, is in one component has initial valencies of $d^{k-1}$ and $d$. The cardinality of the searched approximately varies as the cube of the longest valency, so a binary irregular search may require considerably more computation time than a poly-component regular search in the same tuple.
Limited-underlying tuple set list maximiser
Having addressed the cardinality of the second term of the upper bound on the cardinality of the neighbourhood function of the limited-path-models tuple partition inducer, $|\mathrm{B}(K^{\mathrm{CS}})| \leq \mathrm{bell}(\mathrm{xmax})$ where $K \in \mathrm{tuples}(V,F)$ and $|K^{\mathrm{C}}| \leq \mathrm{xmax}$, by optimising a subset $Q$ of the partitions of a tuple, $Q \subset \mathrm{B}(K^{\mathrm{CS}})$, above, now consider the first term. A subset of the neighbourhood function of the limited-path-models tuple partition inducer, $P_{P,A,A_R,\mathrm{csd},\mathrm{p}}$, may be defined which allows only one partition transform for each tuple, \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{csd},\mathrm{p},\mathrm{u}}(Q) =\\ &&\hspace{2em}\{(F \cup G,I_{\mathrm{csd}}^{ * }((A,A_R,F \cup G))) : \\ &&\hspace{4em}(F,\cdot) \in Q,~\mathrm{layer}(F,\mathrm{der}(F)) < \mathrm{lmax},\\ &&\hspace{4em}B \subseteq \{K : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax}\},\\ &&\hspace{4em}1 \leq |B| \leq \mathrm{bmax},\\ &&\hspace{4em}G \in \{\{P^{\mathrm{T}} : P \in N\} : N \in \prod_{K \in B} (\mathrm{B}(K^{\mathrm{CS}}) \setminus \{\{K^{\mathrm{CS}}\}\})\},\\ &&\hspace{4em}W = \mathrm{der}(F \cup G), ~|W^{\mathrm{C}}| \leq \mathrm{wmax}\} \end{eqnarray} \] In the special case of the fud $F$, defined above, of variable cardinality $|\mathrm{vars}(F)| = \mathrm{lmax} \times \mathrm{bmax}$, the upper bound on the cardinality of the neighbourhood function is \[ \begin{eqnarray} &&|P_{P,A,A_R,\mathrm{csd},\mathrm{p},\mathrm{u}}(\{(F,X_{P,A,A_R,\mathrm{csd},\mathrm{p}}(F))\})|\\ &&\hspace{10em}< ((\mathrm{lmax} \times \mathrm{bmax})^{\underline{\mathrm{kmax}}})^{\underline{\mathrm{bmax}}} \times \mathrm{bell}(\mathrm{xmax})^{\mathrm{bmax}} \end{eqnarray} \] The cardinality of the set of next limited-underlying-volume limited-breadth layers depends on (i) the cardinality of the set of next limited-underlying-volume limited-breadth layer tuple sets, \[ \begin{eqnarray} &&|\{B : B \subseteq \{K : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax}\},~1 \leq |B| \leq \mathrm{bmax}\}| \\ &&\hspace{20em}< ((\mathrm{lmax} \times \mathrm{bmax})^{\underline{\mathrm{kmax}}})^{\underline{\mathrm{bmax}}} \end{eqnarray} \] and (ii) the product of the cardinalities of the sets of partitions of each tuple, \[ \prod_{K \in B} |\mathrm{B}(K^{\mathrm{CS}})| < \mathrm{bell}(\mathrm{xmax})^{\mathrm{bmax}} \] within each layer tuple set, $B$.
