When Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models?
Authors: Nikola Zubić, Qian Li, Yuyi Wang, Davide Scaramuzza
Organizations: Robotics and Perception Group, University of Zurich · Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data · Tengen Intelligence Institute, CRRC Zhuzhou Institute
Modern 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, namely a tape supporting only local read, write, and 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. Selectivity is essential to the construction. In the supplementary material, we make all constants explicit, prove a logarithmic oracle-assisted universal simulation, where O(logB) recurrent bits suffice to simulate any B-state Turing machine, and prove a matching impossibility result.
What formal languages can a recurrent neural language model recognize? Formal results in the literature conflict: some authors report Turing-completeness, while others show equivalence to regular languages. The reason for this discrepancy is that the underlying arithmetic model differs. The paper develops a unified algebraic account of the expressivity of recurrent neural networks, starting with a formal account of various arithmetic models. This account reduces expressivity to an algebraic question, e.g., whether a network's syntactic monoid divides a certain wreath product. As a case study, the paper revisits diagonal state-space models: the same architecture cannot implement an even-modulus counter once floating-point recurrences are enforced, yet realizes every even-modulus counter under unsigned-integer quantization.
Tool-augmented LLM agents commonly rely on step-wise atomic tool calls, where each invocation, observation, and value transfer is exposed in the main reasoning trace. This creates an \emph{execution-granularity mismatch}: locally deterministic tool workflows are unfolded into repeated model-visible decisions, consuming context and forcing the model to manage low-level dataflow in the trace. We introduce \textbf{HyperTool}, a unified executable MCP-style tool interface that changes the model-visible unit of tool execution. A model invokes HyperTool with a code block that can call existing tools through their original schemas, manipulate returned values, and pass intermediate results locally, folding deterministic tool subroutines into a single outer call. To train models to use this interface, we synthesize HyperTool-format trajectories from cross-tool compositional tasks and verify them in real MCP environments. On MCP-Universe, HyperTool improves average accuracy from 15.69% to 35.29% on Qwen3-32B and from 9.93% to 33.33% on Qwen3-8B, and surpass GPT-OSS and Kimi-k2.5 on average accuracy, showing that our HyperTool can substantially improve multi-step tool use.
Chain-of-thought prompting and looped Transformers both give a fixed model more test-time computation, but they differ in what they remember. Chain-of-thought stores intermediate state in generated tokens that remain in the context, whereas a looped Transformer carries state through recurrent hidden activations. We argue that this persistent mutable memory is a central resource for test-time reasoning. We compare three memory regimes, the compressed latent loop, the full sequence-state loop, and the chain-of-thought scratchpad. Our main result shows that a compressed loop is limited by the size of its recurrent state. Running the loop longer adds computation but does not by itself create a growing scratchpad, so a loop with a small recurrent state remains a small-space reasoner even when run for many steps. Under a standard complexity assumption, such loops cannot decide problems that are P-complete under logspace reductions, whereas polynomial-length chain-of-thought can. The separation is specific to compressed loops, as full sequence-state loops carry state at every input position and live in a memory-rich regime closer to explicit scratchpads. Controlled pointer-chasing and associative-recall sweeps illustrate this memory-budget view, with performance sensitive to whether the persistent-state budget matches the task's working-memory demand.