CS 208: Computing Theory

1. CS 208: Computing Theory Assoc. Prof. Dr. Brahim Hnich Faculty of Computer Sciences Izmir University of Economics

2. Computability Theory Decidability

3. Motivation • Turing machines: • a general purpose computer; • Formalizes the notion of an algorithm (Church-Turing thesis) • We now turn our attention into the power of algorithms (i.e. Turing machines) to solve problems • Try to understand the limitation of computers

4. Is studying decidability useful? • An example of a decidable problem: • Is a string a member of a context-free language? • This problem is at the heart of the problem of recognizing and compiling programs in a programming language

5. Preliminaries • Acceptance problem: Does DFAB accept input string w? • For convenience we use languages to represent various computational problems • So, the acceptance problem can be expressed as a language • ADFA = {<B,w>| B is a DFA that accepts input string w}

6. Preliminaries • ADFA = {<B,w>| B is a DFA that accepts input string w} • The problem of testing whether a DFAB accepts a string w is the same as testing whether <B,w> is a member of the language ADFA • Thus, showing that the language is decidable is the same as showing that the computational problem is decidable!

7. Examples of Decidable Languages • ADFA = {<B,w>| B is a DFA that accepts input string w} • Theorem: ADFA is a decidable language • Proof idea: Design a TM M that decides ADFA • How? • On input <B,w> • Simulate B on input w • If simulation ends in accepting state, accept, otherwise reject

8. Examples of Decidable Languages • ANFA = {<B,w>| B is a NFA that accepts input string w} • Theorem: ANFA is a decidable language • Proof idea: Design a TM M that decides ANFA • How? • On input <B,w> • Convert NFA B into equivalent DFA C • Run previous TM on input <C,w> • If that TM accepts, accept, otherwise reject

9. Examples of Decidable Languages • AREX = {<R,w>| R is a regular expression that generates string w} • Theorem: AREX is a decidable language • Proof idea: Design a TM M that decides AREX • How? • On input <R,w> • Convert R into equivalent NFA C • Run previous TM on input <C,w> • If that TM accepts, accept, otherwise reject

10. Examples of Decidable Languages • ACFG = {<G,w>| G is a CFG that generates string w} • Theorem: ACFG is a decidable language • Proof idea: Design a TM M that decides ACFG • How? • (Interested students can read the book p. 156)

11. Examples of Decidable Languages • Theorem: Every CFGis a decidable

12. Relationships among classes of languages Context-free regular enumerable decidable

13. The Haling Problem • One of the most philosophically important theorems of the theory of computation • Computers (and computation) are limited ina very fundamental way • Common, every-day problem are unsolvable • Does a program sort an array of integers? • Both program and specification are precise • But, automating the verification is undecidable • No computer program can perform the task of checking the program against the specification!

14. Halting Problem • Halting problem: Does a Turing machine accept a string? • ATM = {<M,w>| M is a Turing machine that accepts string w} • Theorem: ATM is undecidable

15. Halting Problem • Before proving that ATM is undecidable, note that ATM is enumerable • How? Design a Turing Machine U that recongnizes ATM • On input <M,w> • Simulate M on w • If M ever enters its accept state, accept, and if M ever enters its reject state reject • U is called a universal Turing machine

16. Diagonalization • Diagonalization: a very crucial technique that is useful to prove undecidability ATM • Question: what does it mean to say that two infinite sets are the same size? • Answered by Georg cantor in 1873 • How? Pair them off!

17. Correspondence • Recall a correspondence f: A Bis a bijection: • Injective • Surjective • Question: what does it mean to say that two infinite sets are the same size? • Answer: A and B are the same size if there is a correspondence from A to B

18. Correspondence • Question: in a crowded room, how can we tell if there are more people than chairs, or more chairs than people?

19. Correspondence • Question: in a crowded room, how can we tell if there are more people than chairs, or more chairs than people? • Answer: Establish a correspondence: ask everyone to sit down!

20. Correspondence • Claim: The set of Natural numbers has the same size as the set of even numbers!

21. Correspondence • Claim: The set of Natural numbers has the same size as the set of even numbers! • Proof: Establish a correspondence • Let f(i)=2i Remark: a proper subset of A can be the same size as A!!!!

22. Countable • Definition: A set A is countable iff • Either it is finite, or • It has the same size as N, the set of natural numbers • We have just seen that the set of even numbers is countable • Claim: The set Z of integers is countable • Proof: Define f:NZ by • f(i)=i/2 If i is even • f(i)=-(ceil(i/2) +1) if i is odd

23. Challenge • In Heaven, there is a hotel with a countable number of rooms • One day, the society of Prophets, Oracles, and AI researchers hold a convention that books every room in the hotel • Then one more guest arrives, angrily demanding a room • You are the manager. What to do?

24. Challenge • Then one more guest arrives, angrily demanding a room • You are the manager. What to do? • Answer: Ask the guest in room i to move to room i+1, and put the new comer in room 1!

25. Challenge • Then a countable number of guests arrive, all angrily demanding a room • Now What to do?

26. Challenge • Then a countable number of guests arrive, all angrily demanding a room • Now What to do? • Answer: Ask the guest in room i to move to room 2i, and put the new comers in odd numbered rooms !!!! • (Infinity is such an amazing thing!!)

27. Rational Numbers • Let Q ={m/n | m and n are natural numbers} • Theorem: Q is countable • What is the correspondence between N and Q?

28. Rational Numbers …….. 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 …….. 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 …….. 3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 …….. 4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 . . . . . . . . . . . . . . . . . .

29. Rational Numbers …….. 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 …….. 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 …….. 3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 …….. 4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 Enumerate numbers along Northeast and Southwest diagonals, and skip duplicates

30. Rational Numbers …….. 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 …….. 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 …….. 3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 …….. 4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 Does this mean that every infinite set is countable?

31. The Real Numbers • Theorem: R, the set of reals is uncountable • Cantor introduced the diagonalization method to prove this theorem!

32. The Real Numbers • Theorem: R, the set of reals is uncountable • Proof: By contradiction • Assume there is a correspondence between N and R • Write it down • We show now that there a number x not in the list! • n f(n) • 3.143…… • 55.435…. • 3456.75… • 456.655…

33. n f(n) • 3.143…… • 55.435…. • 3456.75… • 456.655… Diagonalization Proof: By contradiction • Pick x between 0 and 1, so non-zero digits follow decimal point • First fractional digit of f(1) is 1 • Pick first fractional digit of x to be different, say 2 • Second fractional digit of f(2) is 4 • Pick second fractional digit of x to be different, say 6 • And so on …. • X=0.2487…. • Thus x is not the image of any natural which is a contradiction • So, R is uncountable!

34. Important implications • Previous theorem has an important application to the theory of computation • It shows that some languages are not decidable! • Or even Turing machine recognizable • There are languages that are not enumerable • The set of Turing machines is countable • The set of languages is uncountable

35. Important implications • Theorem: The halting problem is undecidable • Proof uses diagonalization technique (see bok those interested) • Theorem: A language is decidable if and only if it is both Turing-recognizable and co-Turing recognizable • In other words, a language is decidable if and only if it and its compliment are Turing-recognizable (enumerable)

36. A non-enumerable language • Corollory: if L is not decidable, then either L or its compliment is not enumerable ???? enumerable Co-enumerable decidable

37. Conclusions Decidable languages Halting problem diagonalization method undecidable example