Showing papers on "ω-automaton published in 2017"

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.

Sampling-based control synthesis for multi-robot systems under global temporal specifications

Abstract: This paper proposes a sampling-based algorithm for multi-robot control synthesis under global Linear Temporal Logic (LTL) formulas. Robot mobility is captured by transition systems whose states represent regions in the environment that satisfy atomic propositions. Existing planning approaches under global temporal goals rely on graph search techniques applied to a synchronous product automaton constructed among the robots. As the number of robots increases, the state-space of the product automaton grows exponentially and, as a result, graph search techniques become intractable. In this paper, we propose a new sampling-based algorithm that builds incrementally a directed tree that approximates the state-space and transitions of the synchronous product automaton. By approximating the product automaton by a tree rather than representing it explicitly, we require much fewer resources to store it and motion plans can be found by tracing the sequence of parent nodes from the leaves back to the root without the need for sophisticated graph search techniques. This significantly increases scalability of our algorithm compared to existing model-checking methods. We also show that our algorithm is probabilistically complete and asymptotically optimal and present numerical experiments that show that it can be used to model-check product automata with billions of states, which was not possible using an off-the-shelf model checker.

Automata for regular expressions with shuffle

Abstract: We generalize the partial derivative automaton and the position automaton to regular expressions with shuffle, and study their state complexity in the worst, as well as in the average case. The number of states of the partial derivative automaton ( A p d ) is, in the worst case, at most 2 m , where m is the number of letters in the expression. The asymptotic average is bounded by ( 4 3 ) m . We define a position automaton ( A p o s ) that is homogeneous, but in which several states can correspond to a same position, and we show that A p d is a quotient of A p o s . The number of states of the position automaton is at most 1 + m ( 2 m − 1 ) , while the asymptotic average is no more than m ( 4 3 ) m .

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.

Index Appearance Record for Transforming Rabin Automata into Parity Automata

Abstract: Transforming deterministic \(\omega \)-automata into deterministic parity automata is traditionally done using variants of appearance records. We present a more efficient variant of this approach, tailored to Rabin automata, and several optimizations applicable to all appearance records. We compare the methods experimentally and find out that our method produces smaller automata than previous approaches. Moreover, the experiments demonstrate the potential of our method for LTL synthesis, using LTL-to-Rabin translators. It leads to significantly smaller parity automata when compared to state-of-the-art approaches on complex formulae.

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.

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.

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.

Bounded determinization of timed automata with silent transitions

Abstract: Deterministic timed automata are strictly less expressive than their non-deterministic counterparts, which are again less expressive than those with silent transitions. As a consequence, timed automata are in general non-determinizable. This is unfortunate since deterministic automata play a major role in model-based testing, observability and implementability. However, by bounding the length of the traces in the automaton, effective determinization becomes possible. We propose a novel procedure for bounded determinization of timed automata. The procedure unfolds the automata to bounded trees, removes all silent transitions and determinizes via disjunction of guards. The proposed algorithms are optimized to the bounded setting and thus are more efficient and can handle a larger class of timed automata than the general algorithms. We show how to apply the approach in a fault-based test-case generation method, called model-based mutation testing, that was previously restricted to deterministic timed automata. The approach is implemented in a prototype tool and evaluated on several scientific examples and one industrial case study. To our best knowledge, this is the first implementation of this type of procedure for timed automata.

Reversible Watson---Crick automata

Abstract: Since 1970, reversible finite automata have generated interest among researcher; but till now, we have not come across a model of reversible read only one-way finite automata which accept all regular languages, In this paper, we introduce a new model of one-way reversible finite automata inspired by the Watson---Crick complementarity relation and show that our model can accept all regular languages. We further show that our model accepts a language which is not accepted by any multi-head deterministic finite automaton.

Minimisation of Multiplicity Tree Automata.

Abstract: We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.

Seminator: A Tool for Semi-Determinization of Omega-Automata

Abstract: We present a tool that transforms nondeterministic omega-automata to semi-deterministic omega-automata. The tool Seminator accepts transition-based generalized Buchi automata (TGBA) as an input and produces automata with two kinds of semi-determinism. The implemented procedure performs degeneralization and semi-determinization simultaneously and employs several other optimizations. We experimentally evaluate Seminator in the context of LTL to semi-deterministic automata translation.

Tight Bounds for Cut-Operations on Deterministic Finite Automata

Abstract: We investigate the state complexity of the cut and iterated cut operation for determin- istic finite automata (DFAs), answering an open question stated in [M. BERGLUND, et al.: Cuts in regular expr ...

Reversible Limited Automata

Abstract: A k-limited automaton is a linear bounded automaton that may rewrite each tape square only in the first k visits, where \(k\ge 0\) is a fixed constant. It is known that these automata accept context-free languages only. We investigate deterministic k-limited automata towards their ability to perform reversible computations, that is, computations in which every configuration has at most one predecessor. A first result is that, for all \(k\ge 0\), sweeping k-limited automata accept regular languages only. In contrast to reversible finite automata, all regular languages are accepted by sweeping 0-limited automata. Then we study the computational power gained in the number k of possible rewrite operations. It is shown that the reversible 2-limited automata accept regular languages only and, thus, are strictly weaker than general 2-limited automata. Furthermore, a proper inclusion between reversible 3-limited and 4-limited automata languages is obtained. The next levels of the hierarchy are separated between every k and \(k+3\) rewrite operations. Finally, it turns out that all k-limited automata accept Church-Rosser languages only, that is, the intersection between context-free and Church-Rosser languages contains an infinite hierarchy of language families beyond the deterministic context-free languages.

Automata and minimization

Abstract: Already in the seventies, strong results illustrating the intimate relationship between category theory and automata theory have been described and are still investigated In this column, we provide a uniform presentation of the basic concepts that underlie minimization results in automata theory We then use this knowledge for introducing a new model of automata that is an hybrid of deterministic finite automata and automata weighted over a field These automata are very natural, and enjoy minimization result by designThe presentation of this paper is indeed categorical in essence, but it assumes no prior knowledge from the reader It is also non-conventional in that it is neither algebraic, nor co-algebraic oriented

An Automata View to Goal-Directed Methods

Abstract: Consequence-based and automata-based algorithms encompass two families of approaches that have been thoroughly studied as reasoning methods for many logical formalisms. While automata are useful for finding tight complexity bounds, consequence-based algorithms are typically simpler to describe, implement, and optimize. In this paper, we show that consequence-based reasoning can be reduced to the emptiness test of an appropriately built automaton. Thanks to this reduction, one can focus on developing efficient consequence-based algorithms, obtaining complexity bounds and other benefits of automata methods for free.

Complexity of a problem concerning reset words for Eulerian binary automata

Abstract: A word is called a reset word for a deterministic finite automaton if it maps all the states of the automaton to a unique state. Deciding about the existence of a reset word of a given length for a given automaton is known to be an NP-complete problem. We prove that it remains NP-complete even if restricted to Eulerian automata with binary alphabets as it has been conjectured by Martyugin (2011).

Independent finite automata on Cayley graphs

Abstract: In the setting of symbolic dynamics on discrete finitely generated infinite groups, we define a model of finite automata with multiple independent heads that walk on Cayley graphs, called group-walking automata, and use it to define subshifts. We characterize the torsion groups (also known as periodic groups) as those on which the group-walking automata are strictly weaker than Turing machines, and those on which the head hierarchy is infinite.

Data Multi-Pushdown Automata

Abstract: We extend the classical model of multi-pushdown systems by considering systems that operate on a finite set of variables ranging over natural numbers The conditions on variables are defined via gap-order constraints that allow to compare variables for equality, or to check that the gap between the values of two variables exceeds a given natural number Furthermore, each message inside a stack is equipped with a data item representing its value When a message is pushed to the stack, its value may be defined by a variable When a message is popped, its value may be copied to a variable Thus, we obtain a system that is infinite in multiple dimensions, namely we have a number of stacks that may contain an unbounded number of messages each of which is equipped with a natural number It is well-known that the verification of any non-trivial property of multi-pushdown systems is undecidable, even for two stacks and for a finite data-domain In this paper, we show the decidability of the reachability problem for the classes of data multi-pushdown system that admit a bounded split-width (or equivalently a bounded tree-width) As an immediate consequence, we obtain decidability for several subclasses of data multi-pushdown systems These include systems with single stacks, restricted ordering policies on stack operations, bounded scope, bounded phase, and bounded context switches

Automata for Unordered Trees

Abstract: We present a framework for defining automata for unordered data trees that is parametrized by the way in which multisets of children nodes are described. Presburger tree automata and alternating Presburger tree automata are particular instances. We establish the usual equivalence in expressiveness of tree automata and MSO for the automata defined in our framework. We then investigate subclasses of automata for unordered trees for which testing language equivalence is in P-time. For this we start from automata in our framework that describe multisets of children by finite automata, and propose two approaches of how to do this deterministically. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers from coNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending on the choice of the order, we obtain different classes of automata, each of which has the same expressiveness as Counting MSO.

Two-dimensional jumping finite automata

Abstract: In this paper, we extend a newly introduced concept called the jumping finite automata for accepting string languages to two-dimensional jumping finite automata for accepting two-dimensional languages. We discuss some of the basic properties of these automata and compare the family of languages accepted by these automata with the family of Siromoney matrix languages and also recognizable picture languages (REC). We also discuss some of the closure properties of these automata along with some of their decidability properties.

On somewhat fuzzy automata continuous functions in fuzzy automata topological spaces

Abstract: In this paper, the concepts of somewhat fuzzy automata continuous functions and somewhat fuzzy automata open functions in fuzzy automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy automata resolvable spaces and fuzzy automata irresolvable spaces are also introduced and their properties are studied.