Sufficient Conditions for Decidability of Boolean Algebras

Leont’eva, M. N.

doi:10.1007/s10958-013-1621-1

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“…All predicates of L are uniformly computable in A. In [11,proof of Thm. 3], it was shown that, in a similar situation, we may calculate β (s), a Δ 0 2 -computable index for an algebraŷ Ψ * λ(s) .…”

Section: Constructionmentioning

confidence: 99%

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Strong Constructivizability of Boolean Algebras of Elementary Characteristic (∞, 0, 0)

Leontieva

2014

Algebra Logic

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We give a complete description of conditions of being strongly constructivizable for Boolean algebras of elementary characteristic (∞, 0, 0) in terms of being computable for a sequence of canonical Ershov-Tarski predicates on Boolean algebras.A model for a finitary language is said to be computable if its universe is a computable set of natural numbers, and operations and relations are computable. A computable model is ncomputable if there exists an algorithm which, given a finitary Σ n -formula and a tuple of elements, decides whether the formula is true on the tuple. A strongly computable model is one for which such an algorithm exists for all formulas of predicate calculus. A model is said to be strongly constructivizable if it has a strongly computable isomorphic copy.We will work with countable Boolean algebras, which, for brevity, are referred to merely as algebras. As a source of preliminary information on the theory of Boolean algebras, we use [1]. Following [2], we will treat Boolean algebras as models for a language Σ BA = {+, ·, −, 0, 1}, where the symbols +, ·, and − stand for union, intersection, and complement, respectively. The expression x 1 , . . . , x n |y means that x 1 + . . . + x n = y and x i · x j = 0 for i = j. The expression x − y is equal to x · (−y), x y = (x − y) + (y − x), and x ≤ y signifies that x + y = y.The set of atoms of an algebra A is denoted by At 0 (A), an ideal of atomless elements by Als 0 (A), and an ideal of atomic elements by Atm 0 (A). We write F 0 (A) for the Frechet ideal (an ideal generated by atoms); E(A) = Als 0 (A) + Atm 0 (A) is the Ershov-Tarski ideal. Let {E n } n∈ω be a sequence of iterated Ershov-Tarski ideals; i.e., E 0 (A) = {0}, E n+1 (A) = (E n • E)(A) =

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Section: Constructionmentioning

confidence: 99%

“…For this case, in particular, the ultimate answer was given in [9]. In [10][11][12][13][14], answers were found for all possible subsets S ⊆ Ψ n+1 , thereby accomplishing such a long-term pursuit.…”

mentioning

confidence: 99%

Strong Constructivizability of Boolean Algebras of Elementary Characteristic (∞, 0, 0)

Leontieva

2014

Algebra Logic

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Sufficient Conditions for Decidability of Boolean Algebras

Cited by 1 publication

References 6 publications

Strong Constructivizability of Boolean Algebras of Elementary Characteristic (∞, 0, 0)

Strong Constructivizability of Boolean Algebras of Elementary Characteristic (∞, 0, 0)

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