Showing papers on "Automata theory published in 2017"

Cache automaton

Abstract: Finite State Automata are widely used to accelerate pattern matching in many emerging application domains like DNA sequencing and XML parsing. Conventional CPUs and compute-centric accelerators are bottlenecked by memory bandwidth and irregular memory access patterns in automata processing. We present Cache Automaton, which repurposes last-level cache for automata processing, and a compiler that automates the process of mapping large real world Non-Deterministic Finite Automata (NFAs) to the proposed architecture. Cache Automaton extends a conventional last-level cache architecture with components to accelerate two phases in NFA processing: state-match and state-transition. State-matching is made efficient using a sense-amplifier cycling technique that exploits spatial locality in symbol matches. State-transition is made efficient using a new compact switch architecture. By overlapping these two phases for adjacent symbols we realize an efficient pipelined design. We evaluate two designs, one optimized for performance and the other optimized for space, across a set of 20 diverse benchmarks. The performance optimized design provides a speedup of 15× over DRAM-based Micron’s Automata Processor and 3840× speedup over processing in a conventional x86 CPU. The proposed design utilizes on an average 1.2 MB of cache space across benchmarks, while consuming 2.3 nJ of energy per input symbol. Our space optimized design can reduce the cache utilization to 0.72 MB, while still providing a speedup of 9× over AP. CCS CONCEPTS • Hardware → Emerging architectures; • Theory of computation → Formal languages and automata theory;

Verification of Multi-agent Systems with Imperfect Information and Public Actions

Abstract: We analyse the verification problem for synchronous, perfect recall multi-agent systems with imperfect information against a specification language that includes strategic and epistemic operators While the verification problem is undecidable, we show that if the agents' actions are public, then verification is 2exptime-complete To illustrate the formal framework we consider two epistemic and strategic puzzles with imperfect information and public actions: the muddy children puzzle and the classic game of battleships

Learning Symbolic Automata

Abstract: Symbolic automata allow transitions to carry predicates over rich alphabet theories, such as linear arithmetic, and therefore extend classic automata to operate over infinite alphabets, such as the set of rational numbers. In this paper, we study the foundational problem of learning symbolic automata. We first present \(\mathrm {\Lambda }^*\), a symbolic automata extension of Angluin’s L\(^*\) algorithm for learning regular languages. Then, we define notions of learnability that are parametric in the alphabet theories of the symbolic automata and show how these notions nicely compose. Specifically, we show that if two alphabet theories are learnable, then the theory accepting the Cartesian product or disjoint union of their alphabets is also learnable. Using these properties, we show how existing algorithms for learning automata over large alphabets nicely fall in our framework. Finally, we implement our algorithm in an open-source library and evaluate it on existing automata learning benchmarks.

Flatten and conquer: a framework for efficient analysis of string constraints

Abstract: We describe a uniform and efficient framework for checking the satisfiability of a large class of string constraints. The framework is based on the observation that both satisfiability and unsatisfiability of common constraints can be demonstrated through witnesses with simple patterns. These patterns are captured using flat automata each of which consists of a sequence of simple loops. We build a Counter-Example Guided Abstraction Refinement (CEGAR) framework which contains both an under- and an over-approximation module. The flow of information between the modules allows to increase the precision in an automatic manner. We have implemented the framework as a tool and performed extensive experimentation that demonstrates both the generality and efficiency of our method.

Computer Aided Synthesis: A Game-Theoretic Approach

Abstract: In this invited contribution, we propose a comprehensive introduction to game theory applied in computer aided synthesis In this context, we give some classical results on two-player zero-sum games and then on multi-player non zero-sum games The simple case of one-player games is strongly related to automata theory on infinite words All along the article, we focus on general approaches to solve the studied problems, and we provide several illustrative examples as well as intuitions on the proofs

Liveness in L/U-Parametric Timed Automata

Abstract: We study timed systems in which some timing features are unknown parameters. Parametric timed automata are a classical formalism for such systems but for which most interesting problems are undecidable. Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achieve decidability for reachability properties by enforcing a separation of parameters used as upper bounds in the automaton constraints, and those used as lower bounds. We further study L/U-PTAs by considering liveness related problems. We prove that: (1) the existence of at least one parameter valuation for which there exists an infinite run in the automaton is PSPACE-complete, (2) the existence of a parameter valuation such that the system has a deadlock is however undecidable, (3) the existence of a valuation for which a run remains in a given set of locations exhibits a very thin border between decidability and undecidability.

Problems on Finite Automata and the Exponential Time Hypothesis

Abstract: We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different type of problems on finite automata relates to aperiodicity and to synchronizing words. We also consider finite automata that work on commutative alphabets and those working on two-dimensional words.

Ambiguity, nondeterminism and state complexity of finite automata

Abstract: The degree of ambiguity counts the number of accepting computations of a nondeterministic finite automaton (NFA) on a given input. Alternatively, the nondeterminism of an NFA can be measured by counting the amount of guessing in a single computation or the number of leaves of the computation tree on a given input. This paper surveys work on the degree of ambiguity and on various nondeterminism measures for finite automata. In particular, we focus on state complexity comparisons between NFAs with quantified ambiguity or nondeterminism.

Ultimate Automizer with an On-Demand Construction of Floyd-Hoare Automata

Abstract: Ultimate Automizer is a software verifier that implements an automata-based approach for the verification of safety and liveness properties. A central new feature that speeded up the abstraction refinement of the tool is an on-demand construction of Floyd-Hoare automata.

Separation for dot-depth two

Abstract: The dot-depth hierarchy of Brzozowski and Cohen is a classification of all first-order definable languages. It rose to prominence following the work of Thomas, who established an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level contains languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to obtain membership algorithms for all levels in this hierarchy.For a fixed level, the membership problem asks whether an input regular language belongs to this level. Despite a significant research effort, membership by itself has only been solved for low levels. Recently, a breakthrough was made by replacing membership with a more general problem called separation. This problem asks whether, for two input languages, there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation for looking at separation is threefold: (1) while more difficult, it is more rewarding; (2) being more general, it provides a more convenient framework, and (3) all recent membership algorithms are actually reductions to separation for lower levels.This paper presents a separation algorithm for dot-depth 2. A crucial point is that while dot-depth 2 is our main application, we prove a much more general theorem. Indeed, dot-depth belongs to a family of hierarchies which all share the same generic construction process: starting from an initial class of languages called the basis, one applies generic operations to build new levels. We prove that for any such hierarchy whose basis is a finite class, level 1 has decidable separation. In the special case of dot-depth, this generic result can easily be lifted to level 2.

Polynomial automata: zeroness and applications

Abstract: We introduce a generalisation of weighted automata over a field, called polynomial automata, and we analyse the complexity of the Zeroness Problem in this model, that is, whether a given automaton outputs zero on all words. While this problem is non-primitive recursive in general, we highlight a subclass of polynomial automata for which the Zeroness Problem is primitive recursive. Refining further, we identify a subclass of affine VAS for which coverability is in 2EXPSPACE. We also use polynomial automata to obtain new proofs that equivalence of streaming string transducers is decidable, and that equivalence of copyless streaming string transducers is in PSPACE.

Bayesian network structure training based on a game of learning automata

Abstract: Bayesian network (BN) is a probabilistic graphical model which describes the joint probability distribution over a set of random variables. Finding an optimal network structure based on an available training dataset is one of the most important challenges in the field of BNs. Since the problem of searching the optimal BN structure belongs to the class of NP-hard problems, typically greedy algorithms are used to solve it. In this paper two novel learning automata-based algorithms are proposed to solve the BNs’ structure learning problem. In both, there is a learning automaton corresponding with each possible edge to determine the appearance and the direction of that edge in the constructed network; therefore, we have a game of learning automata, at each stage of the proposed algorithms. Two special cases of the game of the learning automata have been discussed, namely, the game with a common payoff and the competitive game. In the former, all the automata in the game receive a unique payoff from the environment, but in the latter, each automaton receives its own payoff. As the algorithms proceed, the learning processes focus on the BN structures with higher scores. The use of learning automata has led to design the algorithms with a guided search scheme, which can avoid getting stuck in local maxima. Experimental results show that the proposed algorithms are capable of finding the optimal structure of BN in an acceptable execution time; and compared with other search-based methods, they outperform them.

Groups, Languages and Automata

Abstract: Fascinating connections exist between group theory and automata theory, and a wide variety of them are discussed in this text. Automata can be used in group theory to encode complexity, to represent aspects of underlying geometry on a space on which a group acts, and to provide efficient algorithms for practical computation. There are also many applications in geometric group theory. The authors provide background material in each of these related areas, as well as exploring the connections along a number of strands that lead to the forefront of current research in geometric group theory. Examples studied in detail include hyperbolic groups, Euclidean groups, braid groups, Coxeter groups, Artin groups, and automata groups such as the Grigorchuk group. This book will be a convenient reference point for established mathematicians who need to understand background material for applications, and can serve as a textbook for research students in (geometric) group theory.

Separating regular languages with two quantifier alternations

Abstract: We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes. Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory. Our main theorem is generic: we show that separation is decidable for the level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5/2. In logical terms, this solves separation for $\Sigma_3$: first-order sentences having at most three quantifier blocks starting with an existential one.

Automata theory on sliding windows

Abstract: In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.

The word and order problems for self-similar and automata groups

Abstract: We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.

Open Problems About Regular Languages, 35 Years Later.

Abstract: In 1980, Janusz A. Brzozowski presented a selection of six open problems about regular languages and mentioned two other problems in the conclusion of his article. These problems have been the source of some of the greatest breakthroughs in automata theory over the past 35 years. This survey article summarizes the state of the art on these questions and the hopes for the next 35 years.

Automatic sequences and curves over finite fields

Abstract: We prove that if y = ∑ n=0∞a(n)xn ∈ Fq[[x]] is an algebraic power series of degree d, height h, and genus g, then the sequence a is generated by an automaton with at most qh+d+g−1 states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.

Algebraic state space approach to model and control combined automata

Abstract: A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices (STP), is applied to the automata field. Using the STP, this paper investigates the modeling and controlling problems of combined automata constructed in the ways of parallel, serial and feedback. By representing the states, input and output symbols in vector forms, the transition and output functions are expressed as algebraic equations of the states and inputs. Based on such algebraic descriptions, the control problems of combined automata, including output control and state control, are considered, and two necessary and sufficient conditions are presented for the controllability, by which two algorithms are established to find out all the control strings that make a combined automaton go to a target state or produce a desired output. The results are quite different from existing methods and provide a new angle and means to understand and analyze the dynamics of combined automata.

Parametric Model Checking Timed Automata Under Non-Zenoness Assumption

Abstract: Real-time systems often involve hard timing constraints and concurrency, and are notoriously hard to design or verify. Given a model of a real-time system and a property, parametric model-checking aims at synthesizing timing valuations such that the model satisfies the property. However, the counter-example returned by such a procedure may be Zeno (an infinite number of discrete actions occurring in a finite time), which is unrealistic. We show here that synthesizing parameter valuations such that at least one counterexample run is non-Zeno is undecidable for parametric timed automata (PTAs). Still, we propose a semi-algorithm based on a transformation of PTAs into Clock Upper Bound PTAs to derive all valuations whenever it terminates, and some of them otherwise.

