Abstract
Turing machines form the core of computability theory, or recursion theory as it is also known. This chapter introduces basic notions and results and readers already familiar with them can safely skip it. The exposition is based on standard references [18, 39, 67, 83, 109]. In the discussion that follows, the symbol ℕ will stand for the set of positive integer numbers including zero and ℚ will stand for the set of rational numbers.
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Notes
- 1.
Alternatively known as the free monoid with base A.
- 2.
In a nutshell, hypercomputation is the idea that there are conceptual and real computing machines that transcend the capabilities of the Turing machine. See [126] for an overview of the field of hypercomputation.
- 3.
A random set C in an ordinary set X is characterized by a function C: X → [0, 1], where C(x) denotes the probability that x ∈ X.
- 4.
Readers not familiar with these notions can consult Springer’s “Encyclopedia of Mathematics” web page.
- 5.
A connected component of any vertex is the maximal connected subcomplex that contains this vertex.
- 6.
A multidimensional Turing machine is one in which the tape can be either two or three dimensional. In the case of a two-dimensional machine, the transition function is of the form \(\delta: (Q \setminus H) \times \Gamma \rightarrow Q \times \Gamma \times \{ L,R,D,U,N\}\), where D and U specify movement of the scanning head up and down, respectively.
- 7.
Very roughly, a term yields a value; variables and constants are terms; functions are terms; atoms yield truth values; and each predicate is an atom. An atom is a formula. Given formulas p and q the following are also formulas: ¬p, \(p \vee q\), \(p \wedge q\), p ⇒ q, \(p \equiv q\), \(\forall xp\), and \(\exists xp\). In the last two cases x is said to be a bound variable, while in all other possible cases it is said to be a free variable.
- 8.
The graph of a function f: X → Y is the subset of X ×Y given by \({\bigl \{(x,f(x)) \vert x \in X\bigr \}}\). A total function whose graph is recursively enumerable is a recursive function.
- 9.
Actually, they call this “updated” version of the Church–Turing thesis the Church–Kalmár–Turing thesis, named after Church, László Kalmár, and Turing.
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Syropoulos, A. (2014). A Précis of Classical Computability Theory. In: Theory of Fuzzy Computation. IFSR International Series on Systems Science and Engineering, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8379-3_2
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