A unifying approach to picture grammars

Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using an approach, whereby the right part of a rule is formalized by means of a finite set of permitted tiles. We focus on a simple type of tiling, named regional, and define the corresponding regional tile grammars. They include both Siromoney?s (or Matz?s) Kolam grammars and their generalization by Pr?ša, as well as Drewes?s grid grammars. Regionally defined pictures can be recognized with polynomial-time complexity by an algorithm extending the CKY one for strings. Regional tile grammars and languages are strictly included into our previous tile grammars and languages, and are incomparable with Giammarresi-Restivo tiling systems (or Wang systems).     

Some Bounds on the Computational Power of Piecewise Constant Derivative Systems

We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the (d-2) th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is Σ _d-2 -complete. Received May 1997, and in final form May 1998.     

(DNA) computing by carving

 Inspired by the experiments reported recently in the emerging area of DNA computing, we consider a somewhat unusual type of a computation strategy: generate a (large) set of candidate solutions of a problem, then remove the non-solutions such that what remains is the set of solutions. We call this a computation by carving. This leads both to a speculation with possible important consequences and to interesting theoretical computer science (formal language) questions. The speculation is that in this way we can “compute” non-recursively enumerable languages, because the family of recursively enumerable languages is not closed under complementation. The formal language theory questions concern sequences of languages with certain regularities, needed as languages to be extracted from the total language of candidate solutions of a problem. Specifically, we consider sequences of languages obtained by starting from a given regular language and iteratively applying to it a given finite state sequential transducer (a gsm). Computing by carving with respect to such a sequence of languages can identify all context-sensitive languages and can also lead to non-recursively enumerable languages (but not all recursively enumerable languages can be obtained in this way). In practical circumstances, the carving process should be finite, hence, in general, approximations of the desired language are obtained. We also briefly discuss this aspect.     

Automata for Epistemic Temporal Logic with Synchronous Communication

We suggest that developing automata theoretic foundations is relevant for knowledge theory, so that we study not only what is known by agents, but also the mechanisms by which such knowledge is arrived at. We define a class of epistemic automata, in which agents’ local states are annotated with abstract knowledge assertions about others. These are finite state agents who communicate synchronously with each other and information exchange is ‘perfect’. We show that the class of recognizable languages has good closure properties, leading to a Kleene-type theorem using what we call regular knowledge expressions. These automata model distributed causal knowledge in the following way: each agent in the system has a partial knowledge of the temporal evolution of the system, and every time agents synchronize, they update each other’s knowledge, resulting in a more up-to-date view of the system state. Hence we show that these automata can be used to solve the satisfiability problem for a natural epistemic temporal logic for local properties. Finally, we characterize the class of languages recognized by epistemic automata as the regular consistent languages studied in concurrency theory.     

Unique Decipherability in the Monoid of Languages: An Application of Rational Relations

We attack the problem of deciding whether a finite collection of finite languages is a code, that is, possesses the unique decipherability property in the monoid of finite languages. We investigate a few subcases where the theory of rational relations can be employed to solve the problem. The case of unary languages is one of them and as a consequence, we show how to decide for two given finite subsets of nonnegative integers, whether they are the nth root of a common set, for some n≥1. We also show that it is decidable whether a finite collection of finite languages is a Parikh code, in the sense that whenever two products of these sets are commutatively equivalent, so are the sequences defining these products. Finally, we consider a nonunary special case where all finite sets consist of words containing exactly one occurrence of the specific letter.     

Timed substitutions for regular signal-event languages

In the classical framework of formal languages, a refinement operation is modeled by a substitution and an abstraction by an inverse substitution. These mechanisms have been widely studied, because they describe a change in the specification level, from an abstract view to a more concrete one, or conversely. For timed systems, there is up to now no uniform notion of substitution. In this paper, we study timed substitutions in the general framework of signal-event languages, where both signals and events are taken into account. We prove that regular signal-event languages are closed under substitution and inverse substitution. To obtain these results, we use in a crucial way a “well known” result: regular signal-event languages are closed under intersection. In fact, while this result is indeed easy for languages defined by Alur and Dill’s timed automata, it turns out that the construction is much more tricky when considering the most involved model of signal-event automata. We give here a construction working on finite and infinite signal-event words and taking into account signal stuttering, unobservability of zero-duration τ-signals and Zeno runs. Note that if several constructions have been proposed in particular cases, it is the first time that a general construction is provided.     

