187 – The P vs NP problem

This note is based on lecture notes for the Caltech course Math 6c, prepared with A. Kechris and M. Shulman.

1. Decision problems

Consider a finite alphabet , and “words” on that alphabet (the “alphabet” may consist of digits, of abstract symbols, of actual letters, etc).

We use the notation to indicate the set of all “words” from the alphabet . Here, a word is simply a finite sequence of symbols from . For example, if is the usual alphabet, then

would be a word.

We are also given a set of words, and we say that the words in are valid. ( may be infinite.)

In the decision problem associated to , we are given as input a word in this alphabet. As output we say yes or no, depending on whether the word is in or not (i.e., whether it is “valid”).

We are interested in whether there is an algorithm that allows us to decide the right answer.

Goodstein sequences

Will Sladek, a student at Caltech, wrote an excellent introductory paper on incompleteness in PA, The termite and the tower. While Will was working on his paper, I wrote a short note, Goodstein’s function, on how to compute Goodstein’s function. Please let me know of any comments of corrections to either article.

117b – Some additional remarks on provably recursive functions

There are quite a few examples of natural combinatorial statements not provable in other than the ones discussed in lecture. The references listed below present several of them and provide a guide to the existent literature; I especially recommend the beautiful expository paper by Stephen Simpson and in general the whole collection where that paper appeared. I will ask you to abstain from looking at the Kanamori-McAloon paper for the time being, since its main result is the content of the last homework set.

There are at least two `combinatorial’ methods for showing that a function is not provably recursive. One is to use indicators, this technique, due to Paris, requires a bit of model theory, and is the method of the proof you are asked to provide in the homework.

The other method comes from a careful analysis of which functions are provably total in . For this, one defines a fast-growing hierarchy of recursive functions. This requires some understanding of the concept of an ordinal. The fast-growing (or Grzegorczyk) hierarchy is defined as follows:

• .
• , where the superindex means iterated times.
• , if is limit.

Here, is some (natural) sequence converging to (if you are not aware of ordinals, ignore for the moment what the last clause means).

For example, , , grows like a stack of powers of  and grows like an iterated stack of powers of , there being such iterations. It is easy to see that each is primitive recursive and strictly increasing, that < implies that is eventually dominated by (i.e., that < for all but finitely many values of ), and that every primitive recursive function is eventually dominated by some . Thus, if a function eventually dominates each , it cannot be primitive recursive.

The ordinals provide us with a method for continuing this sequence while preserving that its members are (eventually) increasing and the sequence itself is increasing under eventual domination. Say that a linearly ordered set is well-ordered if it has no infinite descending chains. For our purposes, the ordinals are simply a way to talk about well-ordered sets (classically, ordinals denoted the order types of the well-ordered sets, i.e., the equivalence classes of these sets under the relation of being order-isomorphic. The modern viewpoint is different and we don’t need it here). We use to denote the order type of any linearly ordered set of elements, and use to represent the order type of the natural numbers. We `add’ two order types and by forming the disjoint union of a set of order type  followed by a set of order type . We also say that is smaller than if any set of order type has a (proper) initial segment of order type . We can then continue the sequence of the natural numbers, as follows: 

where we use to represent , etc, to represent and so on, and we can continue even beyond:

Call (epsilon-0) the first ordinal obtained this way such that

(so ).

One can show that any ordinal below admits a unique `pure base ‘ representation, from which there is a natural way of defining the sequence converging to whenever is a limit, which means that it is not of the form for any . For example,

, etc,

are all limits. So, for example, the sequence corresponding to is just and the corresponding function is easily seen to grow as the (diagonal) Ackermann function ; the sequence corresponding to is , etc. As before, each is (eventually) strictly increasing and if < then eventually dominates . Each , for < (and in fact, for many more ) is recursive.

The relation of this sequence with lies in that any function provably recursive in is eventually dominated by one of the with <. So, one purely combinatorial way of showing that a function is not provably recursive is to show that it eventually dominates each such . This kind of analysis was developed by Solovay and Ketonen.

There are also other ways of relating to this number , and proof theorists have a lot more to say about this relation.

Additional references:

• Lecture notes on enormous integers, by H. Friedman. (2001)
• Unprovable theorems,  by H. Friedman. (2005) See www.math.ohio-state.edu/~friedman/manuscripts.html for additional manuscripts and details.
• Unprovable theorems and fast-growing functions, by S. Simpson. In Logic and combinatorics, S. Simpson, ed. AMS (1987).
• On Gödel incompleteness and finite combinatorics, by A. Kanamori and K. McAloon. Annals of Pure and Applied Logic 33 (1987), 23-41.

117b – Undecidability and incompleteness – Lecture 11

Let be r.e. and suppose there is a sentence  such that does not prove . Then  is more powerful than in two ways:  

1. It proves more theorems than ; for example,  .
2. It provides much shorter proofs of theorems of ; for example: For any (total) recursive function there is a sentence  provable in but such that in  there is a proof of  such that no proof of  in is shorter than .

This leads to considering those recursive functions of which can prove that they are total. One calls such functions provably recursive in . It is easy to show that for any -sound , there are recursive functions that are not provably recursive in . The statement that  is total,

,

is easily seen to be and for one can provide natural examples of such recursive but not provably recursive functions , thus providing natural (combinatorial) examples of independent -sentences. Actually, this can also be done for much stronger systems than , like , and Harvey Friedman has provided several such examples.

For , the most famous examples are perhaps the following:

• Goodstein (1944). The pure base- representation of a number is obtained by writting in base , writing the exponents of this representation also in base , writing the exponents of these representations in base , and so on. For let and for let be the result of writing the pure base- representation of , replacing each by , and subtracting . The Goodstein sequence of  is the sequence This sequence eventually converges to 0. The function  that to assigns the number of steps necessary for this to occur is recursive but not provably recursive in . In fact, given any provably recursive in , eventually dominates .
• Kirby, Paris (1982). Define a hydra to be a finite rooted tree. A head of the hydra is any node (other than the head) with only one edge connected to it, together with this edge. Hercules fights the hydra by chopping off its heads one by one by stages. At stage , the hydra grows new heads as follows: from the node that used to be attached to the head just removed, traverse one edge towards the root and from this node just reached, sprout copies of the part of the hydra above the edge just traversed. For any hydra, no matter how Hercules battles it, he eventually wins (i.e., the hydra is reduced to a single node). Moreover, given a hydra there is an absolute bound on the number of stages that it takes Hercules to win. Coding (in any reasonable fashion) hydras by numbers, the function that assigns this number to a given hydra is recursive but eventually dominates all functions provably recursive in .
• Paris, Harrington (1977). Given a set , let denote the collection of its subsets of size . Given a function , say that a subset of is homogeneous for iff restricted to  is constant, and say that it is large if . Ramsey’s theorem, provable in , states that for any there is an such that given any of size , any of size and any , there is a homogeneous for and of size at least . The function that to assigns the least such is well known to be primitive recursive (just about any standard proof of Ramsey theorem in any combinatorics book shows that it roughly grows exponentially). Paris and Harrington showed that if we also require that the homogeneous set be large, then the new statement is also true but the resulting function (clearly recursive) eventually dominates all functions provably recursive in .

117b – Undecidability and incompleteness – Lecture 10

A theory  (r.e., extending ) is reflexive iff it proves the consistency of all its finite subtheories. It is essentially reflexive iff each r.e. extension is reflexive.

Then  is essentially reflexive and therefore no consistent extension of is finitely axiomatizable. This is obtained by showing that, in spite of Tarski’s undefinability of truth theorem, there are (provably in ) –truth predicates for all .

We defined Rosser sentences and showed their undecidability. We also showed Löb’s theorem that if is an r.e. theory and , then . This gives another proof of the second incompleteness theorem.

Finally, we showed that the length of proofs of -sentences is not bounded by any recursive function: For any  r.e. and consistent, and any recursive function , there is a -sentence provable in but such that any proof of  in has length > .

