This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh's works have cristalized and popularized a concern that has appeared in the first half of the cent...

Over the past two decades the notion of a strong monad has found wide applicability in computing. Arising out of a need to interpret products in computational and semantic settings, different approaches to this concept have arisen. In this paper we introduce and investigate the connections between these approaches and also relate the results to monad composition. We also introduce new methods f...

Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m an...

Delfim F. 1t1. Torres [email protected] Departlnent of Mathematics University of Aveiro 3810-193 Aveiro, Portugal We study Smarandache sequences of numbers, and related problems, via a Computer Algebra. Sy::;tem. Solutions are di::;covered, and some conjectures presented. Mathematics Subject Classification 2000. llB83, 11-04, 68VV30.

Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, Löb’s formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence ...

The term parsing, derived from Latin pars orationis (parts of speech), was originally used to denote the grammatical explication of sentences, as practiced in elementary schools. The term was later borrowed into computer science and linguistics, where it has acquired a specialized sense in connection with the theory of formal languages and grammars. However, in practical applications of natural...

We try to explain Tarski’s conception of logical notions, as it emerges from a lecture of his, delivered in 1966 and published posthumously in 1986 (History and Philosophy of Logic, 7, pp. 143-154), a conception based on the idea of invariance. The evaluation of Tarski’s proposal leads us to consider an interesting (and neglected) reply to Skolem in which Tarski hints at his own point of view o...

It is an open problem in the area of computable randomness whether Kolmogorov-Loveland randomness coincides with Martin-Löf randomness. Joe Miller and André Nies suggested some variations of Kolmogorov-Loveland randomness to approach this problem and to provide a partial solution. We show that their proposed notion of partial permutation randomness is still weaker than Martin-Löf randomness.

Let V be a finite set andM a finite collection of subsets of V . ThenM is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V , then the elements of M are called convex sets and the pair (V,M) is called an aligned space or a convexity space. If S ⊆ V , then the convex hull of S, denoted by CH(S), is the smalles...

Embodiment has become an important concept in many areas of cognitive science. There are, however, very different notions of exactly what embodiment is and what kind of body is required for what kind of embodied cognition. This paper identifies and contrasts four different, increasingly restrictive notions of embodiment which can roughly be characterized as (1) ‘structural coupling’ between age...