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Extra resources for A 17/10-approximation algorithm for k -bounded space on-line variable-sized bin packing

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The classes of Pk are Boolean combinations of sets (X)ku-1 for u E k* since a class of Pk is defined by its right contexts. Hence all classes of Pk are I-recognizable. Let c be the number of classes of 0. ' 2 ' Let also r be the right-infinite word r=rorir 2 ... where rn is the class of n mod Pk. For all recurrent factors w of length 2 or r, the set of indices n such that r[n, n + 2] = w is k- and I-recognizable and therefore syndetic. Hence there is an integer d such that any recurrent factor w of length 2 of r has a second occurrence at distance at most d.

PROPOSITION. A set X of integers is k-recognizable iffthere exists afinite alphabetA, 1/0 0/ 0 0/ 0 /21, A=L)1h 1/ 1 0/1 Fig. 26. The division by 3. D. PERRIN 38 a substitution o: A Ak and a fixpoint w of a such that X is a union of sets Xa={nkO0wn=a} for a in A. PROOF (sketch). Let (X)k be recognized by a finite deterministic automaton (Q, i, T) on the alphabet {0, , . , k-iI. We take A = Q and define the substitution a by a(a)=sost . . Sk -1 where si is the state reached from a on input i. We consider the fixpoint w = a'(i).

MCM He has shown that when M is a group, the corresponding identity is deducible from the cyclic identities a* =(I +a+ + a"-')(a")*, iff M is a solvable group. For recent results in this area, see [77]. The notion of a rational relation introduced at the end of Section 3 is a fundamental one. There are several characterizations of rational relations among which one known as Nivat's theorem (see [13]) asserting that X c A* x B* is rational iff there exists an alphabet C, two morphisms f: C* -+A*, g: C* - B* and a rational set R c C* such that X = {(f(r), g(r)) Ir e R}.

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