Quasi Order Research Articles

On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.

Well Quasi Orders in Subclasses of Bounded Treewidth Graphs and Their Algorithmic Applications

We show that three subclasses of bounded treewidth graphs are well quasi ordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertex cover are well quasi ordered by the induced subgraph order, graphs with bounded feedback vertex set are well quasi ordered by the topological-minor order, and graphs with bounded circumference are well quasi ordered by the induced minor order. Our results give algorithms for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.

The quasi order of graphs on an ordinal

For α an ordinal, a graph with vertex set α may be represented by its characteristic function, f : [ α ] 2 → 2 , where f ( { γ , δ } ) = 1 if and only if the pair { γ , δ } is joined in the graph. We call these functions α - colorings. We introduce a quasi order on the α -colorings (graphs) by setting f ≤ g if and only if there is an order-preserving mapping t : α → α such that f ( { γ , δ } ) = g ( { t ( γ ) , t ( δ ) } ) for all { γ , δ } ∈ [ α ] 2 . An α -coloring f is an atom if g ≤ f implies f ≤ g . We show that for α = ω ω below every coloring there is an atom and there are continuum many atoms. For α < ω ω below every coloring there is an atom and there are finitely many atoms.

A Brézis–Browder principle on partially ordered spaces and related ordering theorems

Through a simple extension of Brézis–Browder principle to partially ordered spaces, a very general strong minimal point existence theorem on quasi ordered spaces, is proved. This theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems, and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then, they are applied to prove strong minimal point existence results, generalizations of Bishop–Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion.

On optimization problems with set-valued objective maps

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets in n-dimensional Euclidean space and investigate their properties. Next, by using these orderings, we define the concepts of the convexities to set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and characterize their properties.

Efficient local unfolding with ancestor stacks

AbstractThe most successful unfolding rules used nowadays in the partial evaluation of logic programs are based on well quasi orders (wqo) applied over (covering) ancestors, i.e., a subsequence of the atoms selected during a derivation. Ancestor (sub)sequences are used to increase the specialization power of unfolding while still guaranteeing termination and also to reduce the number of atoms for which the wqo has to be checked. Unfortunately, maintaining the structure of the ancestor relation during unfolding introduces significant overhead. We propose an efficient, practical local unfolding rule based on the notion of covering ancestors which can be used in combination with a wqo and allows a stack-based implementation without losing any opportunities for specialization. Using our technique, certain nonleftmost unfoldings are allowed as long as local unfolding is performed, i.e., we cover depth-first strategies. To deal with practical programs, we propose assertion-based techniques which allow our approach to treat programs that include (Prolog) built-ins and external predicates in a very extensible manner, for the case of leftmost unfolding. Finally, we report on our implementation of these techniques embedded in a practical partial evaluator, which shows that our techniques, in addition to dealing with practical programs, are also significantly more efficient in time and somewhat more efficient in memory than traditional tree-based implementations.

DAKS: AnRPackage for Data Analysis Methods in Knowledge Space Theory

Knowledge space theory is part of psychometrics and provides a theoretical framework for the modeling, assessment, and training of knowledge. It utilizes the idea that some pieces of knowledge may imply others, and is based on order and set theory. We introduce the R package DAKS for performing basic and advanced operations in knowledge space theory. This package implements three inductive item tree analysis algorithms for deriving quasi orders from binary data, the original, corrected, and minimized corrected algorithms, in sample as well as population quantities. It provides functions for computing population and estimated asymptotic variances of and one and two sample Z tests for the diff fit measures, and for switching between test item and knowledge state representations. Other features are a function for computing response pattern and knowledge state frequencies, a data (based on a finite mixture latent variable model) and quasi order simulation tool, and a Hasse diagram drawing device. We describe the functions of the package and demonstrate their usage by real and simulated data examples.

Quasi-Long-Range-Order in Mott State of Bose–Fermi Mixtures on One-Dimensional Optical Lattice

We numerically studied internal structure of Mott insulating state of Bose–Fermi mixture systems on a one-dimensional optical lattice by using quantum Monte Carlo simulations. We looked for Mott state carefully adjusting various parameters to prevent phase separation. From the calculation of density–density correlation functions we found a quasi long-range order in the particle distribution in the Mott state. The order was strongly dependent of the ratio of the number of the bosons to that of the fermions. For example, the bosons and the fermions showed a tendency of aligning alternately when equal number of the bosons and fermions are introduced in the system. The observation of the quasi long-range order in the one-dimensional system indicates the presence of true long-range order in Bose–Fermi mixtures on a three-dimensional optical lattice.

Poverty measurement: the critical comparison value

The Foster–Greer–Thorbecke (FGT) family of poverty-measures is commonly used when comparing income distributions with respect to poverty. Within this framework, the poverty ranking will be sensitive to the choice of poverty aversion parameter (defining a particular FGT poverty-measure). If we content ourselves with applying a few specific parameter values, then we may demand too little in order to claim robustness in our poverty comparisons. On the other hand, we may demand too much if we only work with FGT poverty-measure quasi orderings established by considering every possible parameter value. An alternative approach may be to report the number of parameter values representing possible reversal points in our poverty ranking—what we call critical comparison values—and leave the final evaluative step to the relevant decision-makers. By applying a generalized version of Descartes’ Rule of Signs, we show that the number of critical comparison values depends on the number of times the cumulative distribution functions intersect.

Characterization of certain orders using their associated choice functions

Binary relation and choice functions define two different languages for describing the decision maker's (DM's) preferences over a finite set of alternatives. The study of normal choice functions establishes a link between these two languages. The existence of this kind of function demonstrates the compatibility between the DM's local choices (paired comparisons or binary choices) and the DM's global choices. In this paper, we explore this kind of linkage in depth by studying characterizations of special binary relations (total order, weak order, partial order and quasi order) using certain properties of their associated choice functions.

Dimension of valued relations

The classical notion of dimension of a partial order can be extended to the valued setting, as was indicated in a particular case by Ovchinnikov (1984) (Ovchinnikov, S.V., 1984. Representations of transitive fuzzy relations. In: Skala, H.J., Termini, S., Trillas, E. (Eds.), Aspects of vagueness. Reidel, Boston, pp. 105–118). Relying on Valverde's result (1985) (Valverde, L., 1985. On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17, 313–328) on the transitive closure of a valued relation, we define the dimension of a valued quasi order. Building then on Fodor and Roubens (1995) (Fodor, J., Roubens M., 1995. Structure of transitive valued binary relations. Mathematical Social Sciences, 30, 71–94), we also show that the definition can be generalized to all valued relations by using valued biorders instead of valued weak orders as one-dimensional relations. Interesting, combinatorial questions about the new dimension concept arise and are investigated here. In particular, we aim at a characterization of valued quasi orders of dimension two.

Dimensions of chains of relations

Abstract Any partial order is an intersection of linear orders, generally in many ways. The smallest number of linear orders required in such an intersection is called the dimension of the given partial order. This concept of dimension can be trivially extended to quasi orders (i.e. reflexive and transitive relations), by using weak orders instead of linear orders. A further extension to general relations is obtained by replacing weak orders with biorders (also called Ferrers relations). These notions are now classical, see e.g. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory (1992). In this talk, we introduce several concepts of dimension for chains of relations. The motivation comes from the analysis of valued relations in preference modelling. A forerunner was Ovchinnikov (1984) who however worked in another theoretical setting than ours. Valverde (1985) investigation of the transitive closure of a valued relation paved the way to a fairly general concept of dimension. After having made a choice of definitions among many possible ones, we show that any valued quasi order is an intersection of valued weak orders. Our goal is to study the combinatorics of the resulting dimension concept, which is defined thus for valued quasi-orders, or equivalently for chains of quasi-orders. A similar dimension can also be defined for any valued relation (that is, for any chain of relations) if valued biorders take the place of valued weak orders as the one-dimensional relations. This remark directly follows from Fodor and Roubens (1995). Interesting, combinatorial questions arise about the new concept of dimension for a chain of quasi orders. We stress that the dimension of the chain is not just a plain function of the dimensions of the quasi orders. For instance, if the chain has dimension at most two, all the quasi orders have also dimension at most two; however, the converse does not necessarily hold. Characterizing two-dimensional chains of quasi orders thus requires additional conditions that force the (two-dimensional) quasi orders to be coherent in some way. We describe several such conditions that are inferred from the study of transitive orientations of comparability graphs. Although progress towards a solution is reported, we leave open the problem of characterizing two-dimensional chains. Moreover, the computational complexity of the corresponding decision problem is presently unknown. The talk is based on joint work with Jutta Mitas.

Mean ergodicity on Banach lattices and Banach spaces

We characterize properties of Banach spaces by mean ergodicity of operators belonging to special classes. More precisely, we prove: ¶ (i) The Banach lattice E has order continuous norm iff every power-order-bounded regular Fredholm operator is ergodic. (ii) The countably order complete Banach lattice is a KB-space iff every positive operator which possesses a quasi order bounded attractor is mean ergodic. (iii) The Banach space does not contain c0 if every Fredholm operator is ergodic.

Mixture of Maximal Quasi Orders: a new Approach to Preference Modelling

Normative theories suggest that inconsistencies be pointed out to the Decision Maker who is thus given the chance to modify his/her judgments. In this paper, we suggest that the inconsistencies problem be transferred from the Decision Maker to the Analyst. With the Mixture of Maximal Quasi Orders, rather than pointing out incoherences for the Decision Maker to change, these inconsistencies may be used as new source of information to model his/her preferences.

