On parsing two-level grammars

Summary Making use of the fact that two-level grammars (TLGs) may be thought of as finite specification of context-free grammars (CFGs) with infinite sets of productions, known techniques for parsing CFGs are applied to TLGs by first specifying a canonical CFG G — called skeleton grammar — obtained from the cross-reference of the TLG G. Under very natural restrictions it can be shown that for these grammar pairs (G, G) there exists a 1 — 1 correspondence between leftmost derivations in G and leftmost derivations in G. With these results a straightforward parsing algorithm for restricted TLGs is given.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

Congruence separation of subsets of a monoid with applications to automata

LetF be a monoid and letA={A _i iI} be a set of disjoint subsets ofF, whereI is an index set. A congruence onF is calledA-separating if each of its congruence classes has a non-empty intersection with at most one A_i A The setC ^A of allA-separating congruences onF is a lower semi-lattice with respect to the partial ordering of congruence inclusion. A necessary and sufficient condition forC ^A to be a complete lattice is derived and, in that case, the unique maximalA-separating congruence is characterized. The relation ofA-separating congruences with automata is described. An example of the case that the unique maximalA-separating congruence on a free monoid exists is worked out, and its realization by an incompletely specified finite automaton is given.     

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.     

Finite canonical rewriting systems for congruences generated by concurrency relations

WhenC is a concurrency relation on alphabet , then ^*/= _C is a free partially commutative monoid. Here we show that it is decidable in polynomial time whether or not there exists a finite canonical rewriting systemR on such that the congruences _R ^* generated byR and = _C induced byC coincide. Further, in case such a systemR exists, one such system can be determined in polynomial time.     

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.     

Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems

This paper describes a unified variational theory for design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems for shape, nonshape, material and mechanical properties selection, as well as control problems. The concept of an adjoint system, the principle of virtual work and a Lagrangian-Eulerian formulation to describe the deformations and the design variations are used to develop a unified view point. A general formula for design sensitivity analysis is derived and interpreted for usual performance functionals. Analytical examples are utilized to demonstrate the use of the theory and give insights for application to more complex problems that must be treated numerically.Derivatives The comma notation for partial derivatives is used, i.e. G,_u = G/u. An upper dot represents material time derivative, i.e. ü = ²u/t². A prime implies derivative with respect to the time measured in the reference time-domain, i.e. u = du/d.     

Some results in tree automata

Summary The following three results concerning tree automata are presented in this paper. (1) Rounds has presented the following open problem: For every recognizable setR, can we construct a deterministic finite-state transformation recognizingR? We show that this is not possible, in fact, even for a local set. However, the following is true: For every recognizable setR there is an inverse projectionR effectively obtained such thatR is recognized by a deterministic finite-state transformation. (2) Martin and Vere in their study of tree automata leave open the question of whether Generalized Syntax Directed Transductions (GSDT's) are closed under Arden's transformation or Greibach's transformation, and conjecture that they are not. We prove that this conjecture is true. It is also shown that GSDT's are not closed under transformation to LR(k) grammars. (3) Peters and Ritchie have shown that if, in a grammar where the generative rules are context-free, there are recognition rules which are context-sensitive, the language recognized is still context-free. A tree-automata-oriented proof is given by Rounds. We show that a similar result holds also for right linear grammars, i.e., if the generative rules are right linear, then using context-sensitive rules for recognition, one can still recognize only regular languages. Some other related results concerning context-sensitive extensions of subclasses of context-free languages are also presented.This work was partially supported by NSF Grant GJ27, U.S. Army Research Office, Durham (DA-31-124 ARO(D)-98), and NSF Grant GS-2509.A present on leave at The Institute for Advanced Study, Princeton, N.J.     

