We introduce weak morphisms of higher dimensional automata and use them to define preorder relations for HDAs, among which homeomor-phic abstraction and trace equivalent abstraction. It is shown that homeomor-phic abstraction is essentially always stronger than trace equivalent abstraction. We also define the trace language of an HDA and show that, for a large class of HDAs, it is invariant und...

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category ∆ is replaced by a certain ‘fat’ delta of ‘coloured ordinals’, where the d...

We describe an approach to the verified implementation of transformations on functional programs that exploits the higher-order representation of syntax. In this approach, transformations are specified using the logic of hereditary Harrop formulas (HOHH). On the one hand, these specifications serve directly as implementations, being programs in the language λProlog. On the other hand, they can ...

In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein’s construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and st...

We prove that in the weak topology of measure preserving transformations , a dense G has purely singular continuous spectrum in the orthocomplement of the constant functions. In the uniform topology, a dense G of aperiodic transformations has singular continuous spectrum. We show that a dense G of shift-invariant measures has purely singular continuous spectrum. These results stay true for Z d ...