142:
20:
78:
Terminal symbols are symbols that may appear in the outputs of the production rules of a formal grammar and which cannot be changed using the rules of the grammar. Applying the rules recursively to a source string of symbols will usually terminate in a final output string consisting only of terminal
176:
are those grammars in which the left-hand side of each production rule consists of only a single nonterminal symbol. This restriction is non-trivial; not all languages can be generated by context-free grammars. Those that can are called context-free languages. These are exactly the languages that
375:
166:, a designated member of the set of nonterminals from which all the strings in the language may be derived by successive applications of the production rules. In fact, the language defined by a grammar is precisely the set of
466:
symbol. That is, each production rule maps from one string of symbols to another, where the first string contains at least one nonterminal symbol. In the case that the body consists solely of the
563:
719:
680:
456:
584:
is a notation for expressing certain grammars. For instance, the following production rules in Backus-Naur form are used to represent an integer (which may be signed):
197:(or just 'productions') that specify which symbols may replace which other symbols; these rules may be used to generate strings, or to parse them. Each such rule has a
511:
405:
291:
814:
600:'0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
918:
882:
779:
This example supports strings with leading zeroes like "0056" or "0000", as well as negative zero strings like "-0" and "-00000".
149:
was formed by the grammar defined by the given production rules. This grammar can create strings with any number of the symbol Б
141:
194:
40:
741:
23:
The string "the dog ate the bone" was created using production rules that replaced non-terminal with terminal symbols.
524:
945:
940:
19:
570:
90:
is both a non-terminal symbol and the start symbol. The production rules for creating strings are as follows:
688:
205:, or right-hand side, which consists of a string that may replace it. Rules are often written in the form
122:
is a terminal symbol because no rule exists which would change it into something else. On the other hand,
422:
657:
182:
581:
851:
173:
898:
178:
910:
831:
416:
914:
878:
856:
751:
746:
902:
870:
823:
566:
490:
384:
28:
158:
Nonterminal symbols are those symbols that can be replaced. They may also be called simply
802:
Rosen, K. H. (2012). Discrete mathematics and its applications. McGraw-Hill. pages 847-851
127:
52:
903:
44:
370:{\displaystyle (\Sigma \cup N)^{*}N(\Sigma \cup N)^{*}\rightarrow (\Sigma \cup N)^{*}}
934:
264:
67:
835:
467:
233:
36:
134:
by a particular grammar is the set of strings that can be produced by the grammar
408:
63:) are replaced by groups of terminal symbols according to the production rules.
201:, or left-hand side, which consists of the string that may be replaced, and a
827:
181:. Context-free languages are the theoretical basis for the syntax of most
138:. Diagram 1 illustrates a string that can be produced with this grammar.
812:
Chomsky, Noam (1956). "Three Models for the
Description of Language".
232:
In the classic formalization of generative grammars first proposed by
82:
Consider a grammar defined by two rules. In this grammar, the symbol
140:
66:
The terminals and nonterminals of a particular grammar are in two
18:
875:
Algebraic and automata theoretic properties of formal languages
909:. Reading, Mass.: Addison-Wesley Publishing Company. pp.
126:
has two rules that can change it, thus it is nonterminal. A
521:
A grammar is formally defined as the ordered quadruple
691:
660:
527:
493:
425:
387:
294:
470:, it may be denoted with a special notation (often
713:
674:
557:
505:
450:
399:
369:
558:{\displaystyle \langle N,\Sigma ,P,S\rangle }
8:
552:
528:
565:. Such a formal grammar is often called a
702:
692:
690:
661:
659:
526:
492:
442:
424:
391:
389:
386:
361:
336:
311:
293:
177:can be recognized by a non-deterministic
136:and that consist only of terminal symbols
16:Categories of symbols in formal grammars
795:
763:
905:Introduction to Formal Language Theory
815:IRE Transactions on Information Theory
714:{\displaystyle {\ce {A -> a | ab}}}
240:consists of the following components:
55:defined as part of a formal grammar.
458:represents zero or more symbols, and
7:
675:{\displaystyle {\ce {S -> cAd}}}
645:
641:
451:{\displaystyle (\Sigma \cup N)^{*}}
537:
429:
348:
323:
298:
51:are the elementary symbols of the
14:
170:strings that can be so derived.
33:terminal and nonterminal symbols
877:. North-Holland. pp. 8–9.
770:It contains no symbols at all.
703:
696:
665:
636:In this example, the symbols (
482:) in order to avoid confusion.
439:
426:
358:
345:
342:
333:
320:
308:
295:
162:. A formal grammar includes a
1:
724:In this example, the symbols
742:Alphabet (formal languages)
640:) are terminal symbols and
962:
648:are nonterminal symbols.
732:are nonterminal symbols.
728:are terminal symbols and
86:is a terminal symbol and
828:10.1109/TIT.1956.1056813
586:
571:phrase structure grammar
236:in the 1950s, a grammar
193:A grammar is defined by
68:completely separate sets
487:A distinguished symbol
282:, each rule of the form
39:used in specifying the
849:Chomsky, Noam (1957).
715:
676:
559:
507:
506:{\displaystyle S\in N}
452:
401:
400:{\displaystyle {}^{*}}
371:
150:
145:Diagram 1. The string
24:
716:
677:
638:-,0,1,2,3,4,5,6,7,8,9
560:
508:
453:
402:
372:
183:programming languages
174:Context-free grammars
144:
22:
899:Harrison, Michael A.
852:Syntactic Structures
689:
658:
651:Another example is:
525:
491:
423:
385:
292:
573:in the literature.
250:nonterminal symbols
225:can be replaced by
179:push down automaton
160:syntactic variables
154:Nonterminal symbols
61:syntactic variables
57:Nonterminal symbols
711:
672:
555:
503:
448:
397:
367:
151:
25:
871:Ginsburg, Seymour
752:Recursive grammar
747:Chomsky Hierarchy
709:
701:
695:
670:
664:
213:; e.g., the rule
953:
946:Pattern matching
941:Formal languages
925:
924:
908:
895:
889:
888:
867:
861:
860:
846:
840:
839:
809:
803:
800:
780:
777:
771:
768:
731:
727:
720:
718:
717:
712:
710:
707:
706:
699:
693:
681:
679:
678:
673:
671:
668:
662:
647:
643:
639:
631:
628:
625:
621:
618:
615:
612:
609:
606:
603:
599:
596:
593:
590:
582:Backus–Naur form
567:rewriting system
564:
562:
561:
556:
512:
510:
509:
504:
481:
477:
473:
461:
457:
455:
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414:
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403:
398:
396:
395:
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376:
374:
373:
368:
366:
365:
341:
340:
316:
315:
280:production rules
277:
270:
261:terminal symbols
258:
247:
195:production rules
189:Production rules
148:
125:
121:
114:
110:
104:
101:
97:
89:
85:
74:Terminal symbols
49:Terminal symbols
41:production rules
37:lexical elements
29:formal languages
961:
960:
956:
955:
954:
952:
951:
950:
931:
930:
929:
928:
921:
897:
896:
892:
885:
869:
868:
864:
848:
847:
843:
811:
810:
806:
801:
797:
792:
786:
784:
783:
778:
774:
769:
765:
760:
738:
729:
725:
687:
686:
656:
655:
646:<integer>
637:
634:
633:
629:
626:
623:
619:
616:
613:
610:
607:
604:
601:
597:
594:
591:
588:
579:
523:
522:
489:
488:
479:
475:
471:
459:
438:
421:
420:
412:
388:
383:
382:
357:
332:
307:
290:
289:
275:
268:
256:
245:
221:specifies that
191:
156:
146:
128:formal language
123:
119:
112:
108:
102:
99:
95:
87:
83:
76:
43:constituting a
17:
12:
11:
5:
959:
957:
949:
948:
943:
933:
932:
927:
926:
919:
890:
883:
862:
841:
822:(3): 113–123.
