517:
274:
169:
349:
225:
770:
432:
762:
575:-synchronized sequences exhibit a number of interesting properties. A non-exhaustive list of these properties is presented below.
895:
249:
28:
636:) takes on finitely many terms. This is an immediate consequence of both the above property and the fact that every
426:. Goč, Schaeffer, and Shallit demonstrated that there exists a finite automaton accepting the language
145:
715:
Frougny, C.; Sakarovitch, J. (1993), "Synchronized rational relations of finite and infinite words",
51:
307:
594:
584:
93:
86:
17:
530:
starting within a given block is novel while all other subwords are not. It then verifies that
766:
874:
840:
724:
352:
210:
761:. Lecture Notes in Computer Science. Vol. 7907. Editors Béal MP., Carton O. Berlin:
889:
728:
862:
188:
24:
844:
690:
651:-synchronized sequences is closed under termwise sum and termwise composition.
522:
This automaton guesses the endpoints of every contiguous block of symbols in
415:
878:
39:
831:
Carpi, A.; D'Alonzo, V. (2010), "On factors of synchronized sequences",
548:
is accepted by this automaton, the subword complexity function of the
512:{\displaystyle \{(n,m)_{k}\mid n\geq 0{\text{ and }}m=\rho _{S}(n)\}.}
63:
673:-synchronized sequence, then both the subword complexity of
367:-synchronized sequences was introduced by Carpi and Maggi.
534:
is the sum of the sizes of the blocks. Since the pair (
435:
310:
252:
213:
148:
269:{\displaystyle \mathbb {N} \rightarrow \mathbb {N} }
85:-synchronized sequences lies between the classes of
689:) (similar to subword complexity, but for distinct
640:-regular sequence taking on finitely many terms is
511:
343:
268:
219:
163:
658:-synchronized sequence have a linear growth rate.
863:"On synchronized sequences and their separators"
191:over Σ × ... × Σ, where (
8:
753:Goč, D.; Schaeffer, L.; Shallit, J. (2013).
503:
436:
338:
311:
242: ≥ 0 be a natural number and let
821:Carpi & Maggi (2010), Proposition 2.5
812:Carpi & Maggi (2010), Proposition 2.2
803:Carpi & Maggi (2010), Proposition 2.1
794:Carpi & Maggi (2010), Proposition 2.8
785:Carpi & Maggi (2010), Proposition 2.6
604:-synchronized. To be precise, a sequence
526:and verifies that each subword of length
488:
473:
455:
434:
309:
262:
261:
254:
253:
251:
212:
155:
151:
150:
147:
707:
304:-synchronized if the language of pairs
410:(n) denote the subword complexity of
7:
739:
737:
681:) and the palindromic complexity of
414:; that is, the number of distinct
14:
175:-synchronized if the relation {(
164:{\displaystyle \mathbb {N} ^{r}}
62:, each expressed in some fixed
500:
494:
452:
439:
335:
332:
326:
314:
258:
130:representation of some number
1:
861:Carpi, A.; Maggi, C. (2010),
729:10.1016/0304-3975(93)90230-Q
344:{\displaystyle \{(n,f(n))\}}
54:taking as input two strings
29:theoretical computer science
187:)} is a right-synchronized
912:
867:Theoret. Informatics Appl.
616:-automatic if and only if
583:-synchronized sequence is
15:
845:10.1016/j.tcs.2010.08.005
392:) and an infinite string
138: ≥ 2, a subset
743:Carpi & Maggi (2010)
288:) are expressed in base
120: ≥ 2, and let
112:Let Σ be an alphabet of
16:Not to be confused with
755:Subword complexity and
513:
345:
270:
221:
165:
36:-synchronized sequence
833:Theoret. Comput. Sci.
717:Theoret. Comput. Sci.
514:
346:
276:be a map, where both
271:
222:
166:
50:) characterized by a
896:Sequences and series
839:(44–46): 3932–3937,
552:-automatic sequence
433:
384:-automatic sequence
308:
250:
220:{\displaystyle \in }
211:
146:
90:-automatic sequences
879:10.1051/ita:2001129
697:-regular sequences.
598:-automatic sequence
69:, and accepting if
628:-synchronized and
509:
376:Subword complexity
341:
266:
234:Language-theoretic
217:
161:
97:-regular sequences
18:Synchronizing word
772:978-3-642-38770-8
654:The terms of any
476:
189:rational relation
903:
881:
848:
847:
828:
822:
819:
813:
810:
804:
801:
795:
792:
786:
783:
777:
776:
759:-synchronization
750:
744:
741:
732:
731:
712:
518:
516:
515:
510:
493:
492:
477:
474:
460:
459:
350:
348:
347:
342:
275:
273:
272:
267:
265:
257:
226:
224:
223:
218:
170:
168:
167:
162:
160:
159:
154:
126:denote the base-
81:). The class of
52:finite automaton
911:
910:
906:
905:
904:
902:
901:
900:
886:
885:
860:
857:
852:
851:
830:
829:
825:
820:
816:
811:
807:
802:
798:
793:
789:
784:
780:
773:
752:
751:
747:
742:
735:
714:
713:
709:
704:
570:
564:-synchronized.
