Publisher Summary
This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

CHAPTER 6 – Rewrite Systems

Greobner Bases : an algorithmic method in polynomial ideal theory

Mathematica--A System for Doing Mathematics by Computer.

https://www-polsys.lip6.fr/~jcf/Papers/F99a.pdf

A new efficient algorithm for computing Gröbner bases (F4)

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.

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Most Tensor Problems Are NP-Hard

Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal

Computer algebra is an alternative and complement to numerical mathematics. Its importance is steadily increasing. This volume is the first systematic and complete treatment of computer algebra. It presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature. The volume follows a top-down structure proceeding from very high-level problems which will be well-motivated for most readers to problems whose solution is needed for solving the problems at the higher level. The volume is written as a supplementary text for a traditional algebra course or for a general algorithms course. It also provides the basis for an independent computer algebra course.

Computer algebra: Symbolic and algebraic computation

We present a new criterion that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals. The application of the criterion may drastically reduce the number of reductions of polynomials in the course of the algorithm. Incidentally, the new criterion allows to derive a realistic upper bound for the degrees of the polynomials in the Grobner-bases computed by the algorithm in the case of polynomials in two variables.

A criterion for detecting unnecessary reductions in the construction of Groebner bases

Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems

We define a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of polynomial ideals (e.g. the problem of constructing canonical forms for polynomials) is reducible to the construction of bases of this type. We prove a characterization theorem for these bases which immediately leads to an effective method for their construction.

Bruno Buchberger

Papers

Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal

Computer algebra: Symbolic and algebraic computation

A criterion for detecting unnecessary reductions in the construction of Groebner bases

Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems

A theoretical basis for the reduction of polynomials to canonical forms