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Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize., 26 pages, comments are welcome!

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the $C(4)$ condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in $C(4)$ monoids: the word problem is solvable in $O(\min\{|u|, |v|\})$ time in the size of the input words $u$ and $v$; the uniform word problem for $\langle A|R\rangle$ is solvable in $O(N ^ 2 \min\{|u|, |v|\})$ where $N$ is the sum of the lengths of the words in $R$; and a normal form for any given word $u$ can be found in $O(|u|)$ time. Although Kambites' algorithm for solving the word problem in $C(4)$ monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in $C(4)$ monoids that has time complexity $O(|u| ^ 2)$ for input word $u$, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in $O(N \min\{|u|, |v|\})$ time., Comment: 31 pages, 6 figures (some further minor corrections from the referee are resolved, to appear in Journal of Algebra)

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case- based proofs can now be proved "automatically", using a decision procedure.

We study expression learning problems with syntactic restrictions and introduce the class of finite-aspect checkable languages to characterize symbolic languages that admit decidable learning. The semantics of such languages can be defined using a bounded amount of auxiliary information that is independent of expression size but depends on a fixed structure over which evaluation occurs. We introduce a generic programming language for writing programs that evaluate expression syntax trees, and we give a meta-theorem that connects such programs for finite-aspect checkable languages to finite tree automata, which allows us to derive new decidable learning results and decision procedures for several expression learning problems by writing programs in the programming language.

We develop a probabilistic control algorithm, $\texttt{GTLProCo}$, for swarms of agents with heterogeneous dynamics and objectives, subject to high-level task specifications. The resulting algorithm not only achieves decentralized control of the swarm but also significantly improves scalability over state-of-the-art existing algorithms. Specifically, we study a setting in which the agents move along the nodes of a graph, and the high-level task specifications for the swarm are expressed in a recently-proposed language called graph temporal logic (GTL). By constraining the distribution of the swarm over the nodes of the graph, GTL can specify a wide range of properties, including safety, progress, and response. $\texttt{GTLProCo}$, agnostic to the number of agents comprising the swarm, controls the density distribution of the swarm in a decentralized and probabilistic manner. To this end, it synthesizes a time-varying Markov chain modeling the time evolution of the density distribution under the GTL constraints. We first identify a subset of GTL, namely reach-avoid specifications, for which we can reduce the synthesis of such a Markov chain to either linear or semi-definite programs. Then, in the general case, we formulate the synthesis of the Markov chain as a mixed-integer nonlinear program (MINLP). We exploit the structure of the problem to provide an efficient sequential mixed-integer linear programming scheme with trust regions to solve the MINLP. We empirically demonstrate that our sequential scheme is at least three orders of magnitude faster than off-the-shelf MINLP solvers and illustrate the effectiveness of $\texttt{GTLProCo}$ in several swarm scenarios., Comment: Initial submission to TAC

Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by \'Ciri\'c et al. (2012) as fuzzy relations between the sets of states of the automata. However, they act as a crisp relationship between the automata. In particular, if there exists a (forward) bisimulation between two fuzzy automata, then the fuzzy languages recognized by them are crisply equal. Approximate simulations and bisimulations introduced by Stanimirovi\'c et al. (2020) aim at fuzzifying this phenomenon. However, they are defined only for fuzzy automata over a complete Heyting algebra and do not give the exact relationship between states of the automata. In this article, we introduce and study fuzzy simulations and bisimulations between fuzzy automata over a complete residuated lattice. These notions are novel and have good properties. They are defined for fuzzy automata over any complete residuated lattice. We prove that the fuzzy language recognized by a fuzzy automaton is fuzzily preserved by fuzzy simulations and fuzzily invariant under fuzzy bisimulations. We also prove that the notions of fuzzy simulation and bisimulation have the Hennessy-Milner properties, which are a logical characterization of the greatest fuzzy simulation or bisimulation between two fuzzy automata. In addition, we provide results showing that our notions of fuzzy simulation and bisimulation are more general and refined than the notions of simulation and bisimulation introduced by \'Ciri\'c et al. and the notions of approximate simulation and bisimulation introduced by Stanimirovi\'c et al.

Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local function $\mu :A^S \to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $\tau : A^G \to A^H$, where $H$ is another arbitrary group, via a group homomorphism $\phi : H \to G$. Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When $G=H$, we prove that the group of invertible GCA over $A^G$ is isomorphic to a semidirect product of $\text{Aut}(G)^{op}$ and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid $\text{CA}(G;A)$ consisting of all CA over $A^G$. In particular, we show that every $\phi \in \text{Aut}(G)$ defines an automorphism of $\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $\phi$, and that, when $G$ is abelian, $\text{Aut}(G)$ is embedded in the outer automorphism group of $\text{CA}(G;A)$., Comment: 11 pages

We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper- and lower-bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: we prove (1) that the reachability problem for COCA with global upper- and lower-bound tests is in NC 2 ; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is NP-complete for COCA with parametric counter updates and bound tests.

Coalgebraic bisimilarity minimization generalizes classical automaton minimization to a large class of automata whose transition structure is specified by a functor, subsuming strong, weighted, and probabilistic bisimilarity. This offers the enticing possibility of turning bisimilarity minimization into an off-the-shelf technology, without having to develop a new algorithm for each new type of automaton. Unfortunately, there is no existing algorithm that is fully general, efficient, and able to handle large systems. We present a generic algorithm that minimizes coalgebras over an arbitrary functor in the category of sets as long as the action on morphisms is sufficiently computable. The functor makes at most O ( m log n ) calls to the functor-specific action, where n is the number of states and m is the number of transitions in the coalgebra. While more specialized algorithms can be asymptotically faster than our algorithm (usually by a factor of ( m / n )), our algorithm is especially well suited to efficient implementation, and our tool often uses much less time and memory on existing benchmarks, and can handle larger automata, despite being more generic.

Supervisor localization can be applied to distribute a monolithic supervisor into local supervisors. Performing supervisor localization can be computationally costly. In this work, we consider systems that evolve over time. We study how to reuse the results from a previous supervisor localization, to more efficiently compute local supervisors when the system is adapted. We call this approach transformational supervisor localization, and present algorithms for the procedure. The efficiency of the procedure is experimentally evaluated., Comment: Accepted for IEEE Control Systems Letters (L-CSS) (2023)

It is shown that a higher-dimensional automaton is hhp-bisimilar to the free symmetric HDA generated by it. Consequently, up to hereditary history-preserving bisimilarity, ordinary HDAs and symmetric HDAs are models of concurrency with the same expressive power.

The need for high-level autonomy and robustness of autonomous systems for missions in dynamic and remote environment has pushed developers to come up with new software architectures. A common architecture style is to summarize the capabilities of the robotic system into elementary actions, called skills, on top of which a skill management layer is implemented to structure, test and control the functional layer. However, current available verification tools only provide either mission-specific verification or verification on a model that does not replicate the actual execution of the system, which makes it difficult to ensure its robustness to unexpected events. To that end, a tool, SkiNet, has been developed to transform the skill-based architecture of a system into a Petri net modeling the state-machine behaviors of the skills and the resources they handle. The Petri net allows the use of model-checking, such as Linear Temporal Logic (LTL) or Computational Tree Logic (CTL), for the user to analyze and verify the model of the system., Comment: In Proceedings FMAS2022 ASYDE2022, arXiv:2209.13181

Hyperproperties lift conventional trace-based languages from a set of execution traces to a set of sets of executions. From a formal-language perspective, these are sets of sets of words, namely hyperlanguages. Hyperautomata are based on classical automata models that are lifted to handle hyperlanguages. Finite hyperautomata (NFH) have been suggested to express regular hyperproperties. We study the realizability problem for regular hyperlanguages: given a set of languages, can it be precisely described by an NFH? We show that the problem is complex already for singleton hyperlanguages. We then go beyond regular hyperlanguages, and study context-free hyperlanguages. We show that the natural extension to context-free hypergrammars is highly undecidable. We then suggest a refined model, namely synchronous hypergrammars, which enables describing interesting non-regular hyperproperties, while retaining many decidable properties of context-free grammars., In Proceedings GandALF 2022, arXiv:2209.09333

We present an alternative translation from CCS to an extension of CSP based on m-among-n synchronisation (called CSPmn). This translation is correct up to strong bisimulation. Unlike the g-star renaming approach ([4]), this translation is not limited by replication (viz., recursion with no nested parallel composition). We show that m-among-n synchronisation can be implemented in CSP based on multiway synchronisation and renaming., Comment: In Proceedings EXPRESS/SOS 2022, arXiv:2208.14777

