Deterministic Finite Automata Research Articles

Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. Published by the American Physical Society 2024

Incremental NFA minimization

We tackle the (classic) problem of minimizing (non)deterministic finite automata. The algorithm we put forward has the peculiarity of being incremental, i.e., the minimization proceeds by successive iterations, each producing a partially minimized automaton language-equivalent to the input one.Our algorithm builds upon Almeida et al. from 2014, fixing a minor mistake and generalizing it to the nondeterministic case. It relies on a colouring procedure of a graph associated to the automaton, keeping track of partial information. After dealing with the deterministic case, we extend this idea to the bisimulation-minimization of nondeterministic automata.The algorithms for both the deterministic and the nondeterministic cases run in time O(nm) for an automaton with n states and m transitions. The complexity for the deterministic case matches the complexity claimed by Almeida et al.. The nondeterministic case improves the fastest known incremental algorithm for this problem.We conclude introducing and using a notion of signature of a state, whose aim is to exploit pre-computed information potentially available, to speed-up the process. A signature is used to produce an initial partition of the automaton's states and can be easily integrated in both the incremental and the non-incremental algorithm.

Yet another canonical nondeterministic automaton

Several canonical forms of finite automata have been introduced over the decades. In particular, if one considers the minimal deterministic finite automaton (DFA), the canonical residual finite state automaton (RFSA), and the átomaton of a language, then the átomaton can be seen as the dual automaton of the minimal DFA, but no such dual has been presented for the canonical RFSA so far. We fill this gap by introducing a new canonical automaton that we call the maximized prime átomaton, and study its properties. We also describe how these four automata can be extracted from suitable observation tables used in the automata learning context.

Zombie cheminformatics: extraction and conversion of Wiswesser Line Notation (WLN) from chemical documents

PurposeWiswesser Line Notation (WLN) is a old line notation for encoding chemical compounds for storage and processing by computers. Whilst the notation itself has long since been surpassed by SMILES and InChI, distribution of WLN during its active years was extensive. In the context of modernising chemical data, we present a comprehensive WLN parser developed using the OpenBabel toolkit, capable of translating WLN strings into various formats supported by the library. Furthermore, we have devised a specialised Finite State Machine l, constructed from the rules of WLN, enabling the recognition and extraction of chemical strings out of large bodies of text. Available open-access WLN data with corresponding SMILES or InChI notation is rare, however ChEMBL, ChemSpider and PubChem all contain WLN records which were used for conversion scoring. Our investigation revealed a notable proportion of inaccuracies within the database entries, and we have taken steps to rectify these errors whenever feasible.Scientific contributionTools for both the extraction and conversion of WLN from chemical documents have been successfully developed. Both the Deterministic Finite Automaton (DFA) and parser handle the majority of WLN rules officially endorsed in the three major WLN manuals, with the parser showing a clear jump in accuracy and chemical coverage over previous submissions. The GitHub repository can be found here: https://github.com/Mblakey/wiswesser.

A Learning-Based Approach for Diagnosis and Diagnosability of Unknown Discrete Event Systems.

This article develops a novel active-learning technique for fault diagnosis of an initially unknown finite-state discrete event system (DES). The proposed method constructs a diagnosis tool (termed diagnoser), which is able to detect and identify occurred faults by tracking the observable behaviors of the system under diagnosis. The proposed algorithm utilizes an active-learning mechanism to incrementally collect the information about the system to construct the diagnoser. This is achieved by completing a series of observation tables in a systematic way, resulting in the construction of the diagnoser. It is proven that the proposed algorithm terminates after a finite number of iterations and returns a correctly conjectured diagnoser. The developed diagnoser is a deterministic finite-state automaton. Furthermore, we have proven that the developed diagnoser consists of a minimum number of states. A sufficient condition for diagnosability of the system under diagnosis is derived, which guarantees the diagnosis of faults within a bounded number of observations. The developed method is applied to two case-studies, illustrating the steps of the proposed algorithm and its capability of diagnosing multiple faults.

