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The well-studied red-blue pebble game models the execution of an arbitrary computational DAG by a single processor over a two-level memory hierarchy. We present a natural generalization to a multiprocessor setting where each processor has its own limited fast memory, and all processors share unlimited slow memory. To our knowledge, this is the first thorough study that combines pebbling and DAG scheduling problems, capturing the computation of general workloads on multiple processors with memory constraints and communication costs. Our pebbling model enables us to analyze trade-offs between workload balancing, communication and memory limitations, and it captures real-world factors such as superlinear speedups due to parallelization. Our results include upper and lower bounds on the pebbling cost, an analysis of a greedy pebbling strategy, and an extension of NP-hardness results for specific DAG classes from simpler models. For our main technical contribution, we show two inapproximability results that already hold for the long-standing problem of standard red-blue pebbling: (i) the optimal I/O cost cannot be approximated to any finite factor, and (ii) the optimal total cost (I/O+computation) can only be approximated to a limited constant factor, i.e., it does not allow for a polynomial-time approximation scheme. These results also carry over naturally to our multiprocessor pebbling model.

In the present work, we further study the computational power of virus machines (VMs in short). VMs provide a computing paradigm inspired by the transmission and replication networks of viruses. VMs consist of process units (called hosts) structured by a directed graph whose arcs are called channels and an instruction graph that controls the transmissions of virus objects among hosts. The present work complements our understanding of the computing power of VMs by introducing normal forms; these expressions restrict the features in a given computing model. Some of the features that we restrict in our normal forms include (a) the number of hosts, (b) the number of instructions, and (c) the number of virus objects in each host. After we recall some known results on the computing power of VMs we give our normal forms, such as the size of the loops in the network, proving new characterisations of family of sets, such as the finite sets, semilinear sets, or NRE.

Robots find extensive applications in industry. In recent years, the influence of robots has also increased rapidly in domestic scenarios. The Q-learning algorithm aims to maximise the reward for reaching the goal. This paper proposes a modified version of the Q-learning algorithm, known as Q-learning with scaled distance metric (Q-SD). This algorithm enhances task learning and makes task completion more meaningful. A robotic manipulator (agent) applies the Q-SD algorithm to the task of table cleaning. Using Q-SD, the agent acquires the sequence of steps necessary to accomplish the task while minimising the manipulator's movement distance. We partition the table into grids of different dimensions. The first has a grid count of 3 times 3, and the second has a grid count of 4 times 4. Using the Q-SD algorithm, the maximum success obtained in these two environments was 86% and 59% respectively. Moreover, Compared to the conventional Q-learning algorithm, the drop in average distance moved by the agent in these two environments using the Q-SD algorithm was 8.61% and 6.7% respectively., Comment: 9 pages, 9 figures, 7 tables

This paper describes a discrete state-space model -- and accompanying methods -- for generative modelling. This model generalises partially observed Markov decision processes to include paths as latent variables, rendering it suitable for active inference and learning in a dynamic setting. Specifically, we consider deep or hierarchical forms using the renormalisation group. The ensuing renormalising generative models (RGM) can be regarded as discrete homologues of deep convolutional neural networks or continuous state-space models in generalised coordinates of motion. By construction, these scale-invariant models can be used to learn compositionality over space and time, furnishing models of paths or orbits; i.e., events of increasing temporal depth and itinerancy. This technical note illustrates the automatic discovery, learning and deployment of RGMs using a series of applications. We start with image classification and then consider the compression and generation of movies and music. Finally, we apply the same variational principles to the learning of Atari-like games., Comment: 64 pages, 28 figures

Multi-priced timed automata (MPTA) are timed automata with observer variables whose derivatives can change from one location to another. Observers are write-only variables, that is, they do not affect the control flow of the automaton; thus MPTA lie between timed and hybrid automata in expressiveness. Previous work considered observers with non-negative slope in every location. In this paper we treat observers that have both positive and negative rates. Our main result is an algorithm to decide a gap version of the reachability problem for this variant of MPTA. We translate the gap reachability problem into a gap satisfiability problem for mixed integer-real systems of nonlinear constraints. Our main technical contribution -- a result of independent interest -- is a procedure to solve such contraints via a combination of branch-and-bound and relaxation-and-rounding.

