Hristo Ganchev

Definability via Kalimullin pairs in the structure of the enumeration degrees

We give an alternative definition of the enumeration jump operator. We prove that the class of total enumeration degrees and the class of low enumeration degrees are first order definable in the local structure of the enumeration degrees.

The jump operator on the ω-enumeration degrees☆

Abstract The jump operator on the ω -enumeration degrees was introduced in [I.N. Soskov, The ω -enumeration degrees, J. Logic Computat. 17 (2007) 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure D ω ′ of the ω -enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of D ω ′ and of the enumeration degrees are isomorphic. In the second part of the paper we study the jumps of the ω -enumeration degrees below 0 ω ′ . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω -enumeration degrees below 0 ω ′ which are high n for some n and of the class L of the ω -enumeration degrees below 0 ω ′ which are low n for some n .

Embedding distributive lattices in the Σ02 enumeration degrees

We prove that every countable distributive lattice is embeddable in the Σ02 enumeration degrees via a 0—1 preserving monomorphism. Moreover, we prove that every countable distributive lattice is embeddable below arbitrary Δ02 degree via a 0 preserving monomorphism.

Exact Pair Theorem for the ω -Enumeration Degrees

In the paper the exact pair theorem for theω-enumeration degrees is proved. As a corollary anexact pair theorem involving the jump operation for the enumerationdegrees is obtained. Mathematics subject classification:03D30.

Flexible Noisy Text Correction

We present a new general and language independent approach to the noisy text correction problem developed and implemented in the framework of the CULTURA project. We briefly describe the core candidate generator, REBELS, the complete system concept, its efficient implementation based on functional automata and its immediate applications. The quality of the whole system is empirically established in different experimental settings where language and noise sources are varied.

A note on the enumeration degrees of 1-generic sets

We show that every nonzero

Ivan Soskov: A Life in Computability

{\Delta^{0}_{2}}

The ω-Turing degrees

Δ20 enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.