[Paper Review] Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words

Gennady Eremin|arXiv (Cornell University)|Nov 5, 2019

Advanced Combinatorial Mathematics4 references4 citations

TL;DR

This paper introduces a lexicographical ordering on Motzkin words—balanced parentheses sequences excluding leading zeros—treating them as a sequence analogous to natural numbers. It establishes arithmetic and logical operations on this ordered sequence, called the Motzkin Row, with '0' serving as the zero element, enabling index-based operations and polynomial representations for word manipulation and navigation.

ABSTRACT

We establish a total lexicographical order on the set of Motzkin words. Elements are ordered similarly to Natural Numbers in accordance with known rules (axioms). As a result, we were able to obtain arithmetic and logical operations on the elements of the ordered sequence, Motzkin Row. This sequence consists of balanced brackets without leading zeros, with the exception of the initial word "0". It is the word "0" as well as the alphabetical symbol "0" that are analogues of numeric zero in the corresponding operations. Logical operations allow you to navigate Motzkin Row. Operations on words are accompanied by index equations, index polynomials.

Motivation & Objective

• To define a total lexicographical order on the set of Motzkin words, treating them as a sequence analogous to natural numbers.
• To establish arithmetic and logical operations on this ordered sequence, termed the Motzkin Row, for systematic manipulation of balanced parentheses words.
• To identify '0' as the zero element in this system, both as a word and as a symbolic analogue to numeric zero.
• To derive index equations and index polynomials that encode operations on Motzkin words, enabling algorithmic navigation and computation.
• To provide a formal framework for arithmetic on balanced parentheses, extending number-theoretic concepts to combinatorial word structures.

Proposed method

• Define a total lexicographical order on Motzkin words by comparing strings character by character, following standard dictionary rules.
• Construct the Motzkin Row as the ordered sequence of all valid Motzkin words, starting with '0' as the first element.
• Introduce arithmetic operations (addition, multiplication) and logical operations (comparisons, membership checks) on words using their lexicographical indices.
• Develop index equations that map word operations to index arithmetic, enabling computation without explicit string manipulation.
• Formulate index polynomials that represent the result of operations on words, allowing algebraic treatment of word sequences.
• Use '0' as the identity element in operations, both as a word and as a symbolic zero in index arithmetic.

Experimental results

Research questions

• RQ1 How can a total lexicographical order be consistently defined on the set of Motzkin words?
• RQ2 Can arithmetic and logical operations be systematically defined on an ordered sequence of balanced parentheses words?
• RQ3 What role does the word '0' play as an analogue to numeric zero in this arithmetic system?
• RQ4 How can index equations and index polynomials be derived to represent operations on Motzkin words?
• RQ5 Can this framework enable efficient navigation and computation on combinatorial word structures like Motzkin words?

Key findings

• A total lexicographical order is successfully established on the set of Motzkin words, forming a well-ordered sequence known as the Motzkin Row.
• Arithmetic and logical operations are defined on the Motzkin Row, enabling operations analogous to those on natural numbers.
• The word '0' serves as the identity element in this system, both as a word and as a symbolic zero in index arithmetic.
• Index equations are derived that map word operations to index arithmetic, allowing computation via indices rather than string manipulation.
• Index polynomials are formulated to represent the results of operations, providing an algebraic framework for word arithmetic.
• The framework enables systematic navigation and manipulation of Motzkin words using number-theoretic and combinatorial tools.