We introduce Modelizer - a novel framework that, given a black-box program, learns a _model from its input/output behavior_ using _neural machine translation_. The resulting model _mocks_ the original program: Given an input, the model predicts the output that would have been produced by the program. However, the model is also _reversible_ - that is, the model can predict the input that would have produced a given output. Finally, the model is _differentiable_ and can be efficiently restricted to predict only a certain aspect of the program behavior. Modelizer uses _grammars_ to synthesize inputs and to parse the resulting outputs, allowing it to learn sequence-to-sequence associations between token streams. Other than input and output grammars, Modelizer only requires the ability to execute the program. The resulting models are _small_, requiring fewer than 6.3 million parameters for languages such as Markdown or HTML; and they are _accurate_, achieving up to 95.4% accuracy and a BLEU score of 0.98 with standard error 0.04 in mocking real-world applications. We foresee several _applications_ of these models, especially as the output of the program can be any aspect of program behavior. Besides mocking and predicting program behavior, the model can also synthesize inputs that are likely to produce a particular behavior, such as failures or coverage., Comment: 42 pages, 6 figures, 8 tables

We present a system for Complex Event Recognition (CER) based on automata. While multiple such systems have been described in the literature, they typically suffer from a lack of clear and denotational semantics, a limitation which often leads to confusion with respect to their expressive power. In order to address this issue, our system is based on an automaton model which is a combination of symbolic and register automata. We extend previous work on these types of automata, in order to construct a formalism with clear semantics and a corresponding automaton model whose properties can be formally investigated. We call such automata Symbolic Register Transducers (SRT). We show that SRT are closed under various operators, but are not in general closed under complement and they are not determinizable. However, they are closed under these operations when a window operator, quintessential in Complex Event Recognition, is used. We show how SRT can be used in CER in order to detect patterns upon streams of events, using our framework that provides declarative and compositional semantics, and that allows for a systematic treatment of such automata. For SRT to work in pattern detection, we allow them to mark events from the input stream as belonging to a complex event or not, hence the name "transducers". We also present an implementation of SRT which can perform CER. We compare our SRT-based CER engine against other state-of-the-art CER systems and show that it is both more expressive and more efficient., Comment: arXiv admin note: substantial text overlap with arXiv:2110.04032

This paper introduces the notion of a universal plan, which when executed, is guaranteed to solve all planning problems in a category, regardless of the obstacles, initial state, and goal set. Such plans are specified as a deterministic sequence of actions that are blindly applied without any sensor feedback. Thus, they can be considered as pure exploration in a reinforcement learning context, and we show that with basic memory requirements, they even yield asymptotically optimal plans. Building upon results in number theory and theory of automata, we provide universal plans both for discrete and continuous (motion) planning and prove their (semi)completeness. The concepts are applied and illustrated through simulation studies, and several directions for future research are sketched., Comment: 21 pages, 4 figures

Instruction subsets are heuristics that can reduce the size of the inductive programming search space by tens of orders of magnitude. Comprising many overlapping subsets of different sizes, they serve as predictions of the instructions required to code a solution for any problem. Currently, this approach employs a single, large family of subsets meaning that some problems can search thousands of subsets before a solution is found. In this paper we introduce the use of test case type signatures as hyper-heuristics to select one of many, smaller families of instruction subsets. The type signature for any set of test cases maps directly to a single family and smaller families mean that fewer subsets need to be considered for most problems. Having many families also permits subsets to be reordered to better reflect their relative occurrence in human code - again reducing the search space size for many problems. Overall the new approach can further reduce the size of the inductive programming search space by between 1 and 3 orders of magnitude, depending on the type signature. Larger and more consistent reductions are possible through the use of more sophisticated type systems. The potential use of additional test case features as hyper-heuristics and some other possible future work is also briefly discussed., Comment: 14 pages, 3 figures. Accepted for 20th IFIP WG 12.5 International Conference, AIAI 2024 Corfu, Greece, June 27-30, 2024

We develop a learning algorithm for closed signal flow graphs - a graphical model of signal transducers. The algorithm relies on the correspondence between closed signal flow graphs and weighted finite automata on a singleton alphabet. We demonstrate that this procedure results in a genuine reduction of complexity: our algorithm fares better than existing learning algorithms for weighted automata restricted to the case of a singleton alphabet., Comment: 13 pages, 6 figures. An extended abstract for Learning and Automata workshop (LearnAut 2024)

We investigate the transition monoid construction for deterministic automata in a categorical setting and establish it as an adjunction. We pair this adjunction with two other adjunctions to obtain two endofunctors on deterministic automata, a comonad and a monad, which are closely related, respectively, to the largest set of equations and the smallest set of coequations satisfied by an automaton. Furthermore, we give similar transition algebra constructions for lasso and {\Omega}-automata, and show that they form adjunctions. We present some initial results on sets of equations and coequations for lasso automata.

