I will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then pr...

Why is deciding simulation preorder (and simulation equivalence) computationally harder than deciding bisimulation equivalence on almost all known classes of processes? We try to answer this question by describing two general methods that can be used to construct direct one-to-one polynomial-time reductions from bisimulation equivalence to simulation preorder (and simulation equivalence). These...

This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain applications. Some of the traversals are novel, some have been known in principle but had not been described adequately for any number of dimensions, some of the t...

Reduced precision bounding volume hierarchies and ray traversal can significantly improve the efficiency of ray tracing through low-cost dedicated hardware. A key approach to enabling reduced precision computations during traversal is to translate the ray origin closer to the bounding volume hierarchy node after each traversal step. However, this approach precludes sharing of intersection compu...

We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S ⊂Rd . The algorithm is independent of the origin of coordinates. We show that (1) the graph has one topological end – that is, from any point there is exactly one infinite self-avoiding path; (2) the graph has a unique connected ...

type ref t = id:N{witnessed (has_a_t id t)} let has (id:N) (H h _) = match h id with Used _ _ (Untyped _)→⊤ | _→⊥ abstract type uref = id:N{witnessed (has id)}type uref = id:N{witnessed (has id)} To enforce these invariants on state-manipulating operations, we define a preorder rel on heap, that constrains the heap evolution. It states that every Used identifier remains Used; every Typed refere...

The former method starts by numerically estimating the adequacy and the inadequacy (with respect to the initial data) ofeach statement "the alternative i appears in on aggregate hierarchy on the j-th place* and constructs a preorder which maximizes the ratio of total sum of adequacy values by the total sum of inadequacy values for the associated hierarchy. The latter method involves usual conco...

A preorder on the line-of-sight (LOS) matrices in coherent multipleinput multiple-output (MIMO) Ricean fading channels is introduced. It is demonstrated that under this preorder, the information rate corresponding to zero-mean multivariate circularly symmetric Gaussian inputs of arbitrary covariance matrices is monotonic in the LOS matrix. This result implies the monotonicity of the information...

We present an extension of the classical testing semantics for the case when nondeterminism is unbounded. We deene the corresponding may and must preorders in the new framework. As in the bounded setting the may preorder can be characterized by using the set of nite traces of processes. On the contrary, in order to characterize the must preorder is necessary to record some additional informatio...