Add 2024-09-14-From-AIR-to-Plonkish.md

README.md

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 - [📚 Finite Domain Terminology.md](./_posts/2024-08-04-Finite%20Domain%20Terminology.md)
 - [📚 Swartz-Zippel Lemma.md](./_posts/2024-08-05-Swartz-Zippel%20Lemma.md)
+- [📚 From%20AIR%20to%20Plonkish.md](./_posts/2024-09-14-From%20AIR%20to%20Plonkish.md)

_posts/2024-09-14-From%20AIR%20to%20Plonkish.md

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+
+#zkp 
+
+## From AIR to Plonkish
+### Recap
+
+AIR's **one constraint is bound to one frame of trace**.
+Plonkish flattens the frame and makes **one constraint is bound to one raw of trace**, just like R1CS does.
+
+### e.g. Fibonacci Constraints
+
+![image](/assets/Fib-From-AIR-to-Plonkish.svg)
+
+
+Each Plonkish constraint is bound to a single row. That makes thing simple!
+
+### Plonkish's Copy Constraints and Permutation
+
+In order to enforce that $X_i = Y_{i-1} = Z_{i-2} \quad \text{ when } i \gt 2$, Plonkish introduces **Copy Constraints**. Here is a simplified problem below.
+
+$$\text{Set } \quad \vec{a} = [X_3, X_4, X_5], \quad \vec{b} =[Y_2, Y_3, Y_4]$$
+We aim to enforce that $\vec{a}$ and $\vec{b}$ are **permutation**.
+
+We add a new column $P$
+
+$$
+\begin{aligned}
+& P_0 = 1 \\
+& P_{i+1} = P_i \cdot \frac{X_i + \sigma}{Y_i + \sigma}
+\end{aligned}
+$$
+
+and new copy constraints
+
+$$
+\begin{aligned}
+& P_0 = 0 \\
+& P_{i+1} \cdot (Y_i + \sigma) - P_i \cdot (X_i + \sigma) = 0
+\end{aligned}
+$$
+
+Done! We translate the **permutation** into mathematical constraints!