Automata Language Equivalence vs. Simulations for Model-Based Mutant Equivalence: An Empirical Evaluation

Abstract: Mutation analysis is a popular test assessment method. It relies on the mutation score, which indicates how many mutants are revealed by a test suite. Yet, there are mutants whose behaviour is equivalent to the original system, wasting analysis resources and preventing the satisfaction of the full (100%) mutation score. For finite behavioural models, the Equivalent Mutant Problem (EMP) can be addressed through language equivalence of non-deterministic finite automata, which is a well-studied, yet computationally expensive, problem in automata theory. In this paper, we report on our preliminary assessment of a state-of-the-art exact language equivalence tool to handle the EMP against 3 models of size up to 15,000 states on 1170 mutants. We introduce random and mutation-biased simulation heuristics as baselines for comparison. Results show that the exact approach is often more than ten times faster in the weak mutation scenario. For strong mutation, our biased simulations are faster for models larger than 300 states. They can be up to 1,000 times faster while limiting the error of misclassifying non-equivalent mutants as equivalent to 10% on average. We therefore conclude that the approaches can be combined for improved efficiency.

Deterministic Stack Transducers

Abstract: We introduce and investigate stack transducers, which are one-way stack automata with an output tape. A one-way stack automaton is a classical pushdown automaton with the additional ability to move ...

Permutive One-Way Cellular Automata and the Finiteness Problem for Automaton Groups

Abstract: The decidability of the finiteness problem for automaton groups is a well-studied open question on Mealy automata. We connect this question of algebraic nature to the periodicity problem of one-way cellular automata, a dynamical question known to be undecidable in the general case. We provide a first undecidability result on the dynamics of one-way permutive cellular automata, arguing in favor of the undecidability of the finiteness problem for reset Mealy automata.

Descriptional complexity of limited automata

Abstract: A k-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first k visits, where k ≥ 0 is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity of limited automata. Since the unary languages accepted are necessarily regular, we first study the cost in the number of states when finite automata simulate a unary k-limited automaton. For the conversion of a 4n-state deterministic 1-limited automaton into one-way or two-way deterministic or nondeterministic finite automata, we show a lower bound of n ⋅ F ( n ) states, where F denotes Landau's function. So, even the ability to deterministically rewrite any cell only once gives an enormous descriptional power. For the simulation cost for removing the ability to rewrite each cell k ≥ 1 times, more precisely, the cost for the simulation of sweeping unary k-limited automata by deterministic finite automata, we obtain a lower bound of n ⋅ F ( n ) k . The upper bound of the cost for the simulation by two-way deterministic finite automata is a polynomial whose degree is quadratic in k. If the k-limited automaton is rotating, the upper bound reduces to O ( n k + 1 ) and the lower bound derived is Ω ( n k + 1 ) even for nondeterministic two-way finite automata. So, for rotating k-limited automata, the trade-off for the simulation is tight in the order of magnitude. Finally, we consider the simulation of k-limited automata over general alphabets by pushdown automata. It turns out that the cost is an exponential blow-up of the size. Furthermore, an exponential size is also necessary.

Featured weighted automata

Abstract: A featured transition system is a transition system in which the transitions are annotated with feature expressions: Boolean expressions on a finite number of given features. Depending on its feature expression, each individual transition can be enabled when some features are present, and disabled for other sets of features. The behavior of a featured transition system hence depends on a given set of features. There are algorithms for featured transition systems which can check their properties for all sets of features at once, for example for LTL or CTL properties. Here we introduce a model of featured weighted automata which combines featured transition systems and (semiring-) weighted automata. We show that methods and techniques from weighted automata extend to featured weighted automata and devise algorithms to compute quantitative properties of featured weighted automata for all sets of features at once. We show applications to minimum reachability and to energy properties.

Emptiness of Zero Automata Is Decidable

Abstract: Zero automata are a probabilistic extension of parity automata on infinite trees. The satisfiability of a certain probabilistic variant of MSO, called TMSO+zero, reduces to the emptiness problem for zero automata. We introduce a variant of zero automata called nonzero automata. We prove that for every zero automaton there is an equivalent nonzero automaton of quadratic size and the emptiness problem of nonzero automata is decidable, with complexity co-NP. These results imply that TMSO+zero has decidable satisfiability.