FiniteForm

FiniteForm is a theoretical framework in algebra and computer science for representing and manipulating finite algebraic forms. The framework emphasizes finiteness of the representation to enable exact, finite computation while supporting a broad set of operations.

A FiniteForm consists of a finite set of basis elements, a finite sequence of coefficients from a field or ring, and a collection of operators including addition, multiplication, scalar multiplication, and projection or embedding operations. The space of all forms is closed under these operations, and each form can be encoded as a finite data structure, such as a vector of coefficients along with a specification of the basis. Some variants also incorporate a notion of grading or degree to organize computations.

Canonical forms in FiniteForm are defined via normalization rules that are designed to be confluent and terminating under a given set of reduction rules. This enables equality checking by comparing canonical representations. Variants may extend the framework with sparsity information, symmetry data, or other constraints to optimize storage and processing.

Computational aspects include algorithms for reduction, simplification, and basis changes, as well as methods for determining equivalence of two forms. The finitary nature of FiniteForm supports exact arithmetic and deterministic performance characteristics, which are advantageous for formal verification, symbolic computation, and educational demonstrations of algebraic structures.

Applications span symbolic algebra systems, finite-state verification, and optimization problems involving polynomials of limited degree. FiniteForm is typically implemented in software libraries as a data model for forms and a suite of arithmetic and normalization routines.

See also: polynomial form, quadratic form, boolean form, canonical form, normal form, polynomial algebra, finite-state machines.