On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

Sound and Complete Qualitative Simulation Needs “Quantitative” Filtering

The AI methodology of qualitative reasoning furnishes useful tools to scientists and engineers who need to deal with incomplete system knowledge during design, analysis, or diagnosis tasks. Qualitative simulators have a theoretical soundness guarantee; they cannot overlook any concrete equation implied by their input. On the other hand, the basic qualitative simulation algorithms have been shown to suffer from the incompleteness problem; they may allow non-solutions of the input equation to appear in their output. The question of whether a simulator with purely qualitative input which never predicts spurious behaviors can ever be achieved by adding new filters to the existing algorithm has remained unanswered. In this paper, we show that, if such a sound and complete simulator exists, it will have to be able to handle numerical distinctions with such a high precision that it must contain a component that would better be called a quantitative, rather than qualitative reasoner. This is due to the ability of the pure qualitative format to allow the exact representation of the members of a rich set of numbers.     

Language Simplification through Error-Correcting and Grammatical Inference Techniques

In many language processing tasks, most of the sentences generally convey rather simple meanings. Moreover, these tasks have a limited semantic domain that can be properly covered with a simple lexicon and a restricted syntax. Nevertheless, casual users are by no means expected to comply with any kind of formal syntactic restrictions due to the inherent spontaneous nature of human language. In this work, the use of error-correcting-based learning techniques is proposed to cope with the complex syntactic variability which is generally exhibited by natural language. In our approach, a complex task is modeled in terms of a basic finite state model, F, and a stochastic error model, E. F should account for the basic (syntactic) structures underlying this task, which would convey the meaning. E should account for general vocabulary variations, word disappearance, superfluous words, and so on. Each natural user sentence is thus considered as a corrupted version (according to E) of some simple sentence of L(F). Adequate bootstrapping procedures are presented that incrementally improve the structure of F while estimating the probabilities for the operations of E. These techniques have been applied to a practical task of moderately high syntactic variability, and the results which show the potential of the proposed approach are presented.     

Property preserving abstractions for the verification of concurrent systems

We study property preserving transformations for reactive systems. The main idea is the use of simulations parameterized by Galois connections (, ), relating the lattices of properties of two systems. We propose and study a notion of preservation of properties expressed by formulas of a logic, by a function mapping sets of states of a systemS into sets of states of a systemS'. We give results on the preservation of properties expressed in sublanguages of the branching time -calculus when two systemsS andS' are related via (, )-simulations. They can be used to verify a property for a system by verifying the same property on a simpler system which is an abstraction of it. We show also under which conditions abstraction of concurrent systems can be computed from the abstraction of their components. This allows a compositional application of the proposed verification method.This is a revised version of the papers [2] and [16]; the results are fully developed in [28].This work was partially supported by ESPRIT Basic Research Action REACT.Verimag is a joint laboratory of CNRS, Institut National Polytechnique de Grenoble, Université J. Fourier and Verilog SA associated with IMAG.     

Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Linear Interval Tolerance Problem and Linear Programming Techniques

In this paper, we consider the linear interval tolerance problem, which consists of finding the largest interval vector included in ([A], [b]) = {x R ⁿ | A [A], b [b], Ax = b}. We describe two different polyhedrons that represent subsets of all possible interval vectors in ([A], [b]), and we provide a new definition of the optimality of an interval vector included in ([A], [b]). Finally, we show how the Simplex algorithm can be applied to find an optimal interval vector in ([A], [b]).     

Cardinality problems of compositions of morphisms and inverse morphisms

We show that we cannot effectively determine whether, for morphisms _i , _i ,card (u ₀ ^–1 ₁) =card(u ₀ ^–1 ₁) for all wordsu over the domain alphabets of the two given compositions. In contrast it is decidable for morphisms _i , _i and a regular setR whethercard(u _{0 1} ^–1 ) =card(u _{0 1} ^–1 ) for all wordsu inR. In order to prove the latter result we give a characterization of the multiplicity functions of simple finite automata by using cardinalities of compositions of the above form. Finally, we show that the above decidability result also holds when we consider rational functions rather than morphisms.     

Redesign of a Systems Engineering Language: Formalisation of χ

This paper describes the redesign of a systems engineering language called . This is an engineering language designed to specify and analyse industrial systems. The main objective of this redesign was to enable mathematical reasoning about specifications. We discuss the original language, the requirements and design decisions, and the resulting syntax and semantics of the new version of , called . In particular, we elaborate on semantical aspects of s time model.     

One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers

The language of standard propositional modal logic has one operator ( or ), that can be thought of as being determined by the quantifiers or , respectively: for example, a formula of the form is true at a point s just in case all the immediate successors of s verify .This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of and .     

A Stylometric Analysis of Ya?ar Kemal’s ? nce Memed Tetralogy

We analyze four nce Memed novels of Yaar Kemal using six style markers: most frequent words, syllable counts, word type – or part of speech – information, sentence length in terms of words, word length in text, and word length in vocabulary. For analysis we divide each novel into five thousand word text blocks and count the frequencies of each style marker in these blocks. The style markers showing the best separation are most frequent words and sentence lengths. We use stepwise discriminant analysis to determine the best discriminators of each style marker. We then use these markers in cross validation based discriminant analysis. Further investigation based on multiple analysis of variance (MANOVA) reveals how the attributes of each style marker group distinguish among the volumes.     

Diagrammatic Analysis of Interval Linear Equations - Part I: Basic Notions and the One-Dimensional Case

Different solution sets for the interval linear system A x = b are characterized and classified using diagrammatic tools for interval analysis developed recently. In Part I, a thorough analysis of the basic, one-dimensional expression a x b is conducted, with the help of an appropriate interval space diagram, in which all the needed relations { , , , = } are directly representable. The solution sets are obtained with simple diagrammatic constructions, and are characterized by quotient sequences of a and b. A complete classification of all possible solution types is developed in this way, with various ways of visualizing the structure of the set of these types.     

A mechanical proof of Quadratic Reciprocity

We describe the use of the Boyer-Moore theorem prover in mechanically generating a proof of the Law of Quadratic Reciprocity: for distinct odd primes p and q, the congruences x ² q (mod p) and x ² p (mod q) are either both solvable or both unsolvable, unless pq3 (mod 4). The proof is a formalization of an argument due to Eisenstein, based on a lemma of Gauss. The input to the theorem prover consists of nine function definitions, thirty conjectures, and various hints for proving them. The proofs are derived from a library of lemmas that includes Fermat's Theorem and the Gauss Lemma.     

Regions-of-Interest and Spatial Layout for Content-Based Image Retrieval

To date most content-based image retrieval (CBIR) techniques rely on global attributes such as color or texture histograms which tend to ignore the spatial composition of the image. In this paper, we present an alternative image retrieval system based on the principle that it is the user who is most qualified to specify the query content and not the computer. With our system, the user can select multiple regions-of-interest and can specify the relevance of their spatial layout in the retrieval process. We also derive similarity bounds on histogram distances for pruning the database search. This experimental system was found to be superior to global indexing techniques as measured by statistical sampling of multiple users' satisfaction ratings.     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

The growth of configurations

The transition ruleF of a cellular automaton may sometimes be regarded as a rule of growth of a crystal from a seed. A study is made of the iterates,F,F ² .For certain one-dimensional growth rules, the limiting shapes of the crystals are computed, and an asymptotic formula for the size of the crystal as a function of time is obtained.