A denotational theory of synchronous reactive systems

In this paper, systems which interact permanently with their environments are considered. Such systems are encountered, for instance, in real-time control or signal processing systems, C³-systems, and man-machine interfaces, to mention just a few cases. The design and implementation of such systems require a concurrent programming language which can be used to verify and synthesize the synchronization mechanisms, and to perform transformations of the concurrent source code to match a particular target architecture. Synchronous languages are convenient tools for such a purpose: they rely on the assumptions that: (1) internal actions of synchronous systems are instantaneous, and (2) communication with the environment is performed via instantaneous flashes involving some external stimuli. In this paper, we present a mathematical model of synchronous languages and illustrate its use on the language. This model is denotational, and encompasses both relational and functional styles of specification. It allows us to answer fundamental questions related to synchronous languages, such as “what are the basic constructions which should be provided by such languages?”     

The finiteness of synchronous,tabled picture languages is decidable

In the present paper, synchronous, tabled chain code picture systems based on Lindenmayer systems (sT0L system) are studied with respect to the finiteness of their picture languages. The finiteness is proved to be decidable. Additionally, a method is given for deciding whether or not an sT0L system generates a finite picture language.     

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.     

Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables

Abstract. We consider generalized first-order sentences over < using both ordinary and modular quantifiers. It is known that the languages definable by such sentences are exactly the regular languages whose syntactic monoids contain only solvable groups. We show that any sentence in this logic is equivalent to one using three variables only, and we prove that the languages expressible with two variables are those whose syntactic monoids belong to a particular pseudovariety of finite monoids, namely the wreath product of the pseudovariety DA (which corresponds to the languages definable by ordinary first-order two-variable sentences) with the pseudovariety of finite solvable groups. This generalizes earlier work of Thérien and Wilke on the expressive power of two-variable formulas in which only ordinary quantifiers are present. If all modular quantifiers in the sentence are of the same prime modulus, this provides an algorithm to decide if a regular language has such a two-variable definition.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

HyperFile: A data and query model for documents

Non-quantitative information such as documents and pictures pose interesting new problems in the database world. Traditional data models and query languages do not provide appropriate support for this information. Such data are typically stored in file systems, which do not provide the security, integrity, or query features of database management systems. The hypertext model has emerged as a good interface to this information; however,finding information using hypertext browsing does not scale well. We developed a query interface that serves as an extension of the browsing model of hypertext systems. These queries minimize the repeated user interactions required to locate data in a standard hypertext system. HyperFile is a prototype data server interface. In this article, we describe HyperFile, including a number of issues such as query generation, query processing, and indexing.     

Can Finite Samples Detect Singularities of Real-Valued Functions?

Commonly, in learning theory, the task of the learner is to identify an unknown target object. We consider a variant of this basic task in which the learner is required only to decide whether the unknown target has a certain property. We allow an infinite learning process in which the learner is required to eventually arrive at the correct answer. We say that a problem for which such a learning algorithm exists is Decidable In the Limit (DIL). We analyze the class of DIL problems and provide a necessary and sufficient condition for the membership of a decision problem in this class. We offer an algorithm for any DIL problem, and apply it to several types of learning tasks. We introduce an extension of the usual Inductive Inference learning model—Inductive Inference with a Cheating Teacher. In this model the teacher may choose to present to the learner, not only a language belonging to the agreed-upon family of languages, but also an arbitrary language outside this family. In such a case we require that the learner will be able to eventually detect the faulty choice made by the teacher. We show that such a strong type of learning is possible, and there exist learning algorithms that will fail only on arbitrarily small sets of faulty languages. Furthermore, if an a priori probability distribution P , according to which f is being chosen, is available to the algorithm, then it can be strengthened into a finite algorithm. More precisely, for many distributions P , there exists a polynomial function, l , such that, for every 0 < δ < 1 , there is an algorithm using at most l(log(δ)) many probes that succeeds on more than (1-δ) of the f 's (as measured by P ). We believe that the new approach presented here will be found useful for many further applications. Received February 14, 1997; revised July 6, 1997, and July 18, 1997.     

Polynomial closure and unambiguous product

This article is a contribution to the algebraic theory of automata, but it also contains an application to Büchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL ₀ a ₁ L ₁ ···a _nL_n, where thea _i’s are letters and theL _i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.     

Well limit behaviors of term rewriting systems

The limit behaviors of computations have not been fully explored. It is necessary to consider such limit behaviors when we consider the properties of infinite objects in computer science, such as infinite logic programs, the symbolic solutions of infinite polynomial equations. Usually, we can use finite objects to approximate infinite objects, and we should know what kinds of infinite objects are approximable and how to approximate them effectively. A sequence {R _k : k ε ω} of term rewriting systems has the well limit behavior if under the condition that the sequence has the set-theoretic limit or the distance-based limit, the sequence {Th(R _k ): k ε ω} of corresponding theoretic closures of R _k has the set-theoretic or distance-based limit, and lim _k→∞ Th(R _k ) is equal to the theoretic closure of the limit of {R _k : k ε ω}. Two kinds of limits of term rewriting systems are considered: one is based on the set-theoretic limit, the other is on the distance-based limit. It is proved that given a sequence {R _k : κ ε ω} of term rewriting systems R _k , if there is a well-founded ordering ≺ on terms such that every R _k is ≺-well-founded, and the set-theoretic limit of {R _k : κ ε ω} exists, then {R _k : κ ε ω} has the well limit behavior; and if (1) there is a well-founded ordering ≺ on terms such that every R _k is ≺-well-founded, (2) there is a distance d on terms which is closed under substitutions and contexts and (3) {R _k : k ε ω} is Cauchy under d then {R _κ: κ ε ω} has the well limit behavior. The results are used to approximate the least Herbrand models of infinite Horn logic programs and real Horn logic programs, and the solutions and Gr?bner bases of (infinite) sets of real polynomials by sequences of (finite) sets of rational polynomials.     

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.     

Two-way automata and length-preserving homomorphisms

Closure underlength-preserving homomorphisms is interesting because of its similarity tonondeterminism. We give a characterization of NP in terms of length-preserving homomorphisms and present related complexity results. However, we mostly study the case of two-way finite automata: Let l.p.hom[n state 2DFA] denote the class of languages that are length-preserving homomorphic images of languages recognized byn-state 2DFAs. We give a machine characterization of this class. We show that any language accepted by ann-state two-wayalternating finite automaton (2AFA), or by a l-pebble 2NFA, belongs to l.p.hom[O(n ²) state 2DFA]. Moreover, there are languages in l.p.hom[n state 2DFA] whose smallest accepting 2NFA has at leastc ⁿ states (for some constantc > 1). So for two-way finite automata, the closure under length-preserving homomorphisms is much more powerful than nondeterminism. We disprove two conjectures (of Meyer and Fischer, and of Chrobak) about the state-complexity of unary languages. Finally, we show that the equivalence problems for 2AFAs (resp. 1-pebble 2NFAs) are in PSPACE, and that the equivalence problem for 1-pebble 2AFAs is in ExpSPACE (thus answering a question of Jiang and Ravikumar); it was known that these problems are hard in these two classes. We also give a new proof that alternating 1-pebble machines recognize only regular languages (which was first proved by Goralčíket al.). This research was supported in part by N.S.F. Grant DMS 8702019.     

Temporal-logic property preservation under Z refinement

Formal specification languages such as Z, B and VDM are used in the incremental development of abstract specifications (suitable for establishing required properties) to more concrete specifications (resembling the final implementation). This incremental development process, known as refinement, preserves all observable properties of the original abstract specification. Recent research has looked at applying temporal-logic model checking to such specification languages. While this assists in the establishment of properties of the abstract specification, temporal-logic properties typically refer to state variables which are regarded as non-observable. Hence, such properties are not guaranteed to be preserved by refinement. This paper investigates the classes of temporal-logic properties which are preserved by refinement, and for some of those properties that are not preserved in general, the restrictions on the refinement process under which they are preserved. Results are presented for the temporal logics LTL, CTL and the μ-calculus and the formal specification language Z. They apply equally, however, to related formal specification languages such as B and VDM.