Quantifier-Elimination for the First-Order Theory of Boolean Algebras with Linear Cardinality Constraints

2004 ◽  
pp. 1-21 ◽  
Author(s):  
Peter Revesz
Keyword(s):  
Quantifier Elimination ◽  
Order Theory ◽  
Boolean Algebras ◽  
Cardinality Constraints ◽  
First Order ◽  
First Order Theory

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

On decidable varieties of Heyting algebras

1992 ◽  
Vol 57 (3) ◽  
pp. 988-991 ◽  
Author(s):  
Devdatt P. Dubhashi
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Structure Theory ◽  
Heyting Algebras ◽  
Quantitative Classification ◽  
First Order ◽  
First Order Theory ◽  
Famous Result

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

A remark on Martin's Conjecture

2001 ◽  
Vol 66 (1) ◽  
pp. 401-406
Author(s):  
Su Gao
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
First Order ◽  
First Order Theory ◽  
Elementary Theories

AbstractWe prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

scholarly journals SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

2015 ◽  
Vol 16 (3) ◽  
pp. 447-499 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Algebraic Closure ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Analytic Structure ◽  
Independence Property ◽  
Valued Fields ◽  
First Order ◽  
Witt Vectors ◽  
First Order Theory

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

scholarly journals Automated reasoning-alternative methods

2004 ◽  
Vol 1 (3) ◽  
pp. 15-20
Author(s):  
Aleksandar Perovic ◽  
Nedeljko Stefanovic ◽  
Milos Milosevic ◽  
Dejan Ilic
Keyword(s):  
Automated Reasoning ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Alternative Methods ◽  
Formal Representation ◽  
First Order ◽  
First Order Theory ◽  
Interpretation Method

Our main goal is to describe a potential usage of the interpretation method (i.e. formal representation of one first order theory into another) together with quantifier elimination procedures developed in the GIS.

scholarly journals THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS

2002 ◽  
Vol 02 (02) ◽  
pp. 145-225 ◽  
Author(s):  
STEFFEN LEMPP ◽  
MIKHAIL PERETYAT'KIN ◽  
REED SOLOMON
Keyword(s):  
Boolean Algebra ◽  
Order Theory ◽  
Boolean Algebras ◽  
Quotient Algebra ◽  
First Order ◽  
Finite Models ◽  
First Order Theory ◽  
Recursive Isomorphism ◽  
Atomic Boolean Algebra

In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.

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