Showing papers on "Completeness (order theory) published in 1997"

A unifying construction of orthonormal bases for system identification

Abstract: This paper develops a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of many previously studied orthonormal bases since the common FIR and recently popular Laguerre and two-parameter Kautz model structures are restrictive special cases of the construction presented here. However, in contrast to these special cases, the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject. Results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided.

On the incompleteness of the weil-petersson metric along degenerations of calabi-yau manifolds

Abstract: The classical Weil-Petersson metric on the Teichmuller space of compact Riemann surfaces is a Kahler metric, which is complete only in the case of elliptic curves [Wo]. It has a natural generalization to the deformation spaces of higher dimansional polarized Kahler-Einstein manifolds. It is still Kahler, and in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out to coincide with the Bergman metric of the Hermitian symmetric period domain, hence is in fact “complete” Kahler-Einstein [Sc]. The completeness is an important property for differential geometric reason. Motivated by the above examples, one may naively think that the completeness of the Weil-Petersson metric still holds true for general Calabi-Yau manifolds (compact Kahler manifolds with trivial canonical bundle). However, explicit calculation done by physicists (eg. Candelas et al. [Ca] for some special nodal degenerations of Calabi-Yau 3-folds) indicated that this may not always be the case. The notion of completeness depends on the precise definition the “moduli space”. However, through our analysis, it would become clear that the Weil-Petersson metric is in general incomplete if one sticks

On the parameterized complexity of short computation and factorization

Abstract: A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,BDHW,BFH, DEF,DF1-7,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in $k$ steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word $x$ can be derived in a grammar $G$ in $k$ steps, or whether a permutation has a factorization of length $k$ over a given set of generators. We show hardness and completeness for these problems for various levels of the $W$ hierarchy. In particular, we show that Short TM Computation is complete for $W[1]$ . This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes.

On the complexity of reasoning in Kleene algebra

Abstract: We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E/spl rarr/s=t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *-continuous Kleene algebra, if E contains only commutativity assumptions pq=qp, the problem is II/sub 1//sup 0/-complete; if E contains only monoid equations, the problem is II/sub 2//sup 0/-complete; for arbitrary equations E, the problem is II/sub 1//sup 1/-complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994).

Automatic construction and management of large open webs

Abstract: Many researchers have noted the problems associated with manually created or maintained hyperdocument links, and the consequent need for automated methods. A number of techniques have been applied to the problem, including pattern-matching, information retrieval, and natural language processing. This paper describes a system for the automatic detection and management of structural and referential links. The paper also addresses the issues of link-set soundness and completeness, open link management, and the particular problems engendered by large volatile hyperbases.

Completeness in Abstract Interpretation: A Domain Perspective

Abstract: Completeness in abstract interpretation is an ideal and rare situation where the abstract semantics is able to take full advantage of the power of representation of the underlying abstract domain In this paper, we develop an algebraic theory of completeness in abstract interpretation We show that completeness is an abstract domain property and we prove that there always exist both the greatest complete restriction and the least complete extension of any abstract domain, with respect to continuous semantic functions Under certain hypotheses, a constructive procedure for computing these complete domains is given These methodologies provide advanced algebraic tools for manipulating abstract interpretations, which can be fruitfully used both in program analysis and in semantics design

Regular Polytopes in Ordinary Space

Abstract: The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grunbaum—Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E 3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E 3 . The paper gives a complete classification of the discrete regular polytopes in ordinary space.

On the completeness of object-creating database transformation languages

Abstract: Object-oriented applications of database systems require database transformations involoving nonstandard functionalities such as set manipulation and object creation, that is, the introduction of new domain elements. To deal with thse functionalities, Abiteboul and Kanellakis [1989] introduced the “determinate” transformations as a generalization of the standard domain-preserving transformations. The obvious extensions of complete standard database programming languages, however, are not complete for the determinate transformations. To remedy this mismatch, the “constructive” transformations are proposed. It is shown that the constructive transformations are precisely the transformations that can be expressed in said extensions of complete standard languages. Thereto, a close correspondence between object creation and the construction of hereditarily finite sets is established.A restricted version of the main completeness result for the case where only list manipulations are involved is also presented.

Reducing the complexity of reductions

Abstract: We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a “Gap Theorem”, showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper:¶1. Does the “gap” between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0reductions?¶2. Does a uniform version of the isomorphism theorem hold?¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits?¶We answer all of these questions. In particular, we prove that the Berman—Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class \( {\cal C} \) closed under uniform TC0-computable many-one reductions, the following three theorems hold:¶1. If \( {\cal C} \) contains sets that are complete under a notion of reduction at least as strong as Dlogtime-uniform AC0[mod 2] reductions, then there are such sets that are not complete under (even non-uniform) AC0 reductions.¶2. The sets complete for \( {\cal C} \) under P-uniform AC0 reductions are all isomorphic under isomorphisms computable and invertible by P-uniform AC0circuits of depth-three.¶3. There are sets complete for \( {\cal C} \) under Dlogtime-uniform AC0 reductions that are not isomorphic under any isomorphism computed by (even non-uniform) AC0 circuits of depth two.¶To prove the second theorem, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.)

