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Theorems A and B were used for the basis of the course work of a student of the fourth year, A. A. Karatsuba, "On a problem from the automata theory", which was distinguished by testimonial reference at the competition of student works of the faculty of mechanics and mathematics of
1093:. Notice that this is the source state of the transition. So for each input, the output is already fixed before the input is received, and depends solely on the present state. This is the original definition by E. Moore. It is a common mistake to use the label of state
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occur on the outputs during that brief period while those changes are rippling through the chain, but most systems are designed so that glitches during that brief transition time are ignored or are irrelevant. The outputs then stay the same indefinitely
1217:
chain to decode the current state into the outputs (lambda). The instant the current state changes, those changes ripple through that chain, and almost instantaneously the output gets updated. There are design techniques to ensure that no
2014:
machine, every two states of which are distinguishable from one another, such that the length of the shortest experiments establishing the state of the machine at the end of the experiment is equal to
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A Moore machine with nine states for the above description is shown on the right. The initial state is state A, and the final state is state I. The state table for this example is as follows:
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and react according to its internal configuration at those idealized instants, or else having the state machine wait for a next input symbol (as on a FIFO) and react whenever it arrives.
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equivalent Moore machine, with outputs shifted in time. This is due to the way that state labels are paired with transition labels to form the input/output pairs. Consider a transition
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1205:. Clocked sequential systems are a restricted form of Moore machine where the state changes only when the global clock signal changes. Typically the current state is stored in
327:
519:
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Moore E. F. Gedanken-experiments on
Sequential Machines. Automata Studies, Annals of Mathematical Studies, 34, 129–153. Princeton University Press, Princeton, N.J.(1956).
857:
725:
39:, whose output values are determined both by its current state and by the values of its inputs. Like other finite state machines, in Moore machines, the input typically
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Until the present day (2011), Karatsuba's result on the length of experiments is the only exact nonlinear result, both in automata theory, and in similar problems of
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A sequential network has one input and one output. The output becomes 1 and remains 1 thereafter when at least two 0's and two 1's have occurred as inputs.
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proved the following two theorems, which completely solved Moore's problem on the improvement of the bounds of the experiment length of his "Theorem 8".
333:"Evolution across time" is realized in this abstraction by having the state machine consult the time-changing input symbol at discrete "timer ticks"
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machine, such that every two of its states are distinguishable from one another, then there exists a branched experiment of length at most
43:. Thus the input may indirectly influence subsequent outputs, but not the current or immediate output. The Moore machine is named after
480:
As Moore and Mealy machines are both types of finite-state machines, they are equally expressive: either type can be used to parse a
1209:, and a global clock signal is connected to the "clock" input of the flip-flops. Clocked sequential systems are one way to solve
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However, not every Mealy machine can be converted to an equivalent Moore machine. Some can be converted only to an
730:
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for a Moore machine, or Moore diagram, is a diagram state diagram that associates an output value with each state.
65:
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Karatsuba A. A. Experimente mit
Automaten (German) Elektron. Informationsverarb. Kybernetik, 11, 611–612 (1975).
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Karatsuba A. A. Solution of one problem from the theory of finite automata. Usp. Mat. Nauk, 15:3, 157–159 (1960).
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Another directly following problem is the improvement of the bounds given at the theorems 8 and 9.
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is equivalent to the Mealy machine with the same states and transitions and the output function
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is that in the latter, the output of a transition is determined by the combination of current
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Karatsuba, A. A. (1960). "Solution of one problem from the theory of finite automata".
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At the end of the paper, in
Section "Further problems", the following task is stated:
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More complex Moore machines can have multiple inputs as well as multiple outputs.
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Finite-state machine whose output values are determined only by its current state
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for a Mealy machine, each arc (transition) is labeled with an output value.
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Moore, Edward F (1956). "Gedanken-experiments on
Sequential Machines".
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for a Moore machine, each node (state) is labeled with an output value;
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stay energized, etc.), until the Moore machine changes state again.
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output symbols. Nine theorems are proved about the structure of
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whose current output values are determined only by its current
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is a table listing all the triples in the transition relation
2149:(34). Princeton, N.J.: Princeton University Press: 129–153.
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on 17 December 1958 and was published there in June 1960.
688:{\displaystyle G(s,\sigma )=G_{M}(\delta _{M}(s,\sigma ))}
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in 1958. The paper by
Karatsuba was given to the journal
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problems. A typical electronic Moore machine includes a
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A Moore machine can be regarded as a restricted type of
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mapping a state and the input alphabet to the next state
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Lee, Edward
Ashford; Seshia, Sanjit Arunkumar (2013).
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Simple Moore machines have one input and one output:
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453:{\displaystyle \delta :S\times \Sigma \rightarrow S}
287:{\displaystyle \delta :S\times \Sigma \rightarrow S}
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2147:Automata Studies, Annals of Mathematical Studies
1201:Most digital electronic systems are designed as
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1948:through which one may determine the state of
1168:Types according to number of inputs/outputs.
778:{\displaystyle G_{M}(\delta _{M}(s,\sigma ))}
118:{\displaystyle (S,s_{0},\Sigma ,O,\delta ,G)}
1672:machines" became known as "Moore machines".
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2071:{\displaystyle {\tfrac {(n-1)(n-2)}{2}}+1}
1941:{\displaystyle {\tfrac {(n-1)(n-2)}{2}}+1}
487:The difference between Moore machines and
154:A start state (also called initial state)
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329:mapping each state to the output alphabet
2166:(1.08 ed.). UC Berkeley: Lulu.com.
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1153:{\displaystyle s_{i}\rightarrow s_{j}}
1013:{\displaystyle s_{i}\rightarrow s_{j}}
919:{\displaystyle s_{i}\rightarrow s_{j}}
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391:{\displaystyle t_{0},t_{1},t_{2},...}
1784:{\displaystyle {\tfrac {n(n-1)}{2}}}
1683:Moore's Theorem 8 is formulated as:
1238:Moore machine in combinational logic
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59:A Moore machine can be defined as a
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2218:Regular algebra and finite machines
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322:{\displaystyle G:S\rightarrow O}
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514:{\displaystyle S\times \Sigma }
228:A finite set called the output
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1968:at the end of the experiment.
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1494:on Sequential Machines", the
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1059:{\displaystyle (s_{i},s_{j})}
125:consisting of the following:
2220:. London: Chapman and Hall.
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852:{\displaystyle \delta _{M}}
720:{\displaystyle (s,\sigma )}
35:. This is in contrast to a
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41:influences the next state
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2262:Moore-and-Mealy-Machine
2085:Moscow State University
2007:{\displaystyle (n;m;p)}
1877:{\displaystyle (n;m;p)}
1718:{\displaystyle (n;m;p)}
1632:, and experiments with
1532:automata (or machines)
1525:{\displaystyle (n;m;p)}
1490:In Moore's 1956 paper "
832:'s output function and
403:finite-state transducer
219:{\displaystyle \Sigma }
181:which is an element of
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1254:Example moore machine
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1190:binary adding machine
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1113:{\displaystyle s_{j}}
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174:{\displaystyle s_{0}}
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21:theory of computation
2281:at Wikimedia Commons
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1961:{\displaystyle S}
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1839:{\displaystyle S}
1804:{\displaystyle S}
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1738:{\displaystyle S}
1665:{\displaystyle S}
1645:{\displaystyle S}
1625:{\displaystyle S}
1605:{\displaystyle p}
1585:{\displaystyle m}
1565:{\displaystyle n}
1545:{\displaystyle S}
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825:{\displaystyle M}
609:{\displaystyle M}
574:{\displaystyle G}
561:as the domain of
554:{\displaystyle S}
534:{\displaystyle G}
521:as the domain of
244:{\displaystyle O}
194:{\displaystyle S}
145:{\displaystyle S}
55:Formal definition
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1601:
1581:
1561:
1541:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1487:
1484:
1479:
1476:
1473:
1472:
1469:
1465:
1464:
1461:
1458:
1455:
1451:
1450:
1447:
1443:
1442:
1439:
1436:
1433:
1429:
1428:
1425:
1421:
1420:
1417:
1414:
1411:
1407:
1406:
1403:
1399:
1398:
1395:
1392:
1389:
1385:
1384:
1381:
1377:
1376:
1373:
1370:
1367:
1363:
1362:
1359:
1355:
1354:
1351:
1348:
1345:
1341:
1340:
1337:
1333:
1332:
1329:
1326:
1323:
1319:
1318:
1315:
1311:
1310:
1307:
1304:
1301:
1297:
1296:
1293:
1289:
1288:
1285:
1282:
1279:
1275:
1274:
1271:
1268:
1265:
1244:
1243:Worked Example
1241:
1199:
1198:
1192:
1187:
1173:
1170:
1165:
1162:
1147:
1143:
1139:
1134:
1130:
1107:
1103:
1080:
1076:
1055:
1050:
1046:
1042:
1037:
1033:
1029:
1007:
1003:
999:
994:
990:
967:
963:
940:
936:
913:
909:
905:
900:
896:
868:
846:
842:
821:
799:
795:
774:
771:
768:
765:
762:
759:
754:
750:
746:
741:
737:
716:
713:
710:
707:
704:
684:
681:
678:
675:
672:
669:
664:
660:
656:
651:
647:
643:
640:
637:
634:
631:
628:
625:
605:
594:
593:
590:
570:
550:
530:
510:
507:
504:
489:Mealy machines
477:
474:
465:
462:
449:
446:
443:
440:
437:
434:
431:
415:
412:
410:
407:
387:
384:
381:
378:
373:
369:
365:
360:
356:
352:
347:
343:
331:
330:
318:
315:
312:
309:
306:
295:
283:
280:
277:
274:
271:
268:
265:
251:
240:
226:
215:
201:
190:
168:
164:
152:
141:
114:
111:
108:
105:
102:
99:
96:
93:
90:
85:
81:
77:
74:
71:
56:
53:
15:
9:
6:
4:
3:
2:
2308:
2307:
2296:
2293:
2292:
2290:
2280:
2279:Moore machine
2275:
2271:
2270:
2264:
2263:
2256:
2255:
2250:
2247:
2244:
2241:
2237:
2233:
2229:
2227:0-412-10620-5
2223:
2219:
2215:
2211:
2210:
2197:
2190:
2175:
2173:9780557708574
2169:
2165:
2164:
2156:
2148:
2141:
2139:
2134:
2124:
2121:
2119:
2116:
2114:
2113:Mealy machine
2111:
2109:
2106:
2105:
2099:
2097:
2092:
2090:
2086:
2079:
2065:
2062:
2056:
2049:
2046:
2043:
2034:
2031:
2028:
1998:
1995:
1992:
1989:
1986:
1975:
1969:
1955:
1935:
1932:
1926:
1919:
1916:
1913:
1904:
1901:
1898:
1868:
1865:
1862:
1859:
1856:
1833:
1825:
1820:
1818:
1812:
1798:
1775:
1768:
1765:
1762:
1756:
1732:
1709:
1706:
1703:
1700:
1697:
1684:
1680:
1676:
1673:
1659:
1639:
1619:
1599:
1579:
1559:
1539:
1516:
1513:
1510:
1507:
1504:
1493:
1483:
1470:
1467:
1466:
1459:
1456:
1452:
1448:
1445:
1444:
1437:
1434:
1430:
1426:
1423:
1422:
1415:
1412:
1408:
1404:
1401:
1400:
1393:
1390:
1386:
1382:
1379:
1378:
1371:
1368:
1364:
1360:
1357:
1356:
1349:
1346:
1342:
1338:
1335:
1334:
1327:
1324:
1320:
1316:
1313:
1312:
1305:
1302:
1298:
1294:
1291:
1290:
1283:
1280:
1276:
1272:
1269:
1266:
1264:Current state
1263:
1262:
1259:
1252:
1248:
1236:
1232:
1230:
1226:
1221:
1216:
1212:
1211:metastability
1208:
1204:
1196:
1193:
1191:
1188:
1186:
1182:
1181:edge detector
1179:
1178:
1177:
1169:
1161:
1145:
1141:
1132:
1128:
1105:
1101:
1078:
1074:
1048:
1044:
1040:
1035:
1031:
1005:
1001:
992:
988:
965:
961:
938:
934:
911:
907:
898:
894:
885:
880:
866:
844:
840:
819:
797:
793:
766:
763:
760:
752:
748:
739:
735:
711:
708:
705:
676:
673:
670:
662:
658:
649:
645:
641:
635:
632:
629:
623:
603:
591:
588:
587:
586:
584:
583:state diagram
568:
548:
528:
505:
502:
494:
490:
485:
483:
473:
471:
470:state diagram
461:
447:
438:
435:
432:
429:
421:
406:
404:
399:
385:
382:
379:
376:
371:
367:
363:
358:
354:
350:
345:
341:
316:
310:
307:
304:
296:
281:
272:
269:
266:
263:
256:
253:A transition
252:
238:
231:
227:
206:
202:
188:
166:
162:
153:
139:
132:
128:
127:
126:
109:
106:
103:
100:
97:
94:
88:
83:
79:
75:
72:
62:
52:
50:
46:
42:
38:
37:Mealy machine
34:
30:
26:
25:Moore machine
22:
2260:
2252:
2217:
2214:Conway, J.H.
2195:
2189:
2177:. Retrieved
2162:
2155:
2146:
2093:
2088:
2081:
1973:
1972:
1823:
1822:
1814:
1686:
1682:
1678:
1674:
1489:
1481:
1257:
1246:
1200:
1175:
1167:
883:
881:
595:
486:
479:
467:
417:
400:
332:
58:
24:
18:
926:from state
727:and yields
2236:0231.94041
2129:References
1974:Theorem B.
1824:Theorem A.
1652:. Later, "
1270:Next state
1207:flip-flops
2047:−
2032:−
1917:−
1902:−
1815:In 1957,
1766:−
1229:solenoids
1138:→
998:→
953:to state
904:→
841:δ
767:σ
749:δ
712:σ
677:σ
659:δ
636:σ
509:Σ
506:×
445:→
442:Σ
439:×
430:δ
314:→
279:→
276:Σ
273:×
264:δ
214:Σ
104:δ
92:Σ
2289:Category
2216:(1971).
2102:See also
1725:machine
1572:states,
1220:glitches
1164:Examples
785:, where
255:function
230:alphabet
205:alphabet
1478:Complex
1273:Output
464:Diagram
61:6-tuple
19:In the
2234:
2224:
2179:1 July
2170:
1846:is an
1183:using
1172:Simple
884:almost
131:states
1267:Input
493:state
414:Table
33:state
27:is a
2222:ISBN
2181:2014
2168:ISBN
1225:LEDs
468:The
23:, a
2232:Zbl
1826:If
1185:XOR
859:is
812:is
2291::
2230:.
2137:^
2098:.
1471:I
1463:1
1449:I
1441:0
1427:H
1419:0
1405:F
1397:0
1383:F
1375:0
1361:E
1353:0
1339:C
1331:0
1317:C
1309:0
1295:B
1287:0
1160:.
585:,
484:.
460:.
418:A
405:.
2257:.
2238:.
2183:.
2078:.
2066:1
2063:+
2057:2
2053:)
2050:2
2044:n
2041:(
2038:)
2035:1
2029:n
2026:(
2002:)
1999:p
1996:;
1993:m
1990:;
1987:n
1984:(
1956:S
1936:1
1933:+
1927:2
1923:)
1920:2
1914:n
1911:(
1908:)
1905:1
1899:n
1896:(
1872:)
1869:p
1866:;
1863:m
1860:;
1857:n
1854:(
1834:S
1799:S
1776:2
1772:)
1769:1
1763:n
1760:(
1757:n
1733:S
1713:)
1710:p
1707:;
1704:m
1701:;
1698:n
1695:(
1660:S
1640:S
1620:S
1600:p
1580:m
1560:n
1540:S
1520:)
1517:p
1514:;
1511:m
1508:;
1505:n
1502:(
1468:1
1460:I
1457:0
1454:I
1446:1
1438:H
1435:0
1432:H
1424:1
1416:G
1413:0
1410:G
1402:1
1394:I
1391:0
1388:F
1380:1
1372:H
1369:0
1366:E
1358:1
1350:G
1347:0
1344:D
1336:1
1328:F
1325:0
1322:C
1314:1
1306:E
1303:0
1300:B
1292:1
1284:D
1281:0
1278:A
1223:(
1146:j
1142:s
1133:i
1129:s
1106:j
1102:s
1079:i
1075:s
1054:)
1049:j
1045:s
1041:,
1036:i
1032:s
1028:(
1006:j
1002:s
993:i
989:s
966:j
962:s
939:i
935:s
912:j
908:s
899:i
895:s
867:M
845:M
820:M
798:M
794:G
773:)
770:)
764:,
761:s
758:(
753:M
745:(
740:M
736:G
715:)
709:,
706:s
703:(
683:)
680:)
674:,
671:s
668:(
663:M
655:(
650:M
646:G
642:=
639:)
633:,
630:s
627:(
624:G
604:M
569:G
549:S
529:G
503:S
448:S
436:S
433::
386:.
383:.
380:.
377:,
372:2
368:t
364:,
359:1
355:t
351:,
346:0
342:t
317:O
311:S
308::
305:G
282:S
270:S
267::
239:O
189:S
167:0
163:s
140:S
113:)
110:G
107:,
101:,
98:O
95:,
89:,
84:0
80:s
76:,
73:S
70:(
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