1 Introduction to Computability Theory Lecture14: Recap Prof. Amos Israeli

1.Regular languages – Finite automata. 2.Context free languages – Stack automata. 3.Decidable languages – Turing machines. 4.Undecidability. 5.Reductions. Subjects 2

CFL-s Ex: RL-s Ex: The Language Hierarchy 3 Decidable Ex: Turing recognizable Ex: Non Turing recognizable Ex:

1.Defined DFA-s and their languages. 2.Defined NFA-s and their languages. 3.Defined RE-s and their languages. 4.Showed all three are equivalent. 5.Proved the Pumping lemma and demonstrated its use to prove irregularity. Regular Languages 4

Consider a. Show that L is regular. b. Present an RE for L. Training Problem 1 5

a. Consider Proof 6 a,b a,b, -

b. Consider Proof 7

1.Defined CFG-s and their languages. 2.Defined Stack automata and their languages. 3.Showed that the two classes are equivalent. 4.Proved the Pumping lemma for CFL-s and demonstrated its use to prove languages to be non CFL. Context Free Languages 8

Let Show that L is context free. Proof: Training Problem 2 9

1.Defined Turing machines, decidable languages and Turing recognizable languages. 2.Defined multi-tape TM-s and non deterministic TM-s, and showed their equivalence to ordinary TM-s. 3.Introduced the Church-Turing hypothesis. Decidable Languages 10

Consider a. Show that L is regular by presenting a DFA. b. Show that L is CF by presenting a PDA. c. Show that L is decidable by presenting a TM. Training Problem 3 11

Consider a. Show that L is regular by presenting a DFA. Training Problem ,1

Consider b. Show that L is CF by presenting a PDA. Training Problem ,1 0

Consider c. Show that L is decidable by presenting a TM. Training Problem ,1

1.Defined Cardinality of sets. 2.Showed that the cardinality of the rational numbers is equal to. 3.Used Diagonalization to show that the cardinality of infinite binary sequences is not equal to. Undecidability 15

4.Showed that the cardinality of Turing recognizable languages is equal to. 5.Showed that the cardinality of languages is larger than. 6.Concluded the existence of a non Turing recognizable language. Undecidability (cont.) 16

7.Defined and showed that it is undecidable. Undecidability (cont.) 17

1.Defined reductions. 2.Used reductions to prove that,,,, and many other problems are undecidable. 3.Defined mapping reductions. Reductions 18

Consider the following problem: Show that is undecidable. Training Problem 4 19

We show a reduction from to. Assume TM R is a decider for, let S= “On input where N is a TM 1. Let M be the TM rejecting all its inputs. 2. if R accepts (meaning ) - accept, otherwise reject.” Proof 20

We conclude that S never loops and it accepts iff. In other words: S is a decider for. Since is undecidable, we conclude that is also undecidable. QED Other practice problems: Prove by reduction from and from. Proof 21

Prove or disprove: a. If L is Turing recognizable then L is undecidable. Disprove: A Language L is Turing recognizable if there exists a TM, M, s.t.. If M, halts on every input then L is decidable. In other words: Every decidable language is also Turing recognizable. Training Problem 5 22

Prove or disprove: b. If a Turing machine moves its head only to the right then it must halt. Disprove: Present a state diagram of a TM that goes to the right forever. Training Problem 5 23

Prove or disprove: c. If a language A, is undecidable then its complement is also undecidable. Training Problem 5 24

Prove: Assume towards a contradiction that is decidable and let M be a TM deciding it. Consider TM M’ which is identical to M except that the accepting and rejecting states of M’ are switched. Clearly M’ accepts (rejects resp.) if and only if M rejects (accepts resp.), hence, M’ decides A, a contradiction. QED Training Problem 5 25

An ordinary Turing machine may either change its current cell or leave it unchanged. A changer is a TM that always changes its current cell. Show that every Turing recognizable language is recognizable by a changer TM. Training Problem 6 26

Let L be a Turing recognizable language and let M be a TM recognizing L, namely. Let be M ’s alphabet. Define a TM M’ whose alphabet is, where contains all the “barred” elements of. How should M ’s transition function be changed in order to keep its functionality? Proof 27

Let Consider the following problem: Show that is Turing recognizable. Training Problem 7 28

Consider the following TM: S= “On input where M, N are TM-s 1. Repeat 1.1 Run a single step of M on input w. 1.2 Run a single step of N on input w. 1.3 if either M or N accept - accept, if both reject - reject.” Proof 29

We can conclude: QED Proof 30