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Linear numeration systems of order two. (English) Zbl 0648.68066
The author considers a numeration system \((U_ v,s)\) of the form \[ u_{n+2}=au_{n+1}+bu_ n,\quad u_ 0=1,\quad u_ 1=v\geq 2\quad. \] He shows that the system is complete only if \(v=a+1\) and \(s=a\). For the system \((U_{a+1},a)\) there exists a uniquely defined normal form of a word which may be computed by a composition of two subsequential machines. Addition of integers represented in \((U_{a+1},a)\) may be computed by a left-subsequential machine.
Reviewer: M.Frumkin

MSC:
68Q45 Formal languages and automata
11B37 Recurrences
11A63 Radix representation; digital problems
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[1] Berstel, J., ()
[2] Berstel, J., Fibonacci words—A survey, (), 13-27
[3] Carlitz, L., Fibonacci representations, Fibonacci quart., 6, 4, 193-220, (1968) · Zbl 0167.03901
[4] Choffrut, Ch., Une caractérisation des fonctions séquentielles et des fonctions sous-séquentielles en tant que relations rationnelles, Theoret. comput. sci., 5, 325-337, (1977) · Zbl 0376.94022
[5] Culik, K.; Salomaa, A., Ambiguity and decision problems concerning number systems, (), 137-146
[6] Eilenberg, S., ()
[7] Fraenkel, A.S., Systems of numeration, Amer. math. monthly, 92, 2, 105-114, (1985) · Zbl 0568.10005
[8] Honkala, J., Bases and ambiguity of number systems, Theoret. comput. sci., 31, 61-71, (1984) · Zbl 0546.68066
[9] Huet, G., Confluent reductions: abstract properties and applications to term rewriting systems, J. assoc. comput. Mach., 27, 797-821, (1980) · Zbl 0458.68007
[10] Knuth, D.E., ()
[11] de Luca, A.; Restivo, A., Representations of integers and language theory, (), 407-415
[12] Sakarovitch, J., Easy multiplications, Inform. and comput., 74, 3, 173-197, (1987) · Zbl 0642.20043
[13] Zeckendorf, E., Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. soc. roy. sci. liège, 3-4, 179-182, (1972) · Zbl 0252.10011
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