A general proof rule for procedures in predicate transformer semantics

Summary A proof rule for the procedure call is derived for procedures with value, result and value-result parameters. It is extended to procedures with unrestricted global variables and to recursive procedures. Like D. Gries's proof rule, it is based on the substitution rule for assignment. However, it is more general and much simpler to apply. Assume that {U} S {V} has been proved about the procedure body S. The proof rule for determining whether a call establishes predicate E is based on finding an adaptation A satisfying AV E, where E is derived from E by some substitution of parameters.     

Viscous Lattices

Let E be an arbitrary space, and an extensive dilation of P(E) into itself, with an adjoint erosion . Then, the image [P(E)] of P(E) by is a complete lattice P where the sup is the union and the inf the opening of the intersection according to . The lattice L, named viscous, is not distributive, nor complemented. Any dilation on P(E) admits the same expression in L. However, the erosion in L is the opening according to of the erosion in P(E). Given a connection C on P(E) the image of C under turns out to be a connection C on L as soon as (C)eq C. Moreover, the elementary connected openings _x of C and _(x) are linked by the relation _(x) = _x. A comprehensive class of connection preverving closings is constructed. Two examples, binary and numerical (the latter comes from the heart imaging), prove the relevance of viscous lattices in interpolation and in segmentation problems.Jean Serra obtained the degree of Mining Engineer, in 1962 in Nancy, France, and in 1967 his Ph.D. for a work dealing with the estimation of the iron ore body of Lorraine by geostatistics. In cooperation with Georges Matheron, he laid the foundations of a new method, that he called Mathematical Morphology (1964). Its purpose was to describe quantitatively shapes and textures of natural phenomena, at micro and macro scales. In 1967, he founded with G. Matheron, the Centre de Morphologie Mathematique, at School of Mines of Paris, on the campus of Fontainebleau. Since this time, he has been working in this framework as a Directeur de Recherches. His main book is a two-volume treatise entitled Image Analysis and Mathematical Morphology (Ac. Press, 1982, 1988). He has been Vice President for Europe of the International Society for Stereology from 1979 to 1983. He founded the International Society for Mathematical Morphology in 1993, and was elected his first president. His achievements include several patents of devices for image processing, various awards and titles, such as the first AFCET award, in 1988, or Doctor Honoris Causa of the University of Barcelona (Spain) in 1993. He recently developed a new theory of segmentation, which is based on set connections (2001–2004), and currently works on colour image processing.     

A New Implementation of Automath

This paper presents aut, a modern Automath checker. It is a straightforward re-implementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT-68 and AUT-QE dialects of Automath. This program was written to restore a damaged version of Jutting's translation of Landau's Grundlagen. Some notable features: It is fast. On a 1 GHz machine it will check the full Jutting formalization (736 K of nonwhitespace Automath source) in 0.6 seconds. Its implementation of -terms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. The program can compile an Automath text into one big Automath single line-style -term. It outputs such a term using de Bruijn indices. (These -terms cannot be checked by modern systems like Coq or Agda, because the -typed -calculi of de Bruijn are different from the -typed -calculi of modern type theory.)The source of aut is freely available on the Web at the address .     

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.     

Context-free text grammars

A text is a triple=(, ₁, ₂) such that is a labeling function, and ₁ and ₂ are linear orders on the domain of ; hence may be seen as a word (, ₁) together with an additional linear order ₂ on the domain of . The order ₂ is used to give to the word (, ₁) itsindividual hierarchical representation (syntactic structure) which may be a tree but it may be also more general than a tree. In this paper we introducecontext-free grammars for texts and investigate their basic properties. Since each text has its own individual structure, the role of such a grammar should be that of a definition of a pattern common to all individual texts. This leads to the notion of ashapely context-free text grammar also investigated in this paper.     

Sur les générateurs algébriques et linéaires

Résumé Nous étudions certaines propriétés des générateurs algébriques et linéaires. Nous montrons que le langage algébrique E engendré par la grammaire: S aSbSc + d domine tous les langages algébriques par applications séquentielles fidèles. Nous en déduisons que pour tout langage algébrique L et tout générateur algébrique L, il existe une transduction rationnelle fonctionnelle et fidèle telle que L=(L). Ce résultat, qui n'est pas vérifié pour la famille, Lin, des langages algébriques linéaires, nous permet de montrer qu'aucun générateur algébrique n'appartient à la famille EDTOL. Enfin, nous établissons que si L est un générateur linéaire, L # est un générateur séquentiel pour Lin.

