On computable numbers, with an application to the ENTSCHEIDUNGSPROBLEM

1. On computable numbers, with an application to the ENTSCHEIDUNGSPROBLEM COT 6421 Paresh Gupta by Alan Mathison Turing

2. ENTSCHEIDUNGSPROBLEM? • Decision Problem “ Does there exists a fully automatic procedure or mechanical method which decides whether given mathematical statements from first order logic are universally true or false? ”

3. Posed by • In 17th Century Gottfried Leibnitz • Mechanical machine to decide mathematical statements • 1928 - David Hilbert and Wilhelm Ackermann

4. Index of Turing’s Paper(1/2) • Computing Machines • Definitions Automatic Machines Computing Machines Circle and circle-free Machines Computable sequences and numbers • Examples of computable machines • Abbreviated tables Further examples

5. Index of Turing’s Paper(2/2) • Enumeration of computable sequences • The universal computing machine • Detailed description of universal machine • Application of diagonal process • The extent of the computable numbers • Examples of large classes of numbers which are computable • Application to the Entsheidungsproblem

6. Computable Numbers • Real numbers whose decimal representations are calculable by finite means • Computable numbers does not mean computable functions of real or integral variable, or computable predicates • Class of computable numbers is enumerable

7. Computing Machines • Analogous to humans machine is capable of finitely many conditions to compute a number • “ m-configurations ” • Example- q1, q2, …, qR • Supplied with “ tape ”, divided into sections (called squares) each capable of bearing a symbol • At any moment there is just one symbol S(r) in the r-th square is in the machine • Pair (qR , S(r)) form a configuration

8. Automatic machines • Motion of the machine is completely determined by the configurations • Denoted as a-machine • c-machine are choice machines Computing machines • a-machine which prints figures ( 0 or 1) on its tape and some other tape symbols • Binary symbols left on the tape is called the sequence computed by the machine

9. Circular and circle-free machines • Circular • A computing machine that never writes more than a finite number of symbols ( 0’s or 1’s) • If it reaches a configuration from which there is no possible move, or moves and keeps printing same set symbols • Computable sequence and numbers • A sequence is computable if it can be computed by circle-free machine • A number is computable if it differs by an integer from the number computed by a circle-free machine

10. Enumeration of Computable Sequences (1/3) Transition Table • Standard Description ( S.D) • Concatenation of the expressions qiSjSkLqm separated by semi-colons. • Each qi is replaced by the letter “D” followed by letter “A” repeated i times • Sj is replaced by “D” followed by “C” repeated j times • Entirely made up of letters “A”, “C”, “D”, “L”, “R”, “N”, and “;” • E.g. DADDCRDAA;DAADDRDAAA;DAAADDCCRD;

11. Enumeration of Computable Sequences (2/3) • Description Number (D.N) • Replacing letters “A”, “C”, “D”, “L”, “R”, “N”, and “;” by “1”, “2”, “3”,…, “7” respectively gives a description of the machine in Arabic numeral • The D.N uniquely determines the structure and S.D of a machine • The machine whose D.N is n is described as M(n) • E.g. 31173113353111731113322531111731111335317

12. Enumeration of Computable Sequences (3/3) • To each computable sequence there corresponds at least one D.N, while to no D.N does there correspond more than one computable sequence • Computable sequence and numbers are therefore enumerable Computable Sequences Description Number (D.N) Relationship between computable sequences and D.N

13. The Universal Computing Machine (1/2) • F-squares • The symbols on F-squares form a continuous sequence • E-squares • The symbols on E-squares are liable to be erased • It is possible to invent a single machine which can compute any computable sequence • How does U work? • If this machine U, is supplied with a tape on the beginning of which is written the S.D of the machine M, then U will compute the same sequence as M • Note that S.D of M is nothing but the rules of operation of M

14. The Universal Computing Machine (2/2) • U bears on its tape the symbols е on an F-square and again an е on the next F-square; followed by the S.D of the machine M and a double colon “::” • The instructions of S.D are separated by semi-colons • Each instructions consists of five parts • “D” followed by a sequence of letters “A”. This describes the relevantm-configuration. • “D” followed by a sequence of letters “C”. This describes the scanned symbol. • “D” followed by another sequence of letters “C”. This describes the symbolinto which the scanned symbol is to be changed. • “L”, “R”, “N”, describing whether the machine is to move to left, right, or not atall. • “D” followed by a sequence of letters “A”. This describes the finalm-configuration.

15. Application of diagonal process (1/3) • Turing brings out the significance of circle-free machines by application of diagonal process to prove that a computing machine might never halt • Implies that it is impossible to decide whether a given number is the D.N of a circle-free machine in finite number of steps • Argument that real numbers are not enumerable (by Hobson, Theory of functions of a real variableTheory of functions of a real variable in 1907 ) cannot be used to prove that computable numbers are not enumerable

16. Application of diagonal process (2/3) • Idea is that “ If the computable sequence is enumerable, let αn be the n-th computable sequence, and let Φn(m) be the m-th figure in αn. Let β be the sequence with 1- Φn(n) as its n-th figure. Since β is computable, there exists a number K such that 1- Φn(n)=Φk(n) for all n. Putting n=k, we have 1 = 2 Φk(K), i.e 1 is even. This is impossible therefore computable sequences are not enumerable ” • This argument is incorrect because of the assumption that β is computable. It would be true if we could enumerate computable sequences by finite means

17. Application of diagonal process (3/3)

18. Halting problem (1/4) • Suppose we can invent a machine D which when supplied with the S.D of a machine M will test S.D and if M is circular flag it with the symbol “u” and if it is circle free mark it with “s” • Using D with U we can compute a sequence β1 with Φn(n) as its n-th figure

19. Halting Problem (2/4) • We can construct a Machine H, which has its motion divided into sections • In the first N-1 sections integers 1,2,…,N-1 have been written down and tested by machine D • In the N-th section machine D tests N if N-1 is the D.N of a circle-free machine • If N is satisfactory, then H has found the first N figures of the sequence of which a D.N is N are calculated • If not then H goes on to the (N+1)-th section of its motion

20. Halting problem (3/4) • Note that H is circle-free from its construction • Because each section comes to an end after a finite number of steps ( as per our assumption D takes finite amount of time) • Contradiction –H is circular • Let K be the D.N of H • In the k-th section H tests if K is satisfactory • Since H is circle-free verdict of D for K cannot be circular (“u”) • On the contrary it also cannot be circle-free (“s”) because K-th section would have to compute first R(K-1) figures of the sequence and then R(k)-th figure • But R(k)-th figure would never be found

21. Halting problem (4/4) Therefore H is circular and D cannot exist which can test if given the S.D of a machine M, if M is circular or circle-free in finite amount of time

22. Halting problem- version 2 • There can be no machine ξ, which when applied with the S.D of a machine M,will determine whether Mever prints a symbol ( 0 say ) • If there exists such a machine ξ, then there is a general process for determining whether machine Mprints 0 infinitely often • Let M1 be a machine which prints the same sequence as M, except that it prints Õ where first 0 was printed by M • M2 has prints two symbols 0 replaced by Õ, and so on M3, M4, .. can be created

23. Halting problem- version 2 • Let F be a machine which, when supplied with the S.D of M will write down successively the S.D of M,M1, M2,… • Combining F with machine ξ gives us a machine G • G first writes S.D of M using M and uses ξ to test if it has any 0’s in it • Symbol :0: is written if it is found that M never prints a 0 • Similar actions are taken on M1, M2,… • If G is tested with ξ and if G never prints :0: then M prints 0 infinitely many often • Similarly we can design a general process for determining if M prints 1 infinitely often • By combining all of these machines we have a way of determining whether M prints infinity of figures • This implies we have a machine for determining whether M is circle-free, but that is impossible • Therefore machine ξ cannot exist

24. Application to the Entsheidungsproblem (1/4) • In 1931 Hilbert and Ackermann suggested that there exists a mechanical solution to the Entsheidungsproblem • In 1936 Turing proposed that there can be no general process for determining whether a given formula Ŭ of functional calculus K is provable in a finite amount of time

25. Application to the Entsheidungsproblem (2/4) • Different from Gödel’s Incompleteness Theorem which states that there propositions Ŭ such that neither Ŭ or – Ŭ is provable • Note that if negation of Gödel's Theorem is proved then we have an immediate solution to Decision problem • i.e. For propositions Ŭ, either Ŭ or – Ŭ is provable

26. Application to the Entsheidungsproblem (3/4) • Proof • If symbol S1( i.e. 0 ) appears on the tape in some complete configuration of M, then Un(M) is provable • If Un(M) is provable, then S1 appears on the tape in some complete configuration of M

27. Application to the Entsheidungsproblem (4/4) • If we assume that Entsheidungsproblem can be solved • Then we can determine if Un(M) is provable • This implies that there is a process for determining that Mever prints a symbol 0. • But this is impossible • Hence Entsheidungsproblem cannot be solved

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