1. Francisco Antonio Doria
A Beautiful Theorem
Francisco Antonio Doria

2. But how can a theorem be beautiful
But how can a theorem be beautiful? A theorem may be difficult in its proof. Or easy. And may be useful. Or, perhaps, may be some kind of “mathematical garbage,” something which we can prove but which has no interest… But — a beautiful theorem ? How can a theorem be beautiful?

3. A beautiful theorem.
A beautiful theorem is a mathematical result that unexpectedly connects two seemingly distant subjects. It opens up new landscapes in the mathematical world. New vistas; perhaps even new worlds in that quite mystic region, the universe of mathematics.

4. In our case, the subjects to be connected are: computer science on one side, and economics on the other side. For we will discuss Maymin’s theorem: There are weakly efficient markets if and only if P=NP.

5. Subjects to be connected.
First subject: mathematical economics, the theory of efficient markets (the case of weakly efficient markets).
Second subject: computer science, complexity theory, the P vs. NP question.

6. We start from the side of Computer Science
We start from the side of Computer Science. More precisely, the P vs NP question.

7. The P vs. NP question The P vs NP question discusses problems so that:
Once we have a solution, it is easy to test the solution.
However, it is hard to find a solution — no fast algorithms are known so far.
And all such problems are seen to be equivalent: you settle one of them, you settle them all. The entire class is the class of NP- complete problems; they are connected by “easy,’’ “fast,’’ transformations.

8. In the P vs NP question: “easy’’ means, time-polynomial on the length of the input. “hard’’: time-exponential on the length of the input. The Big Question: are there easy algorithms for these problems?

9. P=NP: there are easy, fast polinomials algorithms.
P≠NP: there aren’t easy, fast algorithms; only exponential algorithms.

10. Now consider a theory S which:
Is consistent.
Contains “enough’’ arithmetic (Peano Arithmetic will be more than enough).
Has a recursively enumerable set of theorems.
Is based on a first-order classical predicate calculus.

11. Is consistent: it doesn’t prove contradictions.
Contains enough arithmetic: “talks’’ about integers and rational numbers.
Its theorems can be generated by a computer program.
Classical predicate calculus: its underlying logic is classical logic.

12. Two results For the negative answer, that is, if P≠NP:
A theory S proves P≠NP (no easy, fast algorithms) if and only if the counterexample function to P=NP is proved total in S.
If a theory S proves P≠NP then there is a recursive (computable) function, which is proved total by S, that dominates the (recursive version of) the counterexample function to P=NP.

13. A wild monster
The counterexample function f to P=NP is a wild monster with many avatars. In its full version f grows — in its peaks — at least as fast as the Busy Beaver function (which is a function that grows faster than any intuitively total computable function). However what counts are the computable versions of f, of which there are infinitely many, and whose growth rate is difficult to assess.

14. An interesting result
If P≠NP is independent of S (that is, S neither proves P=NP nor proves P≠NP) then P≠NP holds true of the standard model for the arithmetic portion of S. That is, S doesn’t prove P≠NP, but P≠NP is true for standard arithmetic — the usual integers we deal with.

15. So, in order that P≠NP holds true in standard (everyday) arithmetic, it suffices to have the sentence “P≠NP” independent of the axioms of our theory S.

16. Now, get ready for Maymin’s Theorem
Now, get ready for Maymin’s Theorem! But first we’ll need a few comments.

18. Weak efficient market hypothesis:
Current prices fully reflect all past publicly available information. (We can specify the nature of the “available information,’’ but for the moment the above condition is enough.)

19. (The publicly available information in the market’s past determines our gain in that market, for instance.)

20. The hypothesis as formulated is vague in an essential point
The hypothesis as formulated is vague in an essential point. We must make it more precise in order to be able to use it. The idea is: the market’s past is processed “as fast as possible” and the output determines its present state.

21. “as fast as possible” translates as “in polynomial time on the size of the input.” That is, information about previous data from the market is transmitted in P-time or poly-time, in the language of computer science, where P (or poly) stands for time polynomial on the length in binary notation of the input.

22. A very simplified model
There are two possible actions on the market, coded 1 and 0 (say, buy and sell).
The market has a memory of length k: a k-bit string (k-step past, a sequence of k 0s and 1s) determines the k+1 digit of the string (the current state).

23. (That k+1-th digit may also code our gain or loss in the market: 0 means “loss” and 1 means “gain.”)

24. For:
There are 2k possible k-bit strings which act upon the market.
They determine the k+1 bit through a mechanical, poly (``easy’’) procedure.
We will now require Post’s Theorem.

25. Post’s Theorem
Identify a binary string of length 2k to a truth table. Then there is a 1 – 1 map between all such 2k bit strings and truth tables for a k- variable Boolean expression; more precisely, 2k bit binary strings mapped onto k-variable Boolean expressions in conjunctive normal form (cnf). We thus code questions about markets as questions about Boolean expressions in cnf.

26. For each 2k bit sequence there is a single, k-variable Boolean cnf expression (adequately defined).
So there is a 1-1 map between the set of all 2k bit sequences and the set of k- variable Boolean cnf expressions.

27. Follows:
There is a 1-1 map between Boolean k- variable cnf forms and k-bit markets.
So we can code memory sequences for markets as Boolean k-variable cnf expressions.

28. Then:
(Maymin’s Theorem.) There are weakly effective markets if and only if P=NP holds.
Sketch of proof: One determines the k+1 digit in polynomial time if and only if P=NP.

29. Follows Maymin’s theorem, as weakly efficient markets are formally “the same as’’ Boolean expressions in cnf.

30. cnf Boolean expressions: Boolean expressions that are the conjunction of disjunctions of Boolean variables (or their negations).

31. Another version of Maymin’s result
Another version of Maymin’s result. I quote it as a paraphrasis: The market reacts fast [= is weakly efficient] if and only if P=NP So, weak market efficiency requires P=NP.

32. More on our requirements
We suppose that information about the market’s past is transmitted as if through a Turing machine to the present state of the market.
Therefore “fast” means “time- polynomial on the size of the input in bits.”

33. A consequence of Maymin’s Theorem
We start from:
If P≠NP isn’t proved by Primitive Recursive Arithmetic, then there is an algorithm which solves NP-complete problems in exponential time, but which at least is bounded by an exponential on the inverse of Ackermann’s function (taken as the total recursive function that tops all primitive recursive functions). That algorithm is O’Donnell’s algorithm.

34. Therefore:
If P≠NP isn’t proved by PRA, then there are almost weakly efficient markets.

35. A positive result We can define:
An almost weakly efficient market is a market as described where information is transmitted exponentially but bounded in time by an exponential at least like the inverse function of f, where f is Ackermann’s function.

36. Such exponential grows very slowly for a long initial stretch.

37. Thus our consequence of Maymin’s Theorem: If P≠NP isn’t proved by Primitive Recursive Arithmetic then there are almost weakly efficient markets.

38. Thank you!

39. Rio de Janeiro-Brazil