Spaces of directed paths on pre-cubical sets

Abstract

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also -sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set K , a CW-complex W (K ) w v that is homotopy equivalent to the space of directed paths between given vertices v, w of K . This construction is functorial with respect to K , and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of W (K ) w v are provided.

Spaces of directed paths on pre-cubical sets II

“…All these constructions are functorial with respect to K, regarded as an object in the category of bi-pointed -sets. This theorems generalize the results of [16].…”

“…. This definition generalizes the earlier definitions for d-paths on d-simplicial complexes [14] and on proper -sets [16].…”

“…Indeed, a cube chain c : ∨n → K determines the sequence (c(u dim(ci) )) l i=1 , and a sequence (c i ) l i=1 determines the unique -map (7.2) c : dim(c1) ∨ · · · ∨ dim(c l ) → K such that c(u dim(ci) ) = c i . This shows that the notion of a cube chain defined here coincides with the definition introduced in [16]. Below, we will identify cube chains c and the corresponding sequences (c 1 , .…”

“…The problem of calculating of the homotopy types of d-path spaces between two vertices of a -set was studied in several papers, eg. [10,11,16,17]. All these results work only for special classes of -sets, like Euclidean complexes or proper -sets, i.e., ones that their triangulations are simplicial complexes.…”

Strictifying and taming directed paths in Higher Dimensional Automata

“…It has been shown by Ziemiański (2017Ziemiański ( , 2020a) that the synchronisation request has no essential significance: The spaces of directed paths and of tame d-paths between two states are always homotopy equivalent. This has two consequences: On the one hand, one may, without global effects, relax the computational model and allow quite general parallel compositions.…”