Discrete-time process algebra with empty process

Abstract: We introduce an ACP-style discrete-time process algebra with relative timing, that features the empty process. Extensions to this algebra are described, and ample attention is paid to the considerations and problems that arise when introducing the empty process. We prove time determinacy, soundness, completeness, and the axioms of standard concurrency. 1991 Mathematics Subject Classification: 68Q10, 68Q22, 68Q55. 1991 CR Categories: D.1.3, D.3.1, F.1.2, F.3.2.

Type and behaviour reconstruction for higher-order concurrent programs

Abstract: In this paper we develop a sound and complete type and behaviour inference algorithm for a fragment of CML (Standard ML with primitives for concurrency). Behaviours resemble terms of a process algebra and yield a concise representation of the communications taking place during execution; types are mostly as usual except that function ypes and ‘delayed communication types’ are labelled by behaviours expressing the communications that will take place if the function is applied or the delayed action is activated. The development of the present paper improves a previously published algorithm in achieving completeness as well as soundness; this is due to an alternative strategy for generalising over types and behaviours.

Completeness and categoricity: Frege, gödel and model theory

Abstract: Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Godel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Godel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory

A Unifying Construction of Orthonormal Bases for System Identification

Abstract: This paper develops a general and very simple construction for complete orthonormal bases for system identification. This con- struction provides a unifying formulation of many previously studied orthonormal bases since the common FIR and recently popular Laguerre and two-parameter Kautz model structures are restrictive special cases of the construction presented here. However, in contrast to these special cases, the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject. Results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided.

Powerset Residuated Algebras and Generalized Lambek Calculus

Abstract: We prove a representation theorem for (abstract) residuated algebras: each residuated algebra is isomorphically embeddable into a powerset residuated algebra. As a consequence, we obtain a completeness theorem for the Generalized Lambek Calculus. We use a Labelled Deductive System which generalizes the one used by Buszkowski [4] and Pankrat'ev [17] in completeness theorems for the Lambek Calculus.

Completeness of real Toda flows and totally positive matrices

Abstract: We give necessary and suucient conditions for real solutions of the nite Toda lattice to be nonsingular. Conditions are formulated in terms of the initial data and the spectrum of the Jacobi matrix in the associated Lax representation.

Isomorphic objects in symmetric monoidal closed categories

Abstract: This paper presents a new and self-contained proof of a result characterizing objects isomorphic in the free symmetric monoidal closed category, i.e., objects isomorphic in every symmetric monoidal closed category. This characterization is given by a finitely axiomatizable and decidable equational calculus, which differs from the calculus that axiomatizes all arithmetical equalities in the language with 1, product and exponentiation by lacking 1c=1 and (a · b)c =ac · bc (the latter calculus characterizes objects isomorphic in the free cartesian closed category). Nevertheless, this calculus is complete for a certain arithmetical interpretation, and its arithmetical completeness plays an essential role in the proof given here of its completeness with respect to symmetric monoidal closed isomorphisms.

A complete calculus for equational deduction in coalgebraic specification

Abstract: The use of coalgebras for the specification of dynamical systems with a hidden state space is receiving more and more attention in the years, as a valid alternative to algebraic methods based on observational equivalences. However, to our knowledge, the coalgebraic framework is still lacking a complete equational deduction calculus which enjoys properties similar to those stated in Birkhoff''s completeness theorem for the algebraic case. In this paper we present a sound and complete equational calculus for coalgebras of a restricted class of polynomial functors. This restriction allows us to borrow some ``algebraic'''' notions for the formalization of the calculus. Aditionally, we discuss the notion of `colours'' as a suitable dualization of variables in the algebraic case. Then the completeness result is extended to the ``non-ground'''' or ``coloured'''' case, which is shown to be expressive enough to deal with equations of hidden sort. Finally we discuss some weaknesses of the proposed results with respect to Birkhoff''s, and we suggest possible future extensions.

On the Finiteness of the Number of Determining Elements for von Karman Evolution Equations

Abstract: We prove the existence of a wide collection of finite sets of functionals that completely determine the long-time behaviour of strong solutions to the von Karman evolution equations. This collection contains finite sets of determining modes, nodes and local volume averages. The approach presented relies on the concept of a completeness defect for a set of linear functionals and it involves some ideas from abstract approximation theory. Our method is general enough and also can be applied to other non-linear partial differential equations which are of second order in time.

Classical action and quantum N-body asymptotic completeness

Abstract: The quantum propagation of N-body systems is asymptotically constrained to Lagrangian manifolds corresponding to particular solutions of the free Hamilton-Jacobi equation. This is used to give a proof of asymptotic completeness for short-range interactions.

A Completeness Proof for a Logic with an Alternative Necessity Operator

Abstract: We show the completeness of a Hilbert-style system LK defined by M Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information The completeness proof uses a variant of Makinson's canonical model construction Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete