Problem 37 - Discrete Mathematics with Applications

Question

Let \(m\) denote the number of \(a^{\prime} s\) in a string. Design an FSA that accepts strings over \(\\{a, b\\}\) which: Begin with \(a a\)

Step-by-Step Solution

Answer

In order to design a Finite State Automaton (FSA) that accepts strings over the alphabet {a, b} and begins with "aa", we define states q0 (initial state), q1 (one 'a' seen), and q2 (two consecutive 'a's seen). Next, specify transitions: 1) from q0 to q1 upon seeing an 'a', 2) from q1 to q2 upon seeing another 'a', 3) from q1 back to q0 upon seeing a 'b', and 4) self-loop transitions on q2 for both 'a' and 'b'. The resulting FSA can be represented by the following state diagram: ``` a ┌───► q1 ──┐ │ │ q0└───► q2 ─ ─┘ a■b ```

Step 1: Identify the alphabet and initial state

The given alphabet is {a, b}, and the initial state is denoted as q0. Initially, we have not seen any input characters.

Step 2: Define states based on the input characters

We need to define states to track the number of 'a's seen so far: - q0: Initial state, no input characters seen yet. - q1: One 'a' has been seen. - q2: Two consecutive 'a's have been seen (requirement fulfilled).

Step 3: Define transitions between the states

Now we will define the transitions that will change the current state depending on the input character seen: 1. Transition from q0 to q1: If we see an 'a', we move from state q0 to q1 (reading the first 'a'). 2. Transition from q1 to q2: If we see another 'a' after q1, we move from state q1 to q2 (reading the second 'a'). 3. Transition from q1 back to q0: If we see a 'b' in state q1 (after seeing the first 'a'), we return to the initial state as the string should start with 'aa'. 4. Self-loop transitions on q2: Once we reach state q2, any further input will be accepted. For this, we will make self-loop transitions for both 'a' and 'b' on state q2. We can summarize the FSA by a transition table or a state diagram as follows (here, states are given in rows, input characters in columns, and entries in the table indicate the new state after receiving input character): Transition Table: ``` State | a | b ------+----+--- q0 | q1 | - q1 | q2 | q0 q2 | q2 | q2 ``` State Diagram: ``` a ┌───► q1 ──┐ │ │ q0└───► q2 ─ ─┘ a■b ```

Key Concepts

State Transitions

In Finite State Automata (FSA), state transitions are crucial. They are the rules that dictate how the automaton moves from one state to another based on input symbols. Consider state transitions like a set of instructions that guide the FSA on its journey through various states.

For example, starting at state q0, if the FSA reads an 'a', it transitions to state q1. This is the first part of the journey. Each input symbol leads the FSA to the next step, depending on these pre-defined transitions.

Transitions are like the gears that keep a machine moving, deciding the path of the automaton until it recognizes the required pattern or structure in the strings.

Alphabet

When working with an FSA, the alphabet is the set of symbols that the automaton can understand and process. For our example, the alphabet consists of two symbols: \( \{a, b\} \). This is a very simple set, but it's essential for defining how the FSA operates.

The alphabet forms the foundation for putting together the FSA's input strings. Each string is made up of sequences of these symbols, guiding the FSA through its states as it detects specific sequences such as 'aa' at the start of a string.

Understanding the alphabet is akin to learning the language spoken by the FSA—it's the dictionary used to instruct its function and transitions.

Transition Table

A transition table is a structured way to display all possible state transitions of an FSA. It's a lot like a roadmap, listing every possible move the automaton can make based on its current state and the input symbol.

For instance, consider the FSA in our problem. The transition table reveals that starting from state q0, an 'a' will move the automaton to state q1. But if a 'b' is encountered, no transition occurs, so the action is undefined (i.e., the string is not accepted).

• From q0, reading 'a' leads to q1.
• From q1, reading 'a' leads to q2, but 'b' returns to q0.
• From q2, both 'a' and 'b' result in remaining in q2, indicating acceptance of further input.

A transition table helps clarify all potential pathways through the FSA's network of states, ensuring a clear understanding of how inputs are processed.

FSA Design

Designing an FSA involves constructing its components: states, transitions, and accepting states. The objective is to capture a specific sequence of symbols that the automaton will recognize.

In our example, the FSA is designed to accept strings that begin with 'aa'. This requires careful crafting of states that reflect progress towards this goal. Initially at state q0, spotting the first 'a' transitions to q1, and recognizing the second 'a' propels it to q2.

These states are connected with transitions that represent input processing. Upon reaching state q2, the FSA should accept any continuation, shown with self-loop transitions. This design process is key to ensuring the FSA completes its task effectively, identifying specific input sequences with precision.