116b- Lecture 20

March 13, 2008

Given any complete, consistent extension T of \mathsf{PA}, we showed that there is a minimal model K_T of T. 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 M\models\mathsf{PA}, let SSy(M), the standard system of M, be the set of those A\subseteq\mathbb{N} coded by elements of M, where a\in M codes A iff A=\{i\in\mathbb{N} : (a)_i\ne0\}. Thus SSy(\mathbb{N}) is the class of finite sets. We showed that if M\models\mathsf{PA} is nonstandard, SSy(M) contains all recursive sets, and that for any non-recursive S there is a nonstandard M\models \mathsf{PA} such that S\notin SSy(M).


116b- Homework 9

March 11, 2008

Homework 9

Due Tuesday March 18 at 1pm.

116b- Lecture 19

March 11, 2008

We 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, 2008

Hilbert’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 \Sigma_1)  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, 2008

Homework 8

Due Tuesday March 11 at the beginning of lecture.

Important update: In problem 4.(a), recall that U^1 is a universal \Sigma_1 predicate for unary formulas, so if x is the Gödel number of a \Sigma_1 formula \psi(v) in one free variable v, then U^1_x(y) holds iff \psi(y) holds. Hence, asking that U^1_x is finite is the same as asking that \{n:\mathbb{N}\models \psi(n)\} is finite.  Actually, this is a serious typo:

\{x:U^1_x \mbox{\ is finite}\} is \Sigma_{\bf 2}-complete. The set \{x \colon U^1_x \mbox{\ is cofinite}\} is \Sigma_3-complete.

Sorry about this. Either ignore 4.(a), or try to show (for extra credit) that the set is \Sigma_2-complete, or (for a much more challenging problem) that the corresponding set with “cofinite” is \Sigma_3-complete.

116b- Lecture 17

March 4, 2008

We proved the Rice, Shapiro, McNaughton theorem characterizing \Sigma_1 index sets.

We also showed that there incomparable Turing degrees below {\bf 0}'.

116b- Lecture 16

March 4, 2008

(Covered by Todor Tsankov)

We defined the analog K_X of the halting problem for any oracle X and showed that any set r.e. in X is many-to-one reducible to K_X (in particular, it is recursive in K_X). Hence, K_X is a complete \Sigma_1(X) set and no such set is recursive in X.

We proved the Smn (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 {\mathbb N} and \emptyset.

116b- Lecture 15

March 4, 2008

We 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 \Sigma_1 graphs, one can see this result as empirical evidence for Church’s thesis, the claim that the intuitive notion of “algorithmically 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 X\subseteq\mathbb{N}, a function f: \mathrm{dom}(f) \subseteq \mathbb{N}^k\to\mathbb{N} is computable by a Turing machine with oracle X iff f is recursive in X (meaning that it is built up from \chi_X and the basic functions by composition, recursion and minimalization) iff f has a graph \Sigma_1 definable in the structure (\mathbb{N},+,\times,0,1,<,X). We then defined the partial order A\le_T B among subsets of \mathbb{N} which holds iff \chi_A is recursive in B.

116b- Homework 7

February 25, 2008

Homework 7

Due Tuesday March 4 at the beginning of lecture.

116b- Lecture 14

February 21, 2008

We defined translations of a language in another and interpretations between theories. These notions allow us to state a more general version of the incompleteness results.

We then defined Turing machines.