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On computable aspects of algebraic and definable closure

Nathanael Ackerman, Cameron Freer, Rehana Patel
2020 Journal of Logic and Computation  
We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas.  ...  We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $\varSigma  ...  Acknowledgements The authors would like to thank Sergei Artemov, Valentina Harizanov, Anil Nerode, and the anonymous referees for helpful comments.  ... 
doi:10.1093/logcom/exaa070 fatcat:s6nnqk36jjdqtdfedfnqhw7wmi

Real algebraic closure of an ordered field

Renaud Rioboo
1992 Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92  
Real algebraic numbers appear in many Computer Algebra pro bJems.  ...  An implementation for the red algebrm"c closure has been done in Ax'om (previously called Scratchpad). P(x) = o. An extension jield K1 of K is said to be algebraic if al! its e!  ...  Every field has an algebraic closure; and if Cl and C2 are two algebraic closures of the same field they are iso- morphic. This enables to speak of the algebraic closure of a field.  ... 
doi:10.1145/143242.143312 dblp:conf/issac/Rioboo92 fatcat:r5tat2ivmbd2pcm3rhvi53aft4

An algebraic formulation of the aggregative closure query

Uchang Park
1996 Theoretical Computer Science  
The aggregative closure problem, a transitive closure problem with aggregations on transitive paths, is formally defined by database terms.  ...  So we can verify the existence of the fixpoint by the suggested conditions. The naive algorithm is proposed as a computational semantics for the aggregative closure problem.  ...  In this paper, the aggregative closure problem is formally defined for the framework and the analysis of fixpoint and its computational semantics.  ... 
doi:10.1016/0304-3975(95)00081-x fatcat:mpayhwnmvzhepkukpsl3xqcthi

Closures in Binary Partial Algebras

Guo-Qiang Zhang
2009 Electronical Notes in Theoretical Computer Science  
Two procedures for computing closures in binary partial algebras (BPA) are introduced: a Fibonacci-style procedure for closures in associative BPAs, and a multistage procedure for closures in associative  ...  , commutative and idempotent BPAs.  ...  We have illustrated how the notions of rank and closure interact with each other, and introduced two algorithms for computing closures in binary partial algebras.  ... 
doi:10.1016/j.entcs.2009.11.023 fatcat:glpcstdg3bg6fakvem5gtkwi24

Constraint Propagation Algorithms for Temporal Reasoning: A Revised Report [chapter]

Marc Vilain, Henry Kautz, Peter van Beek
1990 Readings in Qualitative Reasoning About Physical Systems  
As with the original, this revised document considers computational aspects of intervalbased and point-based temporal representations.  ...  The present paper departs from the original primarily in correcting claims made there about the point algebra, and in presenting some closely related results of van Beek [1989].  ...  and Engineering Research Council of Canada.  ... 
doi:10.1016/b978-1-4832-1447-4.50034-1 fatcat:2ufxmrvykvgxbmkeve7x42bq5e

On the Computation of the Algebraic Closure of Finitely Generated Groups of Matrices [article]

Klara Nosan, Amaury Pouly, Sylvain Schmitz, Mahsa Shirmohammadi, James Worrell
2021 arXiv   pre-print
In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time.  ...  We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices.  ...  Acknowledgements We thank Ehud Hrushovski for an explanation of [Hru02] and other useful suggestions.  ... 
arXiv:2106.01853v3 fatcat:t4fyk56yyjesxmzul6dzdj67pq

ALGORITHMS FOR GALOIS EXTENSIONS OF GLOBAL FUNCTION FIELDS

NICOLE SUTHERLAND
2016 Bulletin of the Australian Mathematical Society  
We investigate the efficient computation of integral closures, or maximal orders, in cyclic extensions of global fields and the determination of Galois groups for polynomials over global function fields  ...  We investigate two of these. Integral closures can be used to compute class groups, unit groups and Galois groups, which are the other three important tasks.  ...  Sutherland [2] of the computation of integral closures can therefore improve the computation of the genus, Riemann-Roch spaces and divisor class groups.  ... 
doi:10.1017/s0004972715001793 fatcat:ex7ucffxorgjxkak3yo5ouirry

Page 6278 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
The paper gives a polymorphic ac- count of relational algebra. It starts by defining type formulas and the notion of a principal-type formula for a relational alge- bra expression.  ...  The paper touches on issues of complexity and then defines the notion of a polymorphic query that it uses to draw attention to the fact that derived operators of the stan- dard relational algebra can become  ... 

