Determinization of weighted finite automata over strong bimonoids

We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, respectively, the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics.     

An improved algorithm for determinization of weighted and fuzzy automata

For a given weighted finite automaton over a strong bimonoid we construct its reduced Nerode automaton, which is crisp-deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics. We show that the reduced Nerode automaton is even smaller than the Nerode automaton, which was previously used in determinization related to this semantics. We determine necessary and sufficient conditions under which the reduced Nerode automaton is finite and provide an efficient algorithm which computes the reduced Nerode automaton whenever it is finite. In determinization of weighted finite automata over semirings and fuzzy finite automata over lattice-ordered monoids this algorithm gives smaller crisp-deterministic automata than any other known determinization algorithm.     

Backward and forward bisimulation minimization of tree automata

We improve on an existing [P.A. Abdulla, J. Högberg, L. Kaati, Bisimulation minimization of tree automata, International Journal of Foundations of Computer Science 18(4) (2007) 699–713] bisimulation minimization algorithm for finite-state tree automata by introducing backward and forward bisimulation and developing minimization algorithms for them. Minimization via forward bisimulation is also effective on deterministic tree automata, faster than the previous algorithm, and yields the minimal equivalent deterministic tree automaton. Minimization via backward bisimulation generalizes the previous algorithm and can yield smaller automata but is just as fast. We demonstrate implementations of these algorithms on a typical task in natural language processing.     

Determinization of fuzzy automata with membership values in complete residuated lattices

In this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices. In comparison with the previous methods, developed by Bělohlávek [R. Bělohlávek, Determinism and fuzzy automata, Information Sciences 143 (2002), 205-209] and Li and Pedrycz [Y.M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005), 68-92], our method always gives a smaller automaton, and in some cases, when the previous methods result in infinite automata, our method can result in a finite one. We also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them, and in particular, to the concept of the Nerode’s fuzzy right congruence of a fuzzy automaton, which we introduce and study here.     

A note on multihead automata and context-sensitive languages

Summary In this note we show that the following two statements are equivalent: (1) Any language over a one-letter alphabet, which is accepted by some nondeterministic multihead automaton can also be accepted by some deterministic multihead automaton. (2) Deterministic and nondeterministic context-sensitive languages are the same.This research was supported by NSF grant GJ-27278.     

Special factors and the combinatorics of suffix and factor automata

The suffix automaton (resp. factor automaton) of a finite word w is the minimal deterministic automaton recognizing the set of suffixes (resp. factors) of w. We study the relationships between the structure of the suffix and factor automata and classical combinatorial parameters related to the special factors of w. We derive formulae for the number of states of these automata. We also characterize the languages L_SA and L_FA of words having respectively suffix automaton and factor automaton with the minimal possible number of states.     

On finite automata with limited nondeterminism

We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by where is the number of states. If the finite automaton is over a one-letter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than . In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACE-hard. Since the problem is already known to be in PSPACE, it is therefore PSPACE-complete. Received: 14 June 1994 / 3 December 1997     

Inequality of finite behaviors of rational weight finite automata is in R

Inequivalence of finite automata accepting finite languages over a non-unary alphabet is NP-complete. However, the inequality of their behaviors does not appear to have been carefully investigated. In the simplest case, the behavior of a finite automaton is the formal series f such that the coefficient f(w) of a word w is the number of distinct accepting computations on w. This notion will be generalized in the paper to finite automata with rational weights. The main result is that inequality of rational weight finite automata with finite behaviors is in R, random polynomial time.     

Efficient implementation of Aho–Corasick pattern matching automata using Unicode

We study different efficient implementations of an Aho–Corasick pattern matching automaton when searching for patterns in Unicode text. Much of the previous research has been based on the assumption of a relatively small alphabet, for example the 7‐bit ASCII. Our aim is to examine the differences in performance arising from the use of a large alphabet, such as Unicode that is widely used today. The main concern is the representation of the transition function of the pattern matching automaton. We examine and compare array, linked list, hashing, balanced tree, perfect hashing, hybrid, triple‐array, and double‐array representations. For perfect hashing, we present an algorithm that constructs the hash tables in expected linear time and linear space. We implement the Aho–Corasick automaton in Java using the different transition function representations, and we evaluate their performance. Triple‐array and double‐array performed best in our experiments, with perfect hashing, hashing, and balanced tree coming next. We discovered that the array implementation has a slow preprocessing time when using the Unicode alphabet. It seems that the use of a large alphabet can slow down the preprocessing time of the automaton considerably depending on the transition function representation used. Copyright © 2006 John Wiley & Sons, Ltd.     

Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.     

Learning automata with continuous input and changing number of actions

The behaviour of a stochastic automaton operating in an S-model environment is described. The environment response takes an arbitrary value in the closed segment [0, 1] (continuous response). The learning automaton uses a reinforcement scheme to update its action probabilities on the basis of the reaction of the environment. The complete set of actions is divided into a collection of non-empty subsets. The action set is changing from instant to instant. Each action set is selected according to a given probability distribution. Convergence and convergence rate results are presented. These results have been derived using quasimartingales theory.     

