$$f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}, \quad s.t. \quad k \geq 1 \quad \mathrm{and} \quad 0 < p < 1$$ |

The Infinity Project is an engineering project, was driven by a desire to understand how all technologies were produced, and could be thought of as trying to **parse** mankind's knowledge.

A video introduction to it is here, and the project lives on inf.li.

On the Infinity Project, we assume that everything that was ever made by people, was driven by people's **work to satisfy their needs**, and that everything that was ever done is part of the equation $\mathbf{F(X)=}\mathrm{Y}$ (where $\mathbf{F}$ - world, $\mathbf{Y}$ - dream, $\mathbf{X}$ - actions), and that it can be logically broken down to the following 7-level decomposition:

**Need** - $\mathrm{y}$

**Goal**- $\mathbf{Y}$

**Idea** - $\mathbf{X}$

**Plan** - $\mathbf{\hat{X}}$

**Step** - $\mathrm{s}$

**Task** - $\mathbf{t}$

**Work** - $\mathbf{\hat{t}}$

We created this decomposition based on common-sense observations, in a hope that if people can see the decompositions of all work this way, it will be natural for them to understand how making of anything was a simple piece of math to solve $\mathbf{F(X) \in Y}$, and the knowledge will be shared.

**Context-free grammars** is a model introduced by the linguist Noam Chomsky to describe the structure of sentences and words in natural language, however, due to their generality, they had become an important formalism in many fields, including computer science, where they are used to model strings by producing *parse trees*.

A context-free grammar G is defined as a 4-tuple $G = (N, \Sigma, R, S)$ where:

- $N$ is a finite set; each element $n\in N$ is called a non-terminal character or a variable. Each variable represents a different type of phrase or clause in the sentence. Variables are also sometimes called syntactic categories. Each variable defines a sub-language of the language defined by $G$.
- $\Sigma$ is a finite set of terminals, disjoint from $N$, which make up the actual content of the sentence. The set of terminals is the alphabet of the language defined by the grammar $G$.
- $R$ is a finite relation from $N$ to $(N\cup\Sigma)^{*}$, where the asterisk represents the Kleene star operation. The members of $R$ are called the (rewrite) rules or productions of the grammar. (also commonly symbolized by a P)
- $S$ is the start variable (or start symbol), used to represent the whole sentence (or program). It must be an element of $N$.

What had we done? It seems, in context of context-free grammars, we had produced a set of non-terminal and terminal symbols $￼(\Sigma \cup N)^{*} = ( \mathrm{y}, \mathbf{Y}, \mathbf{X}, \mathbf{\hat{X}}, \mathrm{s}, \mathbf{t}, \mathbf{\hat{t}},..)$, with respect to which, it could be possible to parse human activities defined in terms of sequences of **works** ($\mathbf{\hat{t}}$).

Moreover, it seems that with probabilistic context-free grammars, we can start learning the production rules of the form $N \to (N \cup \Sigma)^*$ that people use to decompose their needs.