During the last two decades, the decentralized execution of business processes has been one of the main research topics in Business Process Management. Several models (languages) for processes' specification in order to facilitate their distributed execution, have been proposed. LSAWfP is among the most recent in this area: it helps to specify administrative processes with grammatical models indicating, in addition to their fundamental elements, the permissions (reading, writing and execution) of each actor in relation to each of their tasks. In this paper, we present a model for a completely decentralized and artifact-centric execution of administrative processes specified using LSAWfP. The presented model puts particular emphasis on actors' views: it then allows the confidential execution of certain tasks by ensuring that, each actor potentially has only a partial perception of the processes' global execution states. The model thus solves a very important problem in business process execution, which is often sidelined in existing approaches. To accomplish this, the model rely on three projection algorithms allowing to partially replicate the processes' global execution states at a given moment, to consistently update the obtained partial states and to deduce new coherent global states. The proposal of these three algorithms, the proof of underlying mathematical tools' stability and a proposal of their implementation, are this paper's main contributions.

We consider a notion of planarity for two-way finite automata and transducers, inspired by Temperley-Lieb monoids of planar diagrams. We show that this restriction captures star-free languages and first-order transductions., 18 pages, DMTCS submission

This paper deals with the problem of automatically and correctly controlling infinite-state reactive programs to achieve LTL goals. Applications include adapting a program to new requirements, or to repair bugs discovered in the original specification or program code. Existing approaches are able to solve this problem for safety and some reachability properties, but require an a priori template of the solution for more general properties. Fully automated approaches for full LTL exist, reducing the problem into successive finite LTL reactive synthesis problems in an abstraction-refinement loop. However, they do not terminate when the number of steps to be completed depends on unbounded variables. Our main insight is that safety abstractions of the program are not enough -- fairness properties are also essential to be able to decide many interesting problems, something missed by existing automated approaches. We thus go beyond the state-of-the-art to allow for automated reactive program control for full LTL, with automated discovery of the knowledge, including fairness, of the program needed to determine realisability. We further implement the approach in a tool, with an associated DSL for reactive programs, and illustrate the approach through several case studies.

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class $C$, the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of $C$ as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted~contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.

We investigate an operator on classes of languages. For each class $C$, it outputs a new class $FO^2(I_C)$ associated with a variant of two-variable first-order logic equipped with a signature$I_C$ built from $C$. For $C = \{\emptyset, A^*\}$, we get the variant $FO^2(

Different stochastic extensions of hybrid automata have been proposed in the past, with unclear expressivity relations between them. To structure and relate these modeling languages, in this paper we formalize two alternative approaches to extend hybrid automata with stochastic choices of discrete events and their time points. The first approach, which we call decomposed scheduling, adds stochasticity via stochastic races, choosing random time points for the possible discrete events and executing a winner with an earliest time. In contrast, composed scheduling first samples the time point of the next event and then the event to be executed at the sampled time point. We relate the two approaches regarding their expressivity and categorize available stochastic extensions of hybrid automata from the literature., This paper is accepted for publication (without appendix) in the Proceedings of the 2023 International Conference on Quantitative Evaluation of Systems (QEST). The appendix was part of the submission and provides additional material which is not included in the QEST publication

We introduce contextual behavioural metrics (CBMs) as a novel way of measuring the discrepancy in behaviour between processes, taking into account both quantitative aspects and contextual information. This way, process distances by construction take the environment into account: two (non-equivalent) processes may still exhibit very similar behaviour in some contexts, e.g., when certain actions are never performed. We first show how CBMs capture many well-known notions of equivalence and metric, including Larsen's environmental parametrized bisimulation. We then study compositional properties of CBMs with respect to some common process algebraic operators, namely prefixing, restriction, non-deterministic sum, parallel composition and replication., Extended version of a paper accepted for publication in proc. CONCUR 2023

A rational relation is conjugate if every pair of related words are conjugates. It is shown that checking whether a rational relation is conjugate is decidable. For this, we generalise the Lyndon-Sch\"utzenberger's theorem from word combinatorics. A consequence of the generalisation is that a set of pairs generated by a sumfree rational expression is conjugate if and only if there is a word witnessing the conjugacy of all the pairs.

Negotiations, introduced by Esparza et al., are a model for concurrent systems where computations involving a set of agents are described in terms of their interactions. In many situations, it is natural to impose timing constraints between interactions -- for instance, to limit the time available to enter the PIN after inserting a card into an ATM. To model this, we introduce a real-time aspect to negotiations. In our model of local-timed negotiations, agents have local reference times that evolve independently. Inspired by the model of networks of timed automata, each agent is equipped with a set of local clocks. Similar to timed automata, the outcomes of a negotiation contain guards and resets over the local clocks. As a new feature, we allow some interactions to force the reference clocks of the participating agents to synchronize. This synchronization constraint allows us to model interesting scenarios. Surprisingly, it also gives unlimited computing power. We show that reachability is undecidable for local-timed negotiations with a mixture of synchronized and unsynchronized interactions. We study restrictions on the use of synchronized interactions that make the problem decidable., A shorter version appears in FORMATS 2023

In parametric lock-sharing systems processes can spawn new processes to run in parallel, and can create new locks. The behavior of every process is given by a pushdown automaton. We consider infinite behaviors of such systems under strong process fairness condition. A result of a potentially infinite execution of a system is a limit configuration, that is a potentially infinite tree. The verification problem is to determine if a given system has a limit configuration satisfying a given regular property. This formulation of the problem encompasses verification of reachability as well as of many liveness properties. We show that this verification problem, while undecidable in general, is decidable for nested lock usage. We show Exptime-completeness of the verification problem. The main source of complexity is the number of parameters in the spawn operation. If the number of parameters is bounded, our algorithm works in Ptime for properties expressed by parity automata with a fixed number of ranks.

The standard approach to analyzing the asymptotic complexity of probabilistic programs is based on studying the asymptotic growth of certain expected values (such as the expected termination time) for increasing input size. We argue that this approach is not sufficiently robust, especially in situations when the expectations are infinite. We propose new estimates for the asymptotic analysis of probabilistic programs with non-deterministic choice that overcome this deficiency. Furthermore, we show how to efficiently compute/analyze these estimates for selected classes of programs represented as Markov decision processes over vector addition systems with states.

We address the satisfiability problem for string constraints that combine relational constraints represented by transducers, word equations, and string length constraints. This problem is undecidable in general. Therefore, we propose a new decidable fragment of string constraints, called weakly chaining string constraints, for which we show that the satisfiability problem is decidable. This fragment pushes the borders of decidability of string constraints by generalising the existing straight-line as well as the acyclic fragment of the string logic. We have developed a prototype implementation of our new decision procedure, and integrated it into in an existing framework that uses CEGAR with under-approximation of string constraints based on flattening. Our experimental results show the competitiveness and accuracy of the new framework.

Simulations and bisimulations are well-established notions in crisp/fuzzy automata theory and are widely used to compare the behaviors of automata. Their main drawback is that they compare the behaviors of fuzzy automata in a crisp manner. Recently, fuzzy simulations and fuzzy bisimulations have been defined for fuzzy automata as a kind of approximate simulations and approximate bisimulations that compare the behaviors of fuzzy automata in a fuzzy manner. However, they still suffer from serious shortcomings. First, they still cannot correlate all fuzzy automata that are intuitively "more or less" (bi)similar. Second, the currently known algorithms for computing the greatest fuzzy simulation or bisimulation between two finite fuzzy automata have an exponential time complexity when the {\L}ukasiewicz or product structure of fuzzy values is used. This work deals with these problems, providing approximations of fuzzy simulations and fuzzy bisimulations. We define such approximations via a novel notion of decreasing sequences of fuzzy relations whose infima are, under some conditions, fuzzy simulations (respectively, bisimulations). We call such a sequence a depth-bounded fuzzy simulation (respectively, bisimulation), as the $n$th element from the sequence compares the behaviors of fuzzy automata, but only for words with a length bounded by $n$. We further provide a logical characterization of the greatest depth-bounded fuzzy simulation or bisimulation between two fuzzy automata by proving that it satisfies the corresponding Hennessy-Milner property. Finally, we provide polynomial-time algorithms for computing the $n$th component of the greatest depth-bounded fuzzy simulation (respectively, bisimulation) between two finite fuzzy automata.

