Finite automata for testing composition-based reconstructibility of sequences

Symbolic sequences uniquely reconstructible from all their substrings of length k compose a regular factorial language. We thoroughly characterize this language by its minimal forbidden words, and explicitly build up a deterministic finite automaton that accepts it. This provides an efficient on-line algorithm for testing the unique reconstructibility of the sequences.     

A characterization of (regular) circular languages generated by monotone complete splicing systems

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for monotone complete systems, which are finite circular splicing systems with rules of a simpler form. We show that a circular language L is generated by a monotone complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by monotone complete systems. We can also decide whether the language generated by a monotone complete system is regular. Finally, we point out that monotone complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule of a specific type is allowed. From our results on monotone complete systems, it follows that finite simple systems generate a class of languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.     

d-Words and d-languages

Let X be a finite alphabet containing more than one letter. A d-primitive word u overX is a non-overlapping word in the sense that no proper prefix of u is a suffix of u. D(1) is the set of all d-primitive words over X and D is the set of all positive powers of all words in D (1). Every language in D will be called a d-language. In this paper, we study some algebraic properties of d-primitive words and d-languages relative to formal language theory and codes. We show that there are infinitely many cyclic-square-free words over alphabet with three letters. A characterization of three elements codes in D (1) is obtained and we prove that every regular component in D (1) is either a prefix code or a suffix code. Received: 22 September 1997 / 7 January 1998     

Unavoidable Sets of Words of Uniform Length

A set of words X over a finite alphabet A is said to be unavoidable if all but finitely many words in A* have a factor in X. We examine the problem of calculating the cardinality of minimal unavoidable sets of words of uniform length; we correct an error in [8], state a conjecture offering a formula for the minimum size of these so called n-good sets for all values of n, and show that the conjecture is correct in an infinite number of cases.     

Constants and label-equivalence: A decision procedure for reflexive regular splicing languages

A structural characterization of reflexive splicing languages has been recently given in [P. Bonizzoni, C. De Felice, R. Zizza, The structure of reflexive regular splicing languages via Schützenberger constants, Theoretical Computer Science 334 (2005) 71-98] and [P. Bonizzoni, G. Mauri, Regular splicing languages and subclasses, Theoretical Computer Science 340 (2005) 349-363] showing surprising connections between long standing notions in formal language theory, the syntactic monoid and Schützenberger constant and the splicing operation.In this paper, we provide a procedure to decide whether a regular language is a reflexive splicing language, based on the above-mentioned characterization that is given in terms of a finite set of constants for the language. The procedure relies on the notion of label-equivalence that induces a finite refinement of the syntactic monoid of a regular language L. A finite set of representatives for label-equivalent classes of constant words in L is defined and it is proved that such a finite set provides the splice sites of splicing rules generating language L.     

On the Power of Las Vegas for One-Way Communication Complexity, OBDDs, and Finite Automata

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.(i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.(ii) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.(iii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the square root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.     

Regular Model Checking Using Inference of Regular Languages

Regular model checking is a method for verifying infinite-state systems based on coding their configurations as words over a finite alphabet, sets of configurations as finite automata, and transitions as finite transducers. We introduce a new general approach to regular model checking based on inference of regular languages. The method builds upon the observation that for infinite-state systems whose behaviour can be modelled using length-preserving transducers, there is a finite computation for obtaining all reachable configurations up to a certain length n. These configurations are a (positive) sample of the reachable configurations of the given system, whereas all other words up to length n are a negative sample. Then, methods of inference of regular languages can be used to generalize the sample to the full reachability set (or an overapproximation of it). We have implemented our method in a prototype tool which shows that our approach is competitive on a number of concrete examples. Furthermore, in contrast to all other existing regular model checking methods, termination is guaranteed in general for all systems with regular sets of reachable configurations. The method can be applied in a similar way to dealing with reachability relations instead of reachability sets too.     

On condition/event systems with discrete state realizations

This paper introduces condition/event (C/E) systems as a class of continuous-time discrete event dynamic systems (DEDS) with two types of discrete-valued input and output signals:condition signals andevent signals. In applications such as discrete control, C/E systems provide an intuitive continuous-time modeling framework amenable to block diagram representation. In this paper we consider C/E systems with discrete state realizations, and study the relationship between continuous-time C/E systems and untimed models of their sequential inputoutput behavior called C/E languages. We show that C/E systems with discrete state realizations are necessarilytime-change invariant (Theorem 3.1), which means the ensemble of admissible continuous-time input-output behaviors is completely characterized by the C/E language for the system (Theorem 4.1). It is also shown that deterministic C/E systems with discrete state realizations are necessarily discrete-time (clocked) systems (Corollary 3.1), and that finite discrete state realizations exist for a C/E system only if its related C/E language has a finite state generator (Theorem 4.2). Finally, we develop equivalent discrete-state realizations for C/E systems resulting from cascade and feedback interconnections. The paper concludes with a discussion of several directions for future research.Please direct correspondence concerning this paper to B.H. Krogh at the above address.     

One-Unambiguous Regular Languages

The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic meta-language for the definition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions, although only regular expressions that denote words unambiguously, in the sense of the ISO standard, are allowed. In general, a word that is denoted by a regular expression is witnessed by a sequence of occurrences of symbols in the regular expression that match the word. In an unambiguous regular expression as defined by Booket al.(1971,IEEE Trans. Comput.C-20(2), 149–153), each word has at most one witness. But the SGML standard also requires that a witness be computed incrementally from the word with a one-symbol lookahead; we call such regular expressions 1-unambiguous. A regular language is a 1-unambiguouslanguage if it is denoted by some 1-unambiguous regular expression. We give a Kleene theorem for 1-unambiguous languages and characterize 1-unambiguous regular languages in terms of structural properties of the minimal deterministic automata that recognize them. As a result we are able to prove the decidability of whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous regular expression in worst-case optimal time.     

DNA Computing Based on Splicing: The Existence of Universal Computers

We prove that splicing systems with finite components and certain controls on their work are computationally complete (they can simulate any Turing Machine); moreover, there are universal splicing systems (systems with all components fixed which can simulate any given splicing system, when an encoding of the particular system is added—as a program—to the universal system). Splicing systems are based on the splicing operation which is a model for DNA recombination. Informally, a prefix of a word is catenated to a suffix of another word, thus yielding a (possibly) new word. Cutting occurs at specific sites which correspond to specific sequences in DNA strands as they can be recognized by restriction enzymes. When no additional control is assumed, splicing systems with finitely many starting words (axioms) and finitely many splicing rules are known to characterize only regular languages (those recognized by finite automata ). However, when a splicing rule is allowed to be used (1)\hskip .5em only in the presence of certain symbols (``catalyst') or (2)\hskip .5em only in the absence of certain symbols (``inhibitors'), then we can characterize the recursively enumerable languages (recognized by Turing Machines ); the same result is obtained when counting the number of copies of (some of) the words used. From the proofs, we also infer the existence of universal (hence programmable) splicing systems. Received August 1997, and in final form March 1998.     

Fooling a two-way nondeterministic multihead automaton with reversal number restriction

Summary We define a language L and show that it can be recognized by no two-way nondeterministic sensing multihead finite automaton with n ^a reversal bound, where n is the length of input words, and 1/3>a>0 is a real number. Since L is recognized by a two-way deterministic two-head finite automaton working in linear time we obtain, for two-way finite automata, that time, reading heads, and nondeterminism as resources cannot compensate for the reversal number restriction.This work was supported as a part of the SPZV I-1-5/8 grant     

Regular model checking for LTL(MSO)

Regular model checking is a form of symbolic model checking for parameterized and infinite-state systems whose states can be represented as words of arbitrary length over a finite alphabet, in which regular sets of words are used to represent sets of states. We present LTL(MSO), a combination of the logics monadic second-order logic (MSO) and LTL as a natural logic for expressing the temporal properties to be verified in regular model checking. In other words, LTL(MSO) is a natural specification language for both the system and the property under consideration. LTL(MSO) is a two-dimensional modal logic, where MSO is used for specifying properties of system states and transitions, and LTL is used for specifying temporal properties. In addition, the first-order quantification in MSO can be used to express properties parameterized on a position or process. We give a technique for model checking LTL(MSO), which is adapted from the automata-theoretic approach: a formula is translated to a buchi regular transition system with a regular set of accepting states, and regular model checking techniques are used to search for models. We have implemented the technique, and show its application to a number of parameterized algorithms from the literature.     

On the regularity of circular splicing languages: a survey and new developments

Circular splicing has been introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we focus on the relationship between regular circular languages and languages generated by finite circular splicing systems. We survey the known results towards a characterization of the intersection between these two classes and provide new contributions on the open problem of finding this characterization. First, we exhibit a non-regular circular language generated by a circular simple system thus disproving a known result in this area. Then we give new results related to a restrictive class of circular splicing systems, the marked systems. Precisely, we review in a graph theoretical setting the recently obtained characterization of marked systems generating regular circular languages. In particular, we define a slight variant of marked systems, that is the g-marked systems, and we introduce the graph associated with a g-marked system. We show that a g-marked system generates a regular circular language if and only if its associated graph is a cograph. Furthermore, we prove that the class of g-marked systems generating regular circular languages is closed under a complement operation applied to systems. We also prove that marked systems with self-splicing generate only regular circular languages.     

Comparing Verboseness for Finite Automata and Turing Machines

A language is called (m,n)-verbose if there exists a Turing machine that enumerates for any n words at most m possibilities for their characteristic string. This notion is compared with (m,n)-fa-verboseness, where instead of a Turing machine a finite automaton is used. By use of a new diagonalisation method, where finite automata trick Turing machines, it is shown that all (m,n)-verbose languages are (h,k)-verbose iff all (m,n)-fa-verbose languages are (h,k)-fa-verbose. In other words, Turing machines and finite automata behave exactly the same way with respect to inclusion of verboseness classes. This identical behaviour implies that the nonspeedup theorem also holds for finite automata. As an application of the theoretical framework, a lower bound is derived on the number of bits that need to be communicated to finite automata protocol checkers for nonregular protocols.     

On spiking neural P systems

This work deals with several aspects concerning the formal verification of SN P systems and the computing power of some variants. A methodology based on the information given by the transition diagram associated with an SN P system is presented. The analysis of the diagram cycles codifies invariants formulae which enable us to establish the soundness and completeness of the system with respect to the problem it tries to resolve. We also study the universality of asynchronous and sequential SN P systems and the capability these models have to generate certain classes of languages. Further, by making a slight modification to the standard SN P systems, we introduce a new variant of SN P systems with a special I/O mode, called SN P modules, and study their computing power. It is demonstrated that, as string language acceptors and transducers, SN P modules can simulate several types of computing devices such as finite automata, a-finite transducers, and systolic trellis automata.     

On a generalization of Christoffel words: epichristoffel words

Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the extensions of these sequences over a k-letter alphabet, with k≥3, is the episturmian sequences, which generalizes a construction of Sturmian sequences using the palindromic closure operation. There exists a finite version of the Sturmian sequences called the Christoffel words. They have been known since the works of Christoffel and have interested many mathematicians. In this paper, we introduce a generalization of Christoffel words for an alphabet with 3 letters or more, using the episturmian morphisms. We call them the epichristoffel words. We define this new class of finite words and show how some of the properties of the Christoffel words can be generalized naturally or not for this class.     

Non-preserving splicing with delay

In this paper we investigate H systems with strongly non-preserving splicing that exhibit a new feature, namely delay, and introduce a variant of the H system that lies between H systems with strongly non-preserving splicing and H systems with non-reflexively evolving splicing. Informally, the new splicing system behaves as follows: (1) each splicing step is exactly a splicing step in a system with non-reflexively evolving splicing; and (2) the generated language is obtained exactly as in a system with strongly non-preserving splicing. For both H systems with non-reflexively evolving and non-preserving splicing we have a remarkable jump in power between systems with a finite but arbitrarily large delay, and those with infinite delay. The first can generate non-context-free languages, whereas the second do not get beyond the regular limit. Moreover, H systems with null delay generate all recursively enumerable languages.     

Creating word-level language models for large-vocabulary handwriting recognition

We discuss development of a word-unigram language model for online handwriting recognition. First, we tokenize a text corpus into words, contrasting with tokenization methods designed for other purposes. Second, we select for our model a subset of the words found, discussing deviations from an N-most-frequent-words approach. From a 600-million-word corpus, we generated a 53,000-word model which eliminates 45% of word-recognition errors made by a character-level-model baseline system. We anticipate that our methods will be applicable to offline recognition as well, and to some extent to other recognizers, such as speech recognizers and video retrieval systems. Received: November 1, 2001 / Revised version: July 22, 2002     

Supervisory Control of a Class of Concurrent Discrete Event Systems Under Partial Observation

In this paper, we study supervisory control of a class of discrete event systems with simultaneous event occurrences, which we call concurrent discrete event systems, under partial observation. The behavior of the system is described by a language over the simultaneous event set. First, we prove that L_m(G)-closure, controllability, observability, and concurrent well-posedness of a specification language are necessary and sufficient conditions for the existence of a nonblocking supervisor. Next, we synthesize a supervisor that achieves the infimal closed, controllable, observable, and concurrently well-posed superlanguage of a specification language. Finally, we synthesize a supervisor that achieves a maximal closed, controllable, observable, and concurrently well-posed sublanguage of a closed specification language.     

On the supervisory control of multi-agent product systems: Controllability properties

Multi-Agent (MA) product systems were defined in [I. Romanovski, P.E. Caines, On the supervisory control of multi-agent products systems, IEEE Trans. Automated Control 51(5) (2006)] where the notion of an MA controllable vector language specification was introduced and necessary and sufficient conditions for the existence of a supervisor for such a specification were derived. In this paper we present a set of results about the controllability of an MA product system and its components. In particular, we obtain an algorithm for finding the infimal MA-controllable superlanguage inf_MA(K) of a given vector (language) specification K w.r.t. an MA product system G and establish that in fact . It is proven that there is an algorithmic procedure for the recursive construction of MA supervisors when additional automata are added to a system via the MA product. Controllability properties of component structures (such as standard controllability and MA controllability of projections) are considered. Several examples are given to illustrate the results of the paper.