Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D.

Transcription

1 Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman And Introduction to Languages and The by J. C. Martin

2 Basic Mathematical Definitions Numbers: N = natural numbers = {1, 2, 3, } Z = integers = {, -2, -1, 0, 1, 2, } Q = rational numbers ( expressed in ratios, 1/5, 3/7, 2/9 etc.) R = real numbers ( floating point, 1.5, etc. ) C = complex numbers (2+5i, 27-3i etc. ) Countable Set: A set is countable if there is a one-to-one correspondence between the set and N, the natural numbers. It can be enumerated. { a, b, c, d, e, } N, Z and Q are countable. R and C are uncountable. 2

3 Rational Numbers, Q are countable 1/1 1/2 1/3 1/4 1/5 1/6 2/1 2/2 2/3 2/4 2/5 2/6 3/1 3/2 3/3 3/4 3/5 3/6 4/1 4/2 4/3 4/4 4/5 4/6 5/1. 3

4 Uncountable Set: Cantor s Diagonalization s 1 = (0, 0, 0, 0, 0, 0, 0,...) s 2 = (1, 1, 1, 1, 1, 1, 1,...) s 3 = (0, 1, 0, 1, 0, 1, 0,...) s 4 = (1, 0, 1, 0, 1, 0, 1,...) s 5 = (1, 1, 0, 1, 0, 1, 1,...) s 6 = (0, 0, 1, 1, 0, 1, 1,...) s 7 = (1, 0, 0, 0, 1, 0, 0,...). s 0 = (1, 0, 1, 1, 1, 0, 1,...) 4

5 R is uncountable Proof: o Suppose R is countable o List R according to the bijection f: n f(n) _

6 R is uncountable Proof: o Suppose R is countable o List R according to the bijection f: n f(n) _ set x = 0.a 1 a 2 a 3 a 4 where digit a i i th digit after decimal point of f(i) (not 0, 9) e.g. x = x cannot be in the list! 6

7 Turing Machine Decidability Turing decidability L is Turing decidable (or just decidable) if there exists a Turing machine M that accepts all strings in L and rejects all strings not in L. Note that by rejection we mean that the machine halts after a finite number of steps and announces that the input string is not acceptable. Acceptance, as usual, also requires a decision after a finite number of steps. 7

8 Decidability Turing Recognizability L is Turing recognizable if there is a Turing machine M that recognizes L, that is, M should accept all strings in L and M should not accept any strings not in L. This is not the same as decidability because recognizability does not require that M actually reject strings not in L. M may reject some strings not in L but it is also possible that M will simply "remain undecided" on some strings not in L; for such strings, M's computation never halts. 8

9 Recursion and Recursive Functions Enumerable Sets Recursively Enumerable Languages Resursive Languages Non-Recursively Enumerable Languages 9

10 regular languages decidable all languages context free languages RE decidable RE all languages 10

11 A language is recursively enumerable if some Turing machine accepts it. 11

12 A language is recursive if some Turing machine accepts it and halts on any input string. OR A language is recursive if there is a membership algorithm for it. 12

13 A language is recursively enumerable if and only if there is an enumeration procedure for it. If a language L is recursive then there is an enumeration procedure for it 13

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16 Theorem: S is an infinite countable set, the powerset 2 s of S is uncountable Since S is countable, we can write S = { s 1, s 2, s 3, s 4,.. } where S consists of s 1, s 2, s 3, s 4,.. elements and the powerset 2 s is of the form: { {s 1 }, {s 2 }, {s 1, s 2 },.. {s 1, s 2, s 3, s 4 } } 16

17 String Encoding: 17

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21 Coding Turing Machines 21

22 Transition Rules are: Coding Code for M 22

23 Diagonalization Language: 23

24 Theorem: L d is not a recursively enumerable language. That is there is no Turing Machine that accepts L d 24

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30 Decidable Languages about DFA 30

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33 Halting and Acceptance Problems: Acceptance Problem: Does a Turing machine accept an input string? A TM is recursively enumerable. 33

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