804:
794:
793:
791:
788:
782:
781:
772:
762:
761:
759:
756:
755:
754:
749:
744:
737:
734:
722:
721:
705:
698:
683:
682:
667:
587:
578:
575:
554:
551:
548:
545:
542:
539:
536:
533:
530:
519:
518:
502:
499:
496:
484:
483:
445:
441:
437:
434:
431:
428:
394:
379:
378:
377:
364:
360:
356:
353:
350:
347:
344:
339:
335:
331:
328:
325:
322:
319:
314:
310:
306:
303:
300:
297:
284:
283:
272:
253:
190:
187:
155:
152:
116:
115:
105:
75:
72:
45:formal grammar
15:
13:
10:
9:
6:
4:
3:
2:
958:
947:
944:
942:
939:
938:
936:
922:
920:0-201-02955-3
916:
912:
907:
906:
900:
894:
891:
886:
884:0-7204-2506-9
880:
876:
872:
866:
863:
858:
855:. The Hague:
854:
853:
845:
842:
837:
833:
829:
825:
821:
817:
816:
808:
805:
799:
796:
789:
787:
776:
773:
767:
764:
757:
753:
750:
748:
745:
743:
740:
739:
735:
733:
685:
684:
654:
653:
652:
649:
642:<digit>
585:
583:
576:
574:
572:
568:
549:
546:
543:
540:
534:
531:
516:
500:
497:
494:
486:
485:
469:
465:
443:
435:
432:
418:
411:operator and
410:
392:
380:
362:
354:
351:
337:
329:
326:
317:
312:
304:
301:
288:
287:
286:
285:
281:
274:A finite set
273:
266:
262:
255:A finite set
254:
251:
244:A finite set
243:
242:
241:
239:
235:
230:
228:
224:
220:
216:
212:
208:
204:
200:
196:
188:
186:
184:
180:
175:
171:
169:
165:
161:
153:
143:
139:
137:
133:
129:
106:
93:
92:
91:
80:
73:
71:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
21:
904:
893:
874:
865:
850:
844:
819:
813:
807:
798:
785:
775:
766:
723:
650:
635:
580:
520:
515:start symbol
514:
513:that is the
468:empty string
463:
279:
260:
249:
237:
234:Noam Chomsky
231:
226:
222:
218:
214:
210:
206:
202:
198:
192:
172:
167:
164:start symbol
163:
159:
157:
135:
131:
117:
81:
77:
65:
60:
56:
48:
32:
26:
464:nonterminal
409:Kleene star
130:defined or
111:can become
107:The symbol
98:can become
94:The symbol
935:Categories
790:References
462:means one
697:⟶
666:⟶
553:⟩
538:Σ
529:⟨
498:∈
444:∗
433:∪
430:Σ
417:set union
393:∗
363:∗
352:∪
349:Σ
343:→
338:∗
327:∪
324:Σ
313:∗
302:∪
299:Σ
132:generated
79:symbols.
901:(1978).
873:(1975).
836:19519474
736:See also
415:denotes
265:disjoint
263:that is
168:terminal
53:language
35:are the
726:a,b,c,d
605:integer
577:Example
413:∪
407:is the
147:Б Б Б Б
917:
881:
857:Mouton
834:
480:ε
472:Λ
381:where
257:Σ
832:S2CID
758:Notes
627:digit
617:digit
592:digit
569:or a
419:, so
267:from
118:Here
915:ISBN
879:ISBN
644:and
630:>
624:<
620:>
614:<
608:>
602:<
595:>
589:<
211:body
207:head
203:body
199:head
59:(or
824:doi
730:S,A
669:cAd
611:::=
598:::=
478:or
278:of
259:of
248:of
47:.
27:In
937::
913:.
911:13
830:.
818:.
708:ab
632:}
474:,
229:.
217:→
209:→
185:.
70:.
31:,
923:.
887:.
859:.
838:.
826::
820:2
704:|
700:a
694:A
663:S
622:{
550:S
547:,
544:P
541:,
535:,
532:N
517:.
501:N
495:S
476:e
460:N
440:)
436:N
427:(
359:)
355:N
346:(
334:)
330:N
321:(
318:N
309:)
305:N
296:(
276:P
271:.
269:N
252:.
246:N
238:G
227:b
223:a
219:b
215:a
124:Ψ
120:Б
113:Б
109:Ψ
103:Ψ
100:Б
96:Ψ
88:Ψ
84:Б
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