Andrea Sorbi

Some Remarks on the Algebraic Structure of the Medvedev Lattice

This paper investigates the algebraic structure of the Medvedev lattice . We prove that is not a Heyting algebra. We point out some relations between and the Dyment lattice and the Mucnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.

Completeness and Universality of Arithmetical Numberings

We investigate completeness and universality notions, relative to different oracles, and the interconnection between these notions, with applications to arithmetical numberings. We prove that principal numberings are complete; completeness is independent of the oracle; the degree of any incomplete numbering is meet-reducible, uniformly complete numberings exist. We completely characterize which finite arithmetical families have a universal numbering.

Cupping and noncupping in the enumeration degrees of ∑20 sets

Abstract We prove the following three theorems on the enumeration degrees of ∑ 2 0 sets. Theorem A: There exists a nonzero noncuppable ∑ 2 0 enumeration degree. Theorem B: Every nonzero Δ 2 0 enumeration degree is cuppable to 0′ e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ 2 0 enumeration degree with the anticupping property .

Intermediate logics and factors of the Medvedev lattice

We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

We study computably enumerable equivalence relations (ceers), under the reducibility R ≤ S if there exists a computable function f such that x R y if and only if f(x) S f(y), for every x, y. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if R′ ≤ R, where R′ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are Σ3-complete (the former answering an open question of Gao and Gerdes).

Strong Enumeration Reducibilities

We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure

Universal Recursion Theoretic Properties of R.E. Preordered Structures

L(\mathfrak D_s)

Logic and probabilistic systems

of the s-degrees. However,