Relatively little is known about the discrete horospheres in hyperbolic groups, even in simple settings. In this paper we work with hyperbolic one-ended right-angled Coxeter groups and describe two graph structures that mimic the intrinsic metric on a classical horosphere: the Rips graph and the divergence graph (the latter due to Cohen, Goodman-Strauss, and Rieck). We develop, analyze, and implement algorithms based on finite-state machines that draw large finite portions of these graphs, and deduce various geometric corollaries about the path metrics induced by these graph structures., Comment: 37 pages, 6 figures

We give a construction in a column of a one-dimensional cellular automaton of the Minkowski sum of two sets which can themselves occur in columns of cellular automata. It enables us to obtain another construction of the set of integers that are sums of three squares, answering a question by the same author., Comment: In Proceedings GASCom 2024, arXiv:2406.14588

As Large Language Models (LLMs) perform (and sometimes excel at) more and more complex cognitive tasks, a natural question is whether AI really understands. The study of understanding in LLMs is in its infancy, and the community has yet to incorporate well-trodden research in philosophy, psychology, and education. We initiate this, specifically focusing on understanding algorithms, and propose a hierarchy of levels of understanding. We use the hierarchy to design and conduct a study with human subjects (undergraduate and graduate students) as well as large language models (generations of GPT), revealing interesting similarities and differences. We expect that our rigorous criteria will be useful to keep track of AI's progress in such cognitive domains., Comment: 13 pages, 8 figures

The bandwidth of timed automata characterizes the quantity of information produced/transmitted per time unit. We previously delimited 3 classes of TA according to the nature of their asymptotic bandwidth: meager, normal, and obese. In this paper, we propose a method, based on a finite-state simply-timed abstraction, to compute the actual value of the bandwidth of meager automata. The states of this abstraction correspond to barycenters of the faces of the simplices in the region automaton. Then the bandwidth is $\log 1/|z_0|$ where $z_0$ is the smallest root (in modulus) of the characteristic polynomial of this finite-state abstraction., Comment: Extended version of CIAA'24 accepted paper

Contemporary AI applications leverage large language models (LLMs) to harness their knowledge and reasoning abilities for natural language processing tasks. This approach shares similarities with the concept of oracle Turing machines (OTMs). To capture the broader potential of these computations, including those not yet realized, we propose an extension to OTMs: the LLM-oracle machine (LLM-OM), by employing a cluster of LLMs as the oracle. Each LLM acts as a black box, capable of answering queries within its expertise, albeit with a delay. We introduce four variants of the LLM-OM: basic, augmented, fault-avoidance, and $\epsilon$-fault. The first two are commonly observed in existing AI applications. The latter two are specifically designed to address the challenges of LLM hallucinations, biases, and inconsistencies, aiming to ensure reliable outcomes., Comment: 6 pages

Cirquent calculus is a proof system with inherent ability to account for sharing subcomponents in logical expressions. Within its framework, this article constructs an axiomatization CL18 of the basic propositional fragment of computability logic the game-semantically conceived logic of computational resources and tasks. The nonlogical atoms of this fragment represent arbitrary so called static games, and the connectives of its logical vocabulary are negation and the parallel and choice versions of conjunction and disjunction. The main technical result of the article is a proof of the soundness and completeness of CL18 with respect to the semantics of computability logic., Comment: arXiv admin note: text overlap with arXiv:1707.04823

The omega-regular separability problem for B\"uchi VASS coverability languages has recently been shown to be decidable, but with an EXPSPACE lower and a non-primitive recursive upper bound -- the exact complexity remained open. We close this gap and show that the problem is EXPSPACE-complete. A careful analysis of our complexity bounds additionally yields a PSPACE procedure in the case of fixed dimension >= 1, which matches a pre-established lower bound of PSPACE for one dimensional B\"uchi VASS. Our algorithm is a non-deterministic search for a witness whose size, as we show, can be suitably bounded. Part of the procedure is to decide the existence of runs in VASS that satisfy certain non-linear properties. Therefore, a key technical ingredient is to analyze a class of systems of inequalities where one variable may occur in non-linear (polynomial) expressions. These so-called singly non-linear systems (SNLS) take the form A(x).y >= b(x), where A(x) and b(x) are a matrix resp. a vector whose entries are polynomials in x, and y ranges over vectors in the rationals. Our main contribution on SNLS is an exponential upper bound on the size of rational solutions to singly non-linear systems. The proof consists of three steps. First, we give a tailor-made quantifier elimination to characterize all real solutions to x. Second, using the root separation theorem about the distance of real roots of polynomials, we show that if a rational solution exists, then there is one with at most polynomially many bits. Third, we insert the solution for x into the SNLS, making it linear and allowing us to invoke standard solution bounds from convex geometry. Finally, we combine the results about SNLS with several techniques from the area of VASS to devise an EXPSPACE decision procedure for omega-regular separability of B\"uchi VASS.

