Finite representability of semigroups with demonic refinement

The motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

REPRESENTATION CLASS AND GEOMETRICAL INVARIANTS OF QUANTUM STATES UNDER LOCAL UNITARY TRANSFORMATIONS

We investigate the equivalence of bipartite quantum mixed states under local unitary transformations by introducing representation classes from a geometrical approach. It is shown that two bipartite mixed states are equivalent under local unitary transformations if and only if they have the same representation class. Detailed examples are given on calculating representation classes.

Towards New Transitions Among Shape Representations

Four classes of shape representation are dominating nowadays in computer-supported design and modeling of products, (1) point clouds, (2) surface meshes, (3) solid/surface models and (4) design/styling models. To support applications such as high-level shape design, feature-based design, shape modeling, shape analysis, rapid prototyping, feature recognition and shape presentation, it is required that transitions among and within the four representation classes take place. Transitions from a “lower” representation class to “higher” class are far from trivial, and at the same time highly demanded for reverse design purposes. New methods and algorithms are needed to accomplish new transitions. A characterization of the four classes is presented, the most relevant transitions are reviewed and a relatively new transition, from point cloud directly to design/styling model is proposed and experimented. The importance of this transition for new methods of shape reuse and redesign is pointed out and demonstrated.