# Mathematical Logic

DeXe Protocol DAO employs a nonlinear voting system that accounts for more than just the number of tokens held; it also factors in the DAO's level of trust in the voter and a variety of other controlled and customizable parameters to ensure the organization's balanced operation. At the heart of DeXe Protocol DAO's vote calculation are piecewise-defined functions allowing precise control over voting power values across all ranges, minimizing the risk of manipulation. To construct these nonlinear dependencies, the Protocol utilizes polynomial functions of the 4th and 3rd degree.

The graph displays three functions, each illustrating how the voting power increases with the number of tokens for different cases:

The function for regular DAO members — the ordinary token holders.

The function for experts who have been granted the relevant status within the DAO, in case if 100% of the tokens are either on their own balance or delegated from the private balances of regular DAO members. This function assigns more weight to the votes of those selected by the DAO's decision.

The function for experts whose tokens are 100% delegated from the treasury. As shown, tokens delegated from the Treasury confer even greater weight to the experts' votes, reflecting the DAO's exceptional trust in such entities mirrored in the strength of their vote.

In cases where experts have tokens delegated to them both from the balances of regular DAO members and from its treasury, the function for calculating voting power will yield an intermediate value between functions 2 and 3. This value is proportionate to the ratio of tokens delegated from the treasury to those delegated by regular DAO members plus any tokens from the expert's own balance.

## Members’ function

The function for regular DAO members is a piecewise dependence consisting of two ranges: from 0% to 7% of the token's Total Supply, there's a standard linear function where the voting power is equal to the number of tokens; from 7% to 100% of the Total Supply, it's a third-degree polynomial that gradually slows its growth, preventing the concentration of power with large token holdings on a single DAO member's balance.

Unlike the square root function, this dependency prevents potential manipulations with low token balances, avoiding the exploitation of rapid function growth at smaller ranges. Thanks to its piecewise nature, the developed dependence is more balanced and secure.

$V_{m}(t) = \begin{cases} k_m \Bigg(7 \cdot TS \cdot \frac{1}{100} + a \left(\frac{100t}{TS} - 7\right) \\ \quad + b \left(\frac{100t}{TS} - 7\right)^2 + c \left(\frac{100t}{TS} - 7\right)^3\Bigg)\frac{TS}{100}, & \text{for } t \geq 7\% \text{ of } TS \\\\ t, & \text{otherwise} \end{cases}$

**Where:**

— voting power for regular DAO members**V_m**

— token balance of a voter**t**

**TS*** — total supply of a DAO*

**k_m*** —* the slope factor for the function applied to DAO members. By default, the value of *k_m* = 0.97

, **a**

, **b**

— parameters, which are defined to best suit the mathematical and conceptual objectives. By default, there value is:**c**

**a*** = 1.041*

**b*** = -0.007211*

**c*** = 0.00001994*

## Expert function

The function for regular experts is a piecewise-defined 4th-degree polynomial composed of two ranges, each with its own set of coefficients for each partial. The piecewise nature of the functions allows for more precise control over the voting power value.

The basis is the core function **V_exp****(****t****)**. For each expert, this base function will have its own slope coefficient, * k*, within a specific range. By default,

*is within the range [0.92; 1.08]. If all tokens are delegated to the expert from the DAO's treasury, their*

**k***coefficient will be 1.08 — the maximum value. If all tokens are delegated to the expert by regular DAO members or are owned by the expert,*

**k***will be at the minimum value of 0.92.*

**k**In cases where an expert has tokens delegated from both the treasury and regular members, their coefficient * k* is calculated based on the proportion of these tokens, indicated as

**R***(R=1, when 100% tokens are from the treasury and R=0, when 100% tokens are from the members' balances)*.

The formula for the * k* parameter is as follows:

$k = k_{\text{min}} \cdot R + k_{\text{max}} \cdot (1 - R)$

Therefore, the value of the expert function will be determined within the range of the piecewise-defined function *V*exp(*t*) with k_min = 0.92 and *k_*max = 1.08:

$V_{exp}(t) = \begin{cases} k \Bigg(a + b \left(\frac{100t}{TS} - 6.63\right) + c \left(\frac{100t}{TS} - 6.63\right)^2 \\ \quad + d \left(\frac{100t}{TS} - 6.63\right)^3 + e \left(\frac{100t}{TS} - 6.63\right)^4\Bigg)\frac{TS}{100}, & \text{for } t \geq 6.63\% \text{ of } TS \\\\ k \Bigg(f \left(\frac{100t}{TS}\right)^4 + g \left(\frac{100t}{TS}\right)^3 \\ \quad + h \left(\frac{100t}{TS}\right)^2 + i \left(\frac{100t}{TS}\right)\Bigg)\frac{TS}{100}, & \text{otherwise} \end{cases}$

**Where:**

— voting power**V_exp**

— token balance of a voter**t**

**TS*** — total supply of a specific DAO*

**k*** — the slope factor for the function applied to experts of a DAO. By default, the value of k varies from 0,92 to 1,08 depending on the proportion of tokens delegated from the treasury and from members wallets or owned tokens. The greater the proportion of tokens delegated from the treasury, the higher the value of k, and consequently, the higher the values the function will reach.*

**a****, ****b****, ****c**** , ****d**** , ****e**** , ****f**** , ****g**** , ****h**** , ****i***— parameters, which are defined to best suit the mathematical and conceptual objectives. By default, there value is:*

**a*** = 8.83755895036092*

**b*** = 1.130*

**c*** = -0.006086*

**d*** = 0.00004147*

**e*** = -0.000000148*

**f*** = -0.001328*

**g*** = 0.023761*

**h*** = -0.169889*

**i*** = 1.801894*

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