The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is said to be of effective strong measure zero if for any computable sequence {εn}n∈N of positive rationals, a sequence of intervals In of diameter εn covers the set. We observe that a set is of effectiv...

A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure ν on the Cantor space C (the set of all decision problems) and any suitable complexity class C ⊆ C, the theory identifies the subsets of C that are ν-measurable in C and assigns measures to these sets, thereby endowing C with internal measure-theoretic structure. Classes C...

Fuzzy Sets and Systems 161 (2010) 412-432 Elsevier www.elsevier.com/locate/fss A Categorical Semantics for Fuzzy Predicate Logic

does not satisfy (EP), because for x = 0.3, y = 0.5 and z = 0.5 we get I (x, I (y, z)) = 0.7 = 0.5 = I (y, I (x, z)). The corrected last two rows are provided in Table 1 here. In both of these cases the natural negation NI is the classical negation NC(x) = 1 − x. One can easily check that the first function satisfies (I1) and it does not satisfy (EP), because for x = 0.3, y = 1 and z = 0, we ge...

In this work we solve an open problem of U.Höhle [Problem 11, Fuzzy Sets and Systems 145 (2004) 471-479]. We show that the solution gives a characterization of all conditionally cancellative t-subnorms. Further, we give an equivalence condition for a conditionally cancellativite t-subnorm to be a t-norm and hence show that conditionally cancellativite t-subnorms whose natural negations are stro...

We extend Lutz's measure to probabilistic classes, and obtain notions of measure on probabilistic complexity classes C such as BPP, BPE and BPEXP. Unlike former attempts, all our measure notions satisfy all three Lutz's measure axioms, that is every singleton fLg has measure zero in C, the whole space C has measure one in C, and "easy innnite unions" of measure zero sets have measure zero. Fina...

The lattice Lu of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, Kawaguchi and Miyakoshi [Extended t-norms as logical connectives of fuzzy truth values, Multiple-Valued Logic 8(1) (2002) 53–69] showed that this lattice is a complete Heyting algebra. Later, Harding et al. [Lattices of convex, normal functions, Fuzzy Sets an...

• The exterior measure of any set A ⊂ R is μ∗(A) = inf E⊂A μ(E) where the infimum is taken over all elementary sets. • A is Lebesgue measurable if for all > 0 there exists an open set O ⊃ A such that μ∗(O \A) ≤ . • The (Lebesgue) measure of a measurable set is μ(A) = μ∗(A). A function f is measurable if the sets {t | f(t) ≤ a} are measurable for all a ∈ R. Definition 2 (Lebesgue integral) • For...