Mining gradual rules plays a crucial role in many real world applications where huge volumes of complex numerical data must be handled, e.g., biological databases, survey databases, data streams or sensor readings. Gradual rules highlight complex order correlations of the form “The more/less X, then the more/less Y ”. Such rules have been studied since the early 70’s, mostly in the fuzzy logic ...

Session types enable fine-grained static control over communication protocols. Gradual typing is a means to safely integrate statically and dynamically typed program fragments. We propose a calculus for synchronous functional two-party session types, augment this calculus with a dynamically typed fragment as well as coercion operations between statically and dynamically typed parts, and establi...

a4 + 1 a5 + .. . will see that a less wasteful notation, say [ a0 , a1 , a2 , . . . ] , is needed to represent it. Anyone attempting to compute the truncations [ a0 , a1 , . . . , ah ] = ph/qh will be delighted to notice that the definition [ a0 , a1 , . . . , ah ] = a0 + 1/[ a1 , . . . , ah ] immediately implies by induction on h that there is a correspondence ( a0 1 1 0 ) ( a1 1 1 0 ) · · · (...

Given the fundamental importance of the real numbers in mathematics, it is important for mathematicians to have a logically sound description of the real number system. In particular, the adoption of set theory as the basic language for mathematics means that the real numbers need to be described formally in terms of set theory. As indicated in Munkres, two ways of approaching this are as follows: