Problem 43 - Discrete Mathematics with Applications

Question

Let \(m\) denote the number of \(a\) 's in a string. Design an FSA that accepts strings over \(\\{a, b\\}\) which: Have \(m \equiv 2(\bmod 3).\)

Step-by-Step Solution

Answer

The FSA for accepting strings over {a, b} with the number of a's congruent to 2 modulo 3 consists of 3 states, \(q_0\), \(q_1\), and \(q_2\). The accepting state is \(q_2\). Transitions are defined as: \( \begin{cases} (q_0, a) \rightarrow q_1 \\ (q_0, b) \rightarrow q_0 \\ (q_1, a) \rightarrow q_2 \\ (q_1, b) \rightarrow q_1 \\ (q_2, a) \rightarrow q_0 \\ (q_2, b) \rightarrow q_2 \end{cases} \).

Step 1: Identify the states needed for the FSA

In this case, let's define 3 states since there could be 3 distinct scenarios based on the number of a's: 1. \(q_0\) represents when the number of a's encountered so far is 0 modulo 3. 2. \(q_1\) represents when the number of a's encountered so far is 1 modulo 3. 3. \(q_2\) represents when the number of a's encountered so far is 2 modulo 3. Since we need to accept strings with the number of a's \(m \equiv 2 \pmod{3}\), our accepting state should be \(q_2\). Step 2: Define transitions

Step 2: Define input symbol-based transitions for each state

In this step, we will define the state transitions based on the input symbols encountered. There are two input symbols: 'a' and 'b'. We need to define transitions for each of our states. For state \(q_0\): 1. On input 'a', move to state \(q_1\), as the number of a's modulo 3 has incremented from 0 to 1. 2. On input 'b', stay in state \(q_0\) since the number of a's remains unchanged. For state \(q_1\): 1. On input 'a', move to state \(q_2\), as the number of a's modulo 3 has incremented from 1 to 2. 2. On input 'b', stay in state \(q_1\) since the number of a's remains unchanged. For state \(q_2\): 1. On input 'a', move to state \(q_0\), as the number of a's modulo 3 has incremented from 2 to 0. 2. On input 'b', stay in state \(q_2\) since the number of a's remains unchanged. Now, we have defined all the necessary states and transitions. The FSA is represented as a set of states (Q), a set of input symbols (Σ), a transition function (δ), an initial state (q0), and a set of accepting states (F). Formally, 1. Q = { \(q_0\), \(q_1\), \(q_2\) } 2. Σ = { a, b } 3. δ: \( \begin{cases} (q_0, a) \rightarrow q_1 \\ (q_0, b) \rightarrow q_0 \\ (q_1, a) \rightarrow q_2 \\ (q_1, b) \rightarrow q_1 \\ (q_2, a) \rightarrow q_0 \\ (q_2, b) \rightarrow q_2 \end{cases} \) 4. q0 = \(q_0\) 5. F = { \(q_2\) }

Key Concepts

Modular Arithmetic

Modular arithmetic is like clock arithmetic. It involves integers wrapping around after reaching a certain value, known as the modulus. In the context of a finite state automaton (FSA), this concept helps determine state transitions based on input symbols like 'a' or 'b'. For example, when counting occurrences of a particular symbol like 'a', you use modular arithmetic to decide which state the FSA should move to next.

In our exercise, we are interested in strings where the number of 'a's, denoted as \(m\), leaves a remainder of 2 when divided by 3. This is written as \(m \equiv 2 \pmod{3}\). The use of modulo 3 means we focus on the remainders 0, 1, and 2 when dividing the count of 'a's by 3. This idea guides the entire structure of the FSA, including the creation of states and transitions based on the input data.

State Transitions

State transitions in an FSA represent the movement between states based on input symbols. Each state reflects a specific condition of the string being processed. In our example with symbols \(\{a, b\}\), transitions depend heavily on modular arithmetic.

The states are defined as follows:

• \(q_0\): Number of 'a's is 0 modulo 3
• \(q_1\): Number of 'a's is 1 modulo 3
• \(q_2\): Number of 'a's is 2 modulo 3

Transitions occur according to these rules:

• If the current state is \(q_0\) and 'a' is received, transition to \(q_1\).
• If the current state is \(q_0\) and 'b' is received, remain at \(q_0\).
• From \(q_1\), 'a' leads to \(q_2\) while 'b' keeps it at \(q_1\).
• At \(q_2\), receiving 'a' moves it back to \(q_0\), and 'b' keeps it at \(q_2\).

These transitions ensure the FSA tracks the number of 'a's correctly according to modular arithmetic principles, ultimately determining the acceptance of input strings.

Accepting States

In the design of any FSA, the accepting state (or states) is crucial. It defines the condition under which a string is accepted by the automaton. Here, the accepting state is determined by our interest in strings where the total number of 'a's modulo 3 is 2.

For this FSA, \(q_2\) is the accepting state. It means that if the automaton reaches this state after processing the string, the string is accepted. This aligns with our exercise goal: Accept strings where \(m \equiv 2 \pmod{3}\).

To illustrate:

• When the FSA ends at \(q_2\), it indicates that the string constitution fits our acceptance condition.
• This is a direct application of the modular arithmetic used to classify state transitions.
• The set of accepting states is captured as \(F = \{ q_2 \}\), making the condition clear and precise for acceptance.

Understanding accepting states helps clarify design intentions and ensure the FSA achieves the desired outcomes for different string inputs.