cs.FL

arXiv:2503.21906v1 Announce Type: new Abstract: Modern cyber-physical systems (CPS) can consist of various networked components and agents interacting and communicating with each other. In the context of spatially distributed CPS, these connections can be dynamically dependent on the spatial configuration of the various components and agents. In these settings, robust monitoring of the distributed components is vital to ensuring complex behaviors are achieved, and safety properties are maintained. To this end, we look at defining the automaton semantics for the Spatio-Temporal Reach and Escape Logic (STREL), a formal logic designed to express and monitor spatio-temporal requirements over mobile, spatially distributed CPS. Specifically, STREL reasons about spatio-temporal behavior over dynamic weighted graphs. While STREL is endowed with well defined qualitative and quantitative semantics, in this paper, we propose a novel construction of (weighted) alternating finite automata from STREL specifications that efficiently encodes these semantics. Moreover, we demonstrate how this automaton semantics can be used to perform both, offline and online monitoring for STREL specifications using a simulated drone swarm environment.

arXiv:2503.16891v2 Announce Type: replace Abstract: We consider the problem of the verification of an LTL specification $\varphi$ on a system $S$ given some prior knowledge $K$, an LTL formula that $S$ is known to satisfy. The automata-theoretic approach to LTL model checking is implemented as an emptiness check of the product $S\otimes A_{\lnot\varphi}$ where $A_{\lnot\varphi}$ is an automaton for the negation of the property. We propose new operations that simplify an automaton $A_{\lnot\varphi}$ \emph{given} some knowledge automaton $A_K$, to produce an automaton $B$ that can be used instead of $A_{\lnot\varphi}$ for more efficient model checking. Our evaluation of these operations on a large benchmark derived from the MCC'22 competition shows that even with simple knowledge, half of the problems can be definitely answered without running an LTL model checker, and the remaining problems can be simplified significantly.

arXiv:2503.22546v1 Announce Type: cross Abstract: The most developed aspect of the theory of finite semigroups is their classification in pseudovarieties. The main motivation for investigating such entities comes from their connection with the classification of regular languages via Eilenberg's correspondence. This connection prompted the study of various natural operators on pseudovarieties and led to several important questions, both algebraic and algorithmic. The most important of these questions is decidability: given a finite semigroup is there an algorithm that tests whether it belongs to the pseudovariety? Since the most relevant operators on pseudovarieties do not preserve decidability, one often seeks to establish stronger properties. A key role is played by relatively free profinite semigroups, which is the counterpart of free algebras in universal algebra. The purpose of this paper is to give a brief survey of the state of the art, highlighting some of the main developments and problems.

arXiv:2503.22000v1 Announce Type: new Abstract: We introduce a new class of clustered Moore automata (CMA), investigate their temporal behavior, and describe some applications.

arXiv:2503.22558v1 Announce Type: new Abstract: The goal of this paper is to provide exact and terminating algorithms for the formal analysis of deterministic continuous-time control systems with affine input and polynomial state dynamics (in short, polynomial systems). We consider the following semantic properties: zeroness and equivalence, input independence, linearity, and analyticity. Our approach is based on Chen-Fliess series, which provide a unique representation of the dynamics of such systems via their formal generating series. Our starting point is Fliess' seminal work showing how the semantic properties above are mirrored by corresponding combinatorial properties on generating series. Next, we observe that the generating series of polynomial systems coincide with the class of shuffle-finite series, a nonlinear generalisation of Sch\"utzenberger's rational series which has recently been studied in the context of automata theory and enumerative combinatorics. We exploit and extend recent results in the algorithmic analysis of shuffle-finite series (such as zeroness, equivalence, and commutativity) to show that the semantic properties above can be decided exactly and in finite time for polynomial systems. Some of our analyses rely on a novel technical contribution, namely that shuffle-finite series are closed under support restrictions with commutative regular languages, a result of independent interest.

arXiv:2503.05042v1 Announce Type: new Abstract: Automata-conditioned reinforcement learning (RL) has given promising results for learning multi-task policies capable of performing temporally extended objectives given at runtime, done by pretraining and freezing automata embeddings prior to training the downstream policy. However, no theoretical guarantees were given. This work provides a theoretical framework for the automata-conditioned RL problem and shows that it is probably approximately correct learnable. We then present a technique for learning provably correct automata embeddings, guaranteeing optimal multi-task policy learning. Our experimental evaluation confirms these theoretical results.

arXiv:2407.14105v2 Announce Type: replace Abstract: This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings $\alpha$ and $\beta$ such that $\alpha$ is strictly quasi-isometrically reducible to $\beta$, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.

