# Linear Problems in Valued Fields

@article{Sturm2000LinearPI, title={Linear Problems in Valued Fields}, author={Thomas Sturm}, journal={J. Symb. Comput.}, year={2000}, volume={30}, pages={207-219} }

A first-order formula over a valued field is called linear if it contains no products or reciprocals of quantified variables. We give quantifier elimination procedures based on test term ideas for linear formulas in the following classes of valued fields: discretely valued fields, discretely valued fields with a Z -group as the value group over a language containing predicates stating divisibility in the value group, and non-discretely valued fields. From the existence of the elimination…

## 21 Citations

### Weak quantifier elimination for the full linear theory of the integers

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2007

A weak quantifier elimination procedure for the full linear theory of the integers, a generalization of Presburger arithmetic, where the coefficients are arbitrary polynomials in non-quantified variables, is described.

### Parametric Systems of Linear Congruences

- Mathematics
- 2001

Based on an extended quantifier elimination procedure for discretely valued fields, we devise algorithms for solving multivariate systems of linear congruences over the integers. This includes…

### Weak Integer Quantifier Elimination Beyond the Linear Case

- Computer Science, MathematicsCASC
- 2007

This work considers the integers using the language of ordered rings extended by ternary symbols for congruence and incongruence and describes a weak quantifier elimination procedure for univariately nonlinear formulas.

### Quantifier Elimination for Constraint Logic Programming

- Computer Science, MathematicsCASC
- 2005

This work presents an extension of constraint logic programming, where the admissible constraints are arbitrary first-order formulas over various domains: real numbers with ordering,linear constraints over p-adic numbers, complex numbers, linear constraints over the integers with ordering and congruences, quantified propositional calculus, term algebras.

### Thirty Years of Virtual Substitution: Foundations, Techniques, Applications

- Computer ScienceISSAC
- 2018

Virtual substitution has become an established computational tool, which greatly complements cylindrical algebraic decomposition and there are powerful implementations and applications with a current focus on satisfiability modulo theory solving and qualitative analysis of biological networks.

### Generalized Constraint Solving over Dierential Algebras

- Mathematics, Computer Science
- 2004

We describe an algorithm for quantifier elimination over differentially closed fields and its implementation within the computer logic package redlog of the computer algebra system reduce. We give…

### Effective Quantifier Elimination for Presburger Arithmetic with Infinity

- Computer Science, MathematicsCASC
- 2009

An effective quantifier elimination and decision procedure is given for Presburger arithmetic extended by infinity which implies also the completeness of the extension.

### 2 The Notion of Differentially Closed Fields

- Mathematics
- 2007

We describe an algorithm for quantifier elimination over di fferentially closed fields and its implementation within the computer log ic packageREDLOG of the computer algebra system REDUCE. We give…

### On the Existential Theories of Büchi Arithmetic and Linear p-adic Fields

- Mathematics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

The NP upper bound for existential linear arithmetic over p-adic fields resolves an open question and holds despite the fact that satisfying assignments in both theories may have bit-size super-polynomial in the description of the formula.

## References

SHOWING 1-10 OF 30 REFERENCES

### Simplification of Quantifier-Free Formulae over Ordered Fields

- Computer ScienceJ. Symb. Comput.
- 1997

An overview is given over various methods combining elements of field theory, order theory, and logic that do not require a Boolean normal form computation for simplifying intermediate and final results of automatic quantifier elimination by elimination sets.

### Applying Linear Quantifier Elimination

- Computer ScienceComput. J.
- 1993

The linear quantifier elimination algorithm for ordered fields in [15] is applicable to a wide range of examplee involving even non-linear parameters. The Skolem sets of the original algorithm are…

### Diophantine Problems Over Local Fields: III. Decidable Fields

- Mathematics
- 1966

In [3] we gave a complete set of elementary axioms for the valued field of p-adic numbers. In this paper we show how the valued field F((t)) of formal power series over a field F (of characteristic…

### The Complexity of Logical Theories

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1980

### Real Quantifier Elimination in Practice

- MathematicsAlgorithmic Algebra and Number Theory
- 1997

A survey over the knowledge on some structural properties of such rings and fields of invariants as polynomials and rational functions which are invariant under the action of a finite linear group.

### On Definable Subsets of p-Adic Fields

- MathematicsJ. Symb. Log.
- 1976

This paper shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields.

### The computational complexity of logical theories

- Mathematics
- 1979

and background.- Ehrenfeucht games and decision procedures.- Integer addition - An example of an Ehrenfeucht game decision procedure.- Some additional upper bounds.- Direct products of theories.-…

### Reasoning over Networks by Symbolic Methods

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 1999

This work discusses how this quantifier elimination procedure using test point ideas can be used for analysis, sizing, and error diagnosis of physical networks.

### REDLOG: computer algebra meets computer logic

- Computer ScienceSIGS
- 1997

This work illustrates some applications of REDLOG, describes its functionality as it appears to the user, and explains the design issues and implementation techniques.