Computability

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 ofProjections Uin(x1, x2,…,xn) = xi Successor S(x) = x+1 Constant Fc(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. • x0 = 1x(n+1) = xn * 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: Nn and g: Nn+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: q1,q2… • Assign odd numbers starting with 3 to R,L,S0,q1,S1,q2,S2,… • So quadruples q11Rq2 is represented by 9,11,3,13. The instantaneous description: q11111 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(q11Rq2) = 29 * 311* 53* 713 • 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 s1,…sk of expressions representing steps of a TM operating on input • starting with q1 followed by input • sequence, sisi+1, according to TM • sk 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 Vz(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.