arXiv:2409.09769v2 Announce Type: replace Abstract: Humans naturally balance the risks of different concerns while driving, including traffic rule violations, minor accidents, and fatalities. However, achieving the same behavior in autonomous systems remains an open problem. This paper extends a risk metric that has been verified in human-like driving studies to encompass more complex driving scenarios specified by linear temporal logic (LTL) that go beyond just collision risks. This extension incorporates the timing and severity of events into LTL specifications, thereby reflecting a human-like risk awareness. Without sacrificing expressivity for traffic rules, we adopt LTL specifications composed of safety and co-safety formulas, allowing the control synthesis problem to be reformulated as a reachability problem. By leveraging occupation measures, we formulate a linear programming (LP) problem for this LTL-based risk metric. Consequently, the synthesized policy balances different types of risks, including not only collision risks but also traffic rule violations. The effectiveness of the proposed approach is validated by three typical traffic scenarios in the Carla simulator.

arXiv:2503.04762v1 Announce Type: new Abstract: We study the verification problem of stochastic systems under signal temporal logic (STL) specifications. We propose a novel approach that enables the verification of the probabilistic satisfaction of STL specifications for nonlinear systems subject to both bounded deterministic disturbances and stochastic disturbances. Our method, referred to as the STL erosion strategy, reduces the probabilistic verification problem into a deterministic verification problem with a tighter STL specification. The degree of tightening is determined by leveraging recent results on bounding the deviation between the stochastic trajectory and the deterministic trajectory. Our approach can be seamlessly integrated with any existing deterministic STL verification algorithm. Numerical experiments are conducted to showcase the efficacy of our method.

arXiv:2503.05006v1 Announce Type: new Abstract: Markov decision process over vector addition system with states (VASS MDP) is a finite state model combining non-deterministic and probabilistic behavior, augmented with non-negative integer counters that can be incremented or decremented during each state transition. VASS MDPs can be used as abstractions of probabilistic programs with many decidable properties. In this paper, we develop techniques for analyzing the asymptotic behavior of VASS MDPs. That is, for every initial configuration of size \(n\), we consider the number of transitions needed to reach a configuration with some counter negative. We show that given a strongly connected VASS MDP there either exists an integer \(k\leq 2^d\cdot 3^{|T|} \), where \(d \) is the dimension and \(|T|\) the number of transitions of the VASS MDP, such that for all \(\epsilon>0 \) and all sufficiently large \(n\) it holds that the complexity of the VASS MDP lies between \(n^{k-\epsilon} \) and \(n^{k+\epsilon} \) with probability at least \(1-\epsilon \), or it holds for all \(\epsilon>0 \) and all sufficiently large \(n\) that the complexity of the VASS MDP is at least \(2^{n^{1-\epsilon}} \) with probability at least \(1-\epsilon \). We show that it is decidable which case holds and the \(k\) is computable in time polynomial in the size of the considered VASS MDP. We also provide a full classification of asymptotic complexity for VASS Markov chains.

arXiv:2502.10297v2 Announce Type: replace Abstract: Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. While diagonal matrices used in architectures like Mamba, GLA, or mLSTM yield fast runtime, they suffer from severely limited expressivity. To address this, recent architectures such as (Gated) DeltaNet and RWKVv7 adopted a diagonal plus rank-1 structure, allowing simultaneous token-channel mixing, which overcomes some expressivity limitations with only a slight decrease in training efficiency. Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple ($n_h$) steps per token. This naturally leads to diagonal plus rank-$n_h$ state-transition matrices, formed as products of $n_h$ generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency and a stable recurrence. Through extensive experiments, we demonstrate that DeltaProduct achieves superior state-tracking and language modeling capabilities while exhibiting significantly improved length extrapolation compared to DeltaNet. Additionally, we also strengthen the theoretical foundation of DeltaNet's expressivity by proving that it can solve dihedral group word problems in just two layers.

arXiv:2503.05572v1 Announce Type: cross Abstract: We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, CA groups themselves have co-NEXPTIME word problem, and on the lamplighter group (which itself has polynomial-time word problem) we show they can be co-NEXPTIME-hard. We show also two nonembeddability results: the group of cellular automata on a non-cyclic free group does not embed in the group of cellular automata on the integers (this solves a question of Barbieri, Carrasco-Vargas and Rivera-Burgos); and the group of cellular automata in dimension $D$ does not embed in a group of cellular automata in dimension $d$ if $D \geq 3d+2$ (this solves a question of Hochman).

arXiv:2210.02773v3 Announce Type: replace Abstract: In a two-player zero-sum graph game, the players move a token throughout a graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder (called Richman bidding). We focus on discrete-bidding games, in which, motivated by practical applications, the granularity of the players' bids is restricted, e.g., bids must be given in cents. A central quantity in bidding games is threshold budgets: a necessary and sufficient initial budget for winning the game. Previously, thresholds were shown to exist in parity games, but their structure was only understood for reachability games. Moreover, the previously-known algorithms have a worst-case exponential running time for both reachability and parity objectives, and output strategies that use exponential memory. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first is a fixed-point algorithm. It reveals, for the first time, the structure of threshold budgets in parity discrete-bidding games. Based on this structure, we develop a second algorithm that shows that the problem of finding threshold budgets is in NP and coNP for both reachability and parity objectives. Moreover, our algorithm constructs strategies that use only linear memory.

arXiv:2501.11789v1 Announce Type: new Abstract: Nielsen transformations form the basis of a simple and widely used procedure for solving word equations. We make progress on the problem of determining when this procedure terminates in the presence of length constraints. To do this, we introduce extended word equations, a mathematical model of a word equation with partial information about length constraints. We then define extended Nielsen transformations, which adapt Nielsen transformations to the setting of extended word equations. We provide a partial characterization of when repeatedly applying extended Nielsen transformations to an extended word equation is guaranteed to terminate.

arXiv:2501.12302v1 Announce Type: new Abstract: History-deterministic automata are a restricted class of nondeterministic automata where the nondeterminism while reading an input can be resolved successfully based on the prefix read so far. History-deterministic automata are exponentially more succinct than deterministic automata, while still retaining some of the algorithmic properties of deterministic automata, especially in the context of reactive synthesis. This thesis focuses on the problem of checking history-determinism for parity automata. Our main result is the 2-token theorem, due to which we obtain that checking history-determinism for parity automata with a fixed parity index can be checked in PTIME. This improves the naive EXPTIME upper bound of Henzinger and Piterman that has stood since 2006. More precisely, we show that the so-called 2-token game, which can be solved in PTIME for parity automata with a fixed parity index, characterises history-determinism for parity automata. This game was introduced by Bagnol and Kuperberg in 2018, who showed that to decide if a B\"uchi automaton is history-deterministic, it suffices to find the winner of the 2-token game on it. They conjectured that this 2-token game based characterisation of history-determinism extends to parity automata. We prove Bagnol and Kuperberg's conjecture that the winner of the 2-token game characterises history-determinism on parity automata. We also give a polynomial time determinisation procedure for history-deterministic B\"uchi automata, thus solving an open problem of Kuperberg and Skrzypczak from 2015. This result is a consequence of our proof of the 2-token theorem. Finally, we also show NP-hardness for the problem of checking history-determinism for parity automata when the parity index is not fixed. This is an improvement from the lower bound of solving parity games shown by Kuperberg and Skrzypczak in 2015.

arXiv:2212.01679v5 Announce Type: replace Abstract: We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barcel\'o, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $\Pi^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).

arXiv:2501.10981v1 Announce Type: new Abstract: Sequence diagrams are a popular technique for describing interactions between software entities. However, because the OMG group's UML standard is not based on a rigorous mathematical structure, it is impossible to deduce a single interpretation for the notation's semantics, nor to understand precisely how its different fragments interact. While there are a lot of suggested semantics in the literature, they are too mathematically demanding for the majority of software engineers, and often incomplete, especially in dealing with the semantics of lifeline creation and deletion. In this work we describe a simple semantics based on the theory of regular languages, a mathematical theory that is a standard part of the curriculum in every computer science undergraduate degree and covers all the major compositional fragments, and the creation and deletion of lifelines.

arXiv:2501.06579v1 Announce Type: new Abstract: We consider the problem of refuting equivalence of probabilistic programs, i.e., the problem of proving that two probabilistic programs induce different output distributions. We study this problem in the context of programs with conditioning (i.e., with observe and score statements), where the output distribution is conditioned by the event that all the observe statements along a run evaluate to true, and where the probability densities of different runs may be updated via the score statements. Building on a recent work on programs without conditioning, we present a new equivalence refutation method for programs with conditioning. Our method is based on weighted restarting, a novel transformation of probabilistic programs with conditioning to the output equivalent probabilistic programs without conditioning that we introduce in this work. Our method is the first to be both a) fully automated, and b) providing provably correct answers. We demonstrate the applicability of our method on a set of programs from the probabilistic inference literature.

arXiv:2501.05830v1 Announce Type: cross Abstract: We investigate the lengths and starting positions of the longest monochromatic arithmetic progressions for a fixed difference in the Fibonacci word. We provide a complete classification for their lengths in terms of a simple formula. Our strongest results are proved using methods from dynamical systems, especially the dynamics of circle rotations. We also employ computer-based methods in the form of the automatic theorem-proving software Walnut. This allows us to extend recent results concerning similar questions for the Thue-Morse sequence and the Rudin-Shapiro sequence. This also allows us to obtain some results for the Fibonacci word that do not seem to be amenable to dynamical methods.

arXiv:2501.07428v1 Announce Type: new Abstract: The set of finite words over a well-quasi-ordered set is itself well-quasi-ordered. This seminal result by Higman is a cornerstone of the theory of well-quasi-orderings and has found numerous applications in computer science. However, this result is based on a specific choice of ordering on words, the (scattered) subword ordering. In this paper, we describe to what extent other natural orderings (prefix, suffix, and infix) on words can be used to derive Higman-like theorems. More specifically, we are interested in characterizing languages of words that are well-quasi-ordered under these orderings. We show that a simple characterization is possible for the prefix and suffix orderings, and that under extra regularity assumptions, this also extends to the infix ordering. We furthermore provide decision procedures for a large class of languages, that contains regular and context-free languages.