The Church-Turing Thesis

cell containing a left end marker ` (never written over) and extends to infinitely many cells to the right. This machine has a head that can move left or right one cell in each step, and in each step it reads the current symbol it is currently scanning and overwrites it before moving on. Initally, the tape contains the input string, which is finite, on cells numbered 1 onwards (cell 0 contains `). The rest of the tape has a special symbol called the blank symbol t. Machine, initially, starts in state s with its input cell scanning the leftmost cell. In each step it changes its state based on its current state and symbol under its head, and it overwrites this cell, and moves its head left/right before taking the next step. The Turing machine is assumed to have two special halting states — accept state t and reject state r. Like the 2-way DFA, the Turing machine may never halt, if it does not reach either of these two halting states. The formal definition is as follows. Definition 1. A deterministic, one-tape Turing machine is M = (Q,Σ,Γ,`,t, δ, s, t, r) where • Q is a finite set of states, • Σ is a finite input alphabet, used to encode the input string, • Γ is a finite tape alphabet consisting of symbols written and read from the tape; Σ ( Γ, • `∈ Γ \ Σ is the left endmarker, • t ∈ Γ \ Σ is the blank symbol, • s ∈ states is the start state, • t ∈ Q is the unique accepting state, • r ∈ Q (r 6= t) is the unique rejecting state, • δ : (Q \ {t, r})×Γ→ Q×Γ×{L,R} is the transition function that describes the next state, symbol to be written, and direction to move the tape head, given the current state and symbol read; it is assumed that no transitions are enabled from either t or r.