3927:
28:
1937:
1639:
2901:
232:
is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state
5451:
Note however that there is a difference in the interpretation between Kripke structures and Büchi automata. While the former explicitly names every state variable's polarity for every state, the latter just declares the current set of variables holding or not holding true. It says absolutely nothing
1645:
1347:
3829:
3682:
3442:
3014:
1016:
3291:
244:, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize
2466:
is accepting if r' is concatenation of an infinite series of finite segments of 1-states (states with third component 1) and 2-states (states with third component 2) alternatively. There is such a series of segments of r' if r' is accepted by
2252:
4743:
Minimizing deterministic Büchi automata (i.e., given a deterministic Büchi automaton, finding a deterministic Büchi automaton recognizing the same language with a minimal number of states) is an NP-complete problem. This is in contrast to
3914:. Using the definition of ω-regular language and the above closure properties of Büchi automata, it can be easily shown that a Büchi automaton can be constructed such that it recognizes any given ω-regular language. For converse, see
2678:
4673:
of finite state systems can often be translated into various operations on Büchi automata. In addition to the closure operations presented above, the following are some useful operations for the applications of Büchi automata.
1165:
1932:{\displaystyle \scriptstyle \Delta _{2}=\{((q_{A},q_{B},2),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{B}\in F_{B}{\text{ then }}i=1{\text{ else }}i=2\}}
1634:{\displaystyle \scriptstyle \Delta _{1}=\{((q_{A},q_{B},1),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{A}\in F_{A}{\text{ then }}i=2{\text{ else }}i=1\}}
2731:
713:
625:
537:
2105:
1341:
1287:
2005:
2402:
2327:
5205:
2907:
3887:
4411:
3296:
5358:
3567:
882:
3687:
3986:
The class of deterministic Büchi automata does not suffice to encompass all omega-regular languages. In particular, there is no deterministic Büchi automaton that recognizes the language
3493:
3097:
2578:
2524:
831:
3562:
3143:
2724:
1214:
1074:
775:
877:
5318:
4469:
7388:
4092:
4035:
3979:
4254:
4039:, which contains exactly words in which 1 occurs only finitely many times. We can demonstrate it by contradiction that no such deterministic Büchi automaton exists. Let us suppose
5419:
5277:
4336:
2464:
2435:
3524:
3174:
3056:
3179:
5242:
4507:
and vice versa, since both types of automaton are defined as 5-tuple of the same components, only the interpretation of acceptance condition is different. We will show that
214:
151:
4793:
Multiple sets of states in acceptance condition can be translated into one set of states by an automata construction, which is known as "counting construction". Let's say
4682:
Since deterministic Büchi automata are strictly less expressive than non-deterministic automata, there can not be an algorithm for determinization of Büchi automata. But,
4170:
4131:
178:
4657:
4574:
115:
5496:
Büchi presented a doubly exponential complement construction in a logical form. Here, we have his construction in the modern notation used in automata theory. Let
4706:
The language recognized by a Büchi automaton is non-empty if and only if there is a final state that is both reachable from the initial state, and lies on a cycle.
4283:
4201:
2114:
83:
56:
4619:
4536:
7708:
7228:
7381:
4870:
5488:. The first construction was presented by Büchi in 1962. Later, other constructions were developed that enabled efficient and optimal complementation.
2896:{\displaystyle \scriptstyle \Delta '=\Delta _{A}\cup \Delta _{C}\cup \{(q,a,q')|q'\in I_{A}{\text{ and }}\exists f\in F_{C}.(q,a,f)\in \Delta _{C}\}}
233:
is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input.
7254:
2583:
7595:
7374:
7193:
630:
542:
454:
7312:
7026:
4767:
3898:
7520:
7106:
Schewe, Sven (2010). "Minimisation of
Deterministic Parity and Büchi Automata and Relative Minimisation of Deterministic Finite Automata".