Analyzing Robustness of Angluin's L$^*$ Algorithm in Presence of Noise

Angluin's L$^*$ algorithm learns the minimal deterministic finite automaton (DFA) of a regular language using membership and equivalence queries. Its probabilistic approximatively correct (PAC) version substitutes an equivalence query by numerous random membership queries to get a high level confidence to the answer. Thus it can be applied to any kind of device and may be viewed as an algorithm for synthesizing an automaton abstracting the behavior of the device based on observations. Here we are interested on how Angluin's PAC learning algorithm behaves for devices which are obtained from a DFA by introducing some noise. More precisely we study whether Angluin's algorithm reduces the noise and produces a DFA closer to the original one than the noisy device. We propose several ways to introduce the noise: (1) the noisy device inverts the classification of words w.r.t. the DFA with a small probability, (2) the noisy device modifies with a small probability the letters of the word before asking its classification w.r.t. the DFA, (3) the noisy device combines the classification of a word w.r.t. the DFA and its classification w.r.t. a counter automaton, and (4) the noisy DFA is obtained by a random process from two DFA such that the language of the first one is included in the second one. Then when a word is accepted (resp. rejected) by the first (resp. second) one, it is also accepted (resp. rejected) and in the remaining cases, it is accepted with probability 0.5. Our main experimental contributions consist in showing that: (1) Angluin's algorithm behaves well whenever the noisy device is produced by a random process, (2) but poorly with a structured noise, and, that (3) is able to eliminate pathological behaviours specified in a regular way. Theoretically, we show that randomness almost surely yields systems with non-recursively enumerable languages.

Identification of labeled Petri nets from finite automata

Automata and Petri nets are two typical models of discrete event systems. The paper studies the problem of converting a finite automaton into a Petri net that satisfies the specified structural features. More specifically, we identify a labeled Petri net from a nondeterminstic finite automaton such that the reachability graph of the identified net is isomorphic to the given automaton, meaning that the Petri net models exactly the same dynamic system as the automaton. The identified net necessarily satisfies the specified structural requirements, such as the numbers of places and transitions, absence of self-loops, the exact structure of a part of the net, and so on. Since label Petri nets and nondeterminstic finite automata are generalizations of Petri nets and deterministic finite automata, respectively, the proposed approach can identify a larger spectrum of Petri nets and effectively handle the nondeterminstic cases in the given automaton. Four kinds of conditions are first extracted from the given automaton, namely deterministic enabling conditions, nondeterministic enabling conditions (i.e., multiple transitions with the same label are enabled simultaneously at a marking), transition disabling conditions, and marking inequality conditions. Then, an integer linear programming problem is formulated by characterizing these four kinds of conditions using integer linear constraints. A labeled Petri net satisfying the structural requirements is obtained by solving the integer linear programming problem. We also present two examples to show the possible applications of the proposed identification approach.

Performing Regular Operations with 1-Limited Automata

AbstractThe descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines. The situation is different when the operations are applied to deterministic 1-limited automata: while for boolean operations the simulations remain polynomial, for product, star, and reversal they cost exponential in size. The costs for product and star do not reduce if the given machines are sweeping two-way deterministic finite automata. These bounds are tight.

Development of fuzzy search method for creating an efficient information search system in text data

The object of research is the processes of effective search for information in a set of textual data. The subject of the research is the fuzzy search method, which will allow to effectively solve the problem of searching for information in a set of textual data. The paper considers the process of developing a fuzzy search method, which consists of 9 consecutive steps and is required for a quick search for matches in a large set of text data. Based on this method, it is proposed to create a fuzzy search system that will solve the problem of finding the most relevant documents from a set of such documents. The proposed fuzzy search method combines the advantages of algorithms based on deterministic finite automata and algorithms based on dynamic programming for calculating the Damerau-Levenshtein distance. Such a combination allows to implement the symbol similarity table in an optimal way. As part of the work, an approach for creating a symbol similarity table was proposed and an example of such a table was created for symbols from the English alphabet, which allows to find the degree of similarity between two symbols with constant asymptotics and to convert the current symbol into its basic counterpart. For document filtering, a metric was developed to evaluate the correspondence of text data to a search phrase, which simultaneously takes into account the number of found and not found characters and the number of found and not found words. The Damerau-Levenstein algorithm allows to find the edit distance between two words, taking into account the following types of errors: substitution, addition, deletion, and transposition of characters. The work proposed a modification of this algorithm by using a similarity table to more accurately estimate the editing distance between two words. The developed method makes it possible to create a fuzzy search system that will help find the desired results faster and increase the relevance of the obtained results by sorting them according to the values of the proposed test data similarity metric.

State complexity of binary coded regular languages

For the given non-unary input alphabet Σ, a maximal prefix code h mapping strings over Σ to binary strings, and an optimal deterministic finite automaton (DFA) A with n states recognizing a language L over Σ, we consider the problem of how many states we need for an automaton A′ that decides membership in h(L), the binary coded version of L. Namely, A′ accepts binary inputs belonging to h(L) and rejects binary inputs belonging to h(LC), where LC is the complement of L. The outcome on inputs that are not valid binary codes for any string in Σ⁎ can be arbitrary: A′ may accept, reject, or halt in a “don't care” state. We show that every optimal deterministic don't care finite automaton (dcDFA) A′ solving this promise problem uses at most (‖Σ‖−1)⋅n states but at least n states. In addition, we have established a kind of pseudo-isomorphism between such DFAs and dcDFAs. We also show that, for each non-unary input alphabet Σ, there exists a maximal binary prefix code h such that, for each n≥2 and for each N in range from n to (‖Σ‖−1)⋅n, there exists a language L over Σ such that the optimal DFA recognizing L uses exactly n states and every optimal dcDFA for solving the above promise problem uses exactly N states. Thus, we have the complete state hierarchy for deciding membership in the binary coded version of L, with no magic numbers in between the lower and upper bounds.

