Let C be a concept class consisting of 10 concepts c0, . . . , c9, Let the
domain X = {1, 2, . . . , 999}. n is in concept class ci
if the decimal
representation of n has the integer i. What is the VC dimension of
this concept class?

We are tasked with determining the VC (Vapnik-Chervonenkis) dimension of a concept class \( C \) consisting of 10 concepts \( c_0, c_1, \dots, c_9 \). The domain \( X = \{ 1, 2, \dots, 999 \} \), and each concept \( c_i \) represents numbers in the domain whose decimal representation contains the digit \( i \).

### Step-by-step Breakdown:

1. **Understanding the Concepts**:
   - Each concept \( c_i \) consists of all numbers in the domain \( X \) that contain the digit \( i \) somewhere in their decimal representation.
   - For example, \( c_0 \) is the set of all numbers between 1 and 999 that have the digit "0" in their decimal representation, and similarly for the other concepts.

2. **VC Dimension**:
   - The VC dimension is the size of the largest subset of \( X \) that can be "shattered" by the concept class.
   - A subset is "shattered" if, for every possible labeling of the points in the subset (whether they belong to a concept or not), there is some concept in the class that realizes exactly that labeling.

3. **Shattering Power of the Concept Class**:
   - Each concept \( c_i \) can be understood as testing whether the digit \( i \) appears in the decimal representation of a number.
   - To shatter a set of numbers, each number in the set must be distinguishable by the concept class. This happens if each number can be uniquely labeled by the appearance or non-appearance of digits in the decimal system.

4. **Number of Distinguishable Points**:
   - A number in \( X \) is represented by at most 3 digits (since \( X = \{1, 2, \dots, 999\} \)).
   - The concept class is effectively checking for the presence of digits from \( 0 \) to \( 9 \). For any number \( n \), you can test whether a given digit \( i \) is present in \( n \).
   - The maximum number of points that can be shattered depends on how well we can assign all possible binary labelings (present or not present) of the digits to those points.

5. **Bounding the VC Dimension**:
   - Since there are 10 concepts (one for each digit from 0 to 9), and each number can contain up to 3 digits, it is unlikely that any set of more than 3 numbers can be shattered.
   - For any subset of 4 numbers, there will be some overlap in the decimal digits, and not all labelings will be possible. This indicates that the concept class cannot shatter any set larger than 3 numbers.

### Conclusion:

The VC dimension of the given concept class is **3**, because it is the largest set of numbers that can be shattered by the concept class.

---

Would you like further details on the reasoning behind this VC dimension, or have any additional questions?

Here are five questions to expand your understanding:
1. How is the VC dimension related to the shattering of points?
2. Why is it difficult to shatter sets of more than 3 points in this problem?
3. Can the concept class be extended to larger domains while keeping the same VC dimension?
4. What is the role of decimal representation in determining the VC dimension in this case?
5. How would the VC dimension change if the domain \( X \) were larger?

**Tip**: Understanding the VC dimension helps in evaluating the complexity of learning algorithms, as it gives insight into the capacity of the concept class to classify data.

Let C be a concept class consisting of 10 concepts c0, . . . , c9, Let the domain X = {1, 2, . . . , 999}. n is in concept class ci if the decimal representation of n has the integer i. What is the VC dimension of this concept class?

Explore the VC dimension of a concept class where each concept checks if a digit is present in the decimal representation of numbers between 1 and 999. Learn how to determine the largest set that can be shattered and understand the role of binary labeling in the context of VC dimension. Suitable for advanced high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation