Weak quantifier elimination for the full linear theory of the integers

Lasaruk, Aless; Sturm, Thomas

doi:10.1007/s00200-007-0053-x

Cited by 16 publications

(19 citation statements)

References 19 publications

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“…We recall the following theorem, which was proved in [7]: Also see [7] for a discussion on how this relates to similar previously-known results of Weispfenning and Lasaruk-Sturm [11]. Assumption 3.8.…”

Section: Definition 33mentioning

confidence: 77%

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Woods¹,

Goodrick²,

Bogart³

2017

Discreteanal.

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Parametric Presburger arithmetic concerns families of sets S t ⊆ Z d , for t ∈ N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the formRecent results of Chen, Li, Sam; Calegari, Walker; Roune, Woods; and Shen concern specific families in parametric Presburger arithmetic that exhibit quasi-polynomial behavior. For example, |S t | might be a quasi-polynomial function of t or an element x(t) ∈ S t might be specifiable as a function with quasi-polynomial coordinates, for sufficiently large t. Woods conjectured that all parametric Presburger sets exhibit this quasi-polynomial behavior. Here, we prove this conjecture, using various tools from logic and combinatorics.

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Section: Definition 33mentioning

confidence: 77%

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Woods¹,

Goodrick²,

Bogart³

2017

Discreteanal.

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“…Although we are focusing on the reals here, it is noteworthy that extended quantifier elimination is an established concept, which exists also for a variety of other important algebraic theories including the linear theory of valued fields (Sturm, 2000b), Presburger Arithmetic (Lasaruk and Sturm, 2007), initial Boolean algebras (Seidl and Sturm, 2003;Sturm and Zengler, 2010), and certain term algebras (Sturm and Weispfenning, 2002).…”

Section: The Concept Of Extended Quantifier Eliminationmentioning

confidence: 98%

Better answers to real questions

Košta

Sturm

Dolzmann

2016

Journal of Symbolic Computation

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The lack of access to visual information like text labels, icons, and colors can cause frustration and decrease independence for blind people. Current access technology uses automatic approaches to address some problems in this space, but the technology is error-prone, limited in scope, and quite expensive. In this paper, we introduce VizWiz, a talking application for mobile phones that offers a new alternative to answering visual questions in nearly real-time-asking multiple people on the web. To support answering questions quickly, we introduce a general approach for intelligently recruiting human workers in advance called quikTurkit so that workers are available when new questions arrive. A field deployment with 11 blind participants illustrates that blind people can effectively use VizWiz to cheaply answer questions in their everyday lives, highlighting issues that automatic approaches will need to address to be useful. Finally, we illustrate the potential of using VizWiz as part of the participatory design of advanced tools by using it to build and evaluate VizWiz::LocateIt, an interactive mobile tool that helps blind people solve general visual search problems.

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“…3 Note that in the special case most relevant for us, namely that each component of ψ is a different monomial, the assumption made in Lemma [26,27] was originally motivated by the efficient implementation of quantifier elimination based on virtual substitution methods [28][29][30]. Redlog also includes CAD and Hermitian quantifier elimination [31][32][33] for the reals as well as quantifier elimination for various other domains [34] including the integers [35,36]. The development of Redlog was initiated in 1992 by one of the authors (T. Sturm) of this paper and continues until today.…”

Section: Computational Logic Toolsmentioning

confidence: 99%

Detection of Hopf bifurcations in chemical reaction networks using convex coordinates

Errami

Eiswirth

Grigoriev

et al. 2015

Journal of Computational Physics

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We present ecient algorithmic methods to detect Hopf bifurcation xed points in chemical reaction networks with symbolic rate constants, thereby yielding information about the oscillatory behavior of the networks. Our methods use the representations of the systems on convex coordinates that arise from stoichiometric network analysis. One of our methods then reduces the problem of determining the existence of Hopf bifurcation xed points to a rst-order formula over the ordered eld of the reals that can be solved using computational logic packages. The second method uses ideas from tropical geometry to formulate a more ecient method that is incomplete in theory but worked very well for the examples that we have attempted; we have shown it to be able to handle systems involving more than 20 species

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Weak quantifier elimination for the full linear theory of the integers

Cited by 16 publications

References 19 publications

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Better answers to real questions

Detection of Hopf bifurcations in chemical reaction networks using convex coordinates

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