Showing papers on "ω-automaton published in 2015"

Learning regular languages via alternating automata

Abstract: Nearly all algorithms for learning an unknown regular language, in particular the popular L* algorithm, yield deterministic finite automata. It was recently shown that the ideas of L* can be extended to yield non-deterministic automata, and that the respective learning algorithm, NL*, outperforms L* on randomly generated regular expressions. We conjectured that this is due to the existential nature of regular expressions, and NL* might not outperform L* on languages with a universal nature. In this paper we introduce UL* -- a learning algorithm for universal automata (the dual of non-deterministic automata); and AL* -- a learning algorithm for alternating automata (which generalize both universal and non-deterministic automata). Our empirical results illustrate the advantages and trade-offs among L*, NL*, UL* and AL*.

Foundations of active automata learning: an algorithmic perspective

Abstract: The wealth of model-based techniques in software engineering—such as model checking or model-based testing—is starkly contrasted with a frequent lack of formal models in practical settings. Sophisticated static analysis techniques for obtaining models from a sourceor bytecode representation have matured to close this gap to a large extent, yet they might fall short on more complex systems: be it that no sufficiently robust decision procedures are available, or that the system performs calls to external, closed source libraries or even remote web services. Active automata learning has been proposed as a means of overcoming this problem: by executing test cases on a system, finite-state machine models reflecting a portion of the actual runtime behavior of the targeted system can be inferred. This positions active automata learning as an enabler technology, extending the range of application for a whole array of formal, model-based techniques. Its usefulness has been proven in many different subfields of formal methods, such as black-box model checking, test-case generation, interface synthesis, or compositional verification. In a much-noted case study, active automata learning played a key role in analyzing the internal structure of a botnet with the aim of devising countermeasures. One of the major obstacles of applying active automata learning in practice is, however, the fact that it is a rather costly technique: to gain sufficient information for inferring a model, a large number of test cases need to be executed, which is also referred to as “posing queries.” These test cases may be rather heavy-weighted, comprising high-latency operations such as interactions with hardware or remote network services, and learning systems of moderate size may take hours or days even when using algorithms with polynomial query complexities. The costliness of the technique calls for highly efficient algorithms that do not waste any information. The reality is surprisingly different from that ideal: many active automata learning algorithms that are being used in practice—including the well-known L∗ algorithm, which was the first one with a polynomial query complexity—frequently resort to heuristics to ensure certain properties, resulting in an increased overall query complexity. However, it has rarely been investigated why or even if these properties are necessary to ensure correctness, or what violating them entails. Related is the observation that descriptions of active automata learning algorithms are often less-than-formal, and merely focus on somehow arriving at a correctness proof instead of motivating and justifying the single steps. It is one of the stated goals of this thesis to change this situation, by giving a rigorously formal description of an approach to active automata learning that is independent of specific data structures or algorithmic realizations. This formal description allows the identification of a number of properties, some of which are necessary, while others are merely desirable. The connection between these properties, as well as possible reasons for their violation, are investigated. This leads to the observation that, while for each property there is an existing algorithm maintaining it, no algorithm manages to simultaneously maintain all desirable properties. Based on these observations, and exploiting further insights attained through the formalization, a novel active automata learning algorithm, called TTT, is developed. The distinguishing

On the State Minimization of Fuzzy Automata

Abstract: This paper investigates the minimization problem of fuzzy automata, aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one. The decision version of the minimization problem is as follows: Given a fuzzy automaton ${\cal A}$ and a natural number $k$ , i.e., a pair $\langle {\cal A}, k\rangle$ , is there a $k$ -state fuzzy automaton equivalent to ${\cal A}$ ? We prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton. To this end, we introduce the concept of systems of fuzzy polynomial equations, present a procedure for finding solutions of these systems and, finally, reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations. It is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms, the term “minimization” there did not mean state minimization in our sense, since these algorithms did not aim at a minimal fuzzy automaton but found “reasonably” small fuzzy automata.

A succinct canonical register automaton model

Abstract: We present a novel canonical automaton model, based on register automata, that can be used to specify protocol or program behavior. Register automata have a finite control structure and a finite number of registers (variables), and process sequences of terms that carry data values from an infinite domain. We consider register automata that compare data values for equality. A major contribution is the definition of a canonical automaton representation of any language recognizable by a deterministic register automaton, by means of a Nerode congruence. This canonical form is well suited for modeling, e.g., protocols or program behavior. Our model can be exponentially more succinct than previous proposals, since it filters out ‘accidental’ relations between data values. This opens the way to new practical applications, e.g., in automata learning.

Classical automata on promise problems

Abstract: Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.

The Target Discounted-Sum Problem

Abstract: The target discounted-sum problem is the following: Given a rational discount factor 0 < a#x03BB; < 1 and three rational values a, b, and t, does there exist a finite or an infinite sequence a#x03C9; a#x2208;(a, b)* or a#x03C9; a#x2208;(a, b)a#x03C9;, such that a#x3A3;|a#x03C9;| i=0 a#x03C9;(i)a#x03BB;i equals t? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: a#x03B2;-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that a#x03BB; a#x03BB; 1/2 or a#x03BB; = 1/n, for every n a#x2208; N. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value problem for nondeterministic automata over finite words and the universality and inclusion problems for functional automata.

Profile trees for Büchi word automata, with application to determinization

Abstract: The determinization of Buchi automata is a celebrated problem, with applications in synthesis, probabilistic verification, and multi-agent systems. Since the 1960s, there has been a steady progress of constructions: by McNaughton, Safra, Piterman, Schewe, and others. Despite the proliferation of solutions, they are all essentially ad-hoc constructions, with little theory behind them other than proofs of correctness. Since Safra, all optimal constructions employ trees as states of the deterministic automaton, and transitions between states are defined operationally over these trees. The operational nature of these constructions complicates understanding, implementing, and reasoning about them, and should be contrasted with complementation, where a solid theory in terms of automata run dags underlies modern constructions.In 2010, we described a profile-based approach to Buchi complementation, where a profile is simply the history of visits to accepting states. We developed a structural theory of profiles and used it to describe a complementation construction that is deterministic in the limit. Here we extend the theory of profiles to prove that every run dag contains a profile tree with at most a finite number of infinite branches. We then show that this property provides a theoretical grounding for a new determinization construction where macrostates are doubly preordered sets of states. In contrast to extant determinization constructions, transitions in the new construction are described declaratively rather than operationally.

RALib : A LearnLib extension for inferring EFSMs

Abstract: Formal models are often used to describe the behavior of a computer program or component. Behavioral models have many different usages, e.g., in model-based techniques for software development and verification,such as model checking and model based testing.The application of model-based techniques is hampered by the current lack of adequate models for most actual systems, largely due to the significant manual effort typically needed to construct them. To remedy this, automata learning techniques (whereby models can be inferred by performing tests on a component) have been developed for finite automata that capture control flow. However, many usages requiremodels also to capture data flow, i.e., how behavior is affected by data values in method invocations and commands. Unfortunately, techniques are less developed for models that combinecontrol flow with data.In this thesis, we extend automata learning to infer automata models that captureboth control flow and data flow. We base our technique on a corresponding extension of classical automata theoryto capture data.We define a formalism for register automata, a model that extends finite automata by adding registers that can store data values and be used in guards and assignments on transitions. Our formalism is parameterized on a theory, i.e., a set of relations on a data domain. We present a Nerode congruence for the languages that our register automata can recognize, and provide a Myhill-Nerode theorem for constructing canonical register automata, thereby extending classical automata theory to register automata.We also present a learning algorithm for register automata: the new SL* algorithm, which extends the well-known L* algorithm for finite automata. The SL* algorithm is based on our new Nerode congruence, and uses a novel technique to infer symbolic data constraints on parameters. We evaluated our algorithm in a black-box scenario, inferring, e.g., the connection establishment phase of TCP and a priority queue, in addition to a number of smaller models. The SL* algorithm is implemented in a tool, which allows for use in more realistic settings, e.g., where models have both input and output, and for directly inferring Java classes.

From Finite Automata to Regular Expressions and Back—A Summary on Descriptional Complexity

Abstract: The equivalence of finite automata and regular expressions d ates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the pr esent paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (e-transitions) on non-e-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.

Evolvable fashion-based cellular automata for generating cavern systems

Abstract: Cellular automata can be used to rapidly generate complex images. This study introduces fashion-based cellular automata as a new representation for generating cavern-like level maps. Fashion-based automata are defined by a competition matrix that defines the benefit to a given cell state of having a neighbor of each possible cell state. A simple fitness function permits this type of automata to be evolved to produce a variety of level maps. A parameter study is performed and a variety of level maps are evolved with a toroidal grid, ensuring that the level maps tile. The parameter study demonstrates a robustness of the fashion based representation to the variation of parameters. The appearance of a given cavern-like level is encoded in the evolved automaton rule permitting the creation of many levels with a similar character simply by varying initial conditions. The cellular automata rules function in local neighborhoods meaning that the level generation system scales smoothly to any desired level map size.

Nested Weighted Automata

Abstract: Recently there has been a significant effort to handle quantitative properties in formal verification and synthesis. While weighted automata over finite and infinite words provide a natural and flexible framework to express quantitative properties, perhaps surprisingly, some basic system properties such as average response time cannot be expressed using weighted automata, nor in any other know decidable formalism. In this work, we introduce nested weighted automata as a natural extension of weighted automata which makes it possible to express important quantitative properties such as average response time. In nested weighted automata, a master automaton spins off and collects results from weighted slave automata, each of which computes a quantity along a finite portion of an infinite word. Nested weighted automata can be viewed as the quantitative analogue of monitor automata, which are used in run-time verification. We establish an almost complete decidability picture for the basic decision problems about nested weighted automata, and illustrate their applicability in several domains. In particular, nested weighted automata can be used to decide average response time properties.

The Cerny conjecture and 1-contracting automata

Abstract: A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. \v{C}ern\'y conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically $1-$contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the \v{C}ern\'y conjecture holds for aperiodically $1-$contracting automata. As a special case, we prove some results for circular automata.

Proving non-termination by finite automata

Abstract: A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic.

State complexity of deterministic Watson---Crick automata and time varying Watson---Crick automata

Abstract: Watson---Crick automata are finite automata working on double strands. Extensive research work has already been done on non-deterministic Watson---Crick automata and on deterministic Watson---Crick automata. State complexity of Watson---Crick automata has also been discussed. In this paper we analyze the state complexity of deterministic Watson---Crick automata with respect to non-deterministic block automata and non-deterministic finite automata. We also introduce new finite automata combining the properties of Watson---Crick automata and time varying automata. These automata have the unique property that the 1-limited stateless variant of these automata have the power to count. We further discuss the state complexity of time varying automata and time varying Watson---Crick automata.

Complexities of Some Problems Related to Synchronizing, Non-Synchronizing and Monotonic Automata

Abstract: In this study, we first introduce several problems related to finding reset words for deterministic finite automata, and present motivations for these problems for practical applications in areas such as robotics and bio-engineering. We then analyse computational complexities of these problems. Second, we consider monotonic and partially specified automata. Monotonicity is known to be a feature simplyfing the synchronizability problems. On the other hand for partially specified automata, synchronizability problems are known to be harder than the completely specified automata. We investigate the complexity of some synchronizability problems for automata that are both monotonic and partially specified. We show that checking the existence, computing one, and computing a shortest reset word for a monotonic partially specified automaton is NP-hard. We also show that finding a reset word that synchronizes 𝓚 number of states (or maximum number of states) of a given monotonic non-synchronizable automaton to a given set of states is NP-hard.

Automata and rational expressions

Abstract: This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin.

On algebraic study of fuzzy automata

Abstract: This paper is towards the characterization of algebraic concepts such as subautomaton, retrievability and connectivity of a fuzzy automaton in terms of its layers, and to associate upper semilattices with fuzzy automata. Meanwhile, we provide a decomposition of a fuzzy automaton in terms of its layers and propose a construction of a fuzzy automaton corresponding to a given finite partially ordered set (poset). Finally, we establish an isomorphism between the poset of class of subautomata of a fuzzy automaton and an upper semilattice.

What's decidable about recursive hybrid automata?

Abstract: Recursive hybrid automata generalize recursive state machines in a similar way as hybrid automata generalize state machines. Recursive hybrid automata can be considered as collection of classical hybrid automata with special states that correspond to potentially recursive invocation of hybrid automata from the collection. During each such invocation, the semantics of recursive hybrid automata permits optional passing of the continuous variables using either pass-by-value or pass-by-reference mechanism. This model generalizes recursive timed automata model introduced by Trivedi and Wojtczak and dense-timed pushdown automata by Abdulla, Atig, and Stenman. We study natural reachability problem for recursive hybrid automata. Given the undecidability of this problem for hybrid automata, it is not surprising that the problem remains undecidable without further restrictions. We consider various restrictions of recursive hybrid automata and characterize the boundaries between decidable and undecidable variants.

On the Rademacher Complexity of Weighted Automata

Abstract: Weighted automata WFAs provide a general framework for the representation of functions mapping strings to real numbers. They include as special instances deterministic finite automata DFAs, hidden Markov models HMMs, and predictive states representations PSRs. In recent years, there has been a renewed interest in weighted automata in machine learning due to the development of efficient and provably correct spectral algorithms for learning weighted automata. Despite the effectiveness reported for spectral techniques in real-world problems, almost all existing statistical guarantees for spectral learning of weighted automata rely on a strong realizability assumption. In this paper, we initiate a systematic study of the learning guarantees for broad classes of weighted automata in an agnostic setting. Our results include bounds on the Rademacher complexity of three general classes of weighted automata, each described in terms of different natural quantities. Interestingly, these bounds underline the key role of different data-dependent parameters in the convergence rates.

Parallel Induction of Nondeterministic Finite Automata

Abstract: The induction of a minimal nondeterministic finite automaton (NFA) consistent with a given set of examples and counterexamples, which is known to be computationally hard, is discussed. The paper is an extension to the novel approach of transforming the problem of NFA induction into the integer nonlinear programming (INLP) problem. An improved formulation of the problem is proposed along with the two parallel algorithms to solve it. The methods for the distribution of tasks among processors along with distributed termination detection are presented. The experimental results for selected benchmarks are also reported.

On Parallel Versions of Jumping Finite Automata

Abstract: The present paper proposes a new investigation area in automata theory - n-parallel jumping finite automata. These automata further extend recently presented jumping finite automata that are focused on discontinuous reading. The proposed modification uses multiple reading heads that work in parallel and can discontinuously read from the input in several places at once. We also define the more restricted version of these automata which only allows jumping to the right. This restricted version is then further studied, compared with n-parallel right linear grammars, and several of its properties are derived.

A Formalisation of Finite Automata Using Hereditarily Finite Sets

Abstract: Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski’s minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.

The Supports of Weighted Unranked Tree Automata

Abstract: We investigate the supports of weighted unranked tree automata. Our main result states that the support of a weighted unranked tree automaton over a zero-sum free, commutative strong bimonoid is recognizable. For this, we use methods of Kirsten (DLT 2009), in particular, his construction of finite automata recognizing the supports of weighted automata on strings over zero-sum free, commutative semirings. We also get an effective construction of a finite tree automaton recognizing the support of a given weighted unranked tree automaton for zero-sum free, commutative strong bimonoids where Kirsten's zero generation problem is decidable. In addition, we give a translation of nested weighted automata into weighted unranked tree automata for arbitrary commutative strong bimonoids. As a consequence, we derive analogous results for the supports of nested weighted automata. Finally, we give similar results for the supports of weighted pushdown automata.

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.

Equational axioms associated with finite automata for fixed point operations in cartesian categories

Abstract: The axioms of iteration theories, or iteration categories, capture the equational properties of fixed point operations in several computationally significant categories. Iteration categories may be axiomatized by the Conway identities and identities associated with finite automata. We show that in conjunction with the Conway identities, each identity associated with a finite automaton implies the identity associated with any input extension of the automaton. We conclude that the Conway identities and the identities associated with the members of a subclass $\cQ$ of finite automata is complete for iteration categories iff for every finite simple group $G$ there is an automaton $\bQ \in \cQ$ such that $G$ is a quotient of a group in the monoid $M(\bQ)$ of the automaton $\bQ$. We also prove a stronger result that concerns identities associated with finite automata with a distinguished initial state.