Deterministic Finite Automata Research Articles

Verification of Diagnosability for Cyber-Physical Systems via Hybrid Barrier Certificates

In this paper, we provide an automata-based framework for verifying diagnosability property of Cyber-Physical Systems leveraging a notion of so-called hybrid barrier certificates. Concretely, we first construct a so-called (δ,K)-deterministic finite automata ((δ,K)-DFA) associated with the desired diagnosability property, which captures the occurrence of the fault to be diagnosed. Having a (δ,K)-DFA, we show that the verification of diagnosability properties is equivalent to a safety verification problem over a product system between this DFA and the dynamical system of interest. We further show that such a verification problem can be solved via computing hybrid barrier certificates for the product system. To compute the hybrid barrier certificates, we provide a systematic technique leveraging a counter-example guided inductive synthesis framework. Finally, we showcase the effectiveness of our results through a case study.

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.

Real-time, constant-space, constant-randomness verifiers

We study the class of languages that have membership proofs which can be verified by real-time finite-state machines using only a constant number of random bits, regardless of the size of their inputs. Since any further restriction on the verifiers would preclude the verification of nonregular languages, this is the tightest computational budget which allows the checking of externally provided proofs to have meaningful use. We provide a full characterization of this class of languages in terms of a restricted version of the one-way nondeterministic multihead finite automaton model. For any k>0, there exist languages that cannot be recognized by any k-head one-way nondeterministic finite automaton, but that are nonetheless real-time verifiable in this sense. The set of nonpalindromes, which cannot be recognized by any one-way multihead deterministic finite automaton, is also demonstrated to be verifiable within these restrictions.

Improved bit-based filtering algorithm for regular constraint

In this paper, we propose an improved filtering algorithm for regular constraint, which demand that the sequence of all assignments for variables can be accepted by a Deterministic Finite Automaton (DFA) or a regular language. Regular constraint is a type of global constraint in Constraint Satisfaction Problems (CSPs) and a natural expression of deterministic patterns in planning and decision making, and it utilizes the properties of DFA to pre-specify the optional paths for multiple variables. Thus, it is sensible to improve the filtering algorithm that enforces Generalized Arc Consistency (GAC) for regular constraint in order to minimize memory and time consumption. With the goal in mind, we encode DFA using bit vectors and then devise a bit operations algorithm to maintain the domain consistency for regular constraint. Lastly, we carry out an experiment to compare to the state-of-the-art filtering algorithm. The experimental results demonstrates that the proposed algorithm significantly improves both memory and time.

The selection of the control unit optimal signalling sequence in containers stack handling at sea port terminal

The possibility of increasing the productivity of technological operations performed in the storage area of a container terminal due to the introduction of automation elements into mechanized warehouse cargo handling has been researched. As one of the possible automation options, a method for transporting and storing containers with the implementation of a loader control unit in the form of a deterministic finite automaton is proposed. The sequence of actions of the loader is simulated in the control unit by a line of encoded symbols generated by the transition diagram of the Mealy automaton. In this case, the transition diagram is an automatic programming tool allowed mathematically describing the logic of controlling the actions of the loader when it processes a stack of containers. As a result of generation at the output of the machine, several valid loader control lines are got. Each symbol of the control line corresponds to one action from the loader working cycle. It is required to find the control line that will bring the loader to the state, when the picking up of containers from the stack is completed, in the minimum number of actions. The object of the research is a variable sequence of loader actions during partial disassembly of the stack with its corresponding control line. To reduce the complexity of the task being solved, each control line is divided into sections consisting of symbols responsible for the following actions of the loader: moving when changing the disassembled row, disassembling the row with the picking up and movement of target containers, returning blocking containers to the stacks where they were initially placed. As a result of the comparison of the corresponding symbolic sections of different lines, an algorithm that allows choosing the optimal sequence of signals for the control unit of the technological automaton in terms of the number of symbols is developed.

Completely Reachable Automata: An Interplay Between Automata, Graphs, and Trees

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. We characterize such automata in terms of graphs and trees.

