GEC Formal Spec Part 2 - Operators

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

uUnary oOperators

tTrust discounting

fFor trust factors:

𝜏𝑙,𝜏𝑣∈[0,1],$$ \tau_l,\tau_v \in [0,1], $$

define:

Discount⁡(𝑅,𝜏𝑙,𝜏𝑣)=(𝜏𝑙𝑅𝑇,𝜏𝑙𝑅𝐹,𝜏𝑣𝑅𝑉,𝜏𝑣𝑅𝐸).$$ \operatorname{Discount}(R,\tau_l,\tau_v) = (\tau_l R_T,\tau_l R_F,\tau_v R_V,\tau_v R_E). $$

rRepeated discounting composes multiplicatively:

DiscountChain⁡(𝑅,{(𝜏(𝑖)𝑙,𝜏(𝑖)𝑣)})=(𝑅𝑇∏𝑖𝜏(𝑖)𝑙,𝑅𝐹∏𝑖𝜏(𝑖)𝑙,𝑅𝑉∏𝑖𝜏(𝑖)𝑣,𝑅𝐸∏𝑖𝜏(𝑖)𝑣).$$ \operatorname{DiscountChain}(R,\{(\tau_l^{(i)},\tau_v^{(i)})\}) = \left( R_T\prod_i \tau_l^{(i)}, R_F\prod_i \tau_l^{(i)}, R_V\prod_i \tau_v^{(i)}, R_E\prod_i \tau_v^{(i)} \right). $$

vVacuation

iInner vacuation:

VacuateInner⁡(𝑅)=(0,0,𝑅𝑉,𝑅𝐸).$$ \operatorname{VacuateInner}(R)=(0,0,R_V,R_E). $$

oOuter vacuation:

VacuateOuter⁡(𝑅)=(𝑅𝑇,𝑅𝐹,0,0).$$ \operatorname{VacuateOuter}(R)=(R_T,R_F,0,0). $$

tTotal vacuation:

VacuateTotal⁡(𝑅)=(0,0,0,0).$$ \operatorname{VacuateTotal}(R)=(0,0,0,0). $$

vVacuation removes evidence from a channel.

sStrength scaling

fFor:

𝑐𝑙,𝑐𝑣≥0,$$ c_l,c_v \ge 0, $$

define:

Scale⁡(𝑅,𝑐𝑙,𝑐𝑣)=(𝑐𝑙𝑅𝑇,𝑐𝑙𝑅𝐹,𝑐𝑣𝑅𝑉,𝑐𝑣𝑅𝐸).$$ \operatorname{Scale}(R,c_l,c_v) = (c_lR_T,c_lR_F,c_vR_V,c_vR_E). $$

sSpecial cases:

ScaleInner⁡(𝑅,𝑐)=(𝑐𝑅𝑇,𝑐𝑅𝐹,𝑅𝑉,𝑅𝐸)$$ \operatorname{ScaleInner}(R,c)=(cR_T,cR_F,R_V,R_E) $$

ScaleOuter⁡(𝑅,𝑐)=(𝑅𝑇,𝑅𝐹,𝑐𝑅𝑉,𝑐𝑅𝐸).$$ \operatorname{ScaleOuter}(R,c)=(R_T,R_F,cR_V,cR_E). $$

bBlur

bBlur is easiest to understand semantically, but it has an exact rR form.

fFor a generic channel pair (A,B) with N=A+B and blur amount:

𝜌∈[0,1],$$ \rho \in [0,1], $$

define:

BlurPair𝜌⁡(𝐴,𝐵)=((1−𝜌)2𝐴+𝜌(1−𝜌)2𝑁,(1−𝜌)2𝐵+𝜌(1−𝜌)2𝑁).$$ \operatorname{BlurPair}_\rho(A,B) = \left( (1-\rho)^2A+\frac{\rho(1-\rho)}{2}N, (1-\rho)^2B+\frac{\rho(1-\rho)}{2}N \right). $$

tThis is equivalent to:

𝑛′=(1−𝜌)𝑛,𝑝′=𝜌⋅12+(1−𝜌)𝑝.$$ n'=(1-\rho)n, \qquad p'=\rho\cdot\frac{1}{2}+(1-\rho)p. $$

aAt rho=1, the channel becomes vacuous.

iInner blur applies BlurPair to (R_T,R_F). oOuter blur applies it to (R_V,R_E).

pParametric clarification

cClarification acts on the outer channel while preserving the inner channel.

gGiven:

0≤𝑣⋆≤1,Δ𝑛𝑣≥0,$$ 0 \le v^\star \le 1, \qquad \Delta n_v \ge 0, $$

define:

𝑁′𝑣=𝑅𝑉+𝑅𝐸+Δ𝑛𝑣.$$ N_v' = R_V+R_E+\Delta n_v. $$

tThen:

ParametricClarify⁡(𝑅,𝑣⋆,Δ𝑛𝑣)=(𝑅𝑇,𝑅𝐹,𝑁′𝑣𝑣⋆,𝑁′𝑣(1−𝑣⋆)).$$ \operatorname{ParametricClarify}(R,v^\star,\Delta n_v) = (R_T,R_F,N_v'v^\star,N_v'(1-v^\star)). $$

fFor ordinary clarification of an already-vague proposition, implementations usually require:

