scholarly journals Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

2022 ◽  
Vol 73 ◽  
pp. 1-65
Author(s):  
Jan Maly
Keyword(s):  
Linear Order ◽  
Fundamental Problem ◽  
Preference Order ◽  
Not Given ◽  
Linear Orders ◽  
Domain Restriction ◽  
Impossibility Results ◽  
The Family ◽  
Np Complete ◽  
Ranking Sets

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barbera and Pattanaik tell us that some desirable axioms – namely dominance and (strict) independence – are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is \Pi_2^p-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problems can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction. Finally, we show that it is NP-complete to decide for a family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for at least one linear order on the objects.

scholarly journals Lifting Preferences over Alternatives to Preferences over Sets of Alternatives: The Complexity of Recognizing Desirable Families of Sets

2020 ◽  
Vol 34 (02) ◽  
pp. 2152-2159
Author(s):  
Jan Maly
Keyword(s):  
Computational Complexity ◽  
Fundamental Problem ◽  
The Other ◽  
Preference Order ◽  
Not Given ◽  
Linear Orders ◽  
Domain Restriction ◽  
Impossibility Results ◽  
The Family ◽  
Preferences Over Sets

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barberà and Pattanaik tell us that some desirable axioms – namely dominance and (strict) independence – are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is Π2p-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problem can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

scholarly journals On cuts in ultraproducts of linear orders I

2016 ◽  
Vol 16 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Mohammad Golshani ◽  
Saharon Shelah
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Regular Cardinals

For an ultrafilter [Formula: see text] on a cardinal [Formula: see text] we wonder for which pair [Formula: see text] of regular cardinals, we have: for any [Formula: see text]-saturated dense linear order [Formula: see text] has a cut of cofinality [Formula: see text] We deal mainly with the case [Formula: see text]

An Efficient Technique to Ensure the Logical Consistency of Interacting Knowledge Bases

1997 ◽  
Vol 06 (01) ◽  
pp. 27-36 ◽  
Author(s):  
Bertrand Mazure ◽  
Lakhdar Saïs ◽  
Éric Grégoire
Keyword(s):  
Local Search ◽  
Fundamental Problem ◽  
Real Life ◽  
Knowledge Bases ◽  
The Other ◽  
Logical Consistency ◽  
Complete Problem ◽  
Np Complete ◽  
The One ◽  
Cooperative Knowledge

In this paper, we address a fundamental problem in the formalization and implementation of cooperative knowledge bases: the difficulty of preserving consistency while interacting or combining them. Indeed, knowledge bases that are individually consistent can exhibit global inconsistency. This stumbling-block problem is an even more serious drawback when knowledge and reasoning are expressed using logical terms. Indeed, on the one hand, two contradictory pieces of information lead to global inconsistency under complete standard rules of deduction: every assertion and its contrary can be deduced. On the other hand, checking the logical consistency of a propositional knowledge base is an NP-complete problem and is often out of reach for large real-life applications. In this paper, a new practical technique to locate inconsistent interacting pieces of information is presented in the context of cooperative logical knowledge bases. Based on a recently discovered heuristic about the work performed by local search techniques, it can be applied in the context of large interacting knowledge bases.

scholarly journals BUILDING COMMUNICATION BETWEEN FAMILY MEMBERS AS A FORT OF FAMILY RESILIENCE

1970 ◽  
Vol 21 (2) ◽  
pp. 249-262
Author(s):  
Jeri Ariansyah
Keyword(s):  
Qualitative Data ◽  
Fundamental Problem ◽  
Family Members ◽  
Research Literature ◽  
Communication Patterns ◽  
Family Resilience ◽  
Conceptual Approach ◽  
Library Research ◽  
The Social ◽  
The Family

This paper discusses about building communication between family members as a fortress of family resilience. The family is the basis for maintaining diversity, the family is very important to maintain the social understanding of family. As social beings, humans are never separated from communication. Communication is often a fundamental problem in one's family household relations, especially what often becomes a conflict is communication between husband and wife who lacks understanding of the concept of communication patterns in the family in order to maintain family resilience in the household. The purpose of this paper is to provide and express the concepts and principles of communication in the family so that it can be a solution on how to build communication in the family that can fortify family resilience. As for the focus in this paper is how the concept of communication patterns, communication as an ethical value to realize family resilience, communication as the realization of ma'ruf relationships in the family, the theory of ethical values ​​and their relevance to the family communication system and the concept of family resilience. This paper is included in the type of normative legal research literature (library research). By using a conceptual approach and a statutory approach. the type of data in this study using qualitative data. The results of this paper conclude that family resilience is very influential on the concept of communication in the family. By understanding the concepts and principles of communication patterns between family members, they can maintain resilience and strength in the family, so that they can realize the purpose of marriage, namely forming a sakinah, mawaddah and rahmah family as contained in the Qur'an Surah Ar-Rum verse 21. Keywords: communication, family resilience, social

Representations of Well-Founded Preference Orders

1983 ◽  
Vol 35 (3) ◽  
pp. 496-508 ◽  
Author(s):  
Douglas Cenzer ◽  
R. Daniel Mauldin
Keyword(s):  
Utility Function ◽  
Binary Relation ◽  
Linear Order ◽  
Equivalence Classes ◽  
Preference Order ◽  
Real Numbers ◽  
Order Preserving Map ◽  
Preference Orders ◽  
Reasonable Sense ◽  
Natural Relation

A preference order, or linear preorder, on a set X is a binary relation which is transitive, reflexive and total. This preorder partitions the set X into equivalence classes of the form . The natural relation induced by on the set of equivalence classes is a linear order. A well-founded preference order, or prewellordering, will similarly induce a well-ordering. A representation or Paretian utility function of a preference order is an order-preserving map f from X into the R of real numbers (provided with the standard ordering). Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4, 7]. Parametrized versions of this problem have also been studied [1, 7, 8]. Given a continuum of preference orders which vary in some reasonable sense with a parameter t, one would like to obtain a continuum of representations which similarly vary with t.

scholarly journals The Property of Hamiltonian Connectedness in Toeplitz Graphs

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Ayesha Shabbir ◽  
Muhammad Faisal Nadeem ◽  
Tudor Zamfirescu
Keyword(s):  
Hamiltonian Path ◽  
Complete Problem ◽  
Hamiltonian Connected ◽  
The Family ◽  
Np Complete

A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n=3 that Tnp,q is Hamiltonian-connected, while some particular cases of Tnp,q,r for p=1 and q=2,3,4 have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph Tn1,q,r is Hamiltonian-connected for all 1<q<r<n and n≥5r−2.

scholarly journals FURTHER RESULTS ON P SYSTEMS WITH PROMOTERS/INHIBITORS

2006 ◽  
Vol 17 (01) ◽  
pp. 205-221 ◽  
Author(s):  
DRAGOŞ SBURLAN
Keyword(s):  
Upper Bound ◽  
P Systems ◽  
Not Given ◽  
The Family

This paper presents several results regarding P systems with non-cooperative rules and promoters/inhibitors at the level of rules. For the class of P systems using inhibitors, generating families of sets of vectors of numbers, a characterization of the family of Parikh sets of ET0L languages is shown. In the case of P systems with non-cooperative promoted rules even if an upper bound was not given, the inclusion of the family PsET0L was proved. Moreover, a characterization of such systems by means of a particular form of random context grammars, therefore a sequential formal device, is proposed.

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