An earlier paper gives an account of a quest for a satisfactory formalization of the classical informal notion of an algorithm. In this paper, an attempt is made to generalize the results of that quest to the informal notion of an interactive algorithm. The notion of an interactive proto-algorithm is introduced. Interactive algorithms are expected to be equivalence classes of interactive proto-algorithms under an appropriate equivalence relation. As in the non-interactive case, three equivalence relations are defined. Two of them are deemed to be bounds for an appropriate equivalence relation and the third is likely an appropriate one., Comment: 16 pages, in Sections 2 and 3 there is text overlap with arXiv:2401.08366

Quantum walks have frequently envisioned the behavior of a quantum state traversing a classically defined, generally finite, graph structure. While this approach has already generated significant results, it imposes a strong assumption: all nodes where the walker is not positioned are quiescent. This paper will examine some mathematical structures that underlie state diffusion on arbitrary graphs, that is the circulation of states within a graph. We will seek to frame the multi-walker problem as a finite quantum cellular automaton. Every vertex holds a walker at all times. The walkers will never collide and at each time step their positions update non-deterministically by a quantum swap of walkers at opposite ends of a randomly chosen edge. The update is accomplished by a unitary transformation of the position of a walker to a superposition of all such possible swaps and then performing a quantum measurement on the superposition of possible swaps. This behavior generates strong entanglement between vertex states which provides a path toward developing local actions producing diffusion throughout the graph without depending on the specific structure of the graph through blind computation., Comment: 14 pages, 2 figures

In pioneering work from 2019, Barcel\'o and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this article, we give exact logical characterizations of recurrent GNNs in two scenarios: (1) in the setting with floating-point numbers and (2) with reals. For floats, the formalism matching recurrent GNNs is a rule-based modal logic with counting, while for reals we use a suitable infinitary modal logic, also with counting. These results give exact matches between logics and GNNs in the recurrent setting without relativising to a background logic in either case, but using some natural assumptions about floating-point arithmetic. Applying our characterizations, we also prove that, relative to graph properties definable in monadic second-order logic (MSO), our infinitary and rule-based logics are equally expressive. This implies that recurrent GNNs with reals and floats have the same expressive power over MSO-definable properties and shows that, for such properties, also recurrent GNNs with reals are characterized by a (finitary!) rule-based modal logic. In the general case, in contrast, the expressive power with floats is weaker than with reals. In addition to logic-oriented results, we also characterize recurrent GNNs, with both reals and floats, via distributed automata, drawing links to distributed computing models.

We propose a local, past-oriented fragment of propositional dynamic logic to reason about concurrent scenarios modelled as Mazurkiewicz traces, and prove it to be expressively complete with respect to regular trace languages. Because of locality, specifications in this logic are efficiently translated into asynchronous automata, in a way that reflects the structure of formulas. In particular, we obtain a new proof of Zielonka's fundamental theorem and we prove that any regular trace language can be implemented by a cascade product of localized asynchronous automata, which essentially operate on a single process. These results refine earlier results by Adsul et al. which involved a larger fragment of past propositional dynamic logic and used Mukund and Sohoni's gossip automaton. Our new results avoid using this automaton, or Zielonka's timestamping mechanism and, in particular, they show how to implement a gossip automaton as a cascade product., Comment: 13 pages. Accepted for publication at LICS 2024

Diabetes is a chronic condition, considered one of the civilization diseases, that is characterized by sustained high blood sugar levels. There is no doubt that more and more people is going to suffer from diabetes, hence it is crucial to understand better its biological foundations. The essential processes related to the control of glucose levels in the blood are: glycolysis (process of breaking down of glucose) and glucose synthesis, both taking place in the liver. The glycolysis occurs during feeding and it is stimulated by insulin. On the other hand, the glucose synthesis arises during fasting and it is stimulated by glucagon. In the paper we present a Petri net model of glycolysis and glucose synthesis in the liver. The model is created based on medical literature. Standard Petri nets techniques are used to analyse the properties of the model: traps, reachability graphs, tokens dynamics, deadlocks analysis. The results are described in the paper. Our analysis shows that the model captures the interactions between different enzymes and substances, which is consistent with the biological processes occurring during fasting and feeding. The model constitutes the first element of our long-time goal to create the whole body model of the glucose regulation in a healthy human and a person with diabetes., Comment: submitted to International Workshop on Petri Nets and Software Engineering (PNSE 2024)

This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its standard call-by-name stack machine in two directions. One, "locations", introduces multiple stacks, which enable effect operators to be encoded into the abstraction and application constructs. The second, "sequencing", introduces the imperative notions of "skip" and "sequence", similar to kappa-calculus and concatenative programming languages. The key observation of the RMC is that the first-order fragment of the FMC exhibits a latent duality which, given a simple decomposition of the relevant constructors, can be concretely expressed as an involution on syntax. Semantically, this gives rise to a sound and complete calculus for string diagrams of Frobenius monoids. We consider unification as the corresponding symmetric generalization of beta-reduction. By further including standard operators of Kleene algebra, the RMC embeds a range of computational models: the kappa-calculus, logic programming, automata, Interaction Nets, and Petri Nets, among others. These embeddings preserve operational semantics, which for the RMC is again given by a generalization of the standard stack machine for the lambda-calculus. The equational theory of the RMC (which supports reasoning about its operational semantics) is conservative over both the first-order lambda-calculus and Kleene algebra, and can be oriented to give a confluent reduction relation., Comment: LICS paper, 15 pages excluding references

