Finite Cover Automata Research Articles

A model learning based testing approach for spiking neural P systems

This paper proposes a testing approach for spiking neural P systems, significantly different from the past testing research for cell-like P systems. The proposed method provides a solution to the state explosion problem by constructing a series of approximations, using the concept of cover automaton and Angluin-style model learning from queries, more precisely the Ll algorithm for learning a finite cover automaton, adapted to the more general X-machine model. Furthermore, the concept of identifiability, which is an essential prerequisite for the successful application of our method, but also a more general design characteristic inspired from the testing practice, is introduced and investigated in the context of spiking neural P systems. Identifiability of system's components (modules, methods, etc.) is a fundamental criterion used for assessing a system's testability since it allows the components of a system to be identified from the behaviour produced in response to the inputs received and, consequently, maximizes the effectiveness of the testing process.

Two Extensions of Cover Automata

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.

More on deterministic and nondeterministic finite cover automata

Finite languages are an important sub-regular language family, which were intensively studied during the last two decades in particular from a descriptional complexity perspective. An important contribution to the theory of finite languages are the deterministic and the recently introduced nondeterministic finite cover automata (DFCAs and NFCAs, respectively) as an alternative representation of finite languages by ordinary finite automata. We compare these two types of cover automata from a descriptional complexity point of view, showing that these devices have a lot in common with ordinary finite automata. In particular, we study how to adapt lower bound techniques for nondeterministic finite automata to NFCAs such as, e.g., the biclique edge cover technique, solving an open problem from the literature. Moreover, the trade-off of conversions between DFCAs and NFCAs as well as between finite cover automata and ordinary finite automata are investigated. Finally, we present some results on the average size of finite cover automata.

Simplifying Nondeterministic Finite Cover Automata

The concept of Deterministic Finite Cover Automata (DFCA) was introduced at WIA '98, as a more compact representation than Deterministic Finite Automata (DFA) for finite languages. In some cases representing a finite language, Nondeterministic Finite Automata (NFA) may significantly reduce the number of states used. The combined power of the succinctness of the representation of finite languages using both cover languages and non-determinism has been suggested, but never systematically studied. In the present paper, for nondeterministic finite cover automata (NFCA) and l-nondeterministic finite cover automaton (l-NFCA), we show that minimization can be as hard as minimizing NFAs for regular languages, even in the case of NFCAs using unary alphabets. Moreover, we show how we can adapt the methods used to reduce, or minimize the size of NFAs/DFCAs/l-DFCAs, for simplifying NFCAs/l-NFCAs.

Model Learning and Test Generation Using Cover Automata

We propose an approach which, given a state-transition model of a system, constructs, in parallel, an approximate automaton model and a test suite for the system. The approximate model construction relies on a variant of Angluin’s automata learning algorithm, adapted to finite cover automata. A finite cover automaton represents an approximation of the system which only considers sequences of length up to an established upper bound `. Crucially, the size of the cover automaton, which normally depends on `, can be significantly lower than the size of the exact automaton model. Thus, controlling `, the state explosion problem normally associated with constructing and checking state based models can be mitigated. The proposed approach also allows for a gradual construction of the model and of the associated test suite, with complexity and time savings. Moreover, we provide automation of counterexample search, by a combination of black-box and random testing, and metrics to evaluate the quality of the produced results. The approach is presented and implemented in the context of the Event-B modeling language, but its underlying ideas and principles are much more general and can be applied to any system whose behavior can be suitably described by a state-transition model.

Learning finite cover automata from queries

Learning regular languages from queries was introduced by Angluin in a seminal paper that also provides the L ⁎ algorithm. This algorithm, as well as other existing inference methods, finds the exact language accepted by the automaton. However, when only finite languages are used, the construction of a deterministic finite cover automaton ( DFCA) is sufficient. A DFCA of a finite language U is a finite automaton that accepts all sequences in U and possibly other sequences that are longer than any sequence in U. This paper presents an algorithm, called L l , that finds a minimal DFCA of an unknown finite language in polynomial time using membership and language queries, a non-trivial adaptation of Angluinʼs L ⁎ algorithm. As the size of a minimal DFCA of a finite language U may be much smaller than the size of the minimal automaton that accepts exactly U, L l can provide substantial savings over existing automata inference methods.

HYPER-MINIMIZATION IN O(n2)

Two formal languages are f-equivalent if their symmetric difference L1 Δ L2 is a finite set — that is, if they differ on only finitely many words. The study of f -equivalent languages, and particularly the DFAs that accept them, was recently introduced [3]. First, we restate the fundamental results in this new area of research. Second, we introduce our main result, which is a faster algorithm for the natural minimization problem: given a starting DFA D, find the smallest (by number of states) DFA D′ such that L(D) and L(D′) are f -equivalent. Finally, we suggest a technique that combines this hyper-minimization with the well-studied notion of a deterministic finite cover automaton [2, 4, 5], or DFCA, thereby extending the applicability of DFCAs from finite to infinite regular languages.

Finite state based testing of P systems

In this paper, we propose an approach to P system testing based on finite state machine conformance techniques. Of the many variants of P systems that have been defined, we consider cell-like P systems which use non-cooperative transformation and communication rules. We show that a (minimal) deterministic finite cover automaton (DFCA) (a finite automaton that accepts all words in a given finite language, but can also accept words that are longer than any word in the language) provides the right approximation for the computation of a P system. Furthermore, we provide a procedure for generating test sets directly from the P system specification (without explicitly constructing the minimal DFCA model).

AN EFFICIENT ALGORITHM FOR CONSTRUCTING MINIMAL COVER AUTOMATA FOR FINITE LANGUAGES

The concept of cover automata for finite languages was formally introduced in [3]. Cover automata have been studied as an efficient representation of finite languages. In [3], an algorithm was given to transform a DFA that accepts a finite language to a minimal deterministic finite cover automaton (DFCA) with the time complexity O(n4), where n is the number of states of the given DFA. In this paper, we review the basic concept of cover automata and describe a new efficient transformation algorithm with the time complexity O(n2), which is a significant improvement from the previous algorithm.

Minimal cover-automata for finite languages

A cover-automaton A of a finite language L⊆Σ ∗ is a finite deterministic automaton (DFA) that accepts all words in L and possibly other words that are longer than any word in L. A minimal deterministic finite cover automaton (DFCA) of a finite language L usually has a smaller size than a minimal DFA that accept L. Thus, cover automata can be used to reduce the size of the representations of finite languages in practice. In this paper, we describe an efficient algorithm that, for a given DFA accepting a finite language, constructs a minimal deterministic finite cover-automaton of the language. We also give algorithms for the boolean operations on deterministic cover automata, i.e., on the finite languages they represent.