Learning from positive and negative examples: New proof for binary alphabets

One of the most fundamental problems in computational learning theory is the problem of learning a finite automaton A consistent with a finite set P of positive examples and with a finite set N of negative examples. By consistency, we mean that A accepts all strings in P and rejects all strings in N. It is well known that this problem is NP-complete. In the literature, it is stated that NP-hardness holds even in the case of a binary alphabet. As a standard reference for this theorem, the work of Gold from 1978 is either cited or adapted. Nevertheless, the results in Gold's work are stated in terms of Mealy machines, and not in terms of deterministic finite automata (DFAs) as most commonly defined. As Mealy machines are equipped with an output function, they can be more compact than DFAs which accept the same language. We show that the adaptations of Gold's construction for Mealy machines stated in the literature have some issues, and provide a correct proof for the fact that the DFA-consistency problem for binary alphabets is NP-complete.

Automaton-Guided Curriculum Generation for Reinforcement Learning Agents

Despite advances in Reinforcement Learning, many sequential decision making tasks remain prohibitively expensive and impractical to learn. Recently, approaches that automatically generate reward functions from logical task specifications have been proposed to mitigate this issue; however, they scale poorly on long-horizon tasks (i.e., tasks where the agent needs to perform a series of correct actions to reach the goal state, considering future transitions while choosing an action). Employing a curriculum (a sequence of increasingly complex tasks) further improves the learning speed of the agent by sequencing intermediate tasks suited to the learning capacity of the agent. However, generating curricula from the logical specification still remains an unsolved problem. To this end, we propose AGCL, Automaton-guided Curriculum Learning, a novel method for automatically generating curricula for the target task in the form of Directed Acyclic Graphs (DAGs). AGCL encodes the specification in the form of a deterministic finite automaton (DFA), and then uses the DFA along with the Object-Oriented MDP (OOMDP) representation to generate a curriculum as a DAG, where the vertices correspond to tasks, and edges correspond to the direction of knowledge transfer. Experiments in gridworld and physics-based simulated robotics domains show that the curricula produced by AGCL achieve improved time-to-threshold performance on a complex sequential decision-making problem relative to state-of-the-art curriculum learning (e.g, teacher-student, self-play) and automaton-guided reinforcement learning baselines (e.g, Q-Learning for Reward Machines). Further, we demonstrate that AGCL performs well even in the presence of noise in the task's OOMDP description, and also when distractor objects are present that are not modeled in the logical specification of the tasks' objectives.

Learning Interpretable Temporal Properties from Positive Examples Only

We consider the problem of explaining the temporal behavior of black-box systems using human-interpretable models. Following recent research trends, we rely on the fundamental yet interpretable models of deterministic finite automata (DFAs) and linear temporal logic (LTL_f) formulas. In contrast to most existing works for learning DFAs and LTL_f formulas, we consider learning from only positive examples. Our motivation is that negative examples are generally difficult to observe, in particular, from black-box systems. To learn meaningful models from positive examples only, we design algorithms that rely on conciseness and language minimality of models as regularizers. Our learning algorithms are based on two approaches: a symbolic and a counterexample-guided one. The symbolic approach exploits an efficient encoding of language minimality as a constraint satisfaction problem, whereas the counterexample-guided one relies on generating suitable negative examples to guide the learning. Both approaches provide us with effective algorithms with minimality guarantees on the learned models. To assess the effectiveness of our algorithms, we evaluate them on a few practical case studies.

State complexity of finite partial languages

Partial word finite automata are deterministic finite automata that may have state transitions on a special symbol ⋄ which represents an unknown symbol or a hole in the word. Together with a subset of the input alphabet that gives the symbols which may be substituted for the ⋄, a partial word finite automaton (⋄-DFA) represents a regular language. However, this substitution implies a certain form of limited nondeterminism in the computations when the ⋄-transitions are replaced by ordinary transitions. In this paper we consider the state complexity of partial word finite automata accepting finite languages. We study the state complexity of the NFA to ⋄-DFA conversion for finite languages as well as the state complexity of the ⋄-DFA to DFA conversion for finite languages. Then we consider the operational state complexity with respect to complementation, union, reversal, and concatenation of finite languages. It turns out that the upper and lower bounds for all these operations are exponential. Moreover, we establish a state complexity hierarchy on the number of productive ⋄-transitions that may appear in ⋄-DFAs accepting finite languages. The levels of the hierarchy are separated by quadratic state costs.

IDLIQ: An Incremental Deterministic Finite Automaton Learning Algorithm Through Inverse Queries for Regular Grammar Inference.

We present an efficient incremental learning algorithm for Deterministic Finite Automaton (DFA) with the help of inverse query (IQ) and membership query (MQ). This algorithm is an extension of the Identification of Regular Languages (ID) algorithm from a complete to an incremental learning setup. The learning algorithm learns by making use of a set of labeled examples and by posing queries to a knowledgeable teacher, which is equipped to answer IQs along with MQs and equivalence query. Based on the examples (elements of the live complete set) and responses against IQs from the minimally adequate teacher (MAT), the learning algorithm constructs the hypothesis automaton, consistent with all observed examples. The Incremental DFA Learning algorithm through Inverse Queries (IDLIQ) takes time complexity in the presence of a MAT and ensures convergence to a minimal representation of the target DFA with finite number of labeled examples. Existing incremental learning algorithms; the Incremental ID, the Incremental Distinguishing Strings have polynomial (cubic) time complexity in the presence of a MAT. Therefore, sometimes, these algorithms even fail to learn large complex software systems. In this research work, we have reduced the complexity (from cubic to square form) of the DFA learning in an incremental setup. Finally, we prove the correctness and termination of the IDLIQ algorithm.

USE OF THE THEORY OF FINITE AUTOMATA FOR SUPERVISION OF THE OPERATION OF THE COMPRESSOR UNIT

The purpose of the publication is to develop a methodology for presenting the unit "6 kV compressor-synchronous electric motor" in the form of a finite state machine.
 It is hypothesized that the behavior of the compressor unit can be associated with the behavior of an abstract deterministic finite state machine.
 The theory of finite state machines and graphs is applied to analyze the operation of a compressor installation using the example of a compressor of a large-tonnage shop for the production of ammonia. It is shown that the theory of finite state machines allows combining all possible modes of operation and shutdown of compressor units. Coding of reasons for compressor shutdown due to problems in the electrical or mechanical parts of the compressor unit is proposed. A diagram of state transitions has been developed for it. Such a diagram can be useful for compiling a concise description and understanding of the operation of a compressor unit, advanced training of technological personnel during industrial training, for emergency training, etc. 
 A graphical analysis of the obtained diagram was carried out. The optimal (most perceived in terms of energy efficiency) route of the graph is determined.
 The proposed form of abbreviated recording of the compressor operation cycle and the electronic mileage log, which allows you to calculate the remaining resource of the compressor installation and the compressor operation time.
 The implemented methods make it possible to identify bottlenecks in the complex electrical and technological equipment of industrial enterprises.

Synchronizing Boolean networks asynchronously

The asynchronous automaton of a Boolean network f:{0,1}n→{0,1}n, considered in many applications, is the finite deterministic automaton where the set of states is {0,1}n, the alphabet is [n], and the action of letter i on a state x consists in either switching the ith component if fi(x)≠xi or doing nothing otherwise. In this paper, we ask for the existence of synchronizing words for this automaton, and their minimal length, when f is the and-net over an arc-signed digraph G on [n]: for every i∈[n], fi(x)=1 if and only if xj=1 (xj≠0) for every positive (negative) arc from j to i. Our main result is that if G is strongly connected and has no positive cycles, then either there exists a synchronizing word of length at most 10(5+1)n or G is a cycle and there are no synchronizing words. We also give complexity results showing that the situation is much more complex if one of the two hypothesis made on G is removed.

Weight-reducing Turing machines

It is well known that one-tape Turing machines running in linear time are no more powerful than finite automata; namely they recognize exactly the class of regular languages. We prove that it is not decidable if a one-tape machine runs in linear time, even if it is deterministic and restricted to use only the portion of the tape that initially contains the input. This motivates the introduction of a constructive variant of one-tape machines, called a weight-reducing machine, and the investigation of its properties. We focus on the deterministic case. In particular, we show that, paying a polynomial size increase only, each weight-reducing machine can be turned into a halting one that runs in linear time. Furthermore each weight-reducing machine can be converted into equivalent nondeterministic and deterministic finite automata by paying an exponential and doubly-exponential increase in size, respectively. These costs cannot be reduced in the worst case.