What Is Computable? What Is Feasibly Computable? A Physicist's Viewpoint

In this chapter, we show how the questions of what is computable and what is feasibly computable can be viewed from the viewpoint of physics: what is computable within the current physics? what is computable if we assume – as many physicists do – that no final physical theory is possible? what is computable if we consider data processing, i.e., computations based on physical inputs? Our physicsbased analysis of these questions leads to some unexpected answers, both positive and negative. For example, we show that under the no-physical-theory-is-perfect assumption, almost all problems are feasibly solvable – but not all of them. 1 What Is Computable? What Is Feasibly Computable? Different Aspects of These Questions The two main questions of theoretical computer science. One of the main objectives of theoretical computer science is to answer the following two fundamental questions: • The first question is: which tasks are computable in principle? • Once we learned that a task is, in principle, computable, a natural next question is: is this task feasibly computable, i.e., can we perform the corresponding computations in reasonable time? These questions are usually considered from the viewpoint of a computer scientist. From the viewpoint of a computer scientist, computation is a solution to a well-defined task, performed on a well-defined computational devices. As a result, when formulating and analyzing the above problems, computer scientists usually Vladik Kreinovich and Olga Kosheleva University of Texas at El Paso, 500 W. University, El Paso, Texas 79968, USA e-mails: [email protected], [email protected]