Quantifier Elimination over Finite Fields Using Gröbner Bases

Understanding Algebro-Geometric Quantifier Elimination: Part I, Algebraically Closed Fields of Characteristic Zero via Muchnik

In this short expository note, we present a self-contained proof that algebraically closed fields of characteristic zero admit elimination of quantifiers over the elementary language of rings. We do this by fleshing out a method due to Muchnik1 (and possibly Cohen2). This note is the first part of a series on algebro-geometric quantifier elimination: the next note shall use much of the machiner...

Tim Pruss , Priyank Kalla , Senior Member , IEEE , and

Abstraction plays an important role in digital design, analysis and verification. This paper introduces a word-level abstraction of the function implemented by a combinational logic circuit. The abstraction provides a canonical representation of the function as a polynomial Z =F (A) over the finite field F2k , where Z,A represent the k-bit word-level output and input of the circuit, respectivel...

Stability Analysis for Discrete Biological Models Using Algebraic Methods

This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on...

Counting Zeros over Finite Fields Using Gröbner Bases

Complex quantifier elimination in HOL

Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly interesting problems, we also implement a more efficient semidecision procedure for the universal fragment ba...