The powerset algebra as a result of adding programming constructs to the nested relational algebra

Marc Gyssens, Dirk van Gucht
1988 SIGMOD record  
slight extension of the powerset algebra, thus emphasizing both the strength and the naturalness of the powerset algebra as a tool to manipulate nested relations, and, at the same time, lndlcatmg more  ...  In this paper, we discuss augmentations of the nested relatlonal algebra with programmmg constructs, such as while-loops and for-loops We show that the algebras obtruned in this way are equivalent to a  ...  Acknowledgment The first author IS a senior research assistant of the Belgian National Fund of Sclentlfic Research He also wishes to acknowledge the financial support of IBM Belgium, which enabled him  ... 
doi:10.1145/971701.50230 fatcat:zrrh3qarvvac5cdqqjyhn52cii

Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma

Tomas Sander
1991 Journal of Pure and Applied Algebra  
., Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma, Journal of Pure and Applied Algebra 73 (1991) 165-180.  ...  The possibility of such a proof and related results demonstrate fundamental differences between the concepts of real and algebraic closures of fields. 0022-4049/91/$03.511 0 1991 -Elsevier Science Publishers  ...  Friedrichsdorf and A. Prestel.  ... 
doi:10.1016/0022-4049(91)90110-n fatcat:xtlm4vw2ojffzmvix34aqcrkwu

The powerset algebra as a result of adding programming constructs to the nested relational algebra

Marc Gyssens, Dirk van Gucht
1988 Proceedings of the 1988 ACM SIGMOD international conference on Management of data - SIGMOD '88  
slight extension of the powerset algebra, thus emphasizing both the strength and the naturalness of the powerset algebra as a tool to manipulate nested relations, and, at the same time, lndlcatmg more  ...  In this paper, we discuss augmentations of the nested relatlonal algebra with programmmg constructs, such as while-loops and for-loops We show that the algebras obtruned in this way are equivalent to a  ...  Acknowledgment The first author IS a senior research assistant of the Belgian National Fund of Sclentlfic Research He also wishes to acknowledge the financial support of IBM Belgium, which enabled him  ... 
doi:10.1145/50202.50230 dblp:conf/sigmod/GyssensG88 fatcat:nybpv4jzj5cuxont6lwrpd6f6m

Computable valued fields [article]

Matthew Harrison-Trainor
2017 arXiv   pre-print
We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields.  ...  By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal'cev property, we show that they have computable dimension ω.  ...  There is a computable embedding of K into a computable presentation K of its algebraic closure and a computable extension of v to K. Proof.  ... 
arXiv:1602.08408v2 fatcat:e6ujilqvjvazflotjiop3u3szm

Krivine Machine and Taylor Expansion in a Non-uniform Setting

Antoine Allioux
2017 Electronic Proceedings in Theoretical Computer Science  
In the particular case of non-negative real numbers and of algebraic terms M representing probability distributions, the coefficient k gives the probability that the linear head reduction actually uses  ...  These annotations take the form of terms in the resource lambda-calculus. We generalize this resource-driven Krivine machine to the case of the algebraic lambda-calculus.  ...  Given any algebraic closure (M, E) and any stack of algebraic closures Γ 1 , . . . , Γ n with n ≥ 0, we first define T on closures and then extend it to states as follows: T(M, E) = M[T(E(x))/x] x∈Dom(  ... 
doi:10.4204/eptcs.238.3 fatcat:ocborfr3lvgd7chfacg7divqha

Computable valued fields

Matthew Harrison-Trainor
2017 Archive for Mathematical Logic  
We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields.  ...  By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal'cev property, we show that they have computable dimension ω.  ...  Let K be a class of computable structures that admits a r.i.c.e. pregeometry cl. 2 If each M in K of infinite dimension satisfies Conditions G and B, then K has the Mal'cev property.  ... 
doi:10.1007/s00153-017-0589-9 fatcat:npmr6j3wdfb27fiejcamudlmly

Page 3874 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
Comput. Sci. 169 (1996), no. 2, 185-200. The main result of this paper is a topological, algebraic and combinatorial characterization of the polynomial closure of group languages.  ...  Summary: “We define matchings, and show that they capture the essence of context-freeness.  ... 
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