Discrete Calculus for DeFi Math

2022-07-27Eric Forgy

Hello everyone and welcome to the CavalRe! 🤠

In this article, I present a primer on discrete calculus with the intention of helping us analyze DeFi protocols running on blockchains, which are inherently discrete in nature.

Blockchain as a State Machine

The Ethereum yellow paper describes the Ethereum blockchain as a state machine with the state of the world denoted by σt\sigma_t. Given a transaction Tt,t+1T_{t,t+1}, the state is updated to a new state σt+1\sigma_{t+1} by the Ethereum state transition function Υt,t+1\Upsilon_{t,t+1} denoted by

σt+1=Υt,t+1(σt,Tt,t+1).\sigma_{t+1} = \Upsilon_{t,t+1}(\sigma_t,T_{t,t+1}).

It is important to note the inherent discreteness of this process. There is no state between σt\sigma_t and σt+1.\sigma_{t+1}. A transaction takes you from some initial state to some final state with nothing in between and repeats indefinitely with every new transaction.

This discreteness has implications for how we should analyze blockchain protocols such as automated market makers (AMMs) so we will enhance our toolbox for dealing with this discreteness in the remainder of this article.

Node Functions

Every blockchain needs to start with some initial state created by some kind of genesis event. Therefore, it makes sense to restrict tt to be a natural number N\N with the initial state denoted σ0.\sigma_0. A node function is simply a function

f:N→R,f: \N\to\R,

where R\R denotes real numbers. Let et\mathbf{e}^t denote a special function whose value is given by

et(t′)={1if t=t′,0otherwise.\mathbf{e}^t(t') = \begin{cases} 1 &\text{if } t = t',\\ 0 &\text{otherwise.} \end{cases}

Any node function f:N→Rf:\N\to\R can be expressed as a linear combination of these functions, i.e.

f=∑t∈Nftet,f = \sum_{t\in\N} f_t \mathbf{e}^t,

where ft∈Rf_t\in\R is the value of ff at tt. The basis functions can be multiplied

etet′={etif t=t′,0otherwise\mathbf{e}^t \mathbf{e}^{t'} = \begin{cases} \mathbf{e}^t &\text{if } t = t',\\ 0 &\text{otherwise} \end{cases}

which extends to the product of arbitrary node functions f,g:N→Rf,g:\N\to\R intuitively as

fg=∑t∈Nftgtet.f g = \sum_{t\in\N} f_t g_t \mathbf{e}^t.

The product of two node functions is commutative, i.e. fg=gf.f g = g f.

Edge Functions

The natural numbers N\N can be thought of as a simple directed graph. In the context of blockchain, the state σt\sigma_t would then reside on the node t,t, whereas the transaction Tt,t+1T_{t,t+1} would reside on the directed edge between states σt\sigma_t and σt+1\sigma_{t+1} as illustrated below.

Directed Graph

Letting dNd\N denote the set of edges between natural numbers when thinking of N\N as a directed graph, define functions et,t+1:dN→R\mathbf{e}^{t,t+1}: d\N\to\R by

et,t+1(t′,t′+1)={1if t=t′,0otherwise.\mathbf{e}^{t,t+1}(t',t'+1) = \begin{cases} 1 &\text{if } t = t',\\ 0 &\text{otherwise.} \end{cases}

We can then define a general discrete edge function TT as a linear combination

T=∑t∈NTt,t+1et,t+1,T = \sum_{t\in\N} T_{t,t+1} \mathbf{e}^{t,t+1},

where Tt,t+1T_{t,t+1} is the value of TT on the edge (t,t+1).(t,t+1).

Multiplication of an edge function by a node function on the left can be defined in terms of basis functions via

etet′,t′+1={et,t+1if t=t′,0otherwise.\mathbf{e}^t \mathbf{e}^{t',t'+1} = \begin{cases} e^{t,t+1} &\text{if } t = t',\\ 0 &\text{otherwise.} \end{cases}

In other words, the product is zero unless the node coincides with the beginning of the edge. Similarly, multiplication of an edge function by a node function on the right can be defined in terms of basis functions via

et,t+1et′={et,t+1if t+1=t′,0otherwise.\mathbf{e}^{t,t+1} \mathbf{e}^{t'} = \begin{cases} e^{t,t+1} &\text{if } t+1 = t',\\ 0 &\text{otherwise.} \end{cases}

In other words, the product is zero unless the node coincides with the end of the edge. In general we have

fT=∑t∈NftTt,t+1et,t+1f T = \sum_{t\in\N} f_t T_{t,t+1} \mathbf{e}^{t,t+1}

and

Tf=∑t∈Nft+1Tt,t+1et,t+1T f = \sum_{t\in\N} f_{t+1} T_{t,t+1} \mathbf{e}^{t,t+1}

so that the product of node functions and edge functions is noncommutative, i.e. fT≠Tf.f T \neq T f.