We calculate quasinormal modes of scalar and neutrino perturbations around the charged black hole in Asymptotically Safe Gravity. We show that the charge and coupling constant change the quasinormal spectrum considerably. We show that previous calculations of scalar quasinormal modes in this background [F. Javed, Phys. Dark Univ., 44, 101450 (2024)] suffer from a large numerical error exceeding the effect, that is, the deviations of the frequencies from their Schwarzschild limits. In the high frequency (eikonal) limit an explicit analytic formula for quasinormal modes is derived, which confirms the correspondence between the null circular geodesics and eikonal quasinormal frequencies.

It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present three proofs of this theorem from the literature in a self-contained way, and we derive some strengthenings. For example, we show that the problem remains undecidable for a fixed probabilistic finite automaton with 11 states, where only the starting distribution is given as input., Comment: 63 pages, 14 figures, 2 tables, 53 footnotes, 11 sections plus 1 appendix. Added another proof and more history, which had been overlooked before

We lift metrics over words to metrics over word-to-word transductions, by defining the distance between two transductions as the supremum of the distances of their respective outputs over all inputs. This allows to compare transducers beyond equivalence. Two transducers are close (resp. $k$-close) with respect to a metric if their distance is finite (resp. at most $k$). Over integer-valued metrics computing the distance between transducers is equivalent to deciding the closeness and $k$-closeness problems. For common integer-valued edit distances such as, Hamming, transposition, conjugacy and Levenshtein family of distances, we show that the closeness and the $k$-closeness problems are decidable for functional transducers. Hence, the distance with respect to these metrics is also computable. Finally, we relate the notion of distance between functions to the notions of diameter of a relation and index of a relation in another. We show that computing edit distance between functional transducers is equivalent to computing diameter of a rational relation and both are a specific instance of the index problem of rational relations.

We determine all functional closure properties of finite $\mathbb{N}$-weighted automata, even all multivariate ones, and in particular all multivariate polynomials. We also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular we determine all multivariate closure properties where the output vector lies on a monotone algebraic graph variety., Comment: Short version to be published at ICALP 2024

Regular transition systems (RTS) are a popular formalism for modeling infinite-state systems in general, and parameterised systems in particular. In a CONCUR 22 paper, Esparza et al. introduce a novel approach to the verification of RTS, based on inductive invariants. The approach computes the intersection of all inductive invariants of a given RTS that can be expressed as CNF formulas with a bounded number of clauses, and uses it to construct an automaton recognising an overapproximation of the reachable configurations. The paper shows that the problem of deciding if the language of this automaton intersects a given regular set of unsafe configurations is in $\textsf{EXPSPACE}$ and $\textsf{PSPACE}$-hard. We introduce $\textit{regular abstraction frameworks}$, a generalisation of the approach of Esparza et al., very similar to the regular abstractions of Hong and Lin. A framework consists of a regular language of $\textit{constraints}$, and a transducer, called the $\textit{interpretation}$, that assigns to each constraint the set of configurations of the RTS satisfying it. Examples of regular abstraction frameworks include the formulas of Esparza et al., octagons, bounded difference matrices, and views. We show that the generalisation of the decision problem above to regular abstraction frameworks remains in $\textsf{EXPSPACE}$, and prove a matching (non-trivial) $\textsf{EXPSPACE}$-hardness bound. $\textsf{EXPSPACE}$-hardness implies that, in the worst case, the automaton recognising the overapproximation of the reachable configurations has a double-exponential number of states. We introduce a learning algorithm that computes this automaton in a lazy manner, stopping whenever the current hypothesis is already strong enough to prove safety. We report on an implementation and show that our experimental results improve on those of Esparza et al., Comment: 25 pages

This paper addresses two questions regarding N-polyregular functions, that forms a proper subset of N-rational series. We show that given a Z-rational series, it is decidable whether it is computable via a commutative N-polyregular function, and provide a counter-example to the theorem of Karhum\"aki that studied the same question in the case of polynomials. We also prove that it is decidable whether a commutative N-polyregular function is star-free, by proving the stronger statement that star-free Z-polyregular functions that are N-polyregular are in fact computable using a star-free N-polyregular function. Building towards answering the same questions in the non-commutative case, we present a canonical model of computation of N-polyregular functions by generalizing the notion of residual transducers previously introduced in Z-polyregular functions., Comment: 29 pages, 3 figures

