Arithmetic Fuzzy Models

ARITHMETIC-BASED FUZZY CONTROL

Fuzzy control is one of the most important parts of fuzzy theory for which several approaches exist. Mamdani uses $alpha$-cuts and builds the union of the membership functions which is called the aggregated consequence function. The resulting function is the starting point of the defuzzification process. In this article, we define a more natural way to calculate the aggregated consequence funct...

Fuzzy Number Intuitionistic Fuzzy Arithmetic Aggregation Operators

A fuzzy number intuitionistic fuzzy set (FNIFS) is a generalization of intuitionistic fuzzy set. The fundamental characteristic of FNIFS is that the values of its membership function and non-membership function are trigonometric fuzzy numbers rather than exact numbers. In this paper, we define some operational laws of fuzzy number intuitionistic fuzzy numbers, and, based on these operational la...

Fuzzy Arithmetic with Parametric LR Fuzzy Numbers

Cuts and overspill properties in models of bounded arithmetic

In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like $S_{2}^i$ and $T_{2}^i$. In correspondence with polynomial induction, we consider a rather new notion of cut that we call p-cut. We also consider small cuts, i.e. cuts that are bounded above by a small element. We study the basic properties of p-cuts and small cuts. In particula...

Canonical Models of Arithmetic

In 1983 Takeuchi showed that up to conjugation there are exactly 4 arithmetic subgroups of PSL2(R) with signature (1;∞). Shinichi Mochizuki gave a purely geometric characterization of the corresponding arithmetic (1;∞)-curves, which also arise naturally in the context of his recent work on inter-universal Teichmüller theory. Using Bely̆ı maps, we explicitly determine the canonical models of thes...