A Numeration System for Fibonacci-Like Wang Shifts

Labbé, Sébastien; Lepšová, Jana

doi:10.1007/978-3-030-85088-3_9

Cited by 4 publications

(7 citation statements)

References 7 publications

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“…We now consider an extension of the one-sided word f to a two-sided (bi-infinite) word. There are essentially two different ways to do this; see [8,12,18].…”

Section: Negafibonacci Representationmentioning

confidence: 99%

Automatic sequences in negative bases and proofs of some conjectures of shevelev

Shallit

Shan

Yang

2023

RAIRO-Theor. Inf. Appl.

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We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (—k) to express variables, thus permitting quantification over ℤ instead of ℕ. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shut involving bi-infinite binary words.

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“…We now consider an extension of the one-sided word f to a two-sided (bi-infinite) word. There are essentially two different ways to do this; see [8,12,18].…”

Section: Negafibonacci Representationmentioning

confidence: 99%

Automatic sequences in negative bases and proofs of some conjectures of shevelev

Shallit

Shan

Yang

2023

RAIRO-Theor. Inf. Appl.

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“…In this section, we recall the numeration system F introduced in [9] which is defined by the value map val F : Σ(ΣΣ) * → Z given in Equation ( 2) where Σ = {0, 1}. The first observation to make on this value map is given in the next lemma.…”

Section: A Fibonacci Numeration System For Zmentioning

confidence: 99%

“…Motivated by the study of aperiodic tiling of the plane by Wang tiles, a numeration system F representing all integers in Z in a unique way based on Fibonacci numbers was introduced recently [9]. The goal of this contribution is to prove that it is the Fibonacci-equivalent of the two's complement notation.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For every n ∈ Z, it was already proved in [9] that there exists a unique odd-length word w ∈ Σ(ΣΣ) * \ (Σ * 11Σ * ∪ 000Σ * ∪ 101Σ * ) such that n = val F (w) and this unique word is denoted by rep F (n). The numeration system F extends naturally to Z 2 and, in [9], it was used together with a certain automaton A to describe a particular aperiodic Wang shift Ω. Given a position n ∈ Z 2 , the automaton A takes the representation rep F (n) as input and the reached final state in A is the index of the Wang tile at position n ∈ Z 2 in a particular configuration x ∈ Ω, i.e.…”

Section: Introductionmentioning

confidence: 99%

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A Fibonacci's complement numeration system

Labbé¹,

Lepšová²

2022

Preprint

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With the two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. In this work, we consider a Fibonacci-equivalent of the two's complement notation. A theorem of Zeckendorf says that every nonnegative integer can be represented as the sum of distinct nonconsecutive Fibonacci numbers thus leading to a binary representation of nonnegative integers where powers of 2 are replaced by Fibonacci numbers in its evaluation. A transducer provided by Berstel computes the sum of the Zeckendorf binary representation of two nonnegative integers. In this work, we consider a numeration system also based on Fibonacci numbers but representing all integers. As for the two's complement notation, we show that addition of integers represented in this numeration system can be computed with the Berstel transducer with three additional transitions. Whether this can be done more generally is an open question raised by the current work.

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Bounded Wang tilings with integer programming and graph-based heuristics

Tyburec

Zeman

2023

Sci Rep

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Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a 1/2 approximation guarantee for arbitrary tile sets, and a 2/3 guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provide very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic.

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A Numeration System for Fibonacci-Like Wang Shifts

Cited by 4 publications

References 7 publications

Automatic sequences in negative bases and proofs of some conjectures of shevelev

Automatic sequences in negative bases and proofs of some conjectures of shevelev

A Fibonacci's complement numeration system

Bounded Wang tilings with integer programming and graph-based heuristics

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