Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.
For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.
Let $S$ be a reachable image of a strongly connected synchronizing automaton and $\varnothing\ne A\subsetneq S$. We determine the maximum shortest avoid-or-compress length over binary and arbitrary finite alphabets for $m=n-|S|=4$ and $m=5$. The exact values are $\min(n-|A|,11)$ and $\min(n-|A|,14)$, respectively. Uniform binary constructions attain both bounds. For the upper bounds, short-pair graphs and actual-image constraints reduce all parameters to finite necessary domains. Exhaustive checks exclude the domains, apart from a single relaxation eliminated by a multipartite overlap lemma. The largest calculation covers 3,206,988 domains.
Higher-order recursion schemes (HORS) are finite, simply typed recursive programs that generate potentially infinite trees. Their monadic second-order model-checking problem is decidable. We prove that tree equivalence of deterministic HORS is undecidable and, more precisely, $Π^0_1$-complete under total computable many-one reductions. The lower bound holds for safe, productive schemes of order at most four over the fixed ranked alphabet $Σ_2=\{e:0,d:2\}$, consisting of one nullary and one binary symbol. Adapting the counter-guard witness of Boiret, Piorkowski, and Schmude, we generate all candidate executions of a two-counter machine by producing both continuations at every counter test. An integer weight is nonzero exactly on legal prefixes. Functions at fixed simple types represent the polynomial of growing degree. A shift recurrence updates them without storing a separate step counter, allowing the witness to be implemented by safe applicative HORS rules. At candidate halts, the intermediate schemes generate unary observation subtrees whose lengths encode the two natural-number components of the signed weight. Invalid candidates give matching observations; a legal halt produces a finite discrepancy. Fixed constructor contexts encode these intermediate trees over $Σ_2$. For the upper bound, a finite monotone semantics decides head convergence and supports an algorithm computing exact tree prefixes at any prescribed finite depth, including positions labelled $\bot$.
Relational pattern languages generalize Angluin's pattern languages by requiring the strings substituted for repeated occurrences of a variable to stand in a given relation, rather than to be identical. We study erasing pattern languages under the equal-length relation, where all occurrences of a variable must be replaced by strings of the same length. Over alphabets with at least three letters, this class is known to be learnable in the limit from positive data. We show that over a binary alphabet it is not, which resolves an open question from earlier work. Our proof reveals that non-learnability already holds for the restricted subclass of languages generated by patterns that have exactly three maximal substrings of variables. We then show that patterns with at most two maximal substrings of variables are learnable in the limit from positive data for every alphabet of size at least two. If the target pattern is known to have at most one maximal substring of variables, this can even be achieved by an iterative learner, which updates its hypothesis using only the current example and its previous hypothesis.
In this paper we investigate the maximum size of finite semigroups of rational $n \times n$ matrices, with the goal of shedding more light on their structure. Such semigroups provide a rich generalisation of transition monoids of unambiguous (and, in particular, deterministic) finite automata. While in general such semigroups can be arbitrarily large in terms of $n$, a classical result of Schützenberger from 1962 implies an upper bound of $2^{O(n^2 \log n)}$ for irreducible semigroups. A semigroup of rational matrices is called irreducible if the only subspaces of $\mathbb{Q}^n$ that are invariant for all matrices in the semigroup are $\mathbb{Q}^n$ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they can be seen as a generalisation of strongly connected weighted automata. Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to $3^{n^2}$. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every $n$, a finite irreducible semigroup with $3^{\lfloor n^2/4 \rfloor}$ rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold of finite semigroups of rational matrices. The mortality threshold is a number $\ell$ such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most $\ell$ matrices from any subset that generates the semigroup.
We prove a pumping-like lemma for languages accepted by one-register alternating finite-memory automata over an infinite alphabet. Every sufficiently long accepted word has a nonempty factor in a bounded suffix that can be repeated under successive powers of a single alphabet permutation. The proof analyzes accepting runs backwards, counting register values by their state profiles within the remaining suffix and summarizing values outside it. We derive effective semi-linearity of the sets of accepted word lengths and prove non-closure under reversal, concatenation, and Kleene star. We also show that this pumping lemma cannot be extended to alternating automata with two registers.
In the context of learning formal languages, data about an unknown target language L is given in terms of a set of (word,label) pairs, where a binary label indicates whether or not the given word belongs to L. A (polynomial-size) characteristic set for L, with respect to a reference class L of languages, is a set of such pairs that satisfies certain conditions allowing a learning algorithm to (efficiently) identify L within L. In this paper, we introduce the notion of positive characteristic set, referring to characteristic sets of only positive examples. These are of importance in the context of learning from positive examples only. We study this notion for classes of relational pattern languages, which are of relevance to various applications in string processing.
We study the expressive power of First-Order Logic (\FO) over (unordered) infinite trees, with the aim of identifying robust characterisations in terms of branching-time specification formalisms. While such correspondences are well understood in the linear-time setting, the branching-time case presents well-known structural challenges. To this end, we introduce two classes of hesitant tree automata and show that they capture precisely the expressive power of two branching-time temporal logics, namely \PolPCTL and \CTLsf, both of which have been previously shown to be equivalent to \FO over infinite trees. These results provide uniform automata-theoretic characterisations and yield a natural normal form for the latter in terms of a new fragment of \CTLs called \PolCTLs. As a consequence, we identify a fundamental limitation of \FO in this setting: along each branch, it can express only properties that are either safety or co-safety, thereby revealing a sharp expressive boundary for first-order definability over infinite trees.
