A parallel computing model via pictures in an enriched-categorical context

An example of internal-valued non parallel computing via graphical representable elements is given in [1]. We want to use a similar approach to model intrinsically parallel computing, maintaining, at the same time, analogies with the communicating process approach [2]. It has been shown that a suitable topos of (pre)sheaves can provide a good semantics for communicating processes whose computations are labeled from an alphabet of elementary moves L (see e.g. [3]). Alternatively, a semantics for them can be provided by considering an “enrichment” [4] on a category of trees, obtained from the 2-category L built up from L, using concatenation [5]. The two approaches are not in opposition, because one can show that, under suitable conditions, a tree is a presentation of a sheaf over the space defined by its labeling monoid L (the base 2-category) [6]. Nonetheless, this second point of view allows us to consider objects in L as observers labeling computations as well as particular truth-values in the corresponding topos of sheaves. Another nice feature of this second approach is that we can vary the nature of these elements, and all the construction will change parametrically; e. g. L need not be a free meet-semilattice monoid [5]. In the present paper we make a further step and substitute elementary moves by pointed (multidimensional) pictures considered in [7], concatenated via the overlapping operation. As a consequence a single pointed (multidimensional) picture, considered as a coding of some data, becomes an elementary observation/computation move, and a computation will be labeled through a suitable overlapping of such things. The corresponding truth-values system will be of a great flexibility, exploiting the shape of the image in the picture as well as its definiteness degree.