Bounded monotone recursion and multihead automata

On the basis of bounded monotone recursion, a class BMR of word functions over the alphabet {1, 2} is defined. A new type of a computing device is introduced, which is called a multihead nonerasing automaton with output, or an MH automaton. It is proved that the class BMR coincides with the class MHA of word functions computable by MH automata in polynomial time. Numerous examples of word functions from the class BMR are given.     

On dynamics of automata with a stack

In this paper we introduce a new dynamical system of a pushdown automaton, called automaton with a stack (AS). We prove that every AS has a periodic configuration by construction of it. Next, we define a special case of an AS, called AS with finite memory and we prove that the AS has a finite memory if and only if it is positively expansive. Furthermore, we prove that every AS with finite memory has shadowing property. Having these two properties, we set a finite-to-one map between an AS with finite memory and some vertex subshift, which gives us a semi-conjugacy between these two systems. Additionally, we define an algorithm to decide if a given graph G describes some AS with finite memory and to calculate maximal depth of a stack.     

Deciding unique decodability of bigram counts via finite automata

We revisit the problem of deciding by means of a finite automaton whether a string is uniquely decodable from its bigram counts. An efficient algorithm for constructing a polynomial-size Nondeterministic Finite Automaton (NFA) that decides unique decodability is given. This NFA may be simulated efficiently in time and space. Conversely, we show that the minimum deterministic finite automaton for deciding unique decodability has exponentially many states in alphabet size, and compute the correct order of magnitude of the exponent.     

On Boyer-Moore automata

The notion of Boyer-Moore automaton was introduced by Knuth, Morris, and Pratt in their historical paper on fast pattern matching. It leads to an algorithm that requires more preprocessing but is more efficient than the original Boyer-Moore's algorithm. We formalize the notion of Boyer-Moore automaton and we give an efficient building algorithm. Also, bounds on the number of states are presented, and the concept of potential of a transition is introduced to improve the worst-and average-case behavior of these machines. We show that looking at the rightmost unknown character, as suggested by Knuthet al., is not necessarily optimal.R. A. Baeza-Yates gratefully acknowledges the support of Grant C-11001 from Fundación Andes, and C. Choffrut gratefully acknowledges the support of the PRC Mathématiques et Informatique.     

Complementing two-way finite automata

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, for a given automaton (2dfa) with n states, we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a stronger result, by presenting an equivalent 4n-state 2dfa that always halts. For the nondeterministic case, using a variant of inductive counting, we show that the complement of a unary language, accepted by an n-state two-way automaton (2nfa), can be accepted by an O(n⁸)-state 2nfa. Here we also make 2nfa’s halting. This allows the simulation of unary 2nfa’s by probabilistic Las Vegas two-way automata with O(n⁸) states.     

Breaking substitution cyphers using stochastic automata

Let Λ be a finite plaintext alphabet and V be a cypher alphabet with the same cardinality as Λ. In all one-to-one substitution cyphers, there exists the property that each element in V maps onto exactly one element in Λ and vice versa. This mapping of V onto Λ is represented by a function T*, which maps any v∈V onto some λ∈Λ (i.e., T*(v)=λ). The problem of learning the mapping of T* (or its inverse (T *)^-1) by processing a sequence of cypher text is discussed. The fastest reported method to achieve this is a relaxation scheme that utilizes the statistical information contained in the unigrams and trigrams of the plaintext language. A new learning automaton solution to the problem called the cypher learning automaton (CLA) is given. The proposed scheme is fast, and the advantages of the scheme in terms of time and space requirements over the relaxation method have been listed. Simulation results comparing both cypher-breaking techniques are presented     

A note on emptiness for alternating finite automata with a one-letter alphabet

We present a new proof of PSPACE-hardness of the emptiness problem for alternating finite automata with a singleton alphabet. This result was shown by Holzer (1995) who used a proof relying on a series of reductions from several papers. The new proof is simple, direct and self-contained.     

Computability by finite automata and pisot bases

We prove that the function of normalization in base , which maps any -representation of a real number onto its -development, obtained by a greedy algorithm, is a function computable by a finite automaton over any alphabet if and only if is a Pisot number.Christiane Frougny was supported in part by the PRC Mathématiques et Informatique of the Ministère de la Recherche et de l'Espace.     

Learning two-tape automata from queries and counterexamples

We investigate the learning problem of two-tape deterministic finite automata (2-tape DFAs) from queries and counterexamples. Instead of accepting a subset of ∑*, a 2-tape DFA over an alphabet ∑ accepts a subset of ∑* × ∑*, and therefore, it can specify a binary relation on ∑*. In [3] Angluin showed that the class of deterministic finite automata (DFAs) is learnable in polynomial time from membership queries and equivalence queries, namely, from a minimally adequate teacher (MAT). In this article we show that the class of 2-tape DFAs is learnable in polynomial time from MAT. More specifically, we show an algorithm that, given any languageL accepted by an unknown 2-tape DFAM, learns from MAT a two-tape nonde-terministic finite automaton (2-tape NFA)M′ acceptingL in time polynomial inn andl, wheren is the size ofM andl is the maximum length of any counterexample provided during the learning process. This work was supported in part by Grants-in-Aid for Scientific Research No. 04229105 from the Ministry of Education, Science, and Culture, Japan.