## On mass problems of presentability (2006)

Venue: | Li (Eds.): TAMC2006. LNCS 3959 |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Stukachev06onmass,

author = {Alexey Stukachev},

title = {On mass problems of presentability},

booktitle = {Li (Eds.): TAMC2006. LNCS 3959},

year = {2006},

pages = {774--784}

}

### OpenURL

### Abstract

Abstract. We consider the notion of mass problem of presentability for countable structures, and study the relationship between Medvedev and Muchnik reducibilities on such problems and possible ways of syntactically characterizing these reducibilities. Also, we consider the notions of strong and weak presentability dimension and characterize classes of structures with presentability dimensions 1. 1 Basic notions and facts The main problem we consider in this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to [4, 1, 13]. We denote the domains of a structures M, N,... by M, N..... For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of M as a subset, enables us to study effective (computable) properties of M by means of computability theory for admissible sets. The exact definition is as follows: the hereditary finite

### Citations

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Citation Context ...t the relationship between problems of presentability and some other mass problems. Considering the problems of enumerability, in [15] we obtain, by applying results and techniques due to J.F. Knight =-=[5]-=-, the following result, which is in some way analogous to Selman-Rozinas Theorem. Theorem 3. Let M be a structure, and A ⊆ ω, A �= ∅. Then the following are equivalent: 1) EA �w M; 2) EA � (M, ¯m) for... |

33 |
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Citation Context ... some presentation of M. We now distinguish the class of structures N for which the conditions 1, 2 and 3 are equivalent for any structure M. The next important notion was introduced by L. Richter in =-=[16]-=-. A structure M is said to have degree d if d = min{deg T (C) | C is a presentation of M}. The original definition from [16] was formulated with respect to presentations with domains ω only, but it is... |

26 |
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Citation Context ...elation of reducibility �, form a distributive lattice known as the Medvedev lattice [7]. There is another important notion of reducibility between mass problems, which was introduced by A.A. Muchnik =-=[9]-=-. Namely, if A and B are mass problems, then A is said to be weakly reducible to B (denoted by A �w B), if, for any f ∈ B, there is some recursive operator Ψ such that Ψ(f) ∈ A. So the weak (we will a... |

21 | Relative to any nonrecursive set
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Citation Context ...] we show that the requirement that a structure N have a degree in the Theorem 5 is essential and can not be dropped. For this we use the fact ( obtained independently by S. Wehner [17] and T. Slaman =-=[12]-=-) that there exist structures which mass problems of presentability belongs to the least non-zero degree of difficulty in the Medvedev lattice. Now we introduce some class of structures for which, con... |

16 |
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Citation Context ...n this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to =-=[4, 1, 13]-=-. We denote the domains of a structures M, N, . . . by M, N. . . .. For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of... |

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Citation Context ...ation C. 7s8 In [15] we show that the requirement that a structure N have a degree in the Theorem 5 is essential and can not be dropped. For this we use the fact ( obtained independently by S. Wehner =-=[17]-=- and T. Slaman [12]) that there exist structures which mass problems of presentability belongs to the least non-zero degree of difficulty in the Medvedev lattice. Now we introduce some class of struct... |

12 |
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Citation Context ... the atomic diagrams of all presentations of M (recall that, for a set A ⊆ ω, χA(n) = 0 if n ∈ A, and χA(n) is undefined otherwise). Any such set is a partial mass problem in the sense of E.Z. Dyment =-=[3]-=-, and we will call them partial problems of presentability. Such problems, in a different terminology, were considered with respect to classes of finite structures in [2]. In this case enumeration red... |

5 |
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Citation Context ...m in the sense of E.Z. Dyment [3], and we will call them partial problems of presentability. Such problems, in a different terminology, were considered with respect to classes of finite structures in =-=[2]-=-. In this case enumeration reducibility, the main object of study in [2], is no longer equivalent to Turing reducibility.sIn [7] it was introduced a notion of reducibility between mass problems. If A ... |

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Citation Context ... � ω? For such M we necessarily must have Pr-dimw(M) = 1. Indeed, this follows from the inequality Pr-dimw(M) � Pr-dim(M) and the next result observed independently by J.F. Knight [6] and I.N. Soskov =-=[14]-=-: for any M, Pr-dimw(M) is either 1 or uncountable. From this we immediately obtain that, for any M, Pr-dim(M) is either 1 or infinite. Recently I. Kalimullin (personal communication) showed that ther... |

2 |
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Citation Context ...n this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to =-=[4, 1, 13]-=-. We denote the domains of a structures M, N, . . . by M, N. . . .. For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of... |

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Citation Context ...lass of partial problems of presentability. There is a syntactical characterization of these reducibilities in the case of problems of enumerability, which follows from a result obtained by A. Selman =-=[11]-=- and rediscovered by M. Rozinas [10]: for any A, B ⊆ ω, A �e B if and only if, for any X ⊆ ω, the fact that B is X-c.e. implies that A is X-c.e.. From this theorem we directly obtain that, for any A, ... |

1 |
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Citation Context ...n this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to =-=[4, 1, 13]-=-. We denote the domains of a structures M, N, . . . by M, N. . . .. For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of... |

1 |
Degrees of models. Handbook of Recursive Mathematics
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Citation Context ...h that 1 < Pr-dim(M) � ω? For such M we necessarily must have Pr-dimw(M) = 1. Indeed, this follows from the inequality Pr-dimw(M) � Pr-dim(M) and the next result observed independently by J.F. Knight =-=[6]-=- and I.N. Soskov [14]: for any M, Pr-dimw(M) is either 1 or uncountable. From this we immediately obtain that, for any M, Pr-dim(M) is either 1 or infinite. Recently I. Kalimullin (personal communicat... |

1 |
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Citation Context ...el numbering of the atomic formulas of the signature of M. So any presentation, identified with its atomic diagram, can be considered as a subset of ω. A mass problem, as introduced by Yu.T. Medvedev =-=[7]-=-, is any set of total functions from ω to ω. Intuitively, a mass problem can be considered as a set of ”solutions” (in form of functions from ω to ω) of some ”informal problem”. Below we list some exa... |

1 |
Degrees of unsolvability of continuous functions. J.Symbolic Logic. 69
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(Show Context)
Citation Context .... implies that A is X-c.e.. From this theorem we directly obtain that, for any A, B ⊆ ω, EA �w EB ⇐⇒ EA � EB ⇐⇒ A �e B. Besides the syntactical characterization, it implies the fact (observed also in =-=[8]-=-) that Medvedev and Muchnik reducibilities coincide on the class of problems of enumerability. It is clear that (strong) Medvedev reducibility always implies (weak) Muchnik reducibility: for any mass ... |

1 |
On degrees of presentability of structures. I. Algebra and Logic (submitted
- Stukachev
(Show Context)
Citation Context ...reducibilities in the case of problems of presentability We now look at the relationship between problems of presentability and some other mass problems. Considering the problems of enumerability, in =-=[15]-=- we obtain, by applying results and techniques due to J.F. Knight [5], the following result, which is in some way analogous to Selman-Rozinas Theorem. Theorem 3. Let M be a structure, and A ⊆ ω, A �= ... |