On reduction of the decision problem of recursively enumerable sets to the separability problem

AbstractIt is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1 ⊆ Y2 such that Y ⊆ X1, Y2 ⊆ X2 and Y1, ⋃ Y2 = X1 ⋃ X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable.

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When catalytic P systems with one catalyst can be computationally complete

Journal of Membrane Computing ◽

10.1007/s41965-021-00079-x ◽

2021 ◽

Author(s):

Artiom Alhazov ◽

Rudolf Freund ◽

Sergiu Ivanov

Keyword(s):

Computational Complexity ◽

P Systems ◽

Membrane Systems ◽

Computational Completeness ◽

Recursively Enumerable ◽

Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

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Interpretability over peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2586787 ◽

1999 ◽

Vol 64 (4) ◽

pp. 1407-1425

Author(s):

Claes Strannegård

Keyword(s):

Modal Logic ◽

Completeness Theorem ◽

Peano Arithmetic ◽

Embedding Theorem ◽

Compactness Theorem ◽

Partial Answer ◽

Recursively Enumerable ◽

Recursively Enumerable Sets ◽

Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

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