Degrees of partial functions.

Turing Degrees and Muchnik Degrees of Recursively Bounded DNR Functions

Fuzzy type theory with partial functions

This paper is a study of fuzzy type theory (FTT) with partial functions. Out of several possibilities we decided tointroduce a special value ”∗” that represents ”undefined”. In the interpretation of FTT, this value lays outside of thecorresponding domain. In the syntax it can be naturally represented by the description operator acting on the empty(fuzzy) set, because choosing an element from it...

Partial second-order subdifferentials of -prox-regular functions

Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2  functions. The class of prox-regular functions covers all convex functions, lower C2  functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regula...

Degrees of unsolvability of continuous functions

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substr...

The Degrees of Unsolvability of Continuous Functions

Computable analysis provides a standard notion of computability for continuous functions on the real numbers. This notion was first explicitly formulated and studied by Lacombe and Grzegorczyk in the 1950’s, although it can be traced back to Turing and beyond that to Brouwer. However, a satisfactory notion of the degrees of unsolvability of continuous functions has only recently been introduced...