The cardinality of the set of next limited-underlying limited-breadth tuple sets, $|\{B : B \subseteq \{K : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax}\},~1 \leq |B| \leq \mathrm{bmax}\}|$, can be addressed by (i) constructing only a single content alignment optimised next limited-underlying limited-breadth layer tuple set $B_{\mathrm{B}} \subseteq \{K : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax}\}$, and (ii) constructing the tuples of that tuple set, $B_{\mathrm{B}}$, one variable at a time. The tuple set, $B_{\mathrm{B}}$, consists of the maximum breadth, $\mathrm{bmax}$, per maximum tuple derived dimension, $\mathrm{mmax}$, top-most tuples, $|B_{\mathrm{B}}| = \lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor$, if the tuples are uniquely aligned, so that the cardinality of derived variables in the layer optimised by applying the contracted decrementing linear non-overlapping fuds list maximiser, $Z_{P,A,A_R,F,\mathrm{n},-,K}$, to each tuple, $K \in B_{\mathrm{B}}$, is no greater than the maximum breadth, $\mathrm{bmax}$. Define the limited-underlying tuple set list maximiser \[ Z_{P,A,A_R,F,\mathrm{B}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{B}},P_{P,A,A_R,F,\mathrm{B}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{B}}) \] where (i) the optimiser function is \[ \begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{B}} = \\ &&\hspace{2em}\{(K,I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F),A))-I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R))) :\\ &&\hspace{20em}K \in \mathrm{tuples}(V_A,F)\} \end{eqnarray} \] where $\mathrm{his} = \mathrm{histograms} \in \mathcal{F} \to \mathrm{P}(\mathcal{A})$, $\mathrm{apply} \in \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{A}) \times \mathcal{A} \to \mathcal{A}$, and (ii) the neighbourhood function is (Haskell) \[ \begin{eqnarray} &&P_{P,A,A_R,F,\mathrm{B}}(B) = \{(J,X_{P,A,A_R,F,\mathrm{B}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in B,~w \in \mathrm{vars}(F) \cup V_A \setminus K,~J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray} \] and (iii) the initial subset is (Haskell) \[ \begin{eqnarray} R_{P,A,A_R,\emptyset,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,\emptyset,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w,u \in V_A,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \\ R_{P,A,A_R,F,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,F,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \cup V_A ,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray} \] Then the shuffle content alignment optimised next limited-underlying limited-breadth layer tuple set, $B_{\mathrm{B}}$, is (Haskell) \[ \begin{eqnarray} &&B_{\mathrm{B}} = \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})) \in \\ &&\hspace{5em}\{B : B \subseteq \{K : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax}\}\} \end{eqnarray} \] The fud application, $\mathrm{apply} \in \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{V}) \times \mathrm{P}(\mathcal{A}) \times \mathcal{A} \to \mathcal{A}$, traverses from the substrate variables, $V_A$, to the tuple, $K$, via the histograms of the transforms of the fud, $\mathrm{his}(F)$. The fud application is a tractable application equivalent to the application of the fud’s transform’s histogram followed by reduction, $\mathrm{apply}(V_A,K,\mathrm{his}(F),A) = A * \mathrm{histogram}(F^{\mathrm{T}})~\%~K$. The shuffle content fud application alignment, $\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A))-\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R))$, resembles the shuffle content alignment, $\mathrm{algn}(A * G^{\mathrm{T}})-\mathrm{algn}(A_R * G^{\mathrm{T}})$, where $G = \mathrm{depends}(F,K)$, except that fud variables cannot be hidden by being nested in the higher layer depends fud of another fud variable. That is, in some cases $K \neq \mathrm{der}(\mathrm{depends}(F,K))$ and so $\mathrm{apply}(V_A,K,\mathrm{his}(F),A) \neq A * \mathrm{depends}(F,K)^{\mathrm{T}}$.
The tuple set search, $Z_{P,A,A_R,F,\mathrm{B}}$, is optimised by maximising the shuffle content fud application alignment, \[ \mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A))-\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R)) \] not the shuffle content fud application alignment valency-density, \[ (\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A))-\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R)))/y^{1/k} \] where $y = |K^{\mathrm{C}}|$ and $k = |K|$. The maximum alignment varies with dimension, for example a regular histogram of dimension $k$ and valency $d$ has approximate maximum alignment of $z(k-1) \ln d$. Therefore a maximiser function value of shuffle content alignment tends to select tuples of larger dimension. In contrast, the maximum alignment valency-density varies against geometric-average valency, approximately $z(k-1)(\ln d)/d$, and so tends to smaller tuple dimension, $k$, especially where the entropy of valencies, $\mathrm{entropy}(\{(w,|U_A(w)|) : w \in K\})$, is low. The contracted decrementing linear non-overlapping fuds in subsequent applications of contracted decrementing linear non-overlapping fuds list maximisers to each tuple of the tuple set, $\bigcup \{\mathrm{dom}(X_{P,A,A_R,F,\mathrm{n},-,K}) : K \in B_{\mathrm{B}}\} \subset \mathcal{F}_{U_A,\mathrm{P}^{ * }}$, have derived volume less than or equal to the tuple volume, $|\mathrm{der}(H)^{\mathrm{C}}| \leq |K^{\mathrm{C}}|$ where $H \in \mathcal{F}_{U_A,\mathrm{n},-,K}$, so larger tuples can value roll down to smaller subsets of the tuple, but not the reverse. That is, a maximiser function value of shuffle content alignment increases the searched cardinality for each tuple making the neighbourhood function, $P_{P,A,A_R,\mathrm{csd}}$, of the notional list maximiser, $Z_{P,A,A_R,\mathrm{csd}}$, less arbitrary, so increasing the maximum function correlation of the practicable inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, to the tractable inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$.
