Church-Turing Thesis

1. Church-Turing Thesis Chuck Cusack Based (very heavily) on notes by Edward Okie (Radford University) and “Introduction to the Theory of Computation”, 3rd edition, Michael Sipser. (Original version at http://www.radford.edu/~nokie/classes/420/)

2. Hilbert's Tenth Problem • Hilbert's Problems: 1900, 23 challenge problems for the 20th century • Tenth Problem: Find an algorithm to determine if a polynomial has an integral root • Polynomial p has integral roots if the variables of p can be assigned to integer values to make p evaluate to 0 • Example: p(x) = x2 -3x + 2 = (x-1)*(x-2) has integral roots 1 and 2 because p(1) = p(2) = 0 • Problem Solved (1970): Recognizer possible, but no decider

3. 10th Problem: Decidable for One Variable • Problem: language R = {p | p is a string that represents a polynomial of one variable with an integral root} • R is decidable • A TM can be built that decides whether or not a polynomial p is in R (i.e. has an integral root) • R is decidable because for polynomials over one variable, a limit for the root is known to exist • (i.e. the root must be between ± Limit) • Machine M Decides R: • Input is a polynomial p over variable x • Evaluate p with x set successively to 0, 1, -1, 2, -2, ... Limit, -Limit • If p evaluates to 0 for any x, accept. Otherwise reject.

4. 10th Problem: NOT Decidable for Multiple Variables • It's been proven that there is NO limit to the value of the roots for polynomials with more than one variable • Problem: language P = {p | p is a string that represents a polynomial with an integral root} • R is recognizable, but not decidable • Machine M recognizes R: • Input is a polynomial p over variables x, y, ... • Evaluate p with x, y, ... set successively to all combinations of 0, 1, -1, 2, -2, ... • If p evaluates to 0 for any combination, accept • M does not decide R • If p has integral roots, the algorithm will find them • If p has no integral roots, the algorithm runs forever • Thus M is a recognizer but not a decider

5. Algorithm: Informal Definition • Informal definition: • Finite number of steps • Each step doable • Eventually halts • Could in theory be done with pencil and paper • If you have enough of each and enough time • Informal definition long accepted (e.g. Euclid's algorithm for greatest common factor) • But, informal definition not good enough for analysis of what algorithms are possible

6. Algorithm: Formal Definitions • Alonzo Church invented λ-calculus (defining computation with mathematical functions) • Turing invented Turing Machines • λ calculus and Turing Machines have been proven to be equivalent • Anything calculated by one can be calculated by the other • Other equivalent definitions have been developed

7. Church-Turing Thesis • The Church-Turing Thesiscan be stated as: • The informal notion of algorithm is the same as (any of) the formal definitions • Thus, anything that can be computed (in the informal sense) can be computed with a Turing Machine (or with the λ-calculus, or with ...) • Is this a fact? A Theorem? A conjecture? An axiom? • It can’t really be proven • Most people believe it is true • Some argue about how to correctly interpret it

8. Restatements of Church-Turing Thesis • “A function is effectively computable if and only if it is Turing-computable.” [1] • “A certain procedure constitutes a (discrete) algorithm for computing a given function if and only if it can be implemented as a Turing Machine that computes that given function.” [1] • “Whenever there is an effective method (algorithm) for obtaining the values of a mathematical function, the function can be computed by a TM.” [4] • “LCMs [logical computing machines: Turing's expression for Turing machines] can do anything that could be described as 'rule of thumb' or 'purely mechanical'.” [7] • “[T]he 'computable numbers' [the numbers whose decimal representations can be generated progressively by a Turing machine] include all numbers which would naturally be regarded as computable.” [6] • “Intuitive notion of algorithms equals Turing machine algorithms.” [5]

9. Variations of Church-Turing Thesis • Law of Mechanical Computability • Whatever can be calculated by a machine (working on finite data in accordance with a finite program of instructions) is Turing-machine-computable. [3] • “A function is mechanically computable (that is: computable by means of a machine) if and only if it is Turing-computable.” [1] • Strong Church-Turing Thesis • “A TM can do (compute) anything that a computer can do.” [4] • “Any 'reasonable' model of computation can be efficiently simulated on a probabilistic Turing Machine (an efficient simulation is one whose running time is bounded by some polynomial in the running time of the simulated machine). Here we take reasonable to mean in principle physically realizable.” [2]

10. Status of Church-Turing Thesis? • The status of the Church-Turing Thesis is debated by some researchers • In fact, whether or not there is a debate about this is debated • At issue are things like • Interactive computing • Machines/programs/algorithms not designed to halt (e.g. Operating systems, Embedded devices) which makes not fit the formal definitions

11. Unrecognizable Languages • So far we have seen: • Turing-Decidable Languages • Turing-Recognizable Languages • Turing-Recognizable Languages that are not Decidable • In Chapter 4 we will see: • A (sketch of a) proof that languages that are NOT Turing-Recognizable must exist • Some specific languages that are not Turing-recognizable • In the process we see how to prove that a language is not recognizable

12. References • Ben-Amram, The Church-Turing Thesis and its Look-alikes, ACM SIGACT News (2003) 36: 113-114. • Bernstein, E., Vazirani, U. Quantum complexity theory, SIAM Journal on Computing 26(5) (1997) 1411-1473. • Gandy, R. 1980. Church's Thesis and Principles for Mechanisms, In Barwise, J., Keisler, H.J., Kunen, K. (eds) 1980. The Kleene Symposium. Amsterdam: North-Holland. • Goldin, D.,Wegner, P., The Church-Turing Thesis: Breaking the Myth, presented at CiE 2005, Amsterdam, June 2005. • Sipser, M., Introduction to the Theory of Computation, Second Edition, Thomson/Course Technology, 2006. • Turing, A.M., On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, Series 2, 42 (1936-37), pp.230-265. • Turing, A.M., Intelligent Machinery, National Physical Laboratory Report, 1948; In Meltzer, B., Mi