547:
484:
475: and
451:
431:
430:
409:
378:
373:
361:
306:
305:
292:. The sequence
248:
247:
236:
209:
208:
206:
197:
186:
180:
149:
144:
143:
125:
110:
105:
38:is an infinite
21:
12:
11:
5:
909:
907:
899:
898:
888:
887:
884:
883:
873:(6): 513–524,
856:
853:
850:
849:
823:
814:
805:
796:
787:
778:
771:
745:
733:
706:
705:
703:
700:
699:
698:
659:
652:
645:
591:
569:
566:
543:
520:
519:
508:
505:
502:
499:
496:
491:
487:
483:
480:
472:
469:
466:
463:
458:
454:
450:
447:
444:
441:
438:
405:
377:
374:
372:
369:
360:
357:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
264:
260:
256:
235:
232:
216:
202:
195:
182:
176:
158:
153:
121:
116:symbols where
109:
106:
104:
101:
13:
10:
9:
6:
4:
3:
2:
908:
897:
894:
893:
891:
880:
876:
872:
868:
864:
859:
858:
854:
846:
842:
838:
834:
827:
824:
818:
815:
809:
806:
800:
797:
791:
788:
782:
779:
774:
768:
764:
760:
756:
749:
746:
740:
738:
734:
730:
726:
722:
718:
711:
708:
701:
696:
692:
688:
684:
680:
676:
672:
668:
664:
660:
657:
653:
650:
647:The class of
646:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
603:
599:
597:
592:
589:
587:
582:
578:
577:
576:
574:
567:
565:
563:
559:
555:
551:
546:
541:
537:
533:
529:
525:
506:
497:
489:
485:
481:
478:
470:
467:
464:
461:
456:
448:
445:
442:
429:
428:
427:
425:
421:
417:
413:
408:
404:(2)..., let ρ
403:
399:
396: =
395:
391:
387:
383:
375:
370:
368:
366:
363:The class of
358:
356:
354:
329:
323:
320:
317:
303:
299:
295:
291:
287:
283:
279:
245:
241:
233:
231:
229:
214:
205:
201:
194:
190:
185:
179:
174:
156:
141:
137:
133:
129:
124:
119:
115:
107:
102:
100:
98:
96:
91:
89:
84:
80:
76:
73: =
72:
68:
65:
61:
57:
53:
49:
45:
41:
37:
35:
30:
26:
19:
870:
866:
836:
832:
826:
817:
808:
799:
790:
781:
758:
754:
748:
720:
716:
710:
694:
686:
682:
678:
674:
670:
666:
662:
655:
648:
641:
637:
633:
629:
625:
621:
617:
613:
609:
605:
601:
595:
585:
580:
572:
571:
561:
557:
553:
549:
544:
539:
535:
531:
527:
523:
521:
423:
419:
411:
406:
401:
397:
393:
389:
385:
381:
379:
364:
362:
301:
297:
293:
289:
285:
281:
277:
243:
239:
237:
227:
203:
199:
192:
183:
177:
172:
139:
135:
131:
127:
122:
117:
113:
111:
108:As relations
94:
87:
82:
78:
74:
70:
66:
59:
55:
47:
43:
33:
32:
22:
691:palindromes
644:-automatic.
103:Definitions
25:mathematics
855:References
568:Properties
418:of length
723:: 45–82,
486:ρ
468:≥
462:∣
259:→
215:∈
42:of terms
890:Category
763:Springer
588:-regular
416:subwords
380:Given a
134:. Given
40:sequence
669:) is a
538:,
371:Example
359:History
353:regular
198:, ...,
181:, ...,
769:
693:) are
593:Every
579:Every
702:Notes
624:) is
612:) is
560:) is
300:) is
767:ISBN
280:and
238:Let
92:and
64:base
58:and
31:, a
27:and
875:doi
841:doi
837:411
725:doi
721:108
661:If
600:is
422:in
400:(1)
351:is
171:is
142:of
23:In
892::
871:35
869:,
865:,
835:,
765:.
736:^
719:,
355:.
246::
230:.
207:)
99:.
882:.
877::
843::
775:.
757:k
727::
695:k
687:n
685:(
683:s
679:n
677:(
675:s
671:k
667:n
665:(
663:s
656:k
649:k
642:k
638:k
634:n
632:(
630:s
626:k
622:n
620:(
618:s
614:k
610:n
608:(
606:s
602:k
596:k
590:.
586:k
581:k
573:k
562:k
558:n
556:(
554:s
550:k
545:k
542:)
540:m
536:n
532:m
528:n
524:S
507:.
504:}
501:)
498:n
495:(
490:S
482:=
479:m
471:0
465:n
457:k
453:)
449:m
446:,
443:n
440:(
437:{
424:S
420:n
412:S
407:S
402:s
398:s
394:S
390:n
388:(
386:s
382:k
365:k
339:}
336:)
333:)
330:n
327:(
324:f
321:,
318:n
315:(
312:{
302:k
298:n
296:(
294:f
290:k
286:n
284:(
282:f
278:n
263:N
255:N
244:f
240:n
228:R
204:r
200:n
196:1
193:n
184:k
178:k
173:k
157:r
152:N
140:R
136:r
132:n
128:k
123:k
118:k
114:k
95:k
88:k
83:k
79:n
77:(
75:s
71:m
67:k
60:n
56:m
48:n
46:(
44:s
34:k
20:.
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