An upper bound on the expected cardinality of the searched may be computed given the maximum underlying dimension, $\mathrm{kmax}$. The upper bound on the expected cardinality in the first layer, $F = \emptyset$, is \[ \sum_{k \in \{2 \ldots \mathrm{min}(\mathrm{kmax},n)\}} \binom{n}{k} \] where $n = |V_A|$ and $\mathrm{min}=\mathrm{minimum}$. In subsequent layers, $F \neq \emptyset$, the upper bound on the expected cardinality is \[ \sum_{k \in \{2 \ldots \mathrm{min}(\mathrm{kmax},q)\}} \binom{q}{k} - \binom{q-x}{k} \] where $W = \mathrm{vars}(F) \cup V_A$, $q = |W|$, $X = \mathrm{der}(F)$ and $x = |X|$.
The maximum underlying dimension, $\mathrm{kmax}$, may be approximated from the geometric average valency $d = |W^{\mathrm{C}}|^{1/q}$, and the maximum underlying volume, $\mathrm{xmax}$, \[ \mathrm{kmax} = \left\lceil\frac{\ln \mathrm{xmax}}{\ln d}\right\rceil \]
If it is the case that the $\mathrm{topd}(\mathrm{omax})$ optimisation is such that each step of the list has no more than $\mathrm{omax}$ tuples, then in the first layer, $F = \emptyset$, the cardinality of the searched set is constrained \[ |\mathrm{searched}(Z_{P,A,A_R,\emptyset,\mathrm{B}})| < n^2 + (\mathrm{kmax}-2) \times \mathrm{omax} \times n \] In subsequent layers, $F \neq \emptyset$, the cardinality of the searched set is constrained \[ |\mathrm{searched}(Z_{P,A,A_R,F,\mathrm{B}})| < xq + (\mathrm{kmax}-2) \times \mathrm{omax} \times q \]
The limited-underlying tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, has an inclusion function defined $\mathrm{top}(\mathrm{omax}) \in \mathrm{P}(X_{P,A,A_R,F,\mathrm{B}}) \to \mathrm{P}(X_{P,A,A_R,F,\mathrm{B}})$. The application of the $\mathrm{top}(n) \in \mathrm{P}(\mathcal{X} \times \mathcal{Y}) \to \mathrm{P}(\mathcal{X} \times \mathcal{Y})$ aggregation function with parameter $n$ to a relation $R$ may result in a subset of the relation having a cardinality greater than the given parameter, $|\mathrm{top}(n)(R)| > n$, if there are duplicate range values at the $n$-th position of the ordered relation. This might be the case, for example, in a tuple set list maximiser searched set, $\mathrm{searched}(Z_{P,A,A_R,F,\mathrm{B}})$, that contains tuples which contain variables having a partition equal to the self-partition of a singleton underlying variable.
An implementation of the tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, that guarantees no more than $\mathrm{omax}$ tuples at each step of the optimiser list, $\forall B \in \mathrm{set}(\mathrm{list}(Z_{P,A,A_R,F,\mathrm{B}}))~(|B| \leq \mathrm{omax})$, must have additional inclusion order criteria. An example is where the tuples are ordered first by ascending alignment, $X_{P,A,A_R,F,\mathrm{B}}(K)$, and then by descending sum derived variables layer, $-\mathrm{sumlayer}(F,K)$, where $\mathrm{sumlayer} \in \mathcal{F} \times \mathrm{P}(\mathcal{V}) \to \mathbf{N}$ is defined as \[ \mathrm{sumlayer}(F,K) := \sum_{w \in K} \mathrm{layer}(F,\{w\}) \] For example, the tuples $J,K \subset \mathrm{vars}(F)$, such that variable $u \in K$, self partition variable $\{u\}^{\mathrm{CS}\{\}} \in J$ and $J = K \setminus \{u\} \cup \{\{u\}^{\mathrm{CS}\{\}}\}$, have the same alignments, $X_{P,A,A_R,F,\mathrm{B}}(J) = X_{P,A,A_R,F,\mathrm{B}}(K)$, but different sum derived variables layers, $\mathrm{sumlayer}(F,J) = \mathrm{sumlayer}(F,K) + 1$. Ordering by descending sum derived variables layer avoids the addition of extra variables to the model which are merely redundant reframe variables, where these reframe variables are at the inclusion boundary.
There are other order criteria including (i) descending shuffle alignment, $-\mathrm{algn}(\mathrm{apply}(V_A,K,\mathrm{his}(F),A_R))$, and (ii) descending tuple volume, $|K^{\mathrm{C}}|$. Ordering by descending tuple volume tends to prevent tuples from adding mono-effective variables, $|(A\%\{u\})^{\mathrm{F}}| = 1 < |\{u\}^{\mathrm{C}}|$.