Estimating Sentence-like Structure in Synthetic Languages Using Information Topology

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. 2022 Jun 22;24(7):859. doi: 10.3390/e24070859

Algorithm 1 Proposed information topology SLU estimation algorithm

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procedureInfTopSentenceBounds( $Π_{h}$ ) ▹ Estimate contextual SLU bounds
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for all $g \in Π_{h}$ do ▹ Do for all contextual data $Π_{h}$
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$\{u_{v}, u_{z}\} \leftarrow g (Π_{h})$ ▹ Read new set of symbols
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$\{u_{\hat{v}}, u_{\hat{z}}\} \leftarrow \{u_{v}, u_{z}\}$ ▹ Obtain topological set of new unique symbols
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$\{p_{\hat{v}}, p_{\hat{z}}\} \leftarrow f_{s} (\{u_{v}, u_{z}\})$ ▹ Functional spline match distributions
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$H_{D} (\hat{v}, \hat{z}) \leftarrow \{p (\hat{v}), p (\hat{z})\}$ ▹ Normalized relative difference entropy
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$I_{d} (ϕ) \leftarrow I_{0} (ψ) - I_{0} (\bar{ψ} (N_{L}))$ ▹ Incremental information
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$\hat{I} (ϕ; n) \leftarrow I_{d}$ ▹ Cumulative incremental information
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${\tilde{W}}_{d 1} (ϕ; n) = {\tilde{W}}_{1} (ψ; n) - {\tilde{W}}_{1} (\bar{ψ}; n - 1)$ ▹ Incr. norm. Wasserstein-1 distance
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${\hat{W}}_{1} (ϕ; n) \leftarrow {\tilde{W}}_{d 1} (ϕ; n)$ ▹ Cumulative W-1 distance
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${\tilde{θ}}_{1} (ϕ; n) = arctan ({\hat{W}}_{1} (ϕ; n))$ ▹ Curvature of the W-1 distance tangent
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end for
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$\bar{I} (ϕ; n) \leftarrow m e a n (\hat{I} (ϕ; n))$ ▹ Mean cumulative information
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$\bar{θ} (ϕ; n) \leftarrow m e a n ({\tilde{θ}}_{1} (ϕ_{n}; n))$ ▹ Mean incremental W-1 distance
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$Ψ (ϕ_{N}) \leftarrow Σ (\bar{I} (ϕ; n), \bar{θ} (ϕ; n))$ ▹ Covariance of incr. info and W-1 distance
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$ω_{l} = \bar{I} (ϕ; n) - α ψ_{w}$
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$ω_{b} = \bar{θ} (ϕ; n) - α ψ_{h}$
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$ω_{w} = α ψ_{w}$
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$ω_{h} = α ψ_{h}$
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$Ω (ϕ_{N}) = [ω_{l}, ω_{b}, ω_{w}, ω_{h}]$ ▹ Compute decision bounds over $Π_{h}$
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end procedure

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procedureInfTopSentence( $G_{s}, Ω_{ϕ}$ ) ▹ Estimate SLU for current data
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for all $g \in G_{s}$ do ▹ Do for all local data $G_{s}$
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$\{u_{v}, u_{z}\} \leftarrow g (G_{s})$ ▹ Read new set of symbols
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$\{u_{\hat{v}}, u_{\hat{z}}\} \leftarrow \{u_{v}, u_{z}\}$ ▹ Obtain topological set of new unique symbols
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$\{p_{\hat{v}}, p_{\hat{z}}\} \leftarrow f_{s} (\{u_{v}, u_{z}\})$ ▹ Functional spline match distributions
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$H_{D} (\hat{v}, \hat{z}) \leftarrow \{p (\hat{v}), p (\hat{z})\}$ ▹ Norm. relative difference entropy
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$I_{d} (ϕ) \leftarrow I_{0} (ψ) - I_{0} (\bar{ψ} (N_{L}))$ ▹ Incremental information
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$\hat{I} (ϕ; n) \leftarrow I_{d}$ ▹ Cumulative incremental information
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${\tilde{W}}_{d 1} (ϕ; n) = {\tilde{W}}_{1} (ψ; n) - {\tilde{W}}_{1} (\bar{ψ}; n - 1)$ ▹ Incr. norm. W-1 distance
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${\hat{W}}_{1} (ϕ; n) \leftarrow {\tilde{W}}_{d 1} (ϕ; n)$ ▹ Cumulative W-1 distance
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${\tilde{θ}}_{1} (ϕ; n) = arctan ({\hat{W}}_{1} (ϕ; n))$ ▹ Curvature of W-1 distance tangent
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$Ω (ϕ_{N}) = [ω_{l}, ω_{b}, ω_{w}, ω_{h}]$ ▹ Apply SLU decision bounds
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if $(\hat{I} (ϕ; n) \geq ω_{l}) and (\tilde{θ} (ϕ; n) \leq ω_{t})$ then ▹ SLU decision test
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$q (ϕ; n) = 1$ ▹ End of SLU detected
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end if
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end for
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end procedure