Predictive Monitoring against Pattern Regular Languages

While current bug detection techniques for concurrent software focus on unearthing low-level issues such as data races or deadlocks, they often fall short of discovering more intricate temporal behaviours that can arise even in the absence of such low-level issues. In this paper, we focus on the problem of dynamically analysing concurrent software against high-level temporal specifications such as LTL. Existing techniques for runtime monitoring against such specifications are primarily designed for sequential software and remain inadequate in the presence of concurrency — violations may be observed only in intricate thread interleavings, requiring many re-runs of the underlying software in conjunction with the analysis. Towards this, we study the problem of predictive runtime monitoring , inspired by the analogous problem of predictive data race detection studied extensively recently. The predictive runtime monitoring question asks, given an execution σ, if it can be soundly reordered to expose violations of a specification. In general, this problem may become easily intractable when either the specifications or the notion of reorderings used is complex. In this paper, we focus on specifications that are given in regular languages. Our notion of reorderings is trace equivalence , where an execution is considered a reordering of another if it can be obtained from the latter by successively commuting adjacent independent actions. We first show that, even in this simplistic setting, the problem of predictive monitoring admits a super-linear lower bound of O ( n α ), where n is the number of events in the execution, and α is a parameter describing the degree of commutativity, and typically corresponds to the number of threads in the execution. As a result, predictive runtime monitoring even in this setting is unlikely to be efficiently solvable, unlike in the non-predictive setting where the problem can be checked using a deterministic finite automaton (and thus, a constant-space streaming linear-time algorithm). Towards this, we identify a sub-class of regular languages, called pattern languages (and their extension generalized pattern languages ). Pattern languages can naturally express specific ordering of some number of (labelled) events, and have been inspired by popular empirical hypotheses underlying many concurrency bug detection approaches such as the “small bug depth” hypothesis. More importantly, we show that for pattern (and generalized pattern) languages, the predictive monitoring problem can be solved using a constant-space streaming linear-time algorithm. We implement and evaluate our algorithm PatternTrack on benchmarks from the literature and show that it is effective in monitoring large-scale applications.

On Learning Polynomial Recursive Programs

We introduce the class of P-finite automata. These are a generalisation of weighted automata, in which the weights of transitions can depend polynomially on the length of the input word. P-finite automata can also be viewed as simple tail-recursive programs in which the arguments of recursive calls can non-linearly refer to a variable that counts the number of recursive calls. The nomenclature is motivated by the fact that over a unary alphabet P-finite automata compute so-called P-finite sequences, that is, sequences that satisfy a linear recurrence with polynomial coefficients. Our main result shows that P-finite automata can be learned in polynomial time in Angluin's MAT exact learning model. This generalises the classical results that deterministic finite automata and weighted automata over a field are respectively polynomial-time learnable in the MAT model.

Automata for knowledge assessment based on the structure of observed learning outcome taxonomy

The process of adaptive assessment of individuals' knowledge by knowledge space theory has been implemented in deterministic finite automata. However, such automata are not suitable for polytomous situations, which limits the application of automata for knowledge assessment. To overcome this limitation, this paper combines the structure of observed learning outcome (SOLO) taxonomy with the response scale and proposes an automata knowledge assessment process that can assess individuals' knowledge in polytomous situations. Specifically, an automaton suitable for assessing knowledge in the SOLO taxonomy situation is proposed, and a questioning rule is provided based on the quasi-order relation between items. Then, relying on the automaton state update and the questioning rule, an automata knowledge assessment process for knowledge assessment in the SOLO taxonomy situation is given. Finally, empirical research and simulation experiments show that it is necessary to provide a questioning rule for the automaton and that the automata knowledge assessment process proposed in this paper is feasible.

Thermodynamics of deterministic finite automata operating locally and periodically

Real-world computers have operational constraints that cause nonzero entropy production (EP). In particular, almost all real-world computers are ‘periodic’, iteratively undergoing the same physical process; and ‘local’, in that subsystems evolve whilst physically decoupled from the rest of the computer. These constraints are so universal because decomposing a complex computation into small, iterative calculations is what makes computers so powerful. We first derive the nonzero EP caused by the locality and periodicity constraints for deterministic finite automata (DFA), a foundational system of computer science theory. We then relate this minimal EP to the computational characteristics of the DFA. We thus divide the languages recognised by DFA into two classes: those that can be recognised with zero EP, and those that necessarily have non-zero EP. We also demonstrate the thermodynamic advantages of implementing a DFA with a physical process that is agnostic about the inputs that it processes.

