Skip to content

Latest commit

 

History

History
47 lines (26 loc) · 2.61 KB

infinitemonkeysGRADIENT.md

File metadata and controls

47 lines (26 loc) · 2.61 KB
Raw

Infinite Monkey Theorem & The Emergent Language Hypothesis

1. Original Infinite Monkey Theorem:

The Infinite Monkey Theorem posits that a monkey hitting keys at random on a typewriter for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare.

Formal Expression:

Given a single monkey's probability ( p ) of typing a specific character correctly, for a text of length ( l ):

  • Probability that one monkey types the entire text correctly in a time period ( x ): ( p^l )
  • Probability that ( n ) monkeys do not type the text correctly in time ( x ): ( (1-p^l)^n )
  • Therefore, the probability that at least one monkey types the text correctly in time ( x ) is:

[ P(\text{success}) = 1 - (1 - p^l)^n ]

2. Emergent Language Hypothesis:

Building on the original theorem, the emergent language hypothesis posits that by applying a gradient-based weight on the randomly generated text, the emergence of coherent language or structures can be accelerated. This is without any reference to a predefined text, but rather is based on the assumption that language and coherent patterns can emerge spontaneously given the right conditions.

Analogy:

Consider the process of language emergence as similar to the training phase of a neural network. Instead of backpropagating errors based on a predefined target, the network (or monkeys, in this case) is rewarded based on the consistency or similarity of its outputs. This hypothesis draws inspiration from unsupervised learning and generative models in machine learning.

Formal Expression with Gradient Weight:

Let the gradient weight, ( G ), be a function of the "coherence" or "structure" in the generated text, modeled by the consistency or similarity of outputs across the ( n ) monkeys.

[ G = \frac{\text{Number of consistent patterns across monkeys}}{\text{Total number of patterns}} ]

Given this gradient weight:

  • Probability that one monkey types a coherent "work" in a time period ( x ): ( p^l \times G )
  • Probability that ( n ) monkeys do not type a coherent "work" in time ( x ): ( (1 - p^l \times G)^n )
  • Therefore, the probability that at least one monkey types a coherent "work" in time ( x ) is:

[ P(\text{success with gradient}) = 1 - (1 - p^l \times G)^n ]

Using a Poisson distribution with an expectation ( \lambda ) influenced by the gradient weight:

[ \lambda' = n \times G \times \frac{x}{t} ]

The probability of observing coherent language emergence in time ( x ) becomes:

[ P(\text{success with gradient in Poisson}) = 1 - e^{-\lambda'} ]