Probvious

"Probvious" (a portmanteau of the words probabilistic and obvious) is an adjective used to express a high degree of confidence about a mathematical property or statement that is not known to be true. It was introduced by John Conway in an article discussing potentially unproveable statements. The term has been used by bbchallenge contributors to describe the solutions to halting problems for Cryptids such as Bigfoot and Hydra.

Usage

The excerpt from the article by John Conway where "probvious" is introduced.

The word appears in Conway's article a few times as a way of forming conjectures about a Collatz-like function that had already been investigated in the past. This function, denoted $\mu (n)$, is defined as:

$${\begin{array}{lll}\mu (2n)&=&3n\\\mu (4n+1)&=&3n+1\\\mu (4n+3)&=&3n+2\end{array}}\Rightarrow {\begin{array}{lll}\mu ^{-1}(3n)&=&2n\\\mu ^{-1}(3n+1)&=&4n+1\\\mu ^{-1}(3n+2)&=&4n+3\end{array}}$$

Conway first uses "probvious" to describe the idea that the sequences of iterates $(\cdots ,8,\mu (8),\mu ^{2}(8),\cdots )$ and $(\cdots ,14,\mu (14),\mu ^{2}(14),\cdots )$ diverge to infinity. Likewise, there exist Turing machines for which determining whether they halt requires solving a mathematical problem believed to be difficult, oftentimes a Collatz-like problem, but using probabilistic approximations of their functions suggests a clear solution. For example, Bigfoot, Hydra, and others function like biased random walks and treating them as such leads to the conclusion that they are probviously nonhalting. Similarly, machines such as Lucy's Moonlight, Mother of Giants, and others function like very slow-running random number generators that halt once a specific value is reached, which probviously happens with enough attempts.