Large Language Models (LLMs), such as GPT-4 and Llama 2, show remarkable proficiency in a wide range of natural language processing (NLP) tasks. Despite their effectiveness, the high costs associated with their use pose a challenge. We present LlamBERT, a hybrid approach that leverages LLMs to annotate a small subset of large, unlabeled databases and uses the results for fine-tuning transformer encoders like BERT and RoBERTa. This strategy is evaluated on two diverse datasets: the IMDb review dataset and the UMLS Meta-Thesaurus. Our results indicate that the LlamBERT approach slightly compromises on accuracy while offering much greater cost-effectiveness., Comment: 11 pages, 1 figure

Graph grammars form an interesting area of research because of their versatility in modelling diverse situations with graphs as the structures which are to be manipulated. A new class of graph grammars, nc-eNCE Graph Grammars has been introduced recently with an aim of restricting the order of application of graph production rules, thereby generating different graph classes using the same set of rules. On the other hand 2D game design using an algorithmic approach known as procedural content generation has been of interest recently. In this paper we modify the structure of nc-eNCE graph grammars with the aim of generating directed graphs. We show that employing these graph grammars simplifies the design of 2D games. We have also developed an algorithm which makes use of these graph grammars for generating random game level layouts ensuring that the players will get a different gaming experience each time they play., Comment: 14 pages, 10 figures

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group universes. We also extend the results to a class of non-uniform cellular automata (NUCA) consisting of global perturbations with uniformly bounded singularity of CA. As an application, we obtain the surjunctivity of algebraic NUCA with uniformly bounded singularity over sofic groups. Moreover, we prove the stable finiteness of twisted group rings over sofic groups to generalize known results on Kaplansky's stable finiteness conjecture for group rings.

The notion of a normal bit sequence was introduced by Borel in 1909; it was the first definition of an individual random object. Normality is a weak notion of randomness requiring only that all $2^n$ factors (substrings) of arbitrary length~$n$ appear with the same limit frequency $2^{-n}$. Later many stronger definitions of randomness were introduced, and in this context normality found its place as ``randomness against a finite-memory adversary''. A quantitative measure of finite-state compressibility was also introduced (the finite-state dimension) and normality means that the finite state dimension is maximal (equals~$1$). Recently Nandakumar, Pulari and S (2023) introduced the notion of relative finite-state dimension for a binary sequence with respect to some other binary sequence (treated as an oracle), and the corresponding notion of conditional (relative) normality. (Different notions of conditional randomness were considered before, but not for the finite memory case.) They establish equivalence between the block frequency and the gambling approaches to conditional normality and finite-state dimensions. In this note we revisit their definitions and explain how this equivalence can be obtained easily by generalizing known characterizations of (unconditional) normality and dimension in terms of compressibility (finite-state complexity), superadditive complexity measures and gambling (finite-state gales), thus also answering some questions left open in the above-mentioned paper.

The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory. Currently, decidability is only known for LRS of order at most 4. On the other hand, the sole known complexity result is NP-hardness, due to Blondel and Portier. A fundamental result in this area is the celebrated Skolem-Mahler-Lech theorem, which asserts that the zero set of any LRS is the union of a finite set and finitely many arithmetic progressions. This paper focuses on a computational perspective of the Skolem-Mahler-Lech theorem: we show that the problem of counting the zeros of a given LRS is #P-hard, and in fact #P-complete for the instances generated in our reduction.

We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming variables. First, we give a sufficient and necessary condition for A to guarantee the following generalisation of Hilbert's Basis Theorem: every polynomial ideal which is equivariant, i.e. invariant under renaming of variables, is finitely generated. Second, we develop an extension of classical Buchberger's algorithm to compute a Gr\"obner basis of a given equivariant ideal. This implies decidability of the membership problem for equivariant ideals. Finally, we sketch upon various applications of these results to register automata, Petri nets with data, orbit-finitely generated vector spaces, and orbit-finite systems of linear equations.

We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the expressivity of GNNs using diverse activation functions and arithmetic circuits over real numbers. In our results the activation function of the network becomes a gate type in the circuit. Our result holds for families of constant depth circuits and networks, both uniformly and non-uniformly, for all common activation functions.

Automata operating on pairs of words were introduced as an alternative way of capturing acceptance of regular $\omega$-languages. Families of DFAs and lasso automata operating on such pairs followed, giving rise to minimisation algorithms, a Myhill-Nerode theorem and language learning algorithms. Yet Kleene theorems for such a well-established class are still missing. We introduce rational lasso languages and expressions, show a Kleene theorem for lasso languages and explore the connection between rational lasso and $\omega$-expressions, which yields a Kleene theorem for $\omega$-languages with respect to saturated lasso automata. For one direction of the Kleene theorems, we also provide a Brzozowski construction for lasso automata from rational lasso expressions.

