# In this problem's solution, how is the first DFA obtained?

• s3a
In summary, the conversation discusses the implementation of a deterministic finite automaton (DFA) for determining if a binary number is divisible by 5. The first DFA is designed to accept bit strings where the decimal representation is divisible by 5, while the second DFA has reversed arrow directions. The logic behind the first DFA's implementation involves considering the patterns and forms of decimal numbers that are divisible by 5 and designing a DFA based on these. The speaker requests help in understanding this logic.
s3a

## Homework Statement

PDF with problem statement and its solution:
https://www.docdroid.net/BkVhr32/4.pdf

## Homework Equations

State Diagrams for DFAs

## The Attempt at a Solution

My question is what's the logic behind the first DFA's implementation?

I understand that the second DFA just has the arrows' direction reversed, but how does one come up with the first one? I know how DFAs work and have made other ones before; my problem is specifically this binary divisible by 101 stuff.

If someone could help me understand the logic, I would GREATLY appreciate it!

s3a said:
My question is what's the logic behind the first DFA's implementation?

The first DFA is designed to accept bit strings for which ##int(w)## is divisible by 5. You can work with the decimal representations and think about the cases where a positive integer is divisible by 5 or not. What is the form of these numbers? Then start to design a DFA with these forms of numbers. Think about the input and the patterns that binary inputs imply for the decimal representation.

## 1. What is a DFA?

A DFA, or Deterministic Finite Automaton, is a mathematical model used to recognize patterns in strings of characters. It consists of a set of states, a set of input symbols, a transition function, a start state, and a set of accept states.

## 2. How is the first DFA obtained in a problem's solution?

The first DFA is obtained by analyzing the given problem and creating a state diagram that represents the possible transitions between states based on the given input symbols. This diagram is then converted into a table, which is used to construct the DFA.

## 3. What is the purpose of the first DFA in a problem's solution?

The first DFA serves as a visual representation of the problem, allowing us to easily understand and analyze the relationships between different states and input symbols. It also helps in the construction of the final DFA.

## 4. Can the first DFA be modified during the problem-solving process?

Yes, the first DFA can be modified as needed during the problem-solving process. It is common to refine and optimize the DFA to make it more efficient and accurate in recognizing patterns in strings.

## 5. Are there any limitations to using the first DFA in problem-solving?

Yes, the first DFA is a simplified model and may not be able to accurately represent all types of problems. It also assumes that the input symbols are deterministic and may not work well with non-deterministic problems. In such cases, more complex models may be needed.

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