1. Computability Go over homework problems. Godel numbering. Homework: prepare for midterm.

2. TM for Neat addition Create function TM that adds two numbers and returns as answer all 1s starting from the left end of the tape, no blanks. ????

3. TM for multiplication Tape starts with two sets of 1s, recall n+1 1 for number n. Follow example for addition. Consider using markers on tape. ????

4. TM to tidy up… Claim: we can create quintuples to add to any TM to re-format and modify outputs to facilitate composition –output for N is N 1s anywhere on tape. Change to (N+1) 1s. –insert or remove marker symbols separating numbers to be proper inputs.

5. Reprise: Recursive functions The recursive functions are a set of functions defined using a starter set and allowing any functions that can be defined using a finite number of applications of composition, primitive recursion, and minimalization

6. Starter set Identity F(x) = x This is special case of Projections U i n (x 1, x 2,…,x n ) = x i Successor S(x) = x+1 Constant F c (x) = c

7. Composition Given F and G, FG (x) = F(G(x))

8. Primitive recursion Motivation similar to mathematical induction, definition of factorial, exponentiation. –Have definition for the zero case. Have way for building. x 0 = 1 x (n+1) = x n * x

9. Primitive recursion Given functions f: N → N and g: (N,N,N) →N, then define h: (N,N) as follows h(x,0) = f(x) h(x,y+1) = g(x,y,h(x,y)) You will see variations. For example, f: N n and g: N n+2

10. Minimalization Better name MAY be inverse. If f(x) is recursive, then define g to be g(y) = min {x | f(x) = y} if any x exists. Otherwise, g(y) is undefined.

11. Multiplication … is recursive. Can be defined by use of composition, primitive recursion, minimalization of other recursive functions. Can assume addition! ????

12. Subtraction …is recursive. Can be defined by use of composition, primitive recursion, minimalization of other recursive functions. Can assume addition! Hint: use minimalization ????

13. Equivalence of TMs and Recursive functions If a function is recursive, then we can construct a TM for it. –Build TM for starter sets. Assume fix-up function to get things in proper format for next step. –Describe process for doing composition, primitive recursion, and minimalization. Minimalization: given function f for which there is a TM, call it F, then build a TM that calls F in turn for 0, 1, 2, ….

14. TM to Recursive Need to come up with encoding for TM that can be decoded to 'do' the steps as a function. –Gödel Numbering ! Prove set of steps that are recursive that lead to the existence of a computation using the steps of TM

15. Arithmetization of [theory of] Turing machines Symbols of TM are: –symbols available to be on tape: B=S0, S1,… –R and L –states: q 1,q 2 … Assign odd numbers starting with 3 to R,L,S0,q 1,S1,q 2,S2,… So quadruples q 1 1Rq 2 is represented by 9,11,3,13. The instantaneous description: q 1 1111 is represented by 9,11,11,11

16. Arithmetization, cont. Gödel Number of expression represented by sequence of n numbers a1,a2,..,an is the product of the first n primes, each raised to the corresponding ak power! gn(q 1 1Rq 2 ) = 2 9 * 3 11 * 5 3 * 7 13 gn(sequence of n expressions) = product of the first n primes each raised to the gn of each expression Note: gn's are either an expression or a sequence of expressions. gn's are huge! gn's are unique—can get back to expression or sequence.

17. Arithmetization, cont. Take any Turing Machine. (Using quadruples in place of quintuples. The operation of writing is distinct from moving left or right). TM defines by its set of quadruples. Define a gn for a TM is the gn for a sequence of its quadruples. So different orders will produce different gn.

18. Computation A computation is a sequence s 1,…s k of expressions representing steps of a TM operating on input – starting with q1 followed by input – sequence, s i  s i+1, according to TM – s k is terminal (final, no next step)

19. Description of proof With a gn for TM T, show by a set of steps involving what are valid computations starting from expression representing initial state in front of input, that a recursive function exists that has same behavior as T on the input.

20. Predicate A predicate is a function of 1 or more parameters that returns true or false. Proof shows that a certain predicate is primitive recursive (that is, in the set starting with the starting functions and with any functions added using primitive recursion and composition)

21. Predicate T(z,x1, x2, … xn,y) is true if z is the gn for a TM x1, x2,.. xn are the inputs y is a computation of the TM encoded by z T(x,x1,x2, …, xn, y) is false, otherwise, including z or y not being gn for any TM or any computation.

22. Claim By many steps, can show T to be primitive recursive Example, Yield(x,y,z) is defined as the predicate, such that if x and y are gn for instantaneous expressions and z is gn for a TM and x  y for TM z, then true, Otherwise, false

23. Define function in 2 steps Then V z (x1,…xn) defined as min y | T (z,x1, … xn, y) is recursive. Note: at most one such y.

24. Last step U(y), defined as: if y is a computation, number of 1s in last expression of y. Undefined, otherwise. This can be shown to be primitive recursive. Then U(min y | T (z,x1, … xn, y) ) is recursive

25. Comments The predicate T is very powerful. It is false most of the time. It does the job for ALL TMs. For any particular function computed by a particular TM, there would be a more efficient recursive function. The steps are based on the mechanistic feature of TMs. There are other formulations equivalent to these two (TMs and recursive functions).

26. More Topics Implementation of TM (language recognizer or function) in code –there exists on-line examples Gödel work on incompleteness of logical systems Other equivalent formulations for computability computable numbers Work on randomness and computability –Greg Chaitin, others. others cited previously

27. Homework Review midterm guide. Study charts. Come with questions based on your review.