1. The Recursion Theorem Pages 217– 227 1 ADVANCED TOPICS IN C O M P U T A B I L I T Y THEORY

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3. Recursion It concerns the possibility of making machines that can construct replicas of themselves. 1. Living things are machines (operate in a mechanistic way). 2. Living things can self-reproduce (essential characteristic). 3. Machines cannot self-reproduce. 3

4. a machine A that constructs a machine B A must be more complex than B. But a machine cannot be more complex than itself. How can we resolve this paradox? Making machines that reproduce themselves is possible. (The recursion theorem demonstrates how.) 4

5. SELF-REFERENCE Let's begin by making a Turing machine that ignores its input and prints out a copy of its own description. We call this machine SELF. To help describe SELF, we need the following lemma. LEMMA 6.1 5

6. SELF-REFERENCE (algorithm) 6

7. Machines A and B The job of A is to print out a description of B, and conversely the job of B is to print out a description of A. The result is the desired description of SELF. The jobs are similar, but they are carried out differently. Our description of A depends on having a description of B. So we can't complete the description of A until we construct B. 7

8. Machines A and B (cont) For A we use the machine P, described by q( ). q( ) means applying the function q to. If B can obtain, it can apply q to that and obtain. B only needs to look at the tape to obtain. Then after B computes q( ) =, it combines A and B into a single machine and writes its description = on the tape. 8

9. Machines A and B (algorithm) 9

10. TM that prints its own description 10 Suppose that we want to give an English sentence that commands the reader to print a copy of the same sentence. One way to do so is to say: Print out this sentence.

11. Example 2 11

12. TERMINOLOGY FOR THE RECURSION THEOREM recursion theorem in TM - (If you are designing a machine M, you can include the phrase "obtain own description " in the informal description of M's algorithm.) Two ways: 1. use any other computed value 2. simulate. 12

13. Algorithms 13 2. simulate. 1. use any other computed value APPLICATIONS - computer virus

14. THEOREM 6.5 THEOREM 6.6 THEOREM 6.8 14

15. Decidability of logical Theories What is a theorem? What is a proof? What is truth? Can an algorithm decide which statements are true? Are all true statements provable? 15

16. Decidability of logical Theories (cont.) We focus on the problem of: determining whether mathematical statements are true or false and, investigate the decidability of this problem. Can be done - an algorithm to decide truth and another for which this problem is undecidable. 16

17. Statement 1 - infinitely many prime numbers exist - solved. Statement 2 is Fermat' last theorem - solved, and Statement 3 - infinitely many prime pairs 1 exist - unsolved. 1. differ by 2 17 Decidability of logical Theories (cont)

18. let's describe the form of the alphabet of this language: 18 - A formula is a well-formed string over this alphabet - All quantifiers appear in the front of the formula. - A variable that isn't bound within the scope of a quantifier is called a free variable Decidability of logical Theories (cont.)

19. Examples 1. is the universe over which the variables may take values. A universe together with an assignment of relations to relation symbols is called a model. 19

20. Examples (cont) Formally we say that a model M is a tuple (U, P 1,..., P k ), and language of a model is the collection of formulas that use only the relation symbols the model assigns and that use each relation symbol with the correct arity. 20

21. EXAMPLE 6.11 (decidable) let M 2 be the model whose universe is the real numbers R and that assigns the relation PLUS to R 1, where PLUS(a, b, c) = TRUE whenever a + b = c. Then M 2 is a model of Replacing R with N in M2, the sentence would be false. 21

22. A DECIDABLE THEORY Church showed that Th( N, +, x), the theory of this model, is undecidable. 22 PROOF IDEA