Implicit measurements of dynamic complexity properties and splittings of speedable sets

We prove that any speedable computably enumerable set may be split into a disjoint pair of speedable computably enumerable sets. This solves a longstanding question of J.B. Remmel concerning the behavior of computably enumerable sets in Blum's machine independent complexity theory. We specify dynamic requirements and implement a novel way of detecting speedability—by embedding the relevant measurements into the substage structure of the tree construction. Technical difficulties in satisfying the dynamic requirements lead us to implement “local” strategies that only look down the tree. The (obvious) problems with locality are then resolved by placing an isomorphic copy of the entire priority tree below each strategy (yielding a self-similar tree). This part of the construction could be replaced by an application of the Recursion Theorem, but shows how to achieve the same effect with a more direct construction.

An Optimal Disjoint Pair of Additive Spanners for the 3D Grid

A spanner of a connected graph G is a spanning connected subgraph S. If DG(u, v) and DS(u, v) denote the distance between vertices u and v in G and S, respectively, then S is an f(x)-spanner if and only if DS(u, v) ≤ f(DG(u, v)). The value f(x) - x is the delay of the spanner. An additive spanner is one with constant delay. Given a graph G and function f(x), an interesting problem is to partition the edges of G so as to define edge-disjoint f(x)-spanners of G. This paper exhibits a pair of (x + 4)-spanners for the infinite three dimensional grid, and shows that 4 is the least integer K for which there exists an edge-disjoint pair of delay-K spanners. spanner, grid, additive spanner, parallel network.

Double Generativity and Complete Effective Inseparability

§1. Complete Effective Inseparability. A disjoint pair (A1,A2) is by definition recursively inseparable if no recursive superset of A1 is disjoint from A2. This is equivalent to saying that for any disjoint r.e. supersets ωi and ωj of A1 and A2, the set ωi is not the complement of ωj —in other words, there is a number n outside both ωi and ωj. The disjoint pair (A1,A2) is called effectively inseparable—abbreviated E.I.—if there is a recursive function δ(x, y)—called an E.I. function for (A1, A2)—such that for any numbers i and j such that A1⊆ ωi and A2Í ωj. with ωi being disjoint from ωj, the number d(i , j) is outside both a;,- and ωj. We shall call a disjoint pair (A1, A2) completely E.I. if there is a recursive function δ(x, y)—which we call a complete E.I. function for (A1, A2)—such that for any numbers i and j, if A1⊆ ωi and A2Í ωj, then δ(i , j) Í ωi ↔ d(i , j)Í ωj (in other words, d(i, j) is either inside or outside both sets ωi and ωj.). [If ωi and ωj happen to be disjoint, then, of course, d(i, j) is outside both ωi and ωj, so any complete E.I. function for (A1,A2) is also an E.I. function for (A1,A2) In a later chapter, we will prove the non-trivial fact that if (A1, A2) is E.I. and A1 and A2 are both r.e., then (A1,A2) is completely E.I. [The proof of this uses the result known as the double recursion theorem, which we will study in Chapter 9.] Effective inseparability has been well studied in the literature. Complete effective inseparability will play a more prominent role in this volume—especially in the next few chapters. Proposition 1. (1) If (A1,A2) is completely E.I., then so is (A2,A1) —in fact, if d(x,y) is a complete E.I. function for (A1,A2), then d(y,x) is a complete E.I. function for (A2, A1).