2015/02/06 by Vladimir Anashin, Anashin, Vladimir
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Formal Languages and Automata Theory (cs.FL) #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Topological and Geometric Data Analysis #advanced mathematical theories #cs.FL #math-ph #math.DS #math.MP
paper · pdf · doi:10.48550/arxiv.1502.01920
arxiv created 2015/02/06 · openalex publication_date 2015/02/06 · arxiv updated 2015/02/09 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Given an automaton (a letter-to-letter transducer, a dynamical 1-Lipschitz system on the space \mathbb Zp of p-adic integers) \mathfrak A whose input and output alphabets are \mathbb Fp=\0,1,…,p-1\, one visualizes word transformations performed by \mathfrak A by a point set \mathbf P(\mathfrak A) in real plane \mathbb R2. For a finite-state automaton \mathfrak A, it is shown that once some points of \mathbf P(\mathfrak A) constitute a smooth (of a class C2) curve in \mathbb R2, the curve is a segment of a straight line with a rational slope; and there are only finitely many straight lines whose segments are in P(\mathfrak A). Moreover, when identifying \mathbf P(\mathfrak A) with a subset of a 2-dimensional torus \mathbb T2⊂\mathbb R3 (under a natural mapping of the real unit square [0,1]2 onto \mathbb T2) the smooth curves from \mathbf P(\mathfrak A) constitute a collection of torus windings. In cylindrical coordinates either of the windings can be ascribed to a complex-valued function ψ(x)=ei(Ax-2πB(t)) (x∈\mathbb R) for suitable rational A,B(t). Since ψ(x) is a standard expression for a matter wave in quantum theory (where B(t)=tB(t0)), and since transducers can be regarded as a mathematical formalization for causal discrete systems, the paper might serve as a mathematical reasoning why wave phenomena are inherent in quantum systems: This is because of causality principle and the discreteness of matter.