Formal Proof of AGI Incomputability

• Formal Proof of AGI Incomputability is a theoretical framework defining AGI as the ability to generate new functionalities beyond its initial algorithmic closure.
• It employs NAND gate universality and diagonalization techniques to rigorously show that no Turing-computable process can spontaneously innovate beyond its predefined functions.
• The results elucidate the safety–generality tradeoff in AGI, illustrating that systems guaranteeing error-free performance cannot achieve full human-level problem solving.

AGI incomputability concerns the question of whether a fully general, creative intelligence—as formalized in computational terms—can ever be achieved or recognized by any algorithmic process. Recent work rigorously explores whether AGI remains logically or physically inaccessible to Turing-computable systems, elaborates precise technical definitions, and establishes formal incomputability results and boundaries.

1. Formalizations of AGI and Creativity

Multiple recent formulations converge on the notion that AGI fundamentally entails creativity or the ability to introduce genuinely new functional capabilities not derivable from the preexisting system architecture. In "On the Computability of Artificial General Intelligence" (Mappouras et al., 4 Dec 2025), AGI is defined as a system that, for some input $X$, can produce output $Y$ via a function $f$ that is not already contained within its initial set of available functionalities $\mathcal{F}$: $$\text{Definition 1. A system is AGI if and only if for some input } X \text{ it can produce output } Y \text{ where the mapping } f(X) = Y \text{ did not exist in } \mathcal{F} \text{ beforehand}.$$ This "new-functionality" definition is aligned with the core of creative intelligence: innovation by the on-the-fly invention of $f_{n+1}: X \mapsto Y$, $f_{n+1} \notin \mathcal{F}_t$, which is then appended to the future functional set.

In contrast, (Panigrahy et al., 25 Sep 2025) proposes a capability-theoretic notion: an AGI system must be able to solve every instance that a human can solve with provable correctness, setting the standard of AGI as matching the breadth and depth of human problem-solving potential, modulo strict formal definitions of "tasks," "safety," and "trust."

2. Core Incomputability Theorems

The central theorem of (Mappouras et al., 4 Dec 2025) asserts:

AGI Incomputability Theorem:

There exists no Turing-computable process $P$ which, given an initial code base $C$, can, for arbitrary input $X$, produce an output $Y$ by a function $f_{\mathrm{new}}(X) = Y$ that was not already implicit in the set of functions implementable by $C$, with $f_{\mathrm{new}}$ being added on-the-fly by $P$. No finite algorithm can genuinely create functional capabilities beyond its initial closure.

This is instantiated via the NAND-gate universality formalism: no finite NAND-circuit can realize a system that expands its own functional closure outside its original gates, thus definitively precluding algorithmic creativity under this classical computation model.

In (Panigrahy et al., 25 Sep 2025), the incomputability is further sharpened in the context of "safe" and "trusted" AGI: $$\text{Theorem 2.5: } \text{If a system is safe (never produces errors) and trusted (assumed safe), it cannot be AGI—as defined by always matching human solutions—on program verification, planning, or graph reachability tasks.}$$ The proof harnesses variants of Gödel's diagonalization, constructing explicit self-referential problem instances that a safe and trusted system must refuse (to guarantee correctness) even though humans can provably solve them.

3. Proof Strategies and Technical Foundations

(Mappouras et al., 4 Dec 2025) builds its argument from two pillars:

• Universality of NAND circuits: By Church–Turing and NAND universality, every computable function can be implemented with a finite number of NAND gates.
• Minimal-circuit contradiction: Assuming a minimal circuit of size $k$ can instantiate AGI, this circuit can be decomposed into a $(k-1)$-gate subcircuit and a 1-gate subcircuit. One or both must be responsible for the new functionality, contradicting minimality.

Key steps include ruling out trivial and minimal circuits (Lemmas 1–4), carefully distinguishing capabilities that are mere recombinations of existing functions versus truly novel additions, and showing that no bootstrapping of AGI is possible from smaller subcircuits.

(Panigrahy et al., 25 Sep 2025) employs self-reference constructions rooted in classic undecidability proofs (Gödel/Turing-style diagonalization). Example: in the program-verification setting, the carefully constructed "Gödel_program" forces a conflict if the system is both safe and claims to solve every instance provably solvable by humans.

4. Relationship with Other Incomputability Results

The question of AGI incomputability is deeply connected to the incomputability of ideal Bayesian inference and universal reinforcement learning. In particular, Solomonoff induction and AIXI, the archetypal incomputable agents, reside at high levels in the arithmetical hierarchy (Leike et al., 2015):

• Standard AIXI policy is not limit-computable (not $\Delta^0_2$), precluding even approximative implementation on Turing machines.
• $\epsilon$-optimal versions exist at lower arithmetical levels (limit-computable), providing only approximate AGI even with infinite computational resources.

These results imply that any "perfect" Bayesian AGI must transcend Turing computability, highlighting the foundational chasm between theoretical ideals and algorithmic implementability.

5. Decidability of the AGI Trait and Rice’s Theorem

(Fox, 14 Feb 2024) argues that, contrary to some earlier claims, Rice's theorem (which states that any nontrivial semantic property of Turing machines is undecidable) does not directly settle the issue for AGI:

• Semantic vs. syntactic traits: If being an AGI is semantic (depends only on input/output function), Rice's theorem would guarantee undecidability. However, Fox observes that realistic definitions of intelligence embed efficiency constraints (e.g., temporal or resource bounds), leading to syntactic traits.
• For syntactic traits, there exist nontrivial decidable properties, so Rice's theorem is inapplicable. The question remains logically open unless further structural properties (e.g., finite syntactic part with nontrivial semantic core) can be demonstrated.

Formally, there is currently no fully general proof that no algorithm can recognize or decide the AGI trait for arbitrary machines; such a proof would require establishing new trait properties or reductions.

6. Implications for AI Design, Safety, and the Limits of Computation

The incomputability results delineate rigorous boundaries for AI and AGI system capabilities:

• Creativity Limitation: No machine-computable process can introduce genuinely new functionalities beyond its initial specification—AGI, if taken in the strong sense, is unattainable by algorithmic means (Mappouras et al., 4 Dec 2025).
• Safety–Generality Tradeoff: Any iron-clad guarantee of safety precludes generality at the level of human intelligence: safe and trusted systems can always be shown to abstain from some solvable instance (Panigrahy et al., 25 Sep 2025).
• AI Risk Perspective: Concerns about superintelligence runaway scenarios are structurally unfounded if AGI as spontaneous creative generality is algorithmically impossible; future research emphasis may thus pivot more heavily toward specialized systems.

A plausible implication is that explanations for observed human creative intelligence may not be reducible to Turing computation, or, if humans do instantiate true AGI under this definition, physical reality itself may not be fully simulable by any algorithm.

7. Current Open Problems and Research Directions

Despite substantial progress, the logical status of AGI-incomputability is not fully settled for all formalizations:

• For "creativity" definitions (Mappouras et al., 4 Dec 2025), incomputability has been rigorously established under standard computational paradigms.
• For the trait-recognition problem (Fox, 14 Feb 2024), the absence of a settling theorem—due to syntactic resource-limited definitions—means decidability remains open within that formalism.
• Sharp complexity bounds have been proved for universal learning agents (Leike et al., 2015), yet approximative versions exist at strictly lower arithmetical levels, highlighting the role of computational tradeoffs in approaching, but never attaining, theoretical AGI.

Further advances will require either new semantic characterizations linking AGI more tightly to existing incompleteness and undecidability constructs, or demonstrations that practical resource bounds render the AGI trait either decidable or formally intractable.