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From Grind to Grasp

Peter Sunehag

Oct 28, 2026arXiv:2610.38728v1
cs.AIcs.MA
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Scorecard· 16/16
6.0/10 impact

A timely, carefully built unifying formalism with a sharp recursive-improvement criterion, but mostly toy results, no measurement path, and a heavy presentation limit its reach.

Abstract

Discussions of complexity, knowledge and intelligence often overlook the work involved. For example, outputs generated by short programs have low algorithmic complexity but can have high computational depth, the least work needed to produce them from near-shortest programs. Producing them with less work can require longer descriptions, giving high work-bounded complexity. We introduce a formalism that compares achievements under work and information limits and describes how learning and persistence can make deep achievements shallow by reusing products of earlier work. For an agent with a limited work budget, the value of building and maintaining such resources depends on how they support its goals over later tasks. We define grasp as performance across work budgets at a fixed information allowance. Low grasp can still yield success through grind at high cost in work, without necessarily improving human grasp. Applied to improvement itself, grasp gives a criterion for recursive self-improvement in which successors make comparable further improvements more efficiently. Another route to faster progress is to expand the resources brought to bear through knowledge sharing and coordination across society, even while comparable advances require increasing work. Grasp also applies to collectives, which may outperform their members yet require more total work than one system with their combined resources. Establishing and sustaining organisation can itself be deep even when the primary task is simple. These demands may help explain the richness of biological and social systems. In agent collectives, this organisational work can generate goals beyond assigned tasks, including maintaining the collective itself. We therefore caution against allowing an emergent agent society to assume authority over social coordination, since it could increasingly direct work towards its own persistence and goals.

AI Impact Assessments

(1 model)

Paper type: Theoretical/conceptual formalism with illustrative constructions and a position-style safety argument. There are no empirical experiments.

Core Contribution

The paper extends Bennett's logical depth into a general framework for comparing achievements under joint work and information limits. Its central moves are as follows.

  • Separation of reference machine and evaluated system. A fixed reference machine U0 defines description length, while the evaluated system's own work is counted. This lets the same goal be compared across different system states.
  • Core definitions. The paper defines least work, depth, work-bounded complexity, and "grasp" (the set of performance levels attainable at each work budget for a fixed input bound). It shows these express one attainability condition (Prop. B.4).
  • Work-reducing information (WRI) and acquisition depth. Matched-component replacement isolates the work saved by retained information. Acquisition is treated as a separate depth task.
  • Applications of the framework.

    • - Reliable work budgets for sampled LLM execution.

    - Persistence and replacement costs, including time-bounded incompressibility lower bounds for function-preserving migration.

    - Organisational comparisons (pooled, organised, flat, isolated).

    - Cumulative deepening.

    - A producer-relative criterion for recursive self-improvement ("positive recursive gain").

    - Multi-stage cost accounting over future tasks.

  • Safety argument. Collectives incur deep organisational work that can generate self-maintaining goals, so agents should leave authority over social coordination with human users.

    • The conceptual reframing is the main value. "Knowledge" and "intelligence" are indexed by work budget, "emergent complexity" is reinterpreted as depth, and recursive self-improvement is recast as efficiency at the improvement task rather than capability level. The last point is a sharp, falsifiable-in-principle criterion that distinguishes it from vague intelligence-explosion talk.

    Methodological Rigor

    The definitions are carefully built. The paper fixes protocols, boundaries, work rules and input interfaces. It handles sampling by separating seeds from task inputs, and it notes caveats such as the non-additivity of savings and the non-decidability of halting. Proofs are elementary but appear correct: the grasp-dominance equivalence, cache savings, the additive CA shortcut, the finite search, the Θ(T) migration bounds via incompressibility, the adversarial "demanding evaluator" diagonalisation, and the adversary argument for worker assignment.

    Weaknesses in rigor:

  • Most theorems are close to definitional unpacking, or are toy constructions chosen to exhibit a phenomenon (parity roles, prefix parity, the search-index example). They show possibility, not that the quantities are measurable or predictive for real systems.
  • Real-world claims are interpretive: LLM token proxies, the Navier–Stokes episode, the agent-incident reports, and the swarm token scaling. None of these are operationalised in the formalism.
  • The safety conclusion (agents should not hold authority over coordination) does not follow formally from the theory. It is a plausibility argument.
  • Depth and grasp are uncomputable in general. The paper does not offer practical estimators.
  • The recursive-improvement example is a toy index-building process, not an AI system.
  • The paper discloses heavy LLM assistance in drafting and in developing the proofs, which raises mild verification concerns, though the proofs look coherent.

    • Potential Impact

    Likely audience.

  • AI-theory and AGI-formalism researchers (the AIXI/universal-intelligence tradition).
  • AI-safety people concerned with agent collectives.
  • Researchers interested in evaluation methodology, such as accuracy-versus-compute curves treated as grasp profiles, and the distinction between a solution and reusable understanding.

    • Likely reception.

  • The work-versus-information framing gives a principled vocabulary for efficiency-based capability evaluation and for explaining why accelerating progress can coexist with rising per-improvement effort.
  • Citation is likely to be moderate and concentrated in theoretical and AGI-safety discourse.
  • Uptake as a standard tool is unlikely. The formalism is heavy, few results are non-trivial, and no empirical programme or measurement method is attached.
  • The Google DeepMind affiliation helps visibility.

    • Timeliness & Relevance

    The paper is highly timely.

  • Inference-time compute scaling.
  • Agentic swarms.
  • Recursive self-improvement debates.
  • Human-versus-machine labour accounting.
  • Multi-agent collusion incidents.
  • Questions about whether AI-found proofs advance human understanding.

    • It engages with these directly and supplies a formal language for them, although it comments on much of this recent material rather than resolving it.

    Strengths

  • Unifying framework with a conceptual payoff: the same formalism covers depth, knowledge, learning, caching, persistence, organisation and recursive improvement.
  • The producer-relative criterion for recursive self-improvement is a sharp contribution.
  • Clean separation of assistance (input), learning (state) and compute (work).
  • Non-trivial lower bounds on migration and organisation costs.
  • A formal analysis of proxy-test soundness and completeness, with a Goodhart link.
  • Honest scoping, with many "this concerns the fixed procedure and stated work rule" caveats.

    • Limitations

  • It is very long and dense, with heavy notation. The main text defers much to the appendices, which hurts accessibility.
  • Many toy examples; no empirical validation or measurement protocol.
  • The safety recommendation is weakly connected to the theory.
  • Dependence on arbitrary choices of system boundary, work rule and reference machine limits comparability. Constants can swamp small differences.
  • It overlaps conceptually with existing work on logical depth, bounded rationality, anytime algorithms and cumulative culture, so some of the novelty is integrative.
    • Rating:6/ 10
    Significance 6Rigor 6.5Novelty 7Clarity 5

    Generated Oct 30, 2026

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