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Efficient Inference and Computation of Optimal Alternatives for Preference Languages Based On Lexicographic Models

Nic Wilson, Anne-Marie George

TL;DR

It is shown that testing consistency, and thus inference, is polynomial for a specific preference language, which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination.

Abstract

We analyse preference inference, through consistency, for general preference languages based on lexicographic models. We identify a property, which we call strong compositionality, that applies for many natural kinds of preference statement, and that allows a greedy algorithm for determining consistency of a set of preference statements. We also consider different natural definitions of optimality, and their relations to each other, for general preference languages based on lexicographic models. Based on our framework, we show that testing consistency, and thus inference, is polynomial for a specific preference language LpqT, which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination. Computing different kinds of optimal sets is also shown to be polynomial; this is backed up by our experimental results.

Efficient Inference and Computation of Optimal Alternatives for Preference Languages Based On Lexicographic Models

TL;DR

It is shown that testing consistency, and thus inference, is polynomial for a specific preference language, which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination.

Abstract

We analyse preference inference, through consistency, for general preference languages based on lexicographic models. We identify a property, which we call strong compositionality, that applies for many natural kinds of preference statement, and that allows a greedy algorithm for determining consistency of a set of preference statements. We also consider different natural definitions of optimality, and their relations to each other, for general preference languages based on lexicographic models. Based on our framework, we show that testing consistency, and thus inference, is polynomial for a specific preference language LpqT, which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination. Computing different kinds of optimal sets is also shown to be polynomial; this is backed up by our experimental results.

Paper Structure

This paper contains 18 sections, 53 theorems.

Table of Contents

  1. Introduction
  2. Lex Models, Composition and Extension
  3. Lexicographic Inference and Strongly Compositional Preference Statements
  4. Lexicographic Inference
  5. Compositional and Strongly Compositional Preference Statements
  6. Checking Consistency of a Set of Strongly Compositional Preferences
  7. Equivalence w.r.t. Compositional Statements
  8. Further Properties for Inference with (Strongly) Compositional Statements
  9. Decreasing Preference Statements and Sets of Models
  10. Important Instances of Strongly Compositional Preference Statements
  11. Negated Statements
  12. Preference Languages ${\cal L}_{pqT}$ and ${\cal L}'_{pqT}$
  13. The Language ${\cal L}_{pqT}$
  14. Checking Consistency for Subsets of ${\cal L}'_{pqT}$
  15. Checking $\Gamma$-Satisfaction of Maximal $\models^*$-Model
  16. ...and 3 more sections

Key Result

Lemma 1

$\circ$ is an associative operation on ${\cal G}$.

Theorems & Definitions (96)

  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 86 more