𝑣⋆≤𝑣.$$ v^\star \le v. $$

sScheduled clarification is derived from this primitive by choosing a schedule for v_star and Delta n_v.

nNegation

nNegation swaps the inner evidence pair and preserves the outer channel:

Neg⁡(𝑅)=(𝑅𝐹,𝑅𝑇,𝑅𝑉,𝑅𝐸).$$ \operatorname{Neg}(R)=(R_F,R_T,R_V,R_E). $$

tThis corresponds to the proposition-level operation not P. iIt is not a general replacement for triadic logic.

eEdge transform

fFor edge sign:

𝑠∈{+1,−1}$$ s \in \{+1,-1\} $$

and edge strength:

𝑤∈[0,1],$$ w \in [0,1], $$

define:

Edge⁡(𝑅,+1,𝑤)=(𝑤𝑅𝑇,𝑤𝑅𝐹,𝑅𝑉,𝑅𝐸)$$ \operatorname{Edge}(R,+1,w)=(wR_T,wR_F,R_V,R_E) $$

and:

Edge⁡(𝑅,−1,𝑤)=(𝑤𝑅𝐹,𝑤𝑅𝑇,𝑅𝑉,𝑅𝐸).$$ \operatorname{Edge}(R,-1,w)=(wR_F,wR_T,R_V,R_E). $$

pPositive edges scale directional evidence. nNegative edges swap the directional evidence pair and then scale it. eEdge polarity does not modify evaluability evidence.

eEvidence power compression

fFor a generic evidence pair:

(𝑅 +, 𝑅 -) $$ (R_+,R_-) $$

and exponent:

𝜏∈(0,1],$$ \tau \in (0,1], $$

define:

PowerCompress𝜏⁡(𝑅+,𝑅−)=(𝑅𝜏+,𝑅𝜏−).$$ \operatorname{PowerCompress}_\tau(R_+,R_-) = (R_+^\tau,R_-^\tau). $$

eEquivalently:

𝛼′=1+𝑅𝜏+,𝛽′=1+𝑅𝜏−.$$ \alpha'=1+R_+^\tau, \qquad \beta'=1+R_-^\tau. $$

aApply the operator independently to the inner or outer channel:

(𝑅𝑇,𝑅𝐹)↦(𝑅𝜏𝑇,𝑅𝜏𝐹)$$ (R_T,R_F) \mapsto (R_T^\tau,R_F^\tau) $$

or:

(𝑅𝑉,𝑅𝐸)↦(𝑅𝜏𝑉,𝑅𝜏𝐸).$$ (R_V,R_E) \mapsto (R_V^\tau,R_E^\tau). $$

tThis operator applies a power to pseudocount evidence rather than ordinary density power tempering. tThis is not the same as trust discounting. iIf an implementation needs pure weakening of evidence magnitude, use Discount or Scale.

fFusion, pooling, and unfusion

rRegime	pPrimary name	rR-space meaning
aAll sources independent	independent fusion	evidence-coordinate addition
aAll sources share the same evidence base	shared fusion	normalized weighted average in rR-space
pPartially independent / partially shared	mixed fusion	interpolation between independent and shared fusion
eExplicit overlap/provenance data available	overlap-aware fusion	de-duplicating fusion over shared evidence slices
oOverarching weighted family	log pool	arbitrary nonnegative rR-space weights

lLog pool family

fFor evidence vectors:

𝑅𝑖=(𝑅𝑇,𝑖,𝑅𝐹,𝑖,𝑅𝑉,𝑖,𝑅𝐸,𝑖)$$ R_i=(R_{T,i},R_{F,i},R_{V,i},R_{E,i}) $$

and exponents:

𝜆𝑖≥0,$$ \lambda_i \ge 0, $$

define:

𝑅⋆=∑𝑖𝜆𝑖𝑅𝑖.$$ R^\star = \sum_i \lambda_i R_i. $$

cCoordinatewise:

𝑅⋆𝑇=∑𝑖𝜆𝑖𝑅𝑇,𝑖,𝑅⋆𝐹=∑𝑖𝜆𝑖𝑅𝐹,𝑖,$$ R_T^\star=\sum_i \lambda_i R_{T,i}, \qquad R_F^\star=\sum_i \lambda_i R_{F,i}, $$

𝑅⋆𝑉=∑𝑖𝜆𝑖𝑅𝑉,𝑖,𝑅⋆𝐸=∑𝑖𝜆𝑖𝑅𝐸,𝑖.$$ R_V^\star=\sum_i \lambda_i R_{V,i}, \qquad R_E^\star=\sum_i \lambda_i R_{E,i}. $$

tThe semantic chart is recovered after pooling. tThe log pool is the overarching weighted family; named fusion regimes are special cases or structured extensions.

iIndependent fusion

iIndependent fusion assumes each input contributes distinct evidence. iIt is coordinatewise rR addition:

IndependentFuse⁡(𝑅1,…,𝑅𝑚)=𝑚∑𝑖=1𝑅𝑖.$$ \operatorname{IndependentFuse}(R_1,\ldots,R_m) = \sum_{i=1}^m R_i. $$

eEquivalently, it is logPool with:

𝜆𝑖=1forevery 𝑖.$$ \lambda_i=1 \quad \text{for every } i. $$

tThis is exact evidence addition beyond the uniform prior. iIn sSubjective lLogic, this corresponds to cumulative fusion, but independent fusion is the sSlothink/gecGEC name.

iIn bBeta form:

𝛼⋆𝑙=𝑅⋆𝑇+1,𝛽⋆𝑙=𝑅⋆𝐹+1,$$ \alpha_l^\star = R_T^\star+1, \qquad \beta_l^\star = R_F^\star+1, $$

𝛼⋆𝑣=𝑅⋆𝑉+1,𝛽⋆𝑣=𝑅⋆𝐸+1.$$ \alpha_v^\star = R_V^\star+1, \qquad \beta_v^\star = R_E^\star+1. $$

sShared fusion

sShared fusion assumes the inputs are different readings of substantially the same evidence base. fFor normalized weights:

𝑤𝑖≥0,∑𝑖𝑤𝑖=1,$$ w_i \ge 0, \qquad \sum_i w_i=1, $$

define:

SharedFuse𝑤⁡(𝑅1,…,𝑅𝑚)=∑𝑖𝑤𝑖𝑅𝑖.$$ \operatorname{SharedFuse}_w(R_1,\ldots,R_m) = \sum_i w_i R_i. $$

wWhen weights are omitted, use equal weights:

𝑤𝑖=1𝑚.$$ w_i=\frac{1}{m}. $$

tThis is the operation older notes sometimes called geometric fusion or weighted fusion. hHere, both collapse into one named operation: shared fusion.

mMixed fusion

mMixed fusion is the simple adjustable regime for sources that are neither fully independent nor fully shared. fFor an independence parameter:

𝜌∈[0,1],$$ \rho \in [0,1], $$

and normalized shared-fusion weights w, define the baseline mixed operator:

MixedFuse𝜌,𝑤⁡(𝑅1,…,𝑅𝑚)=𝜌IndependentFuse⁡(𝑅1,…,𝑅𝑚)+(1−𝜌)SharedFuse𝑤⁡(𝑅1,…,𝑅𝑚).$$ \operatorname{MixedFuse}_{\rho,w}(R_1,\ldots,R_m) = \rho\,\operatorname{IndependentFuse}(R_1,\ldots,R_m) + (1-\rho)\operatorname{SharedFuse}_w(R_1,\ldots,R_m). $$

bBoundary cases:

𝜌=1⟹MixedFuse𝜌,𝑤=IndependentFuse,$$ \rho=1 \implies \operatorname{MixedFuse}_{\rho,w}=\operatorname{IndependentFuse}, $$

𝜌=0⟹MixedFuse𝜌,𝑤=SharedFuse𝑤⁡.$$ \rho=0 \implies \operatorname{MixedFuse}_{\rho,w}=\operatorname{SharedFuse}_w. $$

tThis is a coarse model; when explicit overlap data is available, use overlap-aware fusion instead.

oOverlap-aware and dependence-aware fusion

oOverlap-aware fusion deduplicates shared evidence slices rather than compressing dependence into a single interpolation parameter.

tThe atom is

𝑅(𝑐)⋆,𝐽=∑𝑎:𝑆𝑎∩𝐽≠∅𝐴(𝑐)𝑎$$ R_{\star,J}^{(c)} = \sum_{a:S_a\cap J\neq\varnothing} A_a^{(c)} $$

per channel c, where each atom is counted once if it appears in any selected source.

tThe practical shared-slice operator will be specified elsewhere. iIt partitions each source into unique slices and shared slices, fuses shared slices by shared fusion, and then adds the unique and de-duplicated shared contributions.

lLet:

Fusedep$$ \operatorname{Fuse}_{\mathrm{dep}} $$

denote the selected dependence-aware fusion policy in a larger pipeline. iIt may choose independent fusion, shared fusion, mixed fusion, or overlap-aware fusion depending on the available dependence information.

tThe requirement is:

Fusedep⁡(⋯)∈ℝ4≥0.$$ \operatorname{Fuse}_{\mathrm{dep}}(\cdots) \in \mathbb{R}_{\ge 0}^4. $$

wWhen no dependence information is supplied, the default is independent fusion.

eExact independent unfusion

iIf:

𝑅⋆=𝑅𝑎+𝑅𝑏,$$ R^\star = R_a + R_b, $$

and R_star and R_a are known, then:

𝑅𝑏=𝑅⋆−𝑅𝑎$$ R_b = R^\star - R_a $$

coordinatewise, provided:

𝑅⋆𝑗−𝑅𝑎,𝑗≥0foreverycoordinate 𝑗.$$ R^\star_j - R_{a,j} \ge 0 \quad \text{for every coordinate } j. $$

tThis is independentUnfuse(fused, known). iIt is exact only for independent fusion. iIf any coordinate would be negative, exact independent unfusion is invalid. aAn implementation must either reject the operation or use an explicit reconciliation routine.