Finite Automata Encoding Piecewise Polynomials

TL;DR

This work introduces a finite automata encoding for piecewise polynomial functions by marrying hierarchical tensor-product B-splines with Kraft's selection mechanism, extending beyond linear encodings to arbitrary smoothness within the spline space $\mathcal{S}_m(\mathcal{T})$. It shows how regular hierarchical meshes yield FA-recognizable representations, enabling polynomial-time verification of nestedness and Assumption B, and provides linear-time value extraction with quadratic-time function evaluation for regular splines. The approach supports unbounded domain support while maintaining a finite-memory automata representation, and demonstrates that regular splines remain regular under regular refinements, preserving decidability and efficiency. This yields a memory-efficient, automata-based framework for spline-based encoding and processing, with potential applications in geometric encoding and image/function compression.

Abstract

Finite automata are used to encode geometric figures, functions and can be used for image compression and processing. The original approach is to represent each point of a figure in $\mathbb{R}^n$ as a convolution of its $n$ coordinates written in some base. Then a figure is said to be encoded as a finite automaton if the set of convolutions corresponding to the points in this figure is accepted by a finite automaton. The only differentiable functions which can be encoded as a finite automaton in this way are linear. In this paper we propose a representation which enables to encode piecewise polynomial functions with arbitrary degrees of smoothness that substantially extends a family of functions which can be encoded as finite automata. Such representation naturally comes from the framework of hierarchical tensor product B-splines, which are piecewise polynomials widely utilized in numerical computational geometry. We show that finite automata provide a suitable tool for solving computational problems arising in this framework when the support of a function is unbounded.

Figures (8)

• Figure 1: The figure on the left shows a portion of infinite domains $\Omega^1$ (bounded by blue line segments) and $\Omega^2$ (bounded by red line segments) satisfying Assumption A. The grid lines of $\mathcal{G}_2 ^0$, $\mathcal{G}_2 ^1$ and $\mathcal{G}_2 ^2$ are depicted as solid, dashed and dotted lines, respectively. The figure on the right shows the corresponding portion of a hierarchical mesh generated by a nested sequence of domains $\Omega^0 = \mathbb{R}^2 \supseteq \Omega^1 \supseteq \Omega^2 \supseteq \Omega^3 = \emptyset$.
• Figure 2: The figure shows a $2$--dimensional cell and its barycentre (a black dot in the centre of the cell).
• Figure 3: The figures show portions of regular $3$--level hierarchical meshes.
• Figure 4: The figure on the left shows the support of $\beta \in B^{\ell}_{2,4}$ with the associated cell $c_\beta$ shaded in gray. The figure on the right shows the support of $\beta \in B^{\ell}_{2,3}$ with the associated cell $c_\beta$ shaded in gray; this cell has the central vertex of $\mathrm{supp}\,\beta$ (shown as a black dot) as its lower left corner.
• Figure 5: The figure on the left shows the support of some tensor product B--spline from $B^{\ell}_{2,4}$ and its intersection with $\mathcal{M}^{\ell}$ shaded in gray which is connected. The figure on the right shows the support of some tensor product B--spline from $B^{\ell}_{2,4}$ and its intersection with $\mathcal{M}^{\ell}$ shaded in gray which is not connected.

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