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Deterministic and Strongly Nondeterministic Decision Trees for Decision Tables from Closed Classes

Azimkhon Ostonov, Mikhail Moshkov

Abstract

In this paper, we consider classes of decision tables with 0-1-decisions closed relative to removal of attributes (columns) and changing decisions assigned to rows. For tables from an arbitrary closed class, we study the dependence of the minimum complexity of deterministic decision trees on various parameters of the tables: the minimum complexity of a test, the complexity of the set of attributes attached to columns, and the minimum complexity of a strongly nondeterministic decision tree. We also study the dependence of the minimum complexity of strongly nondeterministic decision trees on the complexity of the set of attributes attached to columns. Note that a strongly nondeterministic decision tree can be interpreted as a set of true decision rules that cover all rows labeled with the decision 1.

Deterministic and Strongly Nondeterministic Decision Trees for Decision Tables from Closed Classes

Abstract

In this paper, we consider classes of decision tables with 0-1-decisions closed relative to removal of attributes (columns) and changing decisions assigned to rows. For tables from an arbitrary closed class, we study the dependence of the minimum complexity of deterministic decision trees on various parameters of the tables: the minimum complexity of a test, the complexity of the set of attributes attached to columns, and the minimum complexity of a strongly nondeterministic decision tree. We also study the dependence of the minimum complexity of strongly nondeterministic decision trees on the complexity of the set of attributes attached to columns. Note that a strongly nondeterministic decision tree can be interpreted as a set of true decision rules that cover all rows labeled with the decision 1.
Paper Structure (18 sections, 24 theorems, 37 equations, 5 figures)

This paper contains 18 sections, 24 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Definitions and Notation
  3. Decision Tables
  4. Tests
  5. Deterministic and Strongly Nondeterministic Decision Trees
  6. Complexity Measures
  7. Parameters of Decision Trees and Tables
  8. Main Results
  9. Functions $F_{\psi ,A}^{W}$ and $F_{\psi ,A}^{\Theta }$
  10. Function $F_{\psi ,A}$
  11. Function $G_{\psi ,A}$
  12. Family of Closed Classes of Decision Tables
  13. Example of Information System
  14. Auxiliary Statements
  15. Proofs of Theorems \ref{['T1']}, \ref{['T2']}, and \ref{['T3']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\psi$ be a bounded complexity measure and $A$ be a nonempty closed class of decision tables from $\mathcal{M}_{k}^{2}$. Then $F_{\psi ,A}^{W}$ and $F_{\psi ,A}^{\Theta }$ are nondecreasing functions for which one of the following statements holds: (a) If the functions $S_{\psi }$ and $N$ are bo

Figures (5)

  • Figure 1: Decision table from $\mathcal{M}_{2}^{2}$
  • Figure 2: Decision table obtained from the decision table shown in Fig. \ref{['fig1']} by removal of a column and changing of decisions
  • Figure 3: A deterministic decision tree for the decision table shown in Fig. \ref{['fig1']}
  • Figure 4: A strongly nondeterministic decision tree for the decision table shown in Fig. \ref{['fig1']}
  • Figure 5: Decision table $T_{n}$

Theorems & Definitions (60)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 2
  • Definition 5
  • Definition 6
  • Example 3
  • Definition 7
  • ...and 50 more