This paper provides a reference description, in the form of a deduction system, of Earley's (1970) context-free parsing algorithm with various speed-ups. Our presentation includes a known worst-case runtime improvement from Earley's $O (N^3|G||R|)$, which is unworkable for the large grammars that arise in natural language processing, to $O (N^3|G|)$, which matches the runtime of CKY on a binarized version of the grammar $G$. Here $N$ is the length of the sentence, $|R|$ is the number of productions in $G$, and $|G|$ is the total length of those productions. We also provide a version that achieves runtime of $O (N^3|M|)$ with $|M| \leq |G|$ when the grammar is represented compactly as a single finite-state automaton $M$ (this is partly novel). We carefully treat the generalization to semiring-weighted deduction, preprocessing the grammar like Stolcke (1995) to eliminate deduction cycles, and further generalize Stolcke's method to compute the weights of sentence prefixes. We also provide implementation details for efficient execution, ensuring that on a preprocessed grammar, the semiring-weighted versions of our methods have the same asymptotic runtime and space requirements as the unweighted methods, including sub-cubic runtime on some grammars., Main conference long paper at ACL 2023

Solving control tasks in complex environments automatically through learning offers great potential. While contemporary techniques from deep reinforcement learning (DRL) provide effective solutions, their decision-making is not transparent. We aim to provide insights into the decisions faced by the agent by learning an automaton model of environmental behavior under the control of an agent. However, for most control problems, automata learning is not scalable enough to learn a useful model. In this work, we raise the capabilities of automata learning such that it is possible to learn models for environments that have complex and continuous dynamics. The core of the scalability of our method lies in the computation of an abstract state-space representation, by applying dimensionality reduction and clustering on the observed environmental state space. The stochastic transitions are learned via passive automata learning from observed interactions of the agent and the environment. In an iterative model-based RL process, we sample additional trajectories to learn an accurate environment model in the form of a discrete-state Markov decision process (MDP). We apply our automata learning framework on popular RL benchmarking environments in the OpenAI Gym, including LunarLander, CartPole, Mountain Car, and Acrobot. Our results show that the learned models are so precise that they enable the computation of policies solving the respective control tasks. Yet the models are more concise and more general than neural-network-based policies and by using MDPs we benefit from a wealth of tools available for analyzing them. When solving the task of LunarLander, the learned model even achieved similar or higher rewards than deep RL policies learned with stable-baselines3.

The problem of learning a computational model from examples has been receiving growing attention. Models of distributed systems are particularly challenging since they encompass an added succinctness. While positive results for learning some models of distributed systems have been obtained, so far the considered models assume a fixed number of processes interact. In this work we look for the first time (to the best of our knowledge) at the problem of learning a distributed system with an arbitrary number of processes, assuming only that there exists a cutoff. Specifically, we consider fine broadcast protocols, these are broadcast protocols (BPs) with a finite cutoff and no hidden states. We provide a learning algorithm that given a sample consistent with a fine BP, can infer a correct BP, with the help of an SMT solver. Moreover we show that the class of fine BPs is teachable, meaning that we can associate a finite set of words $\mathcal{S}_B$ with each BP $B$ in the class (a so-called characteristic set) so that the provided learning algorithm can correctly infer a correct BP from any consistent sample subsuming $\mathcal{S}_B$. On the negative size we show that (a) characteristic sets of exponential size are unavoidable, (b) the consistency problem for fine BPs is NP hard, and (c) fine BPs are not polynomially predictable., 13 pages, 7 figures, 3 input files of plots

In recent months, Language Models (LMs) have become a part of daily discourse, with focus on OpenAI and the potential of Artificial General Intelligence (AGI). Furthermore, the leaking of LLama's weights to the public has led to an influx of innovations demonstrating the impressive capabilities of generative LMs. While we believe that AGI is still a distant goal, we recognize the potential of LMs in solving tasks such as searching complex documents, compiling reports with basic analysis, and providing assistance in problem-solving. In this paper, we propose formalizing the execution model of language models. We investigate current execution models, to find that this formalism has received little attention, and present our contribution: the first formalized execution model for LMs. We introduce a new algorithm for sampling the predictions of LMs, which we use to build a reliable and inspectable execution model. We introduce a low-level language to write "cognitive program" for this execution model. We hope to shed light on the need for execution models for LMs and encourage further research in this area., Submitted to CGO-24