Using SAT Solvers to Reverse-Engineer FSM Models of Digital Devices

Inferring a functional specification from an existing digital design is a challenge that is suitable with reverse-engineering methods. One of the most widely used functional specification formats is a finite state machine (FSM). This article studies the possibility of blind passive specification mining for a digital device, where the device is treated as a “black box”. The presented approach treats an input and output signal waveform as the transition graph of an incomplete deterministic FSM and learns the FSM through FSM minimization. It employs a Boolean satisfiability problem (SAT) solver to find a minimal FSM that complies with observed object behavior. The known approach to identifying state machines in discrete event systems is adapted to operate with variables in the form of coloring and transition tables. The developed implementation produces a synthesizable specification in hardware description language (HDL) and a state diagram in unified modeling language (UML). The proposed approach for inferring an FSM from a waveform trace can serve as a supplementary tool during reverse engineering to provide developers with meaningful insight regarding the analyzed device. The presented case study defines metrics of successful FSM inference and applies them to a synthetic FSM and a real-world example FSM to demonstrate the applicability of the approach.

Minimal Determinization Algorithm for Fuzzy Automata

The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model of a fuzzy automaton is employed, it is desirable to find its language-equivalent deterministic version because of its computational efficiency. Although many methods have been developed to convert a fuzzy automaton to its language equivalent fuzzy deterministic finite automaton (FDfA), they can be applied only for fuzzy automata defined over specific underlying sets of truth values. For example, recently developed determinization methods employ the concept of maximal factorization, which can be defined only on non-locally finite lattices or the Boolean lattice. In addition, not all such determinization methods result in a minimal FDfA. On the other hand, even though such determinization methods have been developed for fuzzy automata over specific underlying structures, these methods cannot be generalized for fuzzy automata over locally finite lattices. This article focuses on filling this gap and develops a novel method for computing a minimal FDfA for a fuzzy automaton defined over a locally finite and divisible residuated lattice. Our method uses the new concept of a reduction graph that emerges from the strict order relation on the resulting fuzzy states, according to which we can construct all minimal FDfAs equivalent to a given fuzzy automaton.

On the meeting of random walks on random DFA

We consider two random walks evolving synchronously on a random out-regular graph of n vertices with bounded out-degree r≥2, also known as a random Deterministic Finite Automaton (DFA). We show that, with high probability with respect to the generation of the graph, the meeting time of the two walks is stochastically dominated by a geometric random variable of rate (1+o(1))n−1, uniformly over their starting locations. Further, we prove that this upper bound is typically tight, i.e., it is also a lower bound when the locations of the two walks are selected uniformly at random. Our work takes inspiration from a recent conjecture by Fish and Reyzin (2017) in the context of computational learning, the connection with which is discussed.

Risk-informed decision-making and control strategies for autonomous vehicles in emergency situations

This paper proposes risk-informed decision-making and control methods for autonomous vehicles (AVs) under severe driving conditions, where many vehicle interactions occur on slippery roads. We assume that the AV should approach a specific safe zone in case of vehicle malfunctioning. In normal situations, the driving behavior of the AV is based on the deterministic finite-state machine (FSM) that makes an appropriate real-time decision depending on the driving condition, which is efficient when compared with some conventional decision-making algorithms; alternatively, in emergency situations, the AV first has to be prevented the rear-end conflicts while the specific safe zone is simultaneously determined by evaluating the level of risk via two safety level indicators, i.e., Time-To-Collision (TTC) and Deceleration Rate to Avoid the Crash (DRAC). The safe path that guides the AV to avoid car crashes is generated based on the trajectory optimization theory (i.e., Pontryagin maximum principle), and the AV follows it based on the linear time-varying model predictive control (LTV-MPC), which ensures the AV’s lateral stability. We verify the effectiveness of the proposed decision-making and control strategies in various test scenarios, and the results show that the AV behaves appropriately according to the behaviors of surrounding vehicles and road condition.

Synchronizing deterministic push-down automata can be really hard

The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely, that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata.There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.

Bit-Vector Typestate Analysis

Static analyses based on typestates are important in certifying correctness of code contracts. Such analyses rely on Deterministic Finite Automata (DFAs) to specify properties of an object. We target the analysis of contracts in low-latency environments, where many useful contracts are impractical to codify as DFAs and/or the size of their associated DFAs leads to sub-par performance. To address this bottleneck, we present a lightweight compositional typestate analyzer, based on an expressive specification language that can succinctly specify code contracts. By implementing it in the static analyzer Infer , we demonstrate considerable performance and usability benefits when compared to existing techniques. A central insight is to rely on a sub-class of DFAs whose analysis uses efficient bit-vector operations.