Efficient Computation of Base-b Representations for the Golden Ratio and Other Quadratic Irrationals Using Finite Automata

Core Concepts

Finite automata can be used to efficiently compute the base-b representation of the golden ratio and other quadratic irrational numbers.

Abstract

The paper presents a method for using finite automata to compute the base-b representation of the golden ratio and other quadratic irrational numbers. The key insights are: The n'th digit of the base-b representation of the golden ratio can be computed as a finite-state function of the Zeckendorf representation of b^n. This allows the construction of a deterministic finite automaton (DFAO) that, on input the Zeckendorf representation of b^n, outputs the n'th digit of the base-b representation of the golden ratio. Similar results can be proven for any quadratic irrational number. The key is to find an appropriate numeration system that is "tuned" to the particular quadratic irrational, such as the Pell representation for √2 or the Ostrowski representation for other quadratic irrationals. The authors use a SAT solver to prove, in some cases, that the automata they construct are minimal. They find that for the binary digits of the golden ratio, the automaton is unique and minimal. For the ternary digits, they find multiple distinct minimal automata. The authors provide Walnut code to construct automata for computing the base-b digits of various quadratic irrationals, including the "bronze ratio" (√13 + 3)/2 and several Pisot numbers. Overall, the paper demonstrates that finite automata can be used to efficiently compute the base-b representations of quadratic irrational numbers, despite the fact that these representations are not ultimately generated by finite automata in general.

Stats

The paper provides the following key figures and metrics: The Zeckendorf representation of the first few powers of 2 and 3 (Table 1). The Pell representation of the first few powers of 2 and 3 (Table 2). The number of states in the automata constructed for various quadratic irrationals, ranging from 6 states to 16 states (Table 4). The size of the digit set required for the SAT solver to find the minimal automaton, ranging from 27 to 197 digits (Table 4). The time required for the SAT solver to prove minimality, ranging from 0.02 seconds to over 28,000 seconds (Table 4).

Quotes

"Even so, in this paper we show that, using finite automata, one can compute the n'th digit in the base-b representation of the golden ratio φ! At first glance this might seem to contradict the Adamczewski–Bugeaud result. But it does not, since for our theorem the input is not n expressed in base b, but rather bn in an entirely different numeration system, the Zeckendorf representation." "Using a SAT solver, in some cases (such for the binary digits of φ) we prove that the automata constructed is minimal and unique. Interestingly, in other cases (such as for the ternary digits of φ) we were able to prove the minimality of our automaton, but we discovered several distinct automaton with the same number of states computing the same quadratic irrational (at least up to a very high precision)."

Key Insights Distilled From

Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals

by Aaron Barnof... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02727.pdf

Deeper Inquiries

What other numeration systems beyond Zeckendorf, Pell, and Ostrowski could be used to construct efficient finite automata for computing the base-b digits of other classes of irrational numbers

Other numeration systems that could be used to construct efficient finite automata for computing the base-b digits of other classes of irrational numbers include the Fibonacci numeration system, the Lucas numeration system, and the Thue-Morse numeration system. These systems have unique properties that can be leveraged to create automata that efficiently compute the base-b representation of specific classes of irrational numbers. For example, the Fibonacci numeration system is closely related to the Zeckendorf representation and can be used for certain classes of quadratic irrationals. The Lucas numeration system, based on the Lucas numbers, can be applied to other classes of quadratic irrationals or even higher-order algebraic irrationals. The Thue-Morse numeration system, known for its self-similarity and regularity, could be utilized for specific types of transcendental numbers or other complex irrational numbers.

Can the non-uniqueness of minimal automata observed for some quadratic irrationals be explained or characterized more generally

The non-uniqueness of minimal automata observed for some quadratic irrationals can be attributed to the inherent complexity and structure of these numbers. In general, quadratic irrationals have unique properties that can lead to multiple valid automata solutions that compute the base-b digits correctly. The non-uniqueness may stem from the specific characteristics of the irrational number, the chosen numeration system, or the constraints imposed on the automata construction. Additionally, the presence of multiple minimal automata for a given quadratic irrational could indicate underlying symmetries or patterns in the number's representation that allow for different but equally valid computational approaches. Further research and analysis are needed to fully characterize and explain the non-uniqueness of minimal automata for quadratic irrationals.

How do the techniques presented in this paper extend to computing the base-b digits of transcendental numbers, which are known to not be computable by finite automata in general

The techniques presented in the paper can be extended to computing the base-b digits of transcendental numbers by adapting the numeration systems and automata construction methods to suit the unique properties of these numbers. Transcendental numbers, such as π or e, have infinite and non-repeating decimal representations, posing a challenge for finite automata computation. However, by utilizing specialized numeration systems, such as continued fractions or other symbolic representations, and developing tailored automata algorithms, it may be possible to approximate or compute certain digits of transcendental numbers efficiently. The key lies in designing automata that can handle the infinite and non-repeating nature of transcendental numbers while maintaining accuracy and computational efficiency. Further research in this area could lead to novel approaches for computing the base-b digits of transcendental numbers using finite automata.