CDT314 FABER Formal Languages, Automata and Models of Computation MARK BURGIN INDUCTIVE TURING MACHINES

Transcription

1 CDT314 FABER Formal Languages, Automata and Models of Computation MARK BURGIN INDUCTIVE TURING MACHINES

2 Inductive Turing Machines Burgin, M. Inductive Turing Machines, Notices of the Academy of Sciences of the USSR, 1983, v. 270, No. 6, pp (translated from Russian, v. 27, No. 3 ) Introduction of inductive Turing machines and demonstration that they can compute Kolmogorov complexity Burgin, M. How We Know What Technology Can Do, Communications of the ACM, v. 44, No. 11, 2001, pp A popular exposition of super-recursive algorithms and their role in computer science and technology

3 Inductive Turing Machines Burgin, M. Nonlinear Phenomena in Spaces of Algorithms, International Journal of Computer Mathematics, v. 80, No. 12, 2003, pp A complete proof that inductive Turing machine can generate and decide the whole arithmetical hierarchy and demonstration that in contrast to recursive algorithms, such as Turing machines, inductive Turing machines can demonstrate non-linear behavior in the algorithmic universe. Burgin, M. Superrecursive Algorithms, Springer, N. Y., 2005 The most extended description of inductive Turing machines and other Superrecursive Algorithms.

4 The basic problem Is it possible to build models of computation (algorithms) Is it possible to build models of computation (algorithms) that go beyond Turing machines?

5 A Standard Turing Machine C q Rules q i s j q k s h D q - state C control device h head <operating device> Tape h s 3 s 1 s 2 s 1 Cell being scanned

6 The structure of a real computer in a schematic representation ComputerStructure-MarkBurgin.pdf

7 Why the conventional Turing machine does not represent a real computer In the conventional Turing machine, there are no input and output devices. Can we correct this deficiency? Yes! Easily! We add to the conventional Turing machine two more tapes - the input tape and the output tape. It is still a conventional Turing machine. Does it represent a real computer? Yes but only computers at the beginning of computer era.

8 At that time, it was necessary to print out the output to get a result. After printing, computer stopped functioning or began to solve another problem. This exactly corresponds to the Turing machine functioning as it includes an extra condition that after giving a result, algorithm halts. It looks natural because what you have to do more after you have got what you wanted. However, analyzing real computers, we have to change our mind. Now people are working with displays. A computer produces its results on the screen. Those results on the screen exist there only if computer functions. When the computer halts, the result on the screen disappears.

9 In addition, many software systems work without halting, For instance, no computer works without an operating system (OS). Any OS is a program, that is, an algorithm according to the general understanding. di But while arecursive algorithm has to stop to give a result, we cannot say that a result of functioning of OS is obtained when computer stops functioning. On the contrary, when computer is out of service, its OS does not give an expected result. Although, from time to time, OS sends some messages (strings of words) to a user, the real result of OS is reliable functioning of the computer. Stopping computer is only a partial result. The real result of OS is obtained when computer does not stop (at least, potentially). Thus, we come to a conclusion that it is not necessary for algorithm to halt to produce a result.

10 Thus, to provide adequate models for computers and other computational systems, computer scientists introduced various superrecursive algorithms. Inductive Turing machines are superrecursive algorithms that are the closest to the conventional Turing machines.

11 A simple Inductive Turing Machine A l 3 h I q i l 5 Working Tape a 1 Rules l 2 a 7 h w l 1 a 7 Input Tape q i - state C control device h w the Working Tape head <operating device> h I the Input Tape head <input device> h O the Output Tape head <output device> h Output Tape h O s 3 s 1 s 2 s 1 the cell being scanned

12 How a simple inductive Turing machine works - informal description We see that a simple inductive Turing machine has the same structure t and the same rules (instructions) ti as a conventional Turing machine with three tapes. The input tape is a read-only tape and the output tape is a write-only tape. The difference is in output. A Turing machine produces a result only when it halts. The result is a word on the output tape.

13 How a simple inductive Turing machine works - informal description A simple inductive Turing machine produces its results without stopping. It is possible that t in the sequence of computations ti after some step, the word on the output tape is not changing, while the simple inductive Turing machine continues working. This word, which is not changing, is the result. Thus, the simple inductive Turing machine does not halt, producing a result after a finite number of computing operations.

14 How to solve the halting problem A simple inductive Turing machine M that solves this problem contains a universal Turing machine U as a subroutine. Given a word u and description c(t) of a Turing machine T, machine M uses machine U to simulate T with the input x. While U simulates T, machine M produces 0 on the output tape. If machine U stops, and this means that T halts being applied to x, machine M produces 1 on the output t tape.

15 How to solve the halting problem According to the definition, the result of M is equal to 1 when T halts and the result of M is equal to 0 when T never halts. In such a way, M solves the halting problem. This demonstrates t that t a simple inductive Turing machine can accomplish what no Turing machine can do although simple inductive Turing machines are the least powerful in the hierarchy of inductive Turing machines.