A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities U is recursively categorical if every recursive linear order with recursive successivities isomorphic to U is in fact recursively isomorphic to U . We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k 1 + g 1 + k 2 + g 2 +…+ g n -1 (...) + k n where each k n is a finite order type, non-empty for i ϵ{2,…, n -1} and each g i is an order type from among {ω,ω * ,ω+ω * }∪{ k ·η: k. (shrink)

A linear order is -decidable if its universe is and the relations defined by formulas are uniformly computable. This means that there is a computable procedure which, when applied to a formula and a sequence of elements of the linear order, will determine whether or not is true in the structure. A linear order is decidable if the relations defined by all formulas are uniformly computable. These definitions suggest two questions. Are there, for each , -decidable linear orders that are (...) not -decidable? Are there linear orders that are -decidable for all but not decidable? The former was answered in the positive by Moses in 1993. Here we answer the latter, also positively. (shrink)

We prove a number of structural theorems about the honest polynomial m-degrees contingent on the assumption P = NP . In particular, we show that if P = NP , then the topped finite initial segments of Hm are exactly the topped finite distributive lattices, the topped initial segments of Hm are exactly the direct limits of ascending sequences of finite distributive lattices, and all recursively presentable distributive lattices are initial segments of Hm ∩ RE. Additionally, assuming ¦∑¦ = 1, (...) we show that the theory of the hpm-degrees is undecidable. We also show that index sets cannot be minimal. Lastly, we examine an alternative definition of honest m-reduction under which recursive minimal sets can be constructed. (shrink)

The block relation B(x,y) in a linear order is satisfied by elements that are finitely far apart; a block is an equivalence class under this relation. We show that every computable linear order with dense condensation-type (i.e., a dense collection of blocks) but no infinite, strongly η-like interval (i.e., with all blocks of size less than some fixed, finite k ) has a computable copy with the nonblock relation ¬ B(x,y) computably enumerable. This implies that every computable linear order has (...) a computable copy with a computable nontrivial self-embedding and that the long-standing conjecture characterizing those computable linear orders every computable copy of which has a computable nontrivial self-embedding (as precisely those that contain an infinite, strongly η-like interval) holds for all linear orders with dense condensation-type. (shrink)