Devices for Computation

Abstract

To study computational processes we have to fix a computational device first. In this chapter, we study Turing machines and circuits as models of computation. We use the standard notations of formal language theory and represent these notations now briefly. An alphabet is any set A. The elements of an alphabet A are called letters. The concatenation of sets A and B is a set AB consisting of strings formed of any element of A followed by any element of B. Especially, A ^k is the set of strings of length k over A. These strings are also called words. The concatenation w _l w ₂ of words w ₁ and w ₂ is just the word w ₁ followed by w ₂. The length of a word w is denoted by |w| or ℓ(w) and defined as the number of the letters that constitute w. We also define A⁰ to be the set that contains only the empty word e having no letters, and A* = A⁰ ∪ A ^l ∪ A ² ∪ ... is the set of all words over A. Mathematically speaking, A* is the free monoid generated by the elements of A, having the concatenation as the monoid operation and ϵ as the unit element.