Immortal Computation: Turing and Infinite Numerals

• Immortal Computation is a concept defining non-terminating processes using classical Turing models and arithmetic with infinite numerals.
• The topic contrasts the classical halting problem’s undecidability with novel techniques that render infinite iterations computable via the grossone system.
• Applications span cosmology and numerical methods, employing strategies like time cut-off restrictions and continuous observer measures to manage immortal processes.

Immortal computation refers to conceptual, mathematical, and physical models for handling computations or computational processes that do not terminate in finite time. This topic synthesizes two complementary perspectives: the classical theory of computation, in which non-halting Turing machines model "immortality" as undecidable non-termination, and a new applied approach in which infinite processes are rendered computable via arithmetic with explicit infinite numerals and specialized hardware. The central issues span logic, computability, the measure problem in cosmology, and numerical methodologies for infinite and infinitesimal analysis.

1. Formal Definitions

Immortal computation is rigorously framed within the Turing machine formalism and, alternatively, the arithmetic of infinite and infinitesimal numerals.

• Turing Machines and Immortal Computation:

A Turing machine $M = (\Gamma, Q, \delta)$ consists of a tape alphabet $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$, a finite state set $Q$ (including a start state $q_s$ and halting state $q_h$), and a transition function $\delta$. Each machine corresponds to a description number (Gödel number) $\langle M \rangle \in \mathbb{N}$. When initialized with a blank tape, machines that reach $q_h$ after finitely many steps are termed mortal; those that never halt are immortal (Vanchurin, 2015).

• Cosmic Logic and Computational Models:
  • Cosmic Observers (CO): Any Turing machine started on a blank tape, mortal if halting, immortal otherwise.
  • Cosmic Symmetries (CS): $\mathrm{CS}:\mathbb{N}\rightarrow\{0,1\}$, labeling symmetry classes of observers; certain CS machines would, if computable, distinguish mortal from immortal processes.
  • Cosmic Measures (CM): $\mathrm{CM}:\mathbb{N}\rightarrow[0,1]$, outputting observer probabilities.
• Infinite Numerals and the Infinity Computer:

The alternative methodology introduces an infinite base unit, grossone ($$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$0), to represent infinite and infinitesimal quantities. Numbers are represented in positional notation:

$$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$1

where coefficients ("grossdigits") $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$2 and exponents ("grosspowers") $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$3 can be finite, infinite, or infinitesimal. This enables symbolic computation with quantities such as $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$4, with each sum finite in representation but infinite in semantic scope (Sergeyev, 2012).

2. Non-Computability of Immortality in Classical Models

The distinction between mortal and immortal computation in the Turing framework is fundamentally undecidable. For a hypothetical CS*-machine: $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$5 no such algorithm exists (Vanchurin, 2015).

Proof Outline:

• By reduction to the Halting Problem: If $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$6 existed, one could decide for every Turing machine $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$7 and input $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$8 whether $$\Gamma = \{ \blacktriangleright, \mathrm{B}, 0, 1 \}$$9 halts by constructing a special machine $Q$0. This would contradict the undecidability of the Halting Problem.
• Diagonalization: A standard construction shows that any claimed "halting decider" leads to a paradoxical machine, implying nonexistence of such an algorithm.

The undecidability result extends directly to any attempt—cosmological or otherwise—to algorithmically discriminate between finite and infinite computational processes.

3. Immortal Computation in Applied Infinite Numeral Frameworks

The paradigm of infinite and infinitesimal numerals, based on the Infinite Unit Axiom (IUA), enables computation with infinite quantities as bona fide numerals:

• Grossone ($Q$1) serves as the size of $Q$2: for all finite $Q$3, $Q$4 and $Q$5.
• Positional numerals permit arithmetic with finite, infinite, and infinitesimal parts within a single framework.

Infinity Computer Operational Principles:

• Numbers are internally represented as sparse lists of grosspower–grossdigit pairs.
• Arithmetic and comparison operations are defined termwise and extended for grosspowers.
• Symbolic loops: An operation such as "for $Q$6 to $Q$7" is treated as a finite symbolic process, evaluating sums or limits that are classically infinite (Sergeyev, 2012).

This approach renders computations classically taken as "immortal"—e.g., divergent sums, evaluation at infinity, or infinite iterations—as executable and their results accessible in a finite number of computational steps, by interpreting infinity as a symbolic boundary.

4. Cosmological Context and the Measure Problem

In cosmology, immortal computation directly arises in "cosmic logic," particularly in eternal inflation scenarios:

• Cut-off measures frequently assign probability zero to observers (COs) with infinite histories and positive weight to those with finite ones.
• Implementing this dichotomy computably requires an algorithmic test for immortality, which is precluded by the noncomputability of the corresponding CS-machine (Vanchurin, 2015).

Table: Summary of Machine Types in Cosmic Logic

Machine Type	Input	Output
CO	Blank tape	Unrestricted
CM	Description number ($Q$8)	Probability in $Q$9
CS	Description number ($q_s$0)	0 or 1

A plausible implication is that any universal measure reliant on a mortal/immortal cut-off over all possible COs cannot be algorithmically implemented without restriction.

5. Workarounds and Practical Strategies for Immortal Computation

Two principal techniques circumvent non-computability in applications:

1. Time Cut-off Restriction: Impose a hard step limit $q_s$1 and consider only CO machines halting within $q_s$2 steps. The immortal/mortal separation becomes trivial because no process exceeds $q_s$3. This is algorithmically tractable but requires justification for the arbitrary time scale (Vanchurin, 2015).
2. Abandon Global Mortal/Immortal Symmetry: Construct probability measures (via CM machines) that do not depend on distinguishing finite from infinite observer histories. For example, assign weights to partial runs or apply continuous observables, sidestepping the need for immortal discrimination and bypassing the undecidability barrier.

In the infinite numeral framework, all "infinite" operations are grounded in an explicit representation (e.g., the summation $q_s$4), producing mathematically sound, computable results even when the corresponding classical series is divergent or the process is non-terminating (Sergeyev, 2012).

6. Explicit Examples and Key Formulas

Examples of immortal computation by infinite numerals include:

• Summing Divergent Series:

$q_s$5, for both finite $q_s$6 and $q_s$7, yielding $q_s$8.

• Sum of Infinite Geometric Series at Infinity:

$q_s$9, an infinitesimal less than one.

• Cardinality Measures:

$q_h$0, $q_h$1, $q_h$2.

• Symbolic Limits:

$q_h$3 is explicitly $q_h$4.

These results are realized algorithmically on an Infinity Computer, providing finite representations and, for every process, a computable outcome even in contexts that classically involve immortal, non-terminating computation (Sergeyev, 2012).

7. Interpretations, Limitations, and Prospects

The classical Turing approach demonstrates that immortal computation is not just non-terminating, but uncomputable to detect or encapsulate within a symmetry or probability measure over arbitrary systems. This precludes universal cut-off-based approaches in cosmological logic unless some restriction or reformulation is applied (Vanchurin, 2015).

The infinite numeral framework transforms such "immortal" computations into symbolically finite procedures, provided one accepts the axioms underlying grossone arithmetic and the operational semantics of the Infinity Computer. This suggests an alternative paradigm for applications demanding computations with actual infinities, such as symbolic summation, limits, and cardinal arithmetic.

A plausible implication is that the suitability of either approach (classic Turing vs. infinite numerals) depends on the physical, mathematical, or philosophical acceptability of their foundational postulates in the context of the application domain.

References:

• “Cosmic Logic: a Computational Model” (Vanchurin, 2015)
• “A new applied approach for executing computations with infinite and infinitesimal quantities” (Sergeyev, 2012)