# Wacky Math Formula Test

January 15, 2021 - 1 min read

#### Testing the appearance of math formulas in search of bugs.

BREAK

$$\frac{f_{c} + 1}{2}$$

$$\mathcal{D}$$

$$\mathcal{O}(n)$$

$$\mathcal{S}$$

$$\mathcal{1/3}$$

$$2f_{c} + 1$$

$$f_{c} + 1$$

$$r + 1$$

$$f_{c}$$

$$2f_{d} + 1$$

$$f_{d} + 1$$

$$f_{d}$$

$$b_{1}$$

$$2f_{t} + 1$$

$$2f + 1$$

$$f_{t} + 1$$

$$f_{t}$$

$$3f_{t} + 1$$

$$\frac{1}{3}$$

$$S_{r}$$

$$W_{h}$$

$$C_{h}$$

$$\sigma$$

$$\sigma_{h}$$

$$\mathcal{x}$$

$$\mathcal{y}$$

$$\mathcal{x} – \mathcal{y} = 0$$

$$\mathcal{D}_{r}$$

$$\mathcal{x} – \mathcal{yz} \leq \mathcal{D}_{r}$$

(\mathcal{x} – \mathcal{yz} > \mathcal{D}_{r})

$$V_{sphere} = \frac{4}{3}\pi r^3$$

A DORA-CC protocol among $$n$$ nodes $$p_1,p_2,\dots,p_n$$ with each node having inputs $$v_i$$, for a given agreement distance $$\mathcal{D}$$, guarantees that:

1. [Termination] : All honest nodes eventually agree on some value.
2. [Agreement] : The output value $$\mathcal{S}$$ for all the nodes is the same.
3. [Validity] : $$H_{min} – \mathcal{D} \leq \mathcal{S} \leq H_{max}$$, where $$H_{min}$$ and $$H_{max}$$ denote the minimum and maximum values from honest nodes.

Given an agreement distance $$\mathcal{D}$$, we say that two values $$v_1$$ and $$v_2$$ agree with each other, if $$|v_1 – v_2| \leq \mathcal{D}$$. That is, if two values differ at most by the agreement distance, then they are said to agree with each other.

A set of values $$CC$$ is said to form a {\em coherent cluster}, if $$\forall v_1, v_2 \in CC: |v_1 – v_2| \leq \mathcal{D}$$. In other words, a coherent cluster is a set of values where all the values in that set agrees amongst themselves.

Let $$S_r$$ denote the $$S$$-value of round $$r$$. A circuit-breaker function $$\frac{|S_r – S_{r-1}|}{S_{r-1}} \geq thr$$ triggers and breaks the circuit (or halts the trade) when $$S_r$$ deviates from $$S_{r-1}$$ by more than some percentage threshold defined by $$thr$$.

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