Simulation benchmarks are a useful way to measure progress in robotics; typically they provide success rate as the dominant measure of performance. However, success rate reports only whether a rollout completed the task, hiding by what margin it succeeded, how sensitive the success is to goal parameter changes, requirements it may have violated, and whether success persisted. Signal Temporal Logic (STL) provides a formal language for capturing specifications; it can be used to define requirements, and measure how close a robot behavior is to satisfying or violating them. Using STL monitors in a simulator, however, requires exposing simulator state and mapping it to specification-relevant signals. We present Specola, a tool for evaluating robot behaviors in simulation with respect to STL specifications. Specola provides an interface for writing specifications over simulator signals, and creates and manages the relevant monitors while a policy executes. We provide a Python API and command-line interface for integrating monitoring into existing policy execution loops. We provide an adapter interface for connecting new simulators, and provide adapters for nine simulation environments and benchmarks, including LIBERO, RoboCasa, Isaac Lab, and RoboDojo. We provide case studies that illustrate the nuanced evaluation that Specola enables. Code and adapters are available at https://github.com/specola-eval/specola.
Yangge Li, Katherine Braught, Alex Yuan, Sayan Mitra
Verifying cyber-physical systems with many interacting agents remains difficult because interactions are mediated by sensors, individual decisions are constrained by environment (map) structure, and the resulting mode space scales combinatorially. We present Verse, a framework for writing agents, maps, and sensors in a familiar programming language syntax, with a formal semantics for the resulting scenarios. A compiler translates each scenario to a hybrid automaton whose executions correspond exactly to the runs of the scenario. Reachability analysis on this automaton builds an execution tree from two post operators, one for continuous flow over a time step and one for a single agent's mode change, and checks safety assertions at every vertex. The tree expands only the modes that are reached, without enumerating the global mode space. Experiments highlight expressive modeling across cars and drones, 2D/3D maps (including OpenDRIVE imports), noisy sensors, and multiple simultaneous safety assertions. Verse verifies a scenario with 7 agents and 37 modes in 90 seconds on a standard laptop; it is competitive with CORA and JuliaReach on ARCH benchmarks, and on multi-agent scenarios is an order of magnitude faster than JuliaReach at the cost of looser reachable sets.
We study an extension of Zielonka's (fixed) asynchronous automata called reconfigurable asynchronous automata where processes can dynamically change who they communicate with. We show that reconfigurable asynchronous automata are not more expressive than fixed asynchronous automata by giving translations from one to the other. However, going from reconfigurable to fixed comes at the cost of disseminating communication (and knowledge) to all processes in the system. We then show that this is unavoidable by describing a language accepted by a reconfigurable automaton such that in every equivalent fixed automaton, every process must either be aware of all communication or be irrelevant.
Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and $\dim_{\mathbb{K}}V \geq |\mathbb{K}|^{\aleph_0}$. In particular, $|\mathbb{K}|^{\aleph_0}$ is the exact dimension threshold for non-reversibility. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.
Hofstadter's G function is recursively defined via G(0) = 0 and then G(n) = n-G(G(n -1)). Following Hofstadter, we vary the number k of nested recursive calls in this equation and obtain a family of functions (F\_k). In a previous article, we proved that this family is ordered pointwise: for all k and n, we have $F_k(n) $le$ F_{k+1}(n)$. Here we propose a much shorter proof of this theorem, without infinite morphic words. The key ingredient for this direct proof is a new formula expressing iterates of the preimages of $F_k$ . Thanks to this formula, we also prove several conjectures mentioned in our previous article. In particular we obtain that $F_k(n) < F_{k+1}(n)$ when n > (k + 1)(k + 6)/2. We provide a certified formalization of all these results in the Rocq proof assistant.
Decisiveness is central to reachability algorithms for infinite Markov chains, but is a qualitative notion: it guarantees that path-enumeration algorithms terminate, without an a priori bound on the horizon. We study decisiveness quantitatively of Markov chains, and apply our results to the approximate quantitative coverability problem for effective probabilistic well-structured transition systems (pWSTS). We introduce truncation sets, which represent the horizons sufficient to approximate the reachability probability within a precision $\varepsilon$. These sets reformulate decisiveness quantitatively, and their elements can be computed when it is decisive. To obtain such horizons, before any exploration, we introduce geometrically reachable sets, for which the probability of not reaching them within n steps decreases exponentially in n, up to a prefactor depending on the initial state. Horizons valid for all initial states exist iff this prefactor can be chosen uniformly for boundedly coarse chains. For such pWSTS, including probabilistic vector addition systems and Galton-Watson processes, we obtain a priori global horizons growing at most logarithmically in $1/\varepsilon$. In the non-uniform case, we give a drift condition ensuring the existence of a finite geometrically reachable set. Under this condition, we derive a priori horizons growing at most logarithmically in $1/\varepsilon$ and in the weight w(x) measuring the size of the initial state x. In both cases, we compute a horizon function whose evaluation on any initial state and precision requires only a constant number of elementary operations $(+, -, \times, / \text{ and } \log)$. For probabilistic lossy channel systems, this yields, to our knowledge, the first explicit a priori horizons from arbitrary initial states, and it extends to $ν$-pLCS, a new infinitely branching variant with probabilistic emissions.
