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

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 $\log_2 |M| + O(1)$ additional bits, so the augmented system remains finite-state. Second, a single minimal infinite-state tool, namely a tape supporting only local $\mathtt{read}$, $\mathtt{write}$, and $\mathtt{move}$ 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: $\mathrm{EQ}_n$ requires $2^n$ 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 $O(\log B)$ recurrent bits suffice to simulate any $B$-state Turing machine, and prove a matching impossibility result.

Rethinking the Role of Positional Encoding: Sliding-Window Transformers without PE Remain Turing Complete

Positional encoding (PE) is widely viewed as necessary for transformers to process ordered sequences: without them, the next-token map appears permutation-invariant in its context tokens. This intuition underlies all prior universality results, which rely on positional information to prove that transformers with chain-of-thought can perform arbitrary computation, i.e., they are Turing complete. We revisit this belief in the regime most relevant to long-form reasoning, where generation proceeds through a finite sliding context window. Our opening perception is that the window mechanism itself (mildly) breaks the permutation symmetry. To distill and precisely capture the degree of this added expressiveness, we introduce an abstract autoregressive model, the HIST model, in which each update depends only on constant-size internal state and the token-count histogram within the current window. We prove that this HIST model is Turing complete by showing that the evolution of the window can reveal the token that has just left the window, which suffices to simulate Turing-complete Post machines. We then construct a sliding-window transformer over a constant-size token alphabet, without PE, and show that it can simulate the HIST model. Our result demonstrates that positional encodings are not indispensable for transformers to perform universal computation: The window sliding itself already breaks permutation symmetry and captures sufficient positional information.