1 Let S be the set of all objects considered, or universal set. A Function with n arguments (or: n-ary function, or: n-ary mapping, or n-ary operation) is a subset of S n+1 : S 1 ×S 2 × @@@ ×S n+1 , with S i=1, 2, , n+1 f S, é i, n , i # n+1, such that for any two different of its elements a=(a 1 , a 2 ,@@@, a n+1 ) and b=(b 1 , b 2 ,@@@, b n+1 ), we have not simultaneously a 1 =b 1 , a 2 =b 2 ,@@@, a n =b n and a n+1 û b n+1 . The domain of a n-ary function f is the set D f f S n : D f ={(a 1 , a 2 ,@@@, a n ) * þ a n+1 such that (a 1 , a 2 ,@@@, a n , a n+1 ) 0 f}, and the range (or codomain, or image) is the set R f f S: R f ={a n+1 * þ a 1 , a 2 ,@@@, a n such that (a 1 , a 2 ,@@@, a n , a n+1 ) 0 f}; Let this relationship be expressed by f : D f 6 R f . f maps D f onto R f , and maps D f into R, where R f f R f S. Further notice that a function is a special case of a binary relation where each element of the domain can only be related to one element in the range, in other words, one-to-many relations are not functions. Luis Rocha 1 ARTIFICIAL SEMANTICALLY CLOSED OBJECTS LUIS MATEUS ROCHA Los Alamos National Laboratory, MS P990, Los Alamos, NM 87545, USA e-mail: rocha@lanl.gov Communication and Cognition - Artificial Intelligence. Vol. 12, nos. 1-2, pp. 63-90. Special Issue: 6HOI 5HIHUHQFHLQ%LRORJLFDODQG&RJQLWLYH6\VWHPV/5RFKD(G ABSTRACT The notion of computability and the Church -Turing Thesis are discussed in order to establish what is and what is not a computational process. Pattee’s Semantic Closure Principle is taken as the driving idea for building non-computational models of complex systems that avoid the reductionist descent into meaningless component analysis. A slight expansion of Von Foerster’s Cognitive Tiles are then presented as elements observing Semantic Closure and used to model processes at the level of the cell; as a result, a model of a rate-dependent and memory empowered neuron is proposed in the construction of more complex Artificial Neural Networks, where neurons are temporal pattern recognition processors, rather than timeless and memoryless boolean switches. 1 - Computability and the Church-Turing Thesis. Within Mathematics computation or computability are notions unequivocally defined by an array of equivalent descriptions such as Turing machines [Turing, 1936], lambda- calculus (equivalent to general recursive functions [Church, 1936]), the Post [1965] symbol manipulating system, Markov [1951] algorithms, Minsky’s [1967] register machines, etc. The objective of formally describing computability lies in defining those functions 1 for