[Paper Review] Free Fractions: An Invitation to (applied) Free Fields

Konrad Schrempf|arXiv (Cornell University)|Sep 14, 2018

Computability, Logic, AI Algorithms12 references4 citations

TL;DR

This paper presents a computational framework for working with non-commutative rational functions—free fractions—using admissible linear systems (ALS) as a representation. It introduces algorithms for addition, multiplication, inversion, and minimization of these systems, enabling effective manipulation and equivalence testing in the free field, with the key contribution being a practical, algorithmic approach to the word problem in non-commutative rational functions.

ABSTRACT

Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for example 2*3=6=3*2. On the other hand --even before we can write-- we learn to talk (in a language) using words, consisting of purely non-commuting "letters" (or symbols), for example "xy" is not equal to "yx" (with the concatenation as multiplication). Now, if we combine numbers (from a field) with words (from the free monoid of an alphabet) we get non-commutative polynomials which form a ring (with "natural" addition and multiplication), namely the free associative algebra. Adding or multiplying polynomials is easy, for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the integers and the non-commutative polynomials look rather different, they share many properties, for example the unique number of irreducible factors: x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of fractions (aka "free field") of the free associative algebra is highly non-trivial (but really beautiful). Therefore we provide techniques (building on the work of Cohn and Reutenauer) to calculate with free fractions (representing elements in the free field or "skew field of non-commutative rational functions") to be able to explore a fascinating non-commutative world.

Motivation & Objective

• To provide a practical, accessible introduction to free fields and non-commutative rational functions for applied researchers and computer algebra users.
• To address the challenge of the word problem in free fields by developing algorithmic techniques for equivalence testing of free fractions.
• To present a systematic method for minimizing linear representations of non-commutative rational functions, ensuring canonical forms.
• To enable implementation in computer algebra systems through structured algorithms based on Cohn and Reutenauer’s theory.
• To bridge the gap between abstract algebraic constructions and concrete computational tools for non-commutative rational functions.

Proposed method

• Representing elements in the free field via admissible linear systems (ALS), where a system $ As = v $ with invertible $ A $ encodes a rational function.
• Using block-structured ALS matrices with pivot blocks to encode non-commutative rational expressions and enable arithmetic operations.
• Applying left and right minimization steps via invertible transformations $ (P, Q) $ to eliminate linearly dependent components and reduce system size.
• Employing a two-phase minimization algorithm: first checking left family $ s $ for linear dependence, then the right family $ t $, with counter updates to ensure correctness.
• Using the first row of the solution vector $ s_1 $ as the representative of the rational function, with $ s_1 = u^T A^{-1} v $, where $ u = e_1^T $.
• Correcting a flaw in prior algorithms by decrementing the counter $ k $ after successful left minimization when $ k = m > 2 $, ensuring minimality.

Experimental results

Research questions

• RQ1 How can non-commutative rational functions be effectively represented and manipulated in a way suitable for algorithmic computation?
• RQ2 What is the minimal linear representation of a non-commutative rational function, and how can it be computed algorithmically?
• RQ3 How can the word problem—determining whether two free fractions represent the same element—be solved efficiently?
• RQ4 What modifications to existing minimization algorithms are necessary to ensure correctness and minimality in all cases?
• RQ5 How can the theory of free fields be made accessible and implementable in computer algebra systems?

Key findings

• The paper establishes a correct and complete algorithm for minimizing admissible linear systems, with a correction to prior work that prevented minimality in edge cases.
• The minimal ALS for $ ff^{-1} = 1 $ is found to be $ (1, [1], 1) $, confirming correctness of the minimization process.
• The transformation $ (P, Q) $ can be used to eliminate dependent components, reducing the system size while preserving the value of the first component $ s_1 $.
• The left family $ s $ is $ bK $-linearly dependent if there exists a non-trivial solution to $ t^T A = 0 $, which is tested via solving a linear system.
• The right family $ t $ is $ bK $-linearly dependent if a solution exists to $ A^T t = 0 $, with the same test applied to the transpose system.
• The implementation of the algorithm in the FDALG package (since Release 1.3.5) demonstrates feasibility for integration into computer algebra systems.