117b – Undecidability and incompleteness – Lecture 9

Let and be r.e. theories in the language of arithmetic. Assume that is strong enough to binumerate all primitive recursive relations. This suffices to prove the fixed point lemma:

For any formula there is a sentence   such that ,

moreover,  can be chosen of the same complexity as .

The (Löb) Derivability conditions for  are the following three statements:

• For any , if , then .
• . 
• .

For example, suffices for the proof of the fixed point lemma and of the first derivability condition and suffices for the other two.

Gödel incompleteness theorems:

1. Let be r.e. and strong enough to satisfy the fixed point lemma and the first derivability condition. Let be r.e. and consistent. Then there is a true sentence such that does not prove . If is -sound, then does not prove either.
2. If in addition satisfies the other two derivability conditions, then does not prove , the statement asserting the consistency of .

As a corollary, is essentially undecidable: Not only it is undecidable, but any r.e. extension is undecidable as well. Gödel’s original statement replaced -soundness with the stronger assumption of –consistency.

117b – Undecidability and incompleteness – Lecture 8

We introduced Robinson’s arithmetic . It is a weak (and sound) theory that proves all true sentences. We also introduced Peano Arithmetic . is sound and designed to “capture” all true first order statements about . (The incompleteness theorems will show that it does not suceed: There are true statements that does not prove.)

We showed that represents all c.e. sets and binumerates all recursive sets, in particular all primitive recursive functions. actually proves that the representations of primitive recursive functions define functions, so in any (maybe non-standard) model of it makes sense to talk, for example, of the exponential function, or the function enumerating the primes.

We proved the fixed point lemma and deduced from it Tarski’s theorem on the undefinability of truth, thus formally solving the liar’s paradox.

117b – Undecidability and incompleteness – Lecture 7

Define essentially undecidable theories as those theories (of arithmetic) that are recursively axiomatizable but such that any recursive extension is undecidable and therefore incomplete. We showed that if is essentially undecidable then there are continuum many complete extensions of , none of which are c.e. The argument gives an example of a complete binary tree recursive in with no c.e. branches.

We showed that if is then its characteristic function is primitive recursive and that if is a c.e. relation, then it is the range of a primitive recursive function. We used this to prove Craig’s trick showing that a theory admits a c.e. axiomatization iff it admits a primitive recursive axiomatization.

We defined (numeralwise) representability and binumerability of a relation by a formula in a given theory. Next lecture we will show that there is an essentially undecidable finitely axiomatizable theory that represents all c.e. relations and binumerates all recursive relations. This will be the key to the proof of the incompleteness theorems.

117b – Undecidability and incompleteness – Lecture 6

We briefly discussed the problem of generalizing the proof of unsolvability of Hilbert’s tenth problem to other (recursive) rings.

We proved a version of the first incompleteness theorem as a consequence of the unsolvability of the tenth problem: If is any recursive set of axioms strong enough to prove a very weak fragment of arithmetic (the addition and multiplication tables for suffice) then either is not true of , or else is incomplete. The reason is that if is true of , then the set of consequences of is an undecidable r.e. set (by the unsolvability of the tenth problem), but the set of consequences of a complete theory is decidable.

Additional references:

• The incompleteness theorems,  by C. Smorynski. In Handbook of mathematical logic, J. Barwise, ed., North-Holland (1977), 821-865.
• Models of Peano Arithmetic, by R. Kaye. Claredon Press (1991).
• Aspects of incompleteness, by P. Lindström. Springer (1997).
• On formally undecidable sentences of Principia Mathematica and related systems (German: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I.), by K. Gödel. Monatshefte für Mathematik und Physik  38 (1931), 173-198. Reprinted in several places. See for example Collected Works, vol. I, K. Gödel. Oxford University Press (2001).

117b – Undecidability and incompleteness – Lecture 5

We completed the proof of the undecidability of Hilbert’s tenth problem, and discussed some of its consequences, tying up with the general theory of the degrees and leading directly into the second part of this topic: The incompleteness theorems.