A computer program takes as input binary codes of 5 bits and
returns binary sequences of 4 bits. For every input, the program might either
return an output or go into an infinite loop. We call a 5-bit input *valid* if the
program does not hang. Two programs are regarded as the same if they have
the same set of valid inputs and return the same output for each valid input.
• How would you model this mathematically?
• How many distinct programs of this kind exist?

### Modeling the Problem Mathematically:

The given computer program can be viewed as a function that takes a 5-bit binary input and returns either a 4-bit binary output or hangs (goes into an infinite loop). Here’s how we can model it step by step:

1. **Input Space**:
   - A 5-bit binary input means the program has \( 2^5 = 32 \) possible inputs.
   - Each input is represented by a binary number between 00000 and 11111.

2. **Output Space**:
   - For each valid input, the program returns a 4-bit binary output, which means there are \( 2^4 = 16 \) possible outputs for each valid input.

3. **Program Behavior**:
   - For each of the 32 possible inputs, the program can either:
     1. Return one of the 16 possible 4-bit outputs.
     2. Go into an infinite loop (i.e., "hang").

   Therefore, the program’s behavior for each input can be modeled as choosing between 17 options (16 valid outputs + 1 option for hanging).

4. **Valid Inputs**:
   - If we call a 5-bit input *valid* if the program does not hang on that input, we need to account for the fact that the program may hang on some inputs but not others.
   - For each input, the program either hangs or returns a 4-bit binary output, so the set of valid inputs could be any subset of the 32 possible inputs. There are \( 2^{32} \) possible subsets of valid inputs.

5. **Distinguishing Programs**:
   - Two programs are considered the same if they:
     1. Have the same set of valid inputs.
     2. Produce the same output for each valid input.

### Counting the Number of Distinct Programs:

1. **Subsets of Valid Inputs**:
   - The program can decide which subset of the 32 inputs are valid. There are \( 2^{32} \) possible subsets of the 32 inputs.

2. **Outputs for Valid Inputs**:
   - For each valid input, the program must choose one of 16 possible outputs (since a valid input does not hang).
   - For each subset of valid inputs, the program must define a 4-bit output for each valid input. If the number of valid inputs is \( k \), the program has \( 16^k \) ways to assign outputs to the valid inputs.

   Therefore, for a given subset of \( k \) valid inputs, the number of ways to assign outputs is \( 16^k \). Since we’re summing over all possible sizes of valid subsets (from 0 valid inputs to all 32 valid inputs), the total number of distinct programs is:

   \[
   \sum_{k=0}^{32} \binom{32}{k} \cdot 16^k
   \]

   This simplifies to \( (1 + 16)^{32} \) using the binomial theorem. So the total number of distinct programs is:

   \[
   17^{32}
   \]

### Final Answer:
There are \( 17^{32} \) distinct programs of this kind.

---

Let me know if you want more details on any step or have any further questions! Here are 5 related questions to explore this concept further:

1. How would the result change if the program returned 3-bit outputs instead of 4-bit outputs?
2. What is the probability that a randomly chosen program hangs for all inputs?
3. How would you approach this problem if the inputs were 6-bit codes instead of 5-bit codes?
4. What if the program could return multiple types of errors in addition to hanging?
5. How would this problem change if programs could modify their valid inputs dynamically?

**Tip:** For problems involving combinations and distinct behaviors, always consider both the input space and the possible outcomes for each input!

A computer program takes as input binary codes of 5 bits and returns binary sequences of 4 bits. For every input, the program might either return an output or go into an infinite loop. We call a 5-bit input valid if the program does not hang. Two programs are regarded as the same if they have the same set of valid inputs and return the same output for each valid input. • How would you model this mathematically? • How many distinct programs of this kind exist?

Explore how to model a computer program that takes 5-bit binary inputs and returns 4-bit binary outputs, accounting for cases where the program may hang. Learn how combinatorics and the binomial theorem are used to determine that there are 17^32 distinct programs of this kind.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation