GEC Formal Spec Part 1 - Basics

tThis is a formal specification for the basics of gGated eEpistemic cCalculus (gecGEC) for those interested in jumping into the math.

tThe pPrimitive

tThe primitive belief object is:

𝑅 = (𝑅 𝑇, 𝑅 𝐹, 𝑅 𝑉, 𝑅 𝐸) . $$ R = (R_T,R_F,R_V,R_E). $$

R_T: evidence that the proposition is true, conditional on being evaluable
R_F: evidence that the proposition is false, conditional on being evaluable
R_V: evidence that the proposition is vague, ill-posed, or not evaluable as stated
R_E: evidence that the proposition is evaluable as stated

tThe zero vector:

0 𝑅 = (0, 0, 0, 0) $$ 0_R = (0,0,0,0) $$

means no evidence has been supplied on either channel.

tThe inner, directional channel is:

𝑅 𝑙 = (𝑅 𝑇, 𝑅 𝐹) . $$ R_l=(R_T,R_F). $$

tThe outer, evaluability channel is:

𝑅 𝑣 = (𝑅 𝑉, 𝑅 𝐸) . $$ R_v=(R_V,R_E). $$

fFor a generic channel pair:

(𝑅 +, 𝑅 -) $$ (R_+,R_-) $$

define:

𝑁 = 𝑅 + + 𝑅 - . $$ N = R_+ + R_-. $$

tThe corresponding offset-bBeta parameters are:

𝛼=𝑅++1,𝛽=𝑅−+1.$$ \alpha = R_+ + 1,\qquad \beta = R_- + 1. $$

tThe offset by one is part of the canonical model. eEvidence is stored as pseudocount mass beyond the uniform prior.

tThe semantic interpretation is:

𝑏 (𝑅) = (𝑙, 𝑛 𝑙, 𝑣, 𝑛 𝑣) . $$ b(R) = (l,n_l,v,n_v). $$

tThe strengths are:

𝑛𝑙=𝑅𝑇+𝑅𝐹,𝑛𝑣=𝑅𝑉+𝑅𝐸,$$ n_l = R_T+R_F, \qquad n_v = R_V+R_E, $$

tThe coordinates are:

𝑙={𝑅𝑇𝑅𝑇+𝑅𝐹,𝑅𝑇+𝑅𝐹>012,𝑅𝑇+𝑅𝐹=0$$ l = \begin{cases} \dfrac{R_T}{R_T+R_F}, & R_T+R_F>0 \\ \dfrac{1}{2}, & R_T+R_F=0 \end{cases} $$

𝑣={𝑅𝑉𝑅𝑉+𝑅𝐸,𝑅𝑉+𝑅𝐸>012,𝑅𝑉+𝑅𝐸=0.$$ v = \begin{cases} \dfrac{R_V}{R_V+R_E}, & R_V+R_E>0 \\ \dfrac{1}{2}, & R_V+R_E=0. \end{cases} $$

tThe bBeta chart is derived:

𝛼𝑙=𝑅𝑇+1,𝛽𝑙=𝑅𝐹+1,𝛼𝑣=𝑅𝑉+1,𝛽𝑣=𝑅𝐸+1.$$ \alpha_l = R_T+1,\quad \beta_l = R_F+1, \qquad \alpha_v = R_V+1,\quad \beta_v = R_E+1. $$

tThe inner channel is:

𝜃𝑙∼Beta(𝑅𝑇+1,𝑅𝐹+1).$$ \theta_l \sim \mathrm{Beta}(R_T+1,R_F+1). $$

tThe outer channel is:

𝜃𝑣∼Beta(𝑅𝑉+1,𝑅𝐸+1).$$ \theta_v \sim \mathrm{Beta}(R_V+1,R_E+1). $$

tThe two channels can be adjusted independently, but recall that $𝑅 𝐸 = 𝑅 𝑇 + 𝑅 𝐹 $R_E = R_T + R_F$$

tThe predictive channel means are:

¯𝑙=𝑅𝑇+1𝑅𝑇+𝑅𝐹+2,¯𝑣=𝑅𝑉+1𝑅𝑉+𝑅𝐸+2.$$ \bar l = \frac{R_T+1}{R_T+R_F+2}, \qquad \bar v = \frac{R_V+1}{R_V+R_E+2}. $$

gecGEC has three canonical readout layers:

rRepresentative layer $𝑝rep(𝑅)=((1−𝑣)𝑙,(1−𝑣)(1−𝑙),𝑣)$$ p^{\mathrm{rep}}(R)=((1-v)l,(1-v)(1-l),v) $$$
pPredictive layer $𝑝pred(𝑅)=((1−¯𝑣)¯𝑙,(1−¯𝑣)(1−¯𝑙),¯𝑣)$$ p^{\mathrm{pred}}(R)=((1-\bar v)\bar l,(1-\bar v)(1-\bar l),\bar v) $$$
dDensity layer $𝑞dens(𝑅)=Beta(𝑅𝑉+1,𝑅𝐸+1)⊗Beta(𝑅𝑇+1,𝑅𝐹+1)$$ q^{\mathrm{dens}}(R)= \mathrm{Beta}(R_V+1,R_E+1) \otimes \mathrm{Beta}(R_T+1,R_F+1) $$$

tThe predictive projection is a readout, not an automatic evidence generator.

aAny rule that converts a predictive reading into target-side rR evidence must specify an activation or conservation rule.

tThe required invariant is:

𝑅work(𝐴)=0𝑅⟹Act𝑧⁡(𝐴)=0forall 𝑧∈{𝑇,𝐹,𝑉}.$$ R_{\mathrm{work}}(A)=0_R \implies \operatorname{Act}_z(A)=0 \quad \text{for all } z \in \{T,F,V\}. $$