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arXiv2026

Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code

This paper will discuss the relationships among three pillars of mathematics: important research in codes and automata by Marcel-Paul Schutzenberger and others; random walks on finite semigroups by Persi Diaconis and others; and advanced techniques from finite semigroup theory used in proving Krohn-Rhodes complexity c is decidable by Stuart Margolis, Anne Schilling, and myself. Very surprising connections exist between the Fundamental Lemma of Complexity c (epimorphisms between finite semigroups that are one-to-one on subgroups preserve c) and coupling from the past in Markov chains, Diaconis' strong stationary time, and the Crick-Franklin-Watson genes of a finite automaton (which will be defined in the paper).

cs.FL↗
arXiv2026

The Emptiness Problem for Quantum Finite Automata with Classical States

Quantum Finite Automata with Classical states (QFACs) are nondeterministic finite automata over a finite alphabet of quantum operations. We study expressiveness of this model on finite words and the corresponding emptiness problem. We show that regular languages are incomparable with those definable by Quantum Finite Automata (QFAs) and that both are strictly subsumed by QFAC-definable languages. We show that the emptiness problem for a QFAC can be reduced to the emptiness of the language intersection of a QFA and a finite automaton. This intersection is known to be decidable for strict thresholds but undecidable for non-strict cases. Furthermore, we consider the problem for flat QFACs, a restriction where the underlying automata contain no nested loops, and relate it to the higher-dimensional orbit problem, a long-standing open challenge in dynamical systems. Finally, we propose a sound and semi-complete witness searching procedure to verify the non-emptiness of one-loop QFACs, which are sufficiently expressive to represent some prominent quantum algorithms, such as Grover's search and quantum random walks.

cs.FL↗
arXiv2026

CEDAR: Automata as Verifiable Interfaces for Language-Guided Embodied Action

Natural-language tasking of embodied agents is rarely just goal specification: users also impose constraints that must persist while the world changes. Code-generating LLM agents can produce plausible behaviors for such instructions, but their free-form programs provide no stable object to verify, compose with new constraints, or repair from a failing trace. We present CEDAR, a counterexample-guided framework that grounds instructions as regular languages over environment event traces. CEDAR uses a language model for semantic judgments and execution traces for correction, then represents both skills and specifications as deterministic finite automata. This turns constraints into executable finite-state objects: a learned skill can be intersected with a learned sleep at night or stay in this biome specification, yielding a controller that enforces the learned constraint by construction rather than by repeated prompting. In Minecraft, with the same simulator/API observations available to a program-generating baseline, CEDAR maintains temporal and spatial constraints that the baseline fails to preserve and amortizes reuse of learned skills, reducing cumulative LLM queries. These results suggest that regular languages offer a practical verification layer between natural-language instructions and embodied-agent policies.

cs.AI↗
arXiv2026

Stay Within Your Bounds: Distance-Guided Decoding for Guaranteed Context-Free Grammar Compliance

Grammar-constrained decoding helps large language models produce syntactically valid structured outputs, such as code, JSON, and SQL. For context-free grammars, many practical decoders enforce local prefix feasibility: each token must keep the current prefix extendable to some valid completion. Yet, under tokenizer-grammar mismatch and finite token budgets, feasible prefixes may still fail to reach acceptance. We propose a lookahead-guided decoding framework for context-free grammars based on pushdown automata. Offline, we compute bounded pushdown summaries with reachability labels and upper-bound distances to acceptance. Online, these estimates guide horizon-aware pruning and beam search. The resulting decoder is syntactically sound: every output is accepted by the target grammar. Experiments on JSON, SQL, and Linear Temporal Logic (LTL) show both consistent syntactic validity and improved completion quality over existing baselines.

cs.AI↗
arXiv2026

Separating Words with Automata in the Half-adversarial Case

We consider the problem of separating words with deterministic finite automata (DFA) (Goral{č}{í}k and Koubek, 1986). This problem asks: given two distinct words $u,v$ of length at most $n$, what is the size of the smallest DFA that accepts one and rejects the other? The best upper bound on the worst-case over all pairs of words of length at most $n$ is $\tilde{O}(n^{1/3})$ states (Chase, 2021), while the best lower bound is $Ω(\log n)$. In this work, we consider the half-random, half-adversarial case: we show that if $u$ is a uniformly random binary word of length $n$, then with high probability, for any word $v$ not equal to $u$, there is a DFA with $O(\log^{7/3} n \mathrm{poly}\log\log n)$ states that separates $u$ and $v$. Our results are based on a novel analysis that exploits the structural sparsity of random words: we show how to apply block-wise compaction with small deterministic transducers to reduce the separation problem to the case of words with short run-length encodings.

cs.FL↗
arXiv2026

The $\mathsf{AC}^0$-Complexity Of Visibly Pushdown Languages

We study the question of which visibly pushdown languages (VPLs) are in the complexity class $\mathsf{AC}^0$ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in $\mathsf{AC}^0$ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language $L$ is in $\mathsf{AC}^0$, some $m\geq 2$ such that $L$ is $\mathsf{ACC}^0(m)$-hard (implying that $L$ is not in $\mathsf{AC}^0$), or a finite disjoint union of intermediate VPLs that $L$ is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute $k,l\in\mathbb{N}_{>0}$ with $k\not=l$ such that the concrete intermediate VPL $L(S\rightarrow \varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$ is constant-depth reducible to the language $L$. Due to their particular nature we conjecture that either all intermediate VPLs are in $\mathsf{AC}^0$ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in $\mathsf{AC}^0$ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in $\mathsf{AC}^0$ (when restricted to well-matched words). For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et al.), which are closely related to forest algebras (introduced by Bojańczyk and Walukiewicz), and use Green's relations.

cs.FL↗
arXiv2026

Stochastic Estimation of Transduced Language Models

Transduced language models (TLMs) compose a pretrained \emph{source} language model with a functional finite-state transducer to induce a language model over \emph{target} strings. Computing the probability of a target prefix under a TLM amounts to summing the source-model probabilities of all source strings that the transducer maps to target strings beginning with that prefix. This set can be exponentially large or infinite. Prior work uses a computational shortcut based on source prefix probabilities, then approximates the resulting sum with threshold-pruned beam summing. This produces a lower bound with unknown error. Instead, we resample source prefixes without replacement and reweight each selected prefix by the inverse of its inclusion probability. We show that applying this correction recursively gives an unbiased estimator of the target prefix probability and lets us estimate the mass lost by threshold pruning. Our beam-summing algorithm extends the retained source prefixes and samples which prefixes to keep, reducing their number as more probability mass is added to the running estimate. This can save computation and guarantees that the run halts with probability one. We evaluate the method on encyclopedic text and DNA against sequential Monte Carlo baselines that resample with replacement. It achieves a better compute--variance tradeoff on text and lower error at the same maximum number of particles on DNA. On a DNA-to-amino-acid transduction, it reduces runtime by several orders of magnitude relative to threshold-pruned beam summing and makes estimating prefix probabilities for long target strings feasible. Replacing threshold pruning with unbiased sampling in a published reading-time analysis substantially lowers the estimated corpus surprisal but leaves the published conclusions unchanged.

cs.CL↗
arXiv2026

Cycle Counting and Character Expectations Using Alternating Structures

Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.

math.GR↗
arXiv2026

Representing One Letter Weighted Automata Over the Tropical Semiring

We consider weighted automata over the tropical semiring $\mathbb{Z}_\infty(min, +)$. Recently, it was shown that determinisation is decidable; in this paper we focus on the complexity when the alphabet is unary. In 2001, Lombardy showed this problem is decidable, a close inspection of his proof yields a coNP upper bound on the complexity. Earlier Gaubert showed that every weighted automaton in this setting can be effectively turned into an equivalent union of deterministic weighted automata. We prove Gaubert's result efficiently, presenting it as a generalisation of Chrobak's normal form for unary NFA. In particular, we prove that the equivalent union of deterministic weighted automata can be represented by a weighted automaton of quadratic size in the size of the original one, and this representation can be computed in polynomial time. Building on this, we show that determinisation, and even register minimisation (which generalises determinisation), is coNP-complete. We complete the paper with observations that the boundedness problem is also coNP-complete by reductions with determinisation. Lastly, we provide evidence that all of these problems are not FPT (by proving $coW_1$-hardness) when parametrised by the number of deterministic automata in the union.

cs.FL↗
arXiv2026

Exact Accepting-State Spectrum for Reversal of Permutation Automata

We determine the accepting-state spectrum of reversal for permutation automata exactly, thereby proving the Rauch--Holzer conjecture on this operation. For every $m \ge 2$ and every $α\ge 2$, we construct a binary permutation automaton $A_{m,α}$ such that $\operatorname{asc}(L(A_{m,α}))=m$ and $\operatorname{asc}(L(A_{m,α})^R)=α$. Combined with the trivial cases $m=0$ and $m=1$, and with the previously known fact that $1$ is magic for every $m \ge 2$, this yields the exact spectrum $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(0)=\{0\}$, $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(1)=\{1\}$, and $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(m)=\mathbb{N}_{\ge 2}$ for every $m \ge 2$. Thus reversal has, for permutation automata, the simplest possible exact accepting-state spectrum compatible with the single nontrivial obstruction at value $1$. The proof uses a uniform group-theoretic witness family: the states of the forward automaton are the $α$-subsets of $[n]$, where $n=m+α-1$, under the action generated by an $n$-cycle and a transposition, while the accepting states form a single star family. After reversal, the reachable subset-states are exactly the stars. This makes it possible to count the accepting reachable states precisely and to prove minimality of the reachable reverse automaton.

cs.FL↗
arXiv2026

A Unary-to-Nonunary Transition in the Accepting-State Spectrum of Right Quotient for Permutation Automata

This paper resolves the open larger-alphabet quotient case in the accepting-state complexity theory of permutation automata. Rauch and Holzer showed that, in the unary setting, the attainable right-quotient accepting-state complexities are exactly $[1,mn]$. We prove that over arbitrary alphabets the exact spectrum is $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\{0\}$ if $m=0$ or $n=0$, and $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\mathbb{N}_{>0}$ if $m,n\ge 1$. Thus, once both input languages are nonempty, every positive accepting-state complexity is attainable for right quotient, and $0$ is the only unavoidable magic value. The proof has two parts. First, we show that if $m,n\ge 1$, then the quotient language $KL^{-1}$ cannot be empty when $K$ and $L$ are accepted by permutation automata with $\operatorname{asc}(K)=m$ and $\operatorname{asc}(L)=n$; this follows from the bijectivity of the transition action. Second, for every $m,n\ge 1$ and every $α\ge m$, we construct a ternary witness pair $(A^{\mathrm{q}}_{m,α},B^{\mathrm{q}}_{n,α})$ such that $\operatorname{asc}(L(A^{\mathrm{q}}_{m,α}))=m$, $\operatorname{asc}(L(B^{\mathrm{q}}_{n,α}))=n$, and $\operatorname{asc}(L(A^{\mathrm{q}}_{m,α})L(B^{\mathrm{q}}_{n,α})^{-1})=α$. The high-range construction is group-theoretic: the words accepted by $B^{\mathrm{q}}_{n,α}$ induce exactly a point stabilizer in a symmetric group, and the standard quotient construction then saturates the original final set of $A^{\mathrm{q}}_{m,α}$ to a full orbit, yielding a minimal quotient automaton with exactly $α$ final states. Combined with the known unary interval $[1,mn]$, this yields the complete spectrum and resolves the larger-alphabet right-quotient case for permutation automata.

cs.FL↗
arXiv2025

Store Languages of Turing Machines and Counter Machines

The store language of an automaton is the set of store configurations (state and store contents, but not the input) that can appear as an intermediate step in an accepting computation. A one-way nondeterministic finite-visit Turing machine (fvNTM) is a Turing machine with a one-way read-only input tape, and a single worktape, where there is some number $k$ such that in every accepting computation, each worktape cell is visited at most $k$ times. We show that the store language of every fvNTM is a regular language. Furthermore, we show that the store language of every fvNTM augmented by reversal-bounded counters can be accepted by a machine with only reversal-bounded counters and no worktape. Several applications are given to problems in the areas of verification and fault tolerance, and to the study of right quotients. We also continue the investigation of the store languages of one-way and two-way machine models where we present some conditions under which their store languages are recursive or non-recursive.

cs.FL↗
arXiv2024

Proving Properties of $φ$-Representations with the Walnut Theorem-Prover

We revisit a classic theorem of Frougny and Sakarovitch concerning automata for $φ$-representations, and show how to obtain it in a different and more computationally direct way. Using it, we can find simple, induction-free proofs of existing results in the literature about these representations, in a uniform and straightforward manner. In particular, we can easily and "automatically'' recover many of the results of recent papers of Dekking and Van Loon. We also obtain a number of new results on $φ$-representations.

math.NT↗
arXiv2024

Dyck Words, Pattern Avoidance, and Automatic Sequences

We study various aspects of Dyck words appearing in binary sequences, where $0$ is treated as a left parenthesis and $1$ as a right parenthesis. We show that binary words that are $7/3$-power-free have bounded nesting level, but this no longer holds for larger repetition exponents. We give an explicit characterization of the factors of the Thue-Morse word that are Dyck, and show how to count them. We also prove tight upper and lower bounds on $f(n)$, the number of Dyck factors of Thue-Morse of length $2n$.

cs.DM↗