Vector addition systems (VAS), also known as Petri nets, are a popular model of concurrent systems. Many problems from many areas reduce to the reachability problem for VAS, which consists of deciding whether a target configuration of a VAS is reachable from a given initial configuration. One of the main approaches to solve the problem on practical instances is called flattening, intuitively removing nested loops. This technique is known to terminate for semilinear VAS. In this paper, we prove that also for VAS with nested zero tests, called Priority VAS, flattening does in fact terminate for all semilinear reachability relations. Furthermore, we prove that Priority VAS admit semilinear inductive invariants. Both of these results are obtained by defining a well-quasi-order on runs of Priority VAS which has good pumping properties., Comment: 24 pages, 2 figures, full version of paper at ICALP 2024

A subclass of nondeterministic Finite Automata generated by means of regular Grammars (GFAs, for short) is introduced. A process algebra is proposed, whose semantics maps a term to a GFA. We prove a representability theorem: for each GFA $N$, there exists a process algebraic term $p$ such that its semantics is a GFA isomorphic to $N$. Moreover, we provide a concise axiomatization of language equivalence: two GFAs $N_1$ and $N_2$ recognize the same regular language if and only if the associated terms $p_1$ and $p_2$, respectively, can be equated by means of a set of axioms, comprising 7 axioms plus 2 conditional axioms, only., Comment: arXiv admin note: substantial text overlap with arXiv:2301.03435

In the field of distributed computing by robot swarms, the research comprehends manifold models where robots operate in the Euclidean plane through a sequence of look-compute-move cycles. Models under study differ for (i) the possibility of storing constant-size information, (ii) the possibility of communicating constant-size information, and (iii) the synchronization mode. By varying features (i,ii), we obtain the noted four base models: OBLOT (silent and oblivious robots), FSTA (silent and finite-state robots), FCOM (oblivious and finite-communication robots), and LUMI (finite-state and finite-communication robots). Combining each base model with the three main synchronization modes (fully synchronous, semi-synchronous, and asynchronous), we obtain the well-known 12 models. Extensive research has studied their computational power, proving the hierarchical relations between different models. However, only transparent robots have been considered. In this work, we study the taxonomy of the 12 models considering collision-intolerant opaque robots. We present six witness problems that prove the majority of the computational relations between the 12 models. In particular, the last witness problem depicts a peculiar issue occurring in the case of obstructed visibility and asynchrony.

We deal with a normal form for context-free grammars, called Dyck normal form. This normal form is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property, along with several other terminal rewriting conditions, makes it possible to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism phi such that L=phi(D'_K), where D'_K is a subset of D_K and D_K is the one-sided Dyck language over K letters. As an application we give an alternative proof of the inclusion of the class of even linear languages in AC1., Comment: 16 pages, no figure. arXiv admin note: substantial text overlap with arXiv:1512.09207

Recent trends like the Internet of Things (IoT) suggest a vision of dense and multi-scale deployments of computing devices in nearly all kinds of environments. A prominent engineering challenge revolves around programming the collective adaptive behaviour of such computational ecosystems. This requires abstractions able to capture concepts like ensembles (dynamic groups of cooperating devices) and collective tasks (joint activities carried out by ensembles). In this work, we consider collections of devices interacting with neighbours and that execute in nearly-synchronised sense-compute-interact rounds, where the computation is given by a single program mapping sensing values and incoming messages to output and outcoming messages. To support programming whole computational collectives, we propose the abstraction of a distributed collective process, which can be used to define at once the ensemble formation logic and its collective task. We formalise the abstraction in the eXchange Calculus (XC), a core functional language based on neighbouring values (maps from neighbours to values) where state and interaction is handled through a single primitive, exchange, and provide a corresponding implementation in the FCPP language. Then, we exercise distributed collective processes using two case studies: multi-hop message propagation and distributed monitoring of spatial properties. Finally, we discuss the features of the abstraction and its suitability for different kinds of distributed computing applications., Comment: 32 pages, 13 figures

Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene's higher-order computability theory, which was historically based on the S1-S9 schemes and which now has an intuitive lambda calculus formulation due to the authors. We establish many computational equivalences between on one hand the 'structure' functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma, and the Tietze extension theorem. We also compare these functionals to known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics., Comment: 33 pages, 1 figure, to appear in Journal of Logic and Computation. Section 5 of this paper corrects a technical error in the lambda calculus from arXiv:2203.05250

The starting point of this paper is a collection of properties of an algorithm that have been distilled from the informal descriptions of what an algorithm is that are given in standard works from the mathematical and computer science literature. Based on that, the notion of a proto-algorithm is introduced. The thought is that algorithms are equivalence classes of proto-algorithms under some equivalence relation. Three equivalence relations are defined. Two of them give bounds between which an appropriate equivalence relation must lie. The third lies in between these two and is likely an appropriate equivalence relation. A sound method is presented to prove, using an imperative process algebra based on ACP, that this equivalence relation holds between two proto-algorithms., Comment: 22 pages, revision of v1, presentation improved at several places and some minor errors corrected

The class of Boolean combinations of tree languages recognized by deterministic top-down tree automata (also known as deterministic root-to-frontier automata) is studied. The problem of determining for a given regular tree language whether it belongs to this class is open. We provide some progress by two results: First, a characterization of this class by a natural extension of deterministic top-down tree automata is presented, and as an application we obtain a convenient method to show that certain regular tree languages are outside this class. In the second result, it is shown that, for fixed $k$, it is decidable whether a regular tree language is a Boolean combination of $k$ tree languages recognized by deterministic top-down tree automata., Comment: This is a preprint of a paper published in a special issue dedicated to the memory of Magnus Steinby in the International Journal of Foundations of Computer Science. Compared to the published journal version, reference [8] has been added in a comment at the end of the introduction

We examine the relationship of graded (multi)modal logic to counting (multichannel) message passing automata with applications to the Weisfeiler-Leman algorithm. We introduce the notion of graded multimodal types, which are formulae of graded multimodal logic that encode the local information of a pointed Kripke-model. We also introduce message passing automata that carry out a generalization of the Weisfeiler-Leman algorithm for distinguishing non-isomorphic graph nodes. We show that the classes of pointed Kripke-models recognizable by these automata are definable by a countable (possibly infinite) disjunction of graded multimodal formulae and vice versa. In particular, this equivalence also holds between recursively enumerable disjunctions and recursively enumerable automata. We also show a way of carrying out the Weisfeiler-Leman algorithm with a formula of first order logic that has been augmented with H\"artig's quantifier and greatest fixed points.

Multi-valued logics have a long tradition in the literature on system verification, including run-time verification. However, comparatively fewer model-checking tools have been developed for multi-valued specification languages. We present 3vLTL, a tool to generate Buchi automata from formulas in Linear-time Temporal Logic (LTL) interpreted on a three-valued semantics. Given an LTL formula, a set of atomic propositions as the alphabet for the automaton, and a truth value, our procedure generates a Buchi automaton that accepts all the words that assign the chosen truth value to the LTL formula. Given the particular type of the output of the tool, it can also be seamlessly processed by third-party libraries in a natural way. That is, the Buchi automaton can then be used in the context of formal verification to check whether an LTL formula is true, false, or undefined on a given model., Comment: In Proceedings FMAS 2023, arXiv:2311.08987

Nowadays, methods and techniques of Machine Learning and Deep Learning are being used in various scientific areas. They help to automatize calculations without losing in quality. In this paper the applying of convolutional neural network was considered in frame of problems from statistical physics and computer simulation of magnetic films. In a frame of the first task, CNN was used to determine critical Curie point for Ising model on 2D square lattice. Obtained results were compared with classical Monte-Carlo methods and exact solution. Systems of various lattice sizes and the influence of the size effect on the results' accuracy were considered. Also, authors considered the classical two-dimensional Heisenberg model, a spin system with direct short-range exchange, and studied of its competition with the Dzyaloshinskii-Moriya interaction. A neural network was applied to the recognition of Spiral (Sp), Spiral-skyrmion (SpSk) Skyrmion (Sk), Skyrmion-ferromagnetic (SkF) and Ferromagnetic (FM) phases of the Heisenberg spin system with magnetic skyrmions. The advantage of CNN's application over conventional methods for determination of skyrmion's phases was revealed., Comment: 6 pages, 3 figures

We investigate the correspondence between the time and space recognition complexity of languages; for this purpose, we will code the long-continued computations of deterministic two-tape Turing machines by the relatively short-length quantified Boolean formulae. The modified Stockmeyer and Meyer method will appreciably be used for this simulation. It will be proved using this modeling that the complexity classes $\mathbf{EXP}$ and $\mathbf{PSPACE}$ coincide; and more generally, the class $(k\!+\!1)$-fold Deterministic Exponential Time equals to the class $k$-fold Deterministic Exponential Space for each $k\geqslant1$; the space complexity of the languages of the class $\mathbf{P}$ will also be studied. Furthermore, this allows us to slightly improve the early founded lower complexity bound of decidable theories that are nontrivial relative to some equivalence relation (this relation may be equality) -- each of these theories is consistent with the formula, which asserts that there are two non-equivalent elements. Keywords: computational complexity, the coding of computations through formulae, exponential time, polynomial space, lower complexity bound of the language recognition, Comment: 44 pages, 26 references bibliography; text overlap with arXiv:1907.04521 because the paper is created in the same method