Computability

1. Computability

Up to now, we've learned what computers are and how to use algorithms and programs to solve problems. Now, let's explore the boundaries of what computers can and can't compute using today's computers, focusing on theoretical concepts that define their limits.

2. What is computability?

Computability is the study of which problems computers can or can't solve. A problem is computable if there’s a clear algorithm to solve it, and it always produces a result in a finite number of steps. For example, sorting a list is computable because we can define an algorithm, like Merge Sort, that finishes in predictable time. On the other hand, some problems are non-computable, meaning no universal algorithm exists to solve them for every possible input. For some inputs, any attempt to solve the problem might run forever without reaching a conclusion. What’s fascinating is that the boundary between computable and non-computable problems isn't fixed. With advances in algorithmic research, some problems once thought non-computable might eventually become computable.

3. Automata

Automata is a foundational concept in computer science that helps us understand how problems can be modeled and solved by computers. At its core, automata are like imaginary machines or systems that represent how computers process information and make decisions. For example, imagine a traffic light system. Each color—green, yellow, red—represents a state, and there are specific rules, like timing, that dictate how the system transitions from one state to another. Automata work similarly: they have states and rules that determine how they move from one state to another based on inputs or conditions. If a problem can be modeled by one of these automata, it means it can be programmed or solved with an algorithm. By studying these simple 'machines,' we gain insight into what tasks computers can handle and how they make decisions. Next, we’ll look at two key types of automata: Finite Automata and Pushdown Automata.

4. Finite Automata (FA)

Finite Automata are simple machines with a fixed number of states. The system transitions between these states based on predefined rules and no or limited memory. A traffic light has three fixed states: green, yellow, and red, with rules that after a set amount of time, the light changes from green to yellow, then yellow to red, and back to green. No extra memory is required beyond knowing the current state. The traffic light doesn't need to track how many times it has cycled or anything beyond the immediate state.

5. Pushdown Automata (PDA)

Pushdown Automata are like Finite Automata but with a memory, a stack that allows them to handle more complex problems that keeps a history of what its seen. In our traffic light example, imagine it not only switches states but also keeps track of additional information, like the number of cars waiting at the light or if the light had been skipped due to an emergency signal. It can adjust the timing of transitions on the fly. So, the memory helps influence state decisions. These two types of automata help illustrate the difference between simple state machines and more complex systems that rely on memory to handle advanced processes.

6. Summary

In summary, Finite Automata (FA) handle simple tasks with fixed states and no memory, like pattern recognition or password validation. Pushdown Automata (PDA) are more powerful, using stacked memory to process nested structures, like balancing parentheses in code. These models are crucial in computability, defining what problems computers can solve efficiently. If a problem can be modeled by one of these automata, it can be programmed or solved with an algorithm. However, if a problem cannot be modeled by an FA or PDA, it may still be solvable by more powerful models, which we will dive into later.

7. Let's practice!

Let’s look at some automata examples.

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