When Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models?

2026-07-07 · Source: Computation and Language · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Computational Theory · Depth: Expert, quick

Summary

This research provides an exact, architecture-level analysis of how external tool access impacts the computational expressivity of finite-precision recurrent sequence models. The study models these systems as deterministic finite-state controllers interacting with an oracle through a finite command/observation interface. It establishes a sharp dichotomy: tools that are themselves finite-state add negligible expressivity, as a product-state simulation can internalize them with a cost of only λog₂ |M| + O(1) additional bits, keeping the system finite-state. In contrast, a single minimal infinite-state tool, specifically a tape supporting μatthtt{read}, μatthtt{write}, and μatthtt{move} commands, renders the system Turing complete. A controller with O(λog |Q| + λog |Γ|) bits can simulate any single-tape Turing machine, demonstrating an exponential separation where EQₙ requires 2ₙ states without tools but a constant-size controller with the tape tool. This construction is realized by a one-layer finite-precision selective affine SSM controller.

Key takeaway

For AI scientists designing or evaluating agentic sequence models, understand that merely adding finite-state tools offers no fundamental increase in computational power. If your goal is to achieve Turing-complete capabilities, you must integrate tools that possess infinite state, such as a memory tape. This distinction is critical for accurately assessing model expressivity and avoiding overestimating the capabilities gained from simple tool integration.

Key insights

Only infinite-state tools, like a Turing tape, fundamentally increase the computational power of finite-precision recurrent models to Turing completeness.

Principles

• Finite-state tools yield finite-state systems.
• Infinite-state tools enable Turing completeness.
• Selective affine SSMs can realize Turing completeness.

Method

Modeling recurrent models as finite-state controllers interacting with an oracle, then analyzing expressivity based on oracle statefulness.

Topics

• Recurrent Models
• Tool Use
• Computational Expressivity
• Turing Completeness
• Finite-State Automata
• State-Space Models

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Computation and Language.