In a recent article by Chapuy and Perarnau, it was shown that a uniformly chosen automaton on $n$ states with a $2$-letter alphabet has a synchronizing word of length $O(\sqrt{n}\log n)$ with high probability. In this note, we give a new simplified proof of a slightly weaker version of this statement. Our proof is based on two properties of random automata. First, by repeating a fixed character from the alphabet sufficiently many times in a row, the number of possible states reduces to, in expectation, $O(\sqrt{n})$. Second, with high probability, each pair of states can be synchronized by a word of length $O(\log n)$., 7 pages, no figures. comments welcome

We present a large deviation property for the pattern statistics representing the number of occurrences of a symbol in words of given length generated at random according to a rational stochastic model. The result is obtained assuming that in the model the overall weighted transition matrix is primitive. In particular we obtain a rate function depending on the main eigenvalue and eigenvectors of that matrix. Under rather mild conditions, we show that the range of validity of our large deviation estimate can be extended to the interval (0,1), which represents in our context the largest possible open interval of validity of the property.

Using FSA and the construction algorithm, we generated a list of surjective span 6 cellular automata as a modest sample for our FDense program. We wanted to experimentally quantify Mike Boyle's conjecture which states that the jointly periodic points of one-dimensional cellular automata are dense. Furthermore, we wanted to know if the cardinality of cellular automata on N symbols is greater than or equal to the square root of N.

In this paper, we present a proof of the NP-completeness of computing the smallest Deterministic Finite Automaton (DFA) that distinguishes two given regular languages as DFAs. A distinguishing DFA is an automaton that recognizes a language which is a subset of exactly one of the given languages. We establish the NP-hardness of this decision problem by providing a reduction from the Boolean Satisfiability Problem (SAT) to deciding the existence of a distinguishing automaton of a specific size.

Weir has defined a hierarchy of language classes whose second member ($\mathcal{L}_2$) is generated by tree-adjoining grammars (TAG), linear indexed grammars (LIG), combinatory categorial grammars, and head grammars. The hierarchy is obtained using the mechanism of control, and $\mathcal{L}_2$ is obtained using a context-free grammar (CFG) whose derivations are controlled by another CFG. We adapt Weir's definition of a controllable CFG to give a definition of controllable pushdown automata (PDAs). This yields three new characterizations of $\mathcal{L}_2$ as the class of languages generated by PDAs controlling PDAs, PDAs controlling CFGs, and CFGs controlling PDAs. We show that these four formalisms are not only weakly equivalent but equivalent in a stricter sense that we call d-weak equivalence. Furthermore, using an even stricter notion of equivalence called d-strong equivalence, we make precise the intuition that a CFG controlling a CFG is a TAG, a PDA controlling a PDA is an embedded PDA, and a PDA controlling a CFG is a LIG. The fourth member of this family, a CFG controlling a PDA, does not correspond to any formalism we know of, so we invent one and call it a Pushdown Adjoining Automaton., 18 pages, 7 figures. Accepted at ACL 2023

We are interested in connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich's completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise many questions related to both fractals and process calculi.

There is a one-to-one and onto correspondence between the class of numerical semigroups of depth $n$, where $n$ is an integer, and a certain language over the alphabet $\{1,\ldots,n\}$ which we call a Kunz language of depth $n$. The Kunz language associated with the numerical semigroups of depth $2$ is the regular language $\{1,2\}^*2\{1,2\}^*$. We prove that Kunz languages associated with numerical semigroups of larger depth are context-sensitive but not regular., 11 pages

We associate lattices to the sets of unions and intersections of left and right quotients of a regular language. For both unions and intersections, we show that the lattices we produce using left and right quotients are dual to each other. We also give necessary and sufficient conditions for these lattices to have maximal possible complexity., 15 pages

We consider the parameterized verification of arbitrarily large networks of agents which communicate by broadcasting and receiving messages. In our model, the broadcast topology is reconfigurable so that a sent message can be received by any set of agents. In addition, agents have local registers which are initially distinct and may therefore be thought of as identifiers. When an agent broadcasts a message, it appends to the message the value stored in one of its registers. Upon reception, an agent can store the received value or test this value for equality with one of its own registers. We consider the coverability problem, where one asks whether a given state of the system may be reached by at least one agent. We establish that this problem is decidable; however, it is as hard as coverability in lossy channel systems, which is non-primitive recursive. This model lies at the frontier of decidability as other classical problems on this model are undecidable; this is in particular true for the target problem where all processes must synchronize on a given state. By contrast, we show that the coverability problem is NP-complete when each agent has only one register.

chemSKI with tokens is a confluent graph rewrite system where all rewrites are local, which moreover can be used to do SKI calculus reductions. The graph rewrites of chemSKI are made conservative by the use of tokens. We thus achieve several goals: conservative rewrites in a chemical style, a solution to the problem of new edge names in a distributed, decentralized graphical reduction and a new estimation of the cost of a combinatory calculus computation. This formalism can be used either as an artificial chemistry or as a model of a virtual decentralized machine which performs only local reductions. A programs repository and the same article with simulations are available at github at https://mbuliga.github.io/chemski/chemski-with-tokens.html

A safety verification task involves verifying a system against a desired safety property under certain assumptions about the environment. However, these environmental assumptions may occasionally be violated due to modeling errors or faults. Ideally, the system guarantees its critical properties even under some of these violations, i.e., the system is \emph{robust} against environmental deviations. This paper proposes a notion of \emph{robustness} as an explicit, first-class property of a transition system that captures how robust it is against possible \emph{deviations} in the environment. We modeled deviations as a set of \emph{transitions} that may be added to the original environment. Our robustness notion then describes the safety envelope of this system, i.e., it captures all sets of extra environment transitions for which the system still guarantees a desired property. We show that being able to explicitly reason about robustness enables new types of system analysis and design tasks beyond the common verification problem stated above. We demonstrate the application of our framework on case studies involving a radiation therapy interface, an electronic voting machine, a fare collection protocol, and a medical pump device., Full version of CAV23 paper

It is well known that for a given bottom-up tree automaton it can be decided whether or not there exists deterministic top-down tree automaton that recognized the same tree language. Recently it was claimed that such a decision can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their procedure and corresponding property is wrong. Here we correct this mistake and present a correct property which allows to determine in polynomial time whether or not a given tree language can be recognized by a deterministic top-down tree automaton. Furthermore, our new property is stated for arbitrary deterministic bottom-up tree automata, and not for minimal such automata (as before).

This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms — which generalises and unifies the notion of bisimulation to the whole class of weighted automata — and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm.

This paper focuses on Watson-Crick languages inspired by DNA computing, their models, and algorithms for deciding the language membership. It analyzes a recently introduced algorithm called WK-CYK and introduces a state space search algorithm that is based on regular Breadth-first search but uses a number of optimizations and heuristics to be efficient in practical use and able to analyze longer inputs. The key parts are the heuristics for pruning the state space (detecting dead ends) and heuristics for choosing the most promising branches to continue the search. These two algorithms have been tested with 20 different Watson-Crick grammars (40 including their Chomsky normal form versions). While WK-CYK is able to decide the language membership in a reasonable time for inputs of the length of roughly 30-50 symbols and its performance is very consistent for all kinds of grammars and inputs, the state space search is usually (89-98 % of cases) more efficient and able to do the computation for inputs with lengths of hundreds or even thousands of symbols. Thus, the state space search has the potential to be a good tool for practical Watson-Crick membership testing and is a good basis for improvement the efficiency of the algorithm in the future., In Proceedings NCMA 2022, arXiv:208.13015

Watson-Crick (WK) finite automata are working on a Watson-Crick tape, that is, on a DNA molecule. A double stranded DNA molecule contains two strands, each having a 5' and a 3' end, and these two strands together form the molecule with the following properties. The strands have the same length, their 5' to 3' directions are opposite, and in each position, the two strands have nucleotides that are complement of each other (by the Watson-Crick complementary relation). Consequently, WK automata have two reading heads, one for each strand. In traditional WK automata both heads read the whole input in the same physical direction, but in 5'->3' WK automata the heads start from the two extremes and read the input in opposite direction. In sensing 5'->3' WK automata, the process on the input is finished when the heads meet, and the model is capable to accept the class of linear context-free languages. Deterministic variants are weaker, the class named 2detLIN, a proper subclass of linear languages is accepted by them. Recently, another specific variants, the state-deterministic sensing 5'->3' WK automata are investigated in which the graph of the automaton has the special property that for each node of the graph, all out edges (if any) go to a sole node, i.e., for each state there is (at most) one state that can be reached by a direct transition. It was shown that this concept is somewhat orthogonal to the usual concept of determinism in case of sensing 5'->3' WK automata. In this paper a new concept, the quasi-determinism is investigated, that is in each configuration of a computation (if it is not finished yet), the next state is uniquely determined although the next configuration may not be, in case various transitions are enabled at the same time. We show that this new concept is a common generalisation of the usual determinism and the state-determinism, i.e., the class of quasi-deterministic sensing 5'->3' WK automata is a superclass of both of the mentioned other classes. There are various usual restrictions on WK automata, e.g., stateless or 1-limited variants. We also prove some hierarchy results among language classes accepted by various subclasses of quasi-deterministic sensing 5'->3' WK automata and also some other already known language classes., In Proceedings NCMA 2022, arXiv:2208.13015

This paper presents a new approach to regulation of grammars. It divides the derivation trees generated by grammars into two sections-generative and conclusive (the conclusion). The former encompasses generation of symbols up till the moment when the lowest rightmost terminal of the derivation tree is generated, whereas the latter represents the final steps needed to successfully generate a sentence. A control mechanism based on regulating only the conclusion is presented and subsequently applied to tree-controlled grammars, creating conclusive tree-controlled grammars. As the main result, it is shown that the ratio between depths of generative and conclusive sections does not influence the generative power. In addition, it is demonstrated that any recursively enumerable language is generated by these grammars possessing no more than seven nonterminals while the regulating language is union-free., In Proceedings NCMA 2022, arXiv:2208.13015

We investigate the accepting state complexity of deterministic finite automata for regular languages obtained by applying one of the following operations to languages accepted by permutation automata: union, quotient, complement, difference, intersection, Kleene star, Kleene plus, and reversal. The paper thus joins the study of accepting state complexity of regularity preserving language operations which was initiated by the work [J. Dassow: On the number of accepting states of finite automata, J. Autom., Lang. Comb., 21, 2016]. We show that for almost all of the operations, except for reversal and quotient, there is no difference in the accepting state complexity for permutation automata compared to deterministic finite automata in general. For both reversal and quotient we prove that certain accepting state complexities cannot be obtained; these number are called "magic" in the literature. Moreover, we solve the left open accepting state complexity problem for the intersection of unary languages accepted by permutation automata and deterministic finite automata in general., Comment: In Proceedings NCMA 2022, arXiv:2208.13015

We continue the research on the generative capacity of contextual grammars where contexts are adjoined around whole words (externally) or around subwords (internally) which belong to special regular selection languages. All languages generated by contextual grammars where all selection languages are elements of a certain subregular language family form again a language family. We investigate contextual grammars with strictly locally testable selection languages and compare those families to families which are based on finite, monoidal, nilpotent, combinational, definite, suffix-closed, ordered, commutative, circular, non-counting, power-separating, or union-free languages., In Proceedings NCMA 2022, arXiv:2208.13015

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $\Phi$. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying $\Phi$ in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language $L$ if a certain torsion condition is satisfied. This condition holds trivially if $L$ is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet $\Sigma$, and we define a regular set $\mathbb{G}\subseteq \Sigma^*$ such that every nonempty word $w\in \mathbb{G}$ defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over $\Sigma$. Then, we ask whether the automaton $\mathcal{A}$ specifies some graph satisfying a certain property~$\Phi$. Our structural results show that we can answer this question for all "typical" graph properties. In order to show our results, we split $L$ into a finite union of subsets and every subset of this union defines in a natural way a single finite graph $F$ where some edges and vertices are marked. The marked graph in turn defines an infinite graph $F^\infty$ and therefore the family of finite subgraphs of $F^\infty$ where $F$ appears as an induced subgraph. This yields a geometric description of all graphs specified by $L$ based on splitting $L$ into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece., Comment: 25 pages