An F4-Style Involutive Basis Algorithm

How to solve a linear equation system? The echelon form of this system will be obtained by Gaussian elimination then give us the solution. Similarly, Gröbner Basis is the “nice form” of nonlinear equation systems that can span all the polynomials in the given ideal [4]. So we can use Gröbner Basis to analyze the solution of a nonlinear equation system. But how to compute a Gröbner Basis? There exist several ways to do it. Buchberger’s algorithm is the original method [2]. Gebauer-Möller algorithm [6] is a refined Buchberger’s algorithm. The F4 algorithm [5] uses matrix reduction to compute efficiently. Involutive Basis algorithm [8, 1, 12] is an effective method avoiding much ambiguity in the other algorithms. In Chapters 1 and 2 we describe two well-known methods of computing Gröbner Basis called Buchberger’s and F4 algorithm. In Chapter 3 after presenting the definition of involutive division we give a detailed formulation of basic and improved Involutive Basis algorithm. We will see that there exists ambiguity both in Buchberger’s and F4 algorithm. But in the method of Involutive Basis Algorithm, the ambiguity for the choice of prolongation has been avoided. So in Chapter 4 we combine the F4 algorithm and Involutive Basis algorithm in order to obtain a new approach that can reduce polynomials faster as well as avoid ambiguity. The combined algorithm called F4-involutive is a partial result due to its efficiency. More work such as implementing Buchberger’s criteria would be done in the future

An F4-Style Involutive Basis Algorithm

How to solve a linear equation system? The echelon form of this system will be obtained by Gaussian elimination then give us the solution. Similarly, Gröbner Basis is the “nice form” of nonlinear equation systems that can span all the polynomials in the given ideal [4]. So we can use Gröbner Basis to analyze the solution of a nonlinear equation system. But how to compute a Gröbner Basis? There exist several ways to do it. Buchberger’s algorithm is the original method [2]. Gebauer-Möller algorithm [6] is a refined Buchberger’s algorithm. The F4 algorithm [5] uses matrix reduction to compute efficiently. Involutive Basis algorithm [8, 1, 12] is an effective method avoiding much ambiguity in the other algorithms. In Chapters 1 and 2 we describe two well-known methods of computing Gröbner Basis called Buchberger’s and F4 algorithm. In Chapter 3 after presenting the definition of involutive division we give a detailed formulation of basic and improved Involutive Basis algorithm. We will see that there exists ambiguity both in Buchberger’s and F4 algorithm. But in the method of Involutive Basis Algorithm, the ambiguity for the choice of prolongation has been avoided. So in Chapter 4 we combine the F4 algorithm and Involutive Basis algorithm in order to obtain a new approach that can reduce polynomials faster as well as avoid ambiguity. The combined algorithm called F4-involutive is a partial result due to its efficiency. More work such as implementing Buchberger’s criteria would be done in the future

Polynomial Invariants for Affine Programs

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate

Highly Automated Formal Verification of Arithmetic Circuits

This dissertation investigates the problems of two distinctive formal verification techniques for verifying large scale multiplier circuits and proposes two approaches to overcome some of these problems. The first technique is equivalence checking based on recurrence relations, while the second one is the symbolic computation technique which is based on the theory of Gröbner bases. This investigation demonstrates that approaches based on symbolic computation have better scalability and more robustness than state-of-the-art equivalence checking techniques for verification of arithmetic circuits. According to this conclusion, the thesis leverages the symbolic computation technique to verify floating-point designs. It proposes a new algebraic equivalence checking, in contrast to classical combinational equivalence checking, the proposed technique is capable of checking the equivalence of two circuits which have different architectures of arithmetic units as well as control logic parts, e.g., floating-point multipliers