Discrete Differentials

There is a special "unit" node function

1=∑t∈Net1 = \sum_{t\in\N} \mathbf{e}^t

that satisfies 1f=f1=f1 f = f 1 = f and 1T=T1=T.1 T = T 1 = T. Similarly, there is a special "graph" edge function

G=∑t∈Net,t+1.G = \sum_{t\in\N} \mathbf{e}^{t,t+1}.

With the graph edge function, we can define the differential of a node function ff via

df:=[G,f],d f := [G,f],

where [G,f]=Gf−fG[G,f] = G f - f G denotes the commutator, i.e.

df=∑t∈N(Δf)t,t+1et,t+1,d f = \sum_{t\in\N} (\Delta f)_{t,t+1} \mathbf{e}^{t,t+1},

with

(Δf)t,t+1:=ft+1−ft.(\Delta f)_{t,t+1} := f_{t+1} - f_t.

It follows from the properties of the commutator that

[G,fg]=[G,f]g+f[G,g][G,f g] = [G,f] g + f [G,g]

which gives rise to the discrete product rule

d(fg)=(df)g+f(dg).d(f g) = (d f) g + f (d g).

Although the product of node functions on the left-hand side is commutative, the products on the right-hand side are noncommutative so the order they are written matters. We could have also written the discrete product rule as

d(gf)=(dg)f+g(df).d(g f) = (d g) f + g (d f).

This means that when dealing with discrete node and edge functions, as we must with blockchain protocols, we have a degree of freedom in how we decompose terms in the discrete product rule.

Let us introduce a new general notation

{A,B}k=kAB+(1−k)BA,\{A,B\}_k = k A B + (1-k) B A,

where 0≤k≤10\le k\le 1 and we have

{A,B}k={B,A}1−k.\{A,B\}_k = \{B,A\}_{1-k}.

When the product is commutative, as it is with node functions f,g,f,g, we have

{f,g}k=fg=gf\{f,g\}_k = f g = g f

and it is straightforward to show that the general expression for the discrete product rule is given by

d(fg)={df,g}k+{f,dg}k.d(f g) = \{d f,g\}_k + \{f, d g\}_k.

For numerical implementation, we'll need to expand the above into components and we get:

{df,g}k=∑t∈N(Δf)t,t+1(Ekg)t,t+1et,t+1\{d f,g\}_k = \sum_{t\in\N} (\Delta f)_{t,t+1} (E_k g)_{t,t+1} \mathbf{e}^{t,t+1}

and

{f,dg}k=∑t∈N(E1−kf)t,t+1(Δg)t,t+1et,t+1,\{f, d g\}_k = \sum_{t\in\N} (E_{1-k} f)_{t,t+1} (\Delta g)_{t,t+1} \mathbf{e}^{t,t+1},

where

(Ekf)t,t+1=(1−k)ft+kft+1(E_k f)_{t,t+1} = (1-k) f_t + k f_{t+1}

so that the discrete product rule may be expressed in components as

Δ(fg)t,t+1=Δft,t+1Ekgt,t+1+E1−kft,t+1Δgt,t+1.\Delta(f g)_{t,t+1} = \Delta f_{t,t+1} E_k g_{t,t+1} + E_{1-k} f_{t,t+1} \Delta g_{t,t+1}.

The value of kk has no impact on the sum. It merely impacts the way the discrete product rule decomposes into the two terms. As we will see in subsequent articles, this degree of freedom stemming from the inherent discreteness of a blockchain has important implications for AMM design.

Closer look 🧐: Note on Symmetries

One kind of symmetry was already noted above, namely

{A,B}k={B,A}1−k.\{A,B\}_k = \{B,A\}_{1-k}.

There is another related kind of symmetry given by

(Ekf)t,t+1=(E1−kf)t+1,t(E_k f)_{t,t+1} = (E_{1-k} f)_{t+1,t}

and

(Δf)t,t+1=−(Δf)t+1,t.(\Delta f)_{t,t+1} = -(\Delta f)_{t+1,t}.

In other words, if the direction of time is reversed so that tt and t+1t+1 are swapped, then we can simply replace kk with 1−k1-k in the discrete product rule. This means that:

  • The discrete product rule for k=0k=0 is equivalent to the discrete product rule for k=1k=1 with time reversed;
  • The discrete product rule for k=1k=1 is equivalent to the discrete product rule for k=0k=0 with time reversed; and
  • The discrete product rule for k=1/2k=1/2 is the same whether forward in time or time reversed.

We will see this again when looking at self financing in DeFi protocols.

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Eric Forgy