Fuzzy Logic.

Fuzzy logic is the general calculation system that is a superset of traditional logic just Newton’s physics is a special case of Einstein’s physics. It was created in 1960’s by Zadek, a professor at the University of Columbia at Berkley. Togai says the technology will never replace traditional logic, since precise calculations will always be necessary. Fuzzy logic is the best used at the highest level in a system to narrow down the possibilities, relegating traditional digital logic to working out the details.

Fuzzy logic may be able to solve problems that involve controlling and modeling the real world when traditional logic is helpless. Simulating fuzzy logic in software is like emptying a dump track with a spoon. It can be done but it’s very time-consuming.

In traditional logic 0 is always a 0, and 1 is always a 1. In fuzzy logic the basic alphabet of bits is not limited to those 0s and 1s but involves any number of continuous mathematical functions called “membership” functions.

Critics of the theory charge that fuzzy logic provides no clear method of gathering data and combining them to the membership functions. Fuzzy logic-based systems are less susceptible to bugs because a single fault doesn’t crash the whole system. On the latest applications of fuzzy systems is positioning of shuttle by NASA in the USA.

Hilbert’s 10th problem.

The Hilbert’s 10^th problem is to find an effective method to decide if the Diophantine equation has solutions in integers.

Julia Robinson studied at Berkley and that’s logic was built by Tarsky. Julia was in the first seminar he gave in 1943. He had wonderful problem that he would give to students and they would captured by them. Tarsky had suggested that Julia try to prove one thing. Julia couldn’t prove it, so she tried to prove opposite. The outstanding result of her thesis was to show that the logical theory of irrational numbers is just as unsolvable as the theory of integers.

Julia Robinson was exited by combination of ideas from logic and ideas from the theory of numbers. So Martin Davis and Julia met and the first post-war international congress of mathematicians, and discussed their ideas. Both of them had started working on the same problem but had approached it from exactly opposite directions.

Julia Robinson showed that if she could construct a family of polynomial equations in which the solutions size grew exponentially with the size of the coefficients, then the undecidability of exponential Diophantine equations would imply the undecidability of ordinary Diophantine equations. In other words, she would have proved a negative answer to Hilbert’s 10^th problem.

Yuri Matiasevich, the Russian student, also worked on Hilbert’s 10^th problem and he had wrote a lot of paper, but when he was asked to review on of Julia’s papers, so he forced to read it. And when he read her paper he sort of understood immediately how the solution would be. But for proof Yuri used a member theoretic lemma which was not in any American number theory books.