On Axiomatizability of the Multiplicative Theory of Numbers

Salehi, Saeed

doi:10.3233/fi-2018-1665

Cited by 6 publications

(16 citation statements)

References 12 publications

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“…Then in each other α j we replace, say, S k x = u first by S k+m x = S m u which in turn becomes S k t = S m u We now have a formula in which x no longer occurs,so the quantifier may be omitted . [2] Theorem2. The Theory (Z, 0, S, <) admits elimination quantifier, and so has a decidable theory and is finitely axiomatizable.…”

Section: Structuresmentioning

confidence: 99%

“…In each case, we have arrived at a quantifier-free version of the given formula. [2] V. The additive theory of Integer numbers Theorem3.The theory of the structure Z = {+, 0, 1, ≤ , {≡ m } ≥2 }, admits quantifier elimination, and this theory is decidablr theory.…”

Section: Structuresmentioning

confidence: 99%

“…So we showed the theory of the addition of the integer numbers of the language (+, 0, 1, ≤, (≡ n ) n≥2 ) admits quantifier elimination. [2,3]…”

Section: ∃X(mentioning

confidence: 99%

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Structure-Decidable structure-examples of decidable structures with proof-some examples of undecidable structures

Sheikhaleslami

2022

Preprint

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In this paper, decidability of the structures of real, rational, integer and natural numbers will be studied in different languages. Decidability or undecidability of mathematical structures is one of the fundamental and sometimes very difficult problems of mathematical logic, where several examples of problems in this field are still open and unresolved even after decades. One of the goals of Mathematical Logic is the axiomatization of mathematical theoriesTarski has proved the decidability of the theory of real and complex numbers in the language of addition and multiplication, and it is proved that theories of natural, integer, and rational numbers, in the language of addition and multiplication, are undecidable (Theorems of Gödel and Robinson). We will proof The following problemes: The Main Problem 1: ⟨Q; ⊑⟩ is decidable? Problem 2: an explicit axiomatization for ⟨Z; x⟩? and we will study boolean algebras. Boolean algebras are famous mathematical structures. Tarski showed the decidability of the elementary theory of Boolean algebras.In this paper, we study the different kinds of Boolean algebras and their properties. And we present for the first-order theory of atomic Boolean algebras a quantifier elimination algorithm. The subset relation is a partial order and indeed a lattice order, And I will prove that the theory of atomic Boolean lattice orders is decidable, and furthermore admits elimination of quantifiers. So the theory of the subset relation is decidable. And we will study decidability of atomlss boolean algebra.

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Section: Structuresmentioning

confidence: 99%

Section: Structuresmentioning

confidence: 99%

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Structure-Decidable structure-examples of decidable structures with proof-some examples of undecidable structures

Sheikhaleslami

2022

Preprint

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“…) is equivalent with a quantifier -free formula, ψ is equivalent with a formula with one less quantifier; counting this way one can show that ψ equivalent with a formula which has no quantifier. [1] A. Decidability of Structure of Natural Numbers in Different Languages Theorem1. the theory T hN s where N s = (N, 0, s) admits elimination of quantifier.…”

Section: Structure-decidable Structure-examples Of Decidablementioning

confidence: 99%

Structure-Decidable Structure-Examples of Decidable Structures With Proof-Some Examples of Undecidable Structures

Sheikhaleslami¹

2022

Preprint

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Abstract—In this paper, decidability of the structures of real, rational, integer and natural numbers will be studied in different languages. Decidability or undecidability of mathematical structures is one of the fundamental and sometimes very difficult problems of mathematical logic, where several examples of problems in this field are still open and unresolved even after decades. One of the goals of Mathematical Logic is the axiomatization of mathematical theoriesTarski has proved the decidability of the theory of real and complex numbers in the language of addition and multiplication, and it is proved that theories of natural, integer, and rational numbers, in the language of addition and multiplication, are undecidable (Theorems of Gödel and Robinson). We will review The following problemes: The Main Problem 1: ⟨Q; ⊑⟩ is decidable? Problem 2: an explicit axiomatization for ⟨Z; \times⟩? and we will study boolean algebras. Boolean algebras are famous mathematical structures.Tarski showed the decidability of the elementary theory of Boolean algebras.In this paper, we study the different kinds of Boolean algebras and their properties. And we present for the first-order theory of atomic Boolean algebras a quantifier elimination algorithm. The subset relation is a partial order and indeed a lattice order,And I will prove that the theory of atomic Boolean lattice orders is decidable, and furthermore admits elimination of quantifiers. So the theory of the subset relation is decidable. And we will study decidability of atomlss boolean algebra.

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“…2 Tables detailing these complexities can be found together at https://en.wikipedia.org/wiki/Circuits over sets of natural numbers in [32].…”

Section: Related Workmentioning

confidence: 99%

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Glaßer

Jönsson

Martin

2017

Theoretical Computer Science

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On Axiomatizability of the Multiplicative Theory of Numbers

Cited by 6 publications

References 12 publications

Structure-Decidable structure-examples of decidable structures with proof-some examples of undecidable structures

Structure-Decidable structure-examples of decidable structures with proof-some examples of undecidable structures

Structure-Decidable Structure-Examples of Decidable Structures With Proof-Some Examples of Undecidable Structures

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

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