On well quasi orders of free monoids

The paper investigates down-sets associated to well quasi orders. Of particular language-theoretic interest is the quasi order u ⩽ s v (resp. u ⩽ ℘v ) of u being a subword (resp. a Parikh subword) of v, as well as their inverses. We establish a number of results about the regularity and effective regularity of the down-sets, in particular, a general condition for the effective regularity of the down-set associated to an inverse well quasi order (Theorem 7.1 ). It is decidable whether or not an arbitrary regular language results as a down-set from an infinite or bi-infinite word, whereas the same problem is undecidable for context-free language. A quasi order being a well quasi order is connected to arbitrary languages being confluent.

On quasi orders of words and the confluence property

We investigate the confluence property, that is, the property of a language to contain, for any two words of it, one which is bigger, with respect to a given quasi order on the respective free monoid, than each of the former two. This property is investigated mainly for regular and context-free languages. As a consequence of our study, we give an answer to an old open problem raised by Haines concerning the effective regularity of the sets of subwords. Namely, we prove that there are families with a decidable emptiness problem for which the regularity of the sets of subwords is not effective.

Moving glass theory of driven lattices with disorder

We study periodic structures, such as vortex lattices, moving in a random potential. As predicted in [T. Giamarchi, P. Le Doussal Phys. Rev. Lett. 76 3408 (1996)] the periodicity in the direction transverse to motion leads to a new class of driven systems: the Moving Glasses. We analyse using several RG techniques the properties at T=0 and $T>0$: (i) decay of translational long range order (ii) particles flow along static channels (iii) the channel pattern is highly correlated (iv) barriers to transverse motion. We demonstrate the existence of the ``transverse critical force'' at T=0. A ``static random force'' is shown to be generated by motion. Displacements grow logarithmically in $d=3$ and algebraically in $d=2$. The persistence of quasi long range translational order in $d=3$ at weak disorder, or large velocity leads to predict a topologically ordered ``Moving Bragg Glass''. This state continues the static Bragg glass and is stable at $T>0$, with non linear transverse response and linear asymptotic behavior. In $d=2$, or in $d=3$ at intermediate disorder, another moving glass exist (the Moving Transverse Glass) with smectic quasi order in the transverse direction. A phase diagram in $T$ force and disorder for static and moving structures is proposed. For correlated disorder we predict a ``moving Bose glass'' state with anisotropic transverse Meissner effect and transverse pinning. We discuss experimental consequences such as anomalous Hall effect in Wigner crystal and transverse critical current in vortex lattice.

Quasi two-dimensional magnetic order of Tb3+ spins in Pb2Sr2Tb1−x Ca x Cu3O8 (x = 0 and 0.5)

Neutron diffraction experiments were performed to study the magnetic ordering of the Tb+3 sublattice in the high-T c superconductor Pb2Sr2Tb1−x Ca x Cu3O8 (x = 0.5) and the undoped parent compound (x = 0). For the parent compound, a quasi two dimensional (2D) phase with a finite antiferromagnetic correlation along the c-direction and a three dimensional phase with ferromagnetic correlation along the c-direction were found. The coexistence of the two phases is likely to be related to structural imperfections such as stacking faults, strains, oxygen disorder or cation vacancies. The superconductor with a superconducting transition temperature of T c = 71K exhibits a quasi 2D magnetic ordering of the Tb sublattice with a finite ferromagnetic correlation along the c-direction. The correlation lengths along the c-direction for the quasi 2D phases are 32 and 26A for x = 0 and 0.5, respectively. The magnetic saturation moments are with 9.1 and 8.8 μ B in excellent agreement with our mean-field crystal-field calculations. 2D short range correlation could be observed up to about 8T N and is described by a Lorentzian distribution of magnetic intensity perpendicular to the 2D rod in reciprocal space.

A NOVEL AND EFFICIENT METHOD FOR PARSING UNRESTRICTED TEXTS OF QUASI FREE WORD ORDER LANGUAGES

This paper deals with the problems stemming from the parsing of long sentences in quasi free word order languages. Due to the word order freedom of a large category of languages including Greek and the limitations of rule-based grammar parsers in parsing unrestricted texts of such languages, we propose a flexible and effective method for parsing long sentences of such languages that combines heuristic information and pattern-matching techniques in early processing levels. This method is deeply characterized by its simplicity and robustness. Although it has been developed and tested for the Greek language, its theoretical background, implementation algorithm and results are language independent and can be of considerable value for many practical natural language processing (NLP) applications involving parsing of unrestricted texts.

On aggregation of T-transitive fuzzy binary relations

The paper is concerned with the theory of weak orders, quasi orders, and quasi-transitive relations in the framework of group choice in a fuzzy environment. We characterize completely two extreme cases of group choice rules satisfying the Pareto principle. In particular, we prove that a fuzzy binary relation is a fuzzy quasi order if and only if it is an intersection of fuzzy weak orders. In addition, it is a fuzzy quasi-transitive relation if and only if it is a union of fuzzy weak orders.