Theory of Computation

Theory of Computation Decidability Recursive and Recursively Enumerable Languages 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 Theory of Computation by J. C. Martin Prof. Muhammad Saeed

Decidability 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, 2.5674 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. 1 2 3 4 5 Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability 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 … ……. Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability • Uncountable Set: • Cantor’s Diagonalization • s1 = (0, 0, 0, 0, 0, 0, 0, ...) • s2 = (1, 1, 1, 1, 1, 1, 1, ...) • s3 = (0, 1, 0, 1, 0, 1, 0, ...) • s4 = (1, 0, 1, 0, 1, 0, 1, ...) • s5 = (1, 1, 0, 1, 0, 1, 1, ...) • s6 = (0, 0, 1, 1, 0, 1, 1, ...) • s7 = (1, 0, 0, 0, 1, 0, 0, ...) • ………. s0 = (1, 0, 1, 1, 1, 0, 1, ...) Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability R is uncountable Proof: • Suppose Ris countable • List R according to the bijection f: n f(n) _ 1 3.14159… 2 5.55555… 3 0.12345… 4 0.50000… … Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability R is uncountable Proof: • Suppose Ris countable • List R according to the bijection f: n f(n) _ 1 3.14159… 2 5.55555… 3 0.12345… 4 0.50000… … set x = 0.a1a2a3a4… where digit ai ≠ ith digit after decimal point of f(i) (not 0, 9) e.g. x = 0.2312… x cannot be in the list! Dept. of Computer Science & IT, FUUAST Theory of Computation

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. Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability 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. Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability • Recursion and Recursive Functions • Enumerable Sets • Recursively Enumerable Languages • Resursive Languages • Non-Recursively Enumerable Languages Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability decidable all languages regular languages context free languages RE decidable  RE  all languages Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability A language is recursively enumerable if some Turing machine accepts it. Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability 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. Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability 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 Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Theorem: S is an infinite countable set, the powerset 2s of S is uncountable Since S is countable, we can write S = { s1 , s2 , s3 , s4 , …….. } where S consists of s1 , s2 , s3 , s4 , ….. elements and the powerset 2s is of the form: { {s1}, {s2 }, … {s1 , s2 }, ….. {s1 , s2 , s3 , s4 } … } Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability String Encoding: Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Coding Turing Machines Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Coding Transition Rules are: Code for M Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Diagonalization Language: Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Theorem:Ldis not a recursively enumerable language. That is there is no Turing Machine that accepts Ld Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Fig. 10 Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Decidable Languages about DFA Dept. of Computer Science & IT, FUUAST Theory of Computation

Decidability Halting and Acceptance Problems: Acceptance Problem: Does a Turing machine accept an input string? ATM is recursively enumerable. Dept. of Computer Science & IT, FUUAST Theory of Computation

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Theory of Computation