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Algebraic and linear generators

Summary We study some properties of algebraic and linear generators. We show that the algebraic language E generated by the grammar: S aSbSc + d dominates every algebraic language by faithful sequential mappings. We deduce that, for every algebraic language L and every algebraic generator L, there exists a faithful rational function such that L=(L). This result, which does not hold for the family of linear languages, permits us to show that no algebraic generator belongs to the family EDTOL. Also, we prove if L is a linear generator then L # is a sequential generator for Lin.

     

Defaults as restrictions on classical Hilbert-style proofs

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hubert-style proofs of a given logicL. We propose to describe default logic as a logic where the juxtaposition of default proofs is subordinate to a restriction condition . Hence a default logic is a pair (L, ) where properties of the logic , like compactness, can be interpreted through the restriction condition . Different default systems are then given a common characterization through a specific condition on the logicL. We also prove cumulativity for any default logic (L, ) by slightly modifying the notion of default proof. We extend, in fact, the language ofL in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on and called linear logics, which lie between Reiter's and ukaszewicz' versions of default logic.Work carried out in the framework of the agreement between Italian PT Administration and FUBLaforia, Université Paris VI Pierre et Marie Curie, 4 Place Jussieu,Tour 46, 75252 Paris, France     

A two-layer fuzzy pattern matching procedure for the evaluation of conditions involving vague quantifiers

In this note, we are interested in the evaluation of conditions of the form The value of attribute a for Q items of X is in F or more shortly Q items of X are F, where X is a set of items, Q denotes a possibly vague proportion (which may be linguistically expressed, e.g. most), F is a (possibly fuzzy) subset of the attribute domain of a. and where the available knowledge about the value a(x) of the attribute a for any item x may be imprecise or even vague. The evaluation is based on a fuzzy pattern matching procedure repeated two times. Such conditions may be encountered in queries addressed to an incomplete information data base or in the if-part of expert rules.     

Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires (N/d) parallel communication steps in the worst case. This is an improvement upon the (N/d ^3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in (N/logN) steps. The best previously known upper bound was (N). Our algorithm may be practical for smallN (up to about 2¹⁴ nodes).C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.     

Unification in partially commutative semigroups

Unification algorithms have been constructed for semigroups and commutative semigroups. This paper considers the intermediate case of partially commutative semigroups. We introduce classesN and of such semigroups and justify their use. We present an equation-solving algorithm for any member of the classN. This algorithm is relative to having an algorithm to determine all non-negative solutions of a certain class of diophantine equations of degree 2 which we call -equations. The difficulties arising when attempting to solve equations in members of the class are discussed, and we present arguments that strongly suggest that unification in these semigroups is undecidable.     

On the law (a·b'')''=b+a''·b'' in de Morgan algebras and orthomodular lattices

This paper's goal is to prove that the only de Morgan algebras and the only orthomodular lattices in which the law (a·b)=b+a·b, posed by Charles Elkan in [1] holds, are those that are boolean algebras. That is, that among both families of de Morgan algebras and orthomodular lattices, Elkan's formula is only characteristic of boolean algebras.Authors are actually in debt with an anonymous referee that helped them to considerably improve the first version of this paper.This paper has been partially supported by CICYT(Spain) under project TIC2000-1420     

Obvious inferences

The notion of obvious inference in predicate logic is discussed from the viewpoint of proof-checker applications in logic and mathematics education. A class of inferences in predicate logic is defined and it is proposed to identify it with the class of obvious logical inferences. The definition is compared with other approaches. The algorithm for implementing the obviousness decision procedure follows directly from the definition.