• Introduction to Theoretical Computer Science: Between Finite and Infinite - From Mathematical Theory to AI and Autonomous Driving
    • It is generally not possible to prove statements about Infinity with a finite number of verifications.
    • Instead, we use logic to prove them, relying on the computational power of human thinking.
    • Infinity cannot be proven.
      • When we trace the source of the infinite elements in a theorem, we end up with either the universal quantifier () or axioms that define the existence of an infinite number of elements.
        • Even techniques like ε-δ proof are essentially logical formulas that use the universal quantifier ().
      • Infinity is an artificial construct introduced by humans as a starting point for discussions.
      • On the other hand, it is also possible to have discussions that exclude infinity.
      • If you want to know more about this, I recommend researching it yourself.
    • Mathematical Induction provides a clear understanding of both finite and infinite concepts.
    • image
    • Systems like Testing involve testing a finite number of elements from an infinite set, but they do not provide mathematical guarantees.
    • Mathematical Guarantees
      • Hiring smart people to perform these guarantees every time is difficult and expensive.
      • Therefore, we want to use automated methods on computers.
        • One such method is verifying the inclusion relationship of infinite sets of inputs that produce a specific output in automata.
      • There are also research efforts to apply these concepts to the physical world.
        • This approach involves generalizing concepts like Automaton using category theory and applying them to other fields.
        • For example, proving the correctness of Autonomous Driving. #math#mathematical foundations