Hyper(tasks/computation)

I’m particularly interested in super/hyper/ultra tasks & hypercomputation / digital physics / virtual particles.

In Surreal Time & Ultratasks, some interesting ideas are presented:

If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible.

This is similar to what I have been thinking about regarding tiny time, however for that model we use games (pseudo#s).

Day Zero Ultratask

For each ordinal $\alpha$, an $\alpha$-task could be done in a finite-length interval of moment-less time so long as the interval contains an $\alpha$-length sequence of pairwise non-overlapping subintervals. Perhaps there is a pointless analog of the system of surreal numbers, along with other pointless continua that contain infinitesimal intervals.

We have $+_0=\uparrow$ & $+_{on}=tiny$. So we can (at least in theory) have an $\mathbb{On}$ length sequence on day $0$ (maybe w/ $over=\{|over\}$ as the “event horizon” of Day $1$ 🤔).

In Hypercomputation: computing more than the Turing machine they present a figure (7 - pg.31 ) which looks identical to the game tree for $over=\{0|over\}$ 👀


I was also thinking about loop(y) time again, this time inspired by affine games (they give other forms born on Day $0$ besides $0$!). We could have a game tree w/ $0$ as the root & $on/off$ as the first steps. Maybe $on/off$ can be seen as types of ultratasks (maybe every loopy game can) 🤷‍♀️

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A hypertask that includes one task for each ordinal number is called an ultratask.

Super-ultratask A hypertask that includes a countably infinite number of tasks for each ordinal number.

Hyper-ultratask A hypertask that includes an uncountably infinite number of tasks for each ordinal number.

A-Task* A hypertask that includes an $\mathbb{O}n$ length number of tasks for each ordinal number.

Ultratask Hiearchy

  • $0$-task $\mapsto 0$ (No task)
  • $1$-task $\mapsto$ Ultratask $\equiv\mathbb{O}n$
  • $2$-task $\mapsto$ A-Task* $\equiv\mathbb{O}n^2$
  • $\omega$-task $\mapsto \mathbb{O}n^\omega$
  • $\mathbb{O}n$-task $\mapsto \mathbb{O}n ^\land \mathbb{O}n$
  • $\Omega$-task $\mapsto \mathbb{O}n ^\land \mathbb{O}n ^\land \mathbb{O}n ^\land ...$

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