Introduction to Computation Across Models

Computation is defined in various models of computation, each offering unique insights into the nature of computation and its limitations. This article delves into how we define and understand computation across different models, from classical to quantum, and the underlying principles that govern these models.

Computation Across Models

At its core, a computer is any physical system that follows rules to modify data. These rules can be as simple as in a Turing Machine (TM) or as complex as in a quantum system. In all models, a computer consists of rules for modifying data, which can change depending on the system's operations on the data.

The Human Brain as a Computer

The human brain is a prime example of a non-traditional computer. It is a large neural network, composed of neurons that can modify the way future signals travel through them, a concept known as neural plasticity. This ability allows the brain to learn from data, effectively making it a computer that is not built out of silicon but made out of organic material.

Quantum Systems and Computers

Quantum systems represent another form of computation where data and rules are based on the entangled states of quantum particles. These systems follow the rules of quantum mechanics, leading to unique and often unpredictable behaviors. In this model, the data changes based on the quantum states of the system, providing a different perspective on computation.

Turing Machines and Universal Computing

Turing Machines (TMs) introduced by Alan Turing, are foundational in the field of computation. A TM follows a set of very simple rules to modify data, making it one of the simplest yet most powerful models of computation. The concept extends beyond TMs to Universal Turing Machines (UTMs), which are capable of simulating any other TM when provided with the former's rules and data. This leads to the Church-Turing-Deutsch Thesis, which suggests that any computation achievable by a Turing Machine can also be performed by any other model of computation, including quantum systems.

The Church-Turing-Deutsch Principle

According to the Church-Turing-Deutsch Principle, the computational capabilities of any system in the universe can be perfectly simulated by a Turing Machine, assuming the universe operates according to the principles of quantum mechanics. This principle is one of the most profound and far-reaching ideas in the field of computation, suggesting that the physical universe and its computational capabilities can be fully described using a theoretical computer model.

Limits of Computation

Not all problems can be solved by TMs or by extension, by any computer. Turing proved that many problems are unsolvable by TMs and thus by any computer. These problems are classified based on their complexity, which is often exponential as the size of the data increases. For instance, some problems may require exponentially more computational power to solve as the input size grows. This highlights the importance of finding approximate solutions for problems that cannot be solved exactly.

Complexity and Approximation

While some problems may be unsolvable, approximate solutions can often be found for important problems. Moreover, some problems, while solvable, can be computationally expensive. Techniques such as approximation algorithms are used to find solutions that are close to optimal within a reasonable amount of time. Understanding the complexity of problems is crucial for the development of efficient algorithms and computational systems.

Conclusion

This article has explored various models of computation, from classical to quantum, and the principles that govern these models. The Church-Turing-Deutsch Principle provides a unifying framework for understanding the computational capabilities of the universe, while the limitations of computation highlight the importance of developing efficient and approximate solutions for complex problems.