A dot product means something only when both vectors share a coordinate system. Content words have one; a fresh entity does not: rotating its coordinates alone changes nothing. We take this per-entity frame freedom as a symmetry and ask what it forces in attention. For orthogonal transports, invariant scores depend only on norms and the transported inner product $\langle x_i,T_{ij}x_j\rangle$; equivariant messages lie in the span of $x_i$ and $T_{ij}x_j$. Endpoint channel count bounds transport rank, so with one vector per entity relations must be supplied; standard attention is the flat sector of a shared frame. Representation theory attaches an integer to each mechanism; the readout decides which. With orthogonal transitions, a convex-cell readout composes a finite group exactly only at or above its smallest faithful real representation dimension. At this minimum, exactness forces a faithful representation. An even readout can decode a cover at smaller width, and the gap is unbounded. Below its threshold, the part of the group a mechanism cannot see sets an accuracy ceiling computable before training. 17/17 fitting seeds at minimal width passed tests through length 1000, versus 7/18 at twice the width; 12 models carry correctness certificates to explicit lengths, from the weights alone. On $C_3\times A_4$ (faithful width 5), descent at width 4 learned the 72-state double cover under sign-blind attention (6/48 seeds; monotone 0/48); on a frozen cover the cross-entropy-optimal affine head is uniform, a quadratic head exact. On parsed CLUTRR, transported composition beats a matched transformer by 41 points at 4-10 hops, but forcing isometries costs 65 points; on StepGame's commutative algebra, isometries help. Static embeddings gave each word one vector; contextual models gave each occurrence its own; per-entity frames make each relation an operator, the unit in which relations compose.
A certified split point lets a parallel lexer cut unlexed input at a single byte with the serial token stream provably preserved, but several conventional token sets in the predecessor's controlled study certify no byte once string, comment, or whitespace-run forms are included (arXiv:2608.03473v2). We generalize from a byte to a bounded window, a byte string after which the position where the current token began is known, regardless of surrounding context. We certify the directly usable form of that recovery, namely that the token covering the window's final byte begins at the reported origin. The certificate is conditional on occurrence and may be vacuous; every applicability figure counts only windows carrying an asserted completely tokenizable occurrence witness. We give a conservative model of a maximal-munch scanner's possible histories across a window, prove it sound, and decide reachability in that model exactly by exhausting a finite quotient of its reachable configurations. Within the flat, completely tokenizable scope, model-positive answers are semantic certificates; negatives are relative to the model, which deliberately refuses some windows a greedy scanner would allow. Whether the model certifies any window, plain or witnessed, is PSPACE-complete, and a shortest window can need $2^{Ω(\sqrt{n})}$ bytes over n live states. Two smaller searches decide plain model existence, the second shipped in release 2.2.0 of munch. In a sample of 400 random token sets, 322 of the 337 sets certifying no byte gain a witnessed window, with zero inconclusive searches, and every exact-empty row of the predecessor's study gains a witnessed window of two to four bytes. The analysis runs once after automaton construction, using only the compiled tables and no input. A boundary profile marks each gap of a window forced, crossed or open, and a window-local cut is sound exactly at forced gaps.
Table-driven DFA lexing is sequential, since each transition depends on the previous byte's state. Scanning one input in parallel needs each chunk's entry state, which existing methods recover by simulation, speculation, prescanning, or overlap. We give two conditions under which none is needed. For a longest-match scanner restarting from q0 at every token boundary, a byte b is a certified split symbol when no reachable state other than q0 has a b-transition whose target can reach acceptance, and q0 is not re-entrant if it has one. Every occurrence of such a byte in completely tokenizable input begins a token, so chunks starting there reproduce the serial sequence of kinds and lengths. The condition is necessary and sufficient, but one string, comment, or whitespace run can remove every useful certificate. Comments and whitespace are usually discarded, so we weaken the guarantee to equality after deleting a declared discarded set and give a second condition, a precomputed constant-time one-bit test. It is sound but incomplete and more permissive, strictly so on suitable token sets and discarded sets. It recovers newline for a conventional C-like tokenization and tab, newline and carriage return for JSON, and refuses newline where block comments are unrestricted. Safety modulo the discarded set is decidable in PSPACE for every token automaton and in polynomial time for literal vocabularies, and every unsafe byte has a replayable counterexample. On the fifteen token sets studied the second condition is already exact. Splitting at exact certificates in munch reaches 92.6-95.3% parallel efficiency at eight threads on a restricted CPU set and a 512 MiB corpus, and a 3.46-3.94x end-to-end speedup at four threads, on one machine. It turns delimiter-based parallel lexing from a language-specific assumption into a property a compiler checks.