The obscure and tedious way of mathematical language was designed to be unambiguous.
> It's difficult to talk about mathematical statements which are inconsistent, > in a consistent fashion. That’s because whenever a statement is inconsistent in mathematics, it gets tossed. A contradiction is a sentence together with its negation, and a theory is inconsistent if it includes a contradiction. Consider also the logical principle ex contradictione quodlibet (ECQ) (from a contradiction every proposition may be deduced--also recently called explosion). Unless you really do want a way to talk about inconsistencies—then you might try Paraconsistent Logic. The idea of paraconsistency is that coherence is possible even without consistency. Put another way, a paraconsistent logician can say that a theory is inconsistent without meaning that the theory is incoherent, or absurd. (perhaps expressly designed for those that are pro-life and pro capital punishment) Lets break down the statement “if we could solve problems whose complexity grows exponentially, then EMH would be true.” If P then Q P: > In 1965, Jack Edmonds [13] gave an efficient algorithm to solve this matching > problem and suggested a formal definition of “efficient computation” (runs in > time a fixed poly- nomial of the input size). The class of problems with > efficient solutions would later become known as P for “Polynomial Time”. > Computational complexity describes the time or space it takes to run an algorithm. > ...the known algorithms for many basic problems within P, including Frechet > distance, edit distance, string matching, k-dominating set, orthogonal > vectors, stable marriage for low dimensional ordering functions, and many > others, are essentially optimal. > We were thinking about distinguishing very hard problems, such as NP-complete problems, from relatively easy problems, such as those in P. Anyway the equalities of complexity classes translate upwards. For example, if P=NP, then EXP=NEXP. >>>>>> Since we can't But maybe we can. A computer algorithm might use brute force to go through all available options. Humans automatically search for a solution that intuitively feels right. Several NP-complete problems have exponential algorithms. Peter Shor’s algorithm finds the prime factors of an integer P. (effectively breaking RSA.) Previously the runtime was exponential—given call to Quantum computer—it would be polynomial. Shor's Algorithm in Quantum Computing - Topcoder > https://www.topcoder.com/blog/shors-algorithm-in-quantum-computing/ > <https://www.topcoder.com/blog/shors-algorithm-in-quantum-computing/> But what we need is an algorithm for an NP-complete problem that runs in polynomial time. We do not have that yet and it appears beyond the currently known techniques. Current approaches to the P vs NP problem are disguised forms of problems in cryptanalysis. In order to prove a function f is not in a complexity class C, exhibit some combinatorial property of f that provably prevents it from being in the class C. So let’s say in future we did have a solution. If P then Q P then Q Except on what basis did we establish P then Q ? Nowhere. P is independent of Q. P(A|B)=P(A)--the occurrence of B has no effect on the likelihood of A. Whether or not the event A has occurred is independent of the event B. So evaluate EMH as independent. The newer definition of efficient financial markets is that such markets do not allow investors to earn above-average returns without accepting above-average risks. Whatever patterns or irrationalities in the pricing of individual stocks that have been discovered in a search of historical experience are unlikely to persist and will not provide investors with a method to obtain extraordinary returns. Ah—risk adjusted. Ah—all patterns disappear and can’t be exploited. If that is the case—EMH is true because it is a tautology. > tautology | tɔːˈtɒlədʒi | > noun (plural tautologies) [mass noun] > the saying of the same thing twice over in different words, generally > considered to be a fault of style (e.g. they arrived one after the other in > succession). > • [count noun] a phrase or expression in which the same thing is said twice > in different words. > • Logic a statement that is true by necessity or by virtue of its logical > form. Donna Y [email protected] > On Sep 5, 2019, at 6:06 PM, Raul Miller <[email protected]> wrote: > > On Thu, Sep 5, 2019 at 6:03 PM Jose Mario Quintana > <[email protected]> wrote: >> That is why the claim, >> >>>>> In other words if we could solve problems whose complexity grows >>>>> exponentially, then EMH would be true. Since we can't, we can at least >>>>> classify EMH as not always being useful in the general case. >> >> Does not make any sense to me despite the expounding below (even assuming >> that the claim "if we could solve problems whose complexity grows >> exponentially, then EMH would be true" holds, which I regard as very >> misleading), > > Yeah, that carries a lot of assumptions with it. > > It's difficult to talk about mathematical statements which are > inconsistent, in a consistent fashion. > > Thanks, > > -- > Raul > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
