[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81952-en":3,"doc-seo-81952-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81952,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","When Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models","Modern sequence models increasingly act as agents that interleave token generation with calls to external tools. The work provides an exact, architecture-level account of when tool access increases computational expressivity. Fixed finite-precision recurrent models are formalized as deterministic finite-state controllers interacting with an oracle via a finite command/observation interface. A sharp dichotomy is proved: finite-state tools add essentially nothing, while a minimal unbounded tape tool yields Turing completeness.","arXiv :2607 .06 155v 1 [ cs .FL] 7 Jul 2026  \nWhen Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models?  \nNikola Zubi´c [zubic@ifi.uzh.ch](zubic@ifi.uzh.ch)  \nRobotics and Perception Group, University of Zurich  \nQian Li [liqian.ict@gmail.com](liqian.ict@gmail.com)  \nShenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data  \nYuyi Wang [yuyiwang920@gmail.com](yuyiwang920@gmail.com)  \nTengen Intelligence Institute, CRRC Zhuzhou Institute  \nDavide Scaramuzza [sdavide@ifi.uzh.ch](sdavide@ifi.uzh.ch)  \nRobotics and Perception Group, University of Zurich  \nAbstract  \nModern sequence models are increasingly deployed as agents that interleave token generation with calls to external tools. We give an exact, architecture-level account of when such tool access increases computational expressivity. We model any fixed finite-precision recurrent sequence model, including finite-precision state-space models (SSMs) with B bits of internal state, as a deterministic finite-state controller interacting with an oracle through a finite command/observation interface. Our results form a sharp dichotomy. First, tools that are themselves finite-state add essentially nothing: a product-state simulation internalizes any finite-state bounded-interface oracle with finite memory set M at a cost of only log2 |M|+O(1) additional bits, so the augmented system remains finite-state. Second, a single minimal infinite-state tool (a tape supporting only local read/write/move commands) makes the system Turing complete: for every single-tape Turing machine with state set Q and tape alphabet Γ, a controller with O(log |Q| + log |Γ|) bits of internal memory simulates it, and we exhibit a concrete exponential separation (EQn requires 2n states without tools but a single constant-size controller with the tape tool) . Third, we show that this construction is realized exactly by a natural one-layer finite-precision selective affine SSM controller with binary one-hot hidden states, {0, 1} transition matrices, and zero biases.  \nSelectivity is essential to the construction. In the supplementary material, we make all constants explicit, prove a logarithmic oracle-assisted universal simulation (O(log B) recurrent bits suffice to simulate any B-state Turing machine), and prove a matching impossibility result: without external memory, directly realizing an arbitrary B-state transition map in one exact affine recurrent update requires dimension exactly B − 1, for any number of layers. Together, these results give a precise resource-accounting picture of tool-augmented recurrent computation.  \nKeywords: expressivity; state-space models; tool use; oracle machines; Turing completeness; finite precision; recurrent models; computational complexity  \n© 2026 N. Zubi´c, Q. Li, Y. Wang & D. Scaramuzza.  \nZubi´c Li Wang Scaramuzza  \n1. Introduction  \nSequence models are no longer used only as sequence-to-sequence maps: they are deployed as agents that interleave output generation with calls to external tools such as calculators, code interpreters, retrieval systems, and scratchpads (Schick et al. , 2023; Yao et al. , 2023; Nye et al. , 2021) . This raises a basic theoretical question:  \nWhich tools increase the computational expressivity of a fixed finite-precision sequence model, by how much, and at what internal memory cost?  \nThe starting point is that any fixed finite-precision recurrent model is, from the standpoint of expressivity, a finite-state machine: a model with B bits of internal state has at most 2B configurations, and hence, without external interaction, recognizes only regular languages (Weiss et al. , 2018; Merrill, 2019; Zubic et al. , 2025) . This covers, in particular, finite-precision structured state-space models (SSMs) (Gu et al. , 2022; Gu and Dao, 2024), whose formal-language limitations have been analyzed in detail (Merrill et al. , 2024; Sarrofet al. , 2024; Zubic et al. , 2025; Zu","cbCaigfdLULkWVFx","https://ap.wps.com/l/cbCaigfdLULkWVFx","pdf",364496,6,1,24,"English","en",105,"# Introduction\n## Our model\n## Contributions\n# Exact expressivity dichotomy\n## Finite-state tools\n## Unbounded tape tools\n# Exact realization with selective SSM controllers\n## Selectivity and controller construction","[{\"question\":\"What question does the paper address about tool use in sequence models?\",\"answer\":\"It asks which tools increase the computational expressivity of a fixed finite-precision sequence model, how much they increase it, and what internal memory cost is required.\"},{\"question\":\"Why do finite-state tools add essentially nothing to expressivity?\",\"answer\":\"A finite-state bounded-interface oracle can be internalized by simulating the product state, keeping the augmented system finite-state with only log2|M|+O(1) extra bits.\"},{\"question\":\"What kind of tool makes the system Turing complete?\",\"answer\":\"A minimal infinite-state tape tool that supports local read/write/move commands enables a controller with O(log|Q|+log|Γ|) bits of memory to simulate any single-tape Turing machine.\"}]","When Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models | 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question does the paper address about tool use in sequence models?","Question",{"text":77,"@type":78},"It asks which tools increase the computational expressivity of a fixed finite-precision sequence model, how much they increase it, and what internal memory cost is required.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"Why do finite-state tools add essentially nothing to expressivity?",{"text":82,"@type":78},"A finite-state bounded-interface oracle can be internalized by simulating the product state, keeping the augmented system finite-state with only log2|M|+O(1) extra bits.",{"name":84,"@type":75,"acceptedAnswer":85},"What kind of tool makes the system Turing complete?",{"text":86,"@type":78},"A minimal infinite-state tape tool that supports local read/write/move commands enables a controller with O(log|Q|+log|Γ|) bits of memory to simulate any single-tape Turing 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