For anything past March 18, 2008, see here.

## (Yes, we moved)

April 25, 2014## 116b- Lecture 20

March 13, 2008Given any complete, consistent extension of , we showed that there is a* minimal* model of . This model is unique up to (unique) isomorphism, it is rigid (i.e., it has no automorphisms other than the identity), and has no proper elementary substructures.

Given a model , let , the *standard system* of , be the set of those coded by elements of , where codes iff . Thus is the class of finite sets. We showed that if is nonstandard, contains all recursive sets, and that for any non-recursive there is a nonstandard such that .

## 116b- Homework 9

March 11, 2008Due Tuesday March 18 at 1pm.

## 116b- Lecture 19

March 11, 2008We showed that Exponentiation is Diophantine, completing the proof of the Davis-Matiyasevich-Putnam-Robinson theorem. The result follows from a careful examination of certain second order linear recurrence relations.

## 116b- Lecture 18

March 10, 2008Hilbert’s tenth problem asks whether there is an algorithm that given a polynomial with integer coefficients (in an arbitrary number of variables) determines whether it has integer roots. A celebrated theorem of Davis, Matiyasevich, Putnam and Robinson shows that this is not the case. Their result shows that the class of *Diophantine* sets coincides with the *a priori* larger class of r.e. (or ) sets.

We proved this result under the assumption that exponentiation is Diophantine. This is the key result, and will be dealt with in the following lecture.

## 116b- Homework 8

March 5, 2008Due Tuesday March 11 **at the beginning of lecture**.

**Important update:** In problem 4.(a), recall that is a universal predicate for unary formulas, so if is the Gödel number of a formula in one free variable , then holds iff holds. Hence, asking that is finite is the same as asking that is finite. Actually, this is a serious typo:

is -complete. The set is -complete.

Sorry about this. Either ignore 4.(a), or try to show (for extra credit) that the set is -complete, or (for a much more challenging problem) that the corresponding set with “cofinite” is -complete.

## 116b- Lecture 17

March 4, 2008We proved the Rice, Shapiro, McNaughton theorem characterizing index sets.

We also showed that there incomparable Turing degrees below .

## 116b- Lecture 16

March 4, 2008(Covered by Todor Tsankov)

We defined the analog of the halting problem for any oracle and showed that any set r.e. in is many-to-one reducible to (in particular, it is recursive in ). Hence, is a complete set and no such set is recursive in .

We proved the –– (or index function) theorem and Kleene’s recursion (or fixed point) theorem. Finally, we introduced the notion of an *index set *and proved Rice’s theorem that the only recursive index sets are and .

## 116b- Lecture 15

March 4, 2008We proved that the class of functions computable by means of Turing machines coincides with the class of recursive functions. Together with the characterization of the recursive functions as those admitting graphs, one can see this result as empirical evidence for Church’s thesis, the claim that the intitive notion of “algorithymically computable” is correctly formalized in the notion of recursive. An immediate consequence of this result is the existence of universal Turing machines.

We defined machines with oracles and stated the corresponding result: Given an oracle , a function is computable by a Turing machine with oracle iff is recursive in (meaning that it is built up from and the basic functions by composition, recursion and minimalization) iff has a graph definable in the structure . We then defined the partial order among subsets of which holds iff is recursive in .

## 116b- Homework 7

February 25, 2008Due Tuesday March 4 at the beginning of lecture.