7610:
2011:
4886:
A given Muller automaton can be transformed into an equivalent Büchi automaton with the following automata construction. Let's suppose
1293:
7535:
6098:
to any of the states in Q'. Since the regular languages are closed under finite intersections and under relative complements, every α
5477:
recognized by the given Büchi automaton. Existence of algorithms for this construction proves that the set of ω-regular languages is
1221:
7082:
5851:
249:
2336:
2261:
1945:
7703:
7688:
1087:
418:, the transition function δ is replaced with a transition relation Δ that returns a set of states, and the single initial state
7564:
3009:{\displaystyle \scriptstyle {\text{ if }}I_{C}\cap F_{C}{\text{ is empty then }}I'=I_{C}{\text{ otherwise }}I'=I_{C}\cup I_{A}}
5163:
4735:
Each of the steps of this algorithm can be done in time linear in the automaton size, hence the algorithm is clearly optimal.
85:, the former of which is the start state and the latter of which is accepting. Its inputs are infinite words over the symbols
245:
4785:
4772:
3437:{\displaystyle \scriptstyle \Delta '=\Delta _{C}\cup \{(q_{\text{new}},a,q')|\exists q\in I_{C}.(q,a,q')\in \Delta _{C}\}.}
3843:
4345:
7581:
7506:
4718:
3677:{\displaystyle \scriptstyle A''=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta '',\{q_{\text{new}}\},\{q_{\text{new}}\})}
1011:{\displaystyle \scriptstyle (Q_{A}\cup Q_{B},\Sigma ,\Delta _{A}\cup \Delta _{B},I_{A}\cup I_{B},{F}_{A}\cup {F}_{B}).}
7678:
5470:
3824:{\displaystyle \scriptstyle \Delta ''=\Delta '\cup \{(q,a,q_{\text{new}})|\exists q'\in F_{C}.(q,a,q')\in \Delta '\}.}
5324:
429:
of initial states. Generally, the term Büchi automaton without qualifier refers to non-deterministic Büchi automata.
5017:, since initial states are same and ∆' contains ∆. At some non-deterministically chosen position in the input word,
7574:
3061:
2541:
2480:
794:
7499:
4586:
to visit final states infinitely often. Thus, infinitely many finite prefixes of this ω-word will be accepted by
3529:
3110:
2683:
1170:
1030:
731:
836:
7755:
7651:
7646:
5283:
4422:
4687:
4048:
3991:
3935:
7732:
Any language in each category is generated by a grammar and by an automaton in the category in the same line.
4476:
The class of languages recognizable by deterministic Büchi automata is characterized by the following lemma.
3447:
7750:
7154:
7130:
5082:
4762:
4683:
4212:
265:
5397:
7662:
7600:
7525:
7353:
257:
5248:
4290:
3286:{\displaystyle \scriptstyle A'=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta ',\{q_{\text{new}}\},F_{C}),}
7605:
7553:
7366:
4861:
2440:
2411:
276:
6985:
6931:
5474:
4757:
3915:
3911:
3498:
3148:
3030:
7698:
7673:
7530:
7491:
3145:
is constructed in two stages. First, we construct a finite automaton A' such that A' also recognizes
402:
accepts exactly those runs in which at least one of the infinitely often occurring states is in
7358:
5211:
183:
120:
6903:. The r' obtained from this process will have infinitely many run segments containing states from
6270:
3926:
3526:
does not contain the empty string. Second, we will construct the Büchi automaton A" that recognize
7683:
7625:
7569:
7107:
7088:
7036:
4725:
4148:
4109:
156:
4873:. And, a translation from a generalized Büchi automaton to a Büchi automaton is presented above.
7418:
7308:
7191:(1987), "The complementation problem for Büchi automata with applications to temporal logic",
7149:
7078:
7022:
5473:
a Büchi automaton, i.e., constructing another automaton that recognizes the complement of the
2247:{\displaystyle \scriptstyle r'=(q_{A}^{0},q_{B}^{0},i^{0}),(q_{A}^{1},q_{B}^{1},i^{1}),\dots }
4869:
A translation from a Linear temporal logic formula to a generalized Büchi automaton is given
3058:
does not contain the empty word then there is a Büchi automaton that recognizes the language
88:
7667:
7620:
7587:
7433:
7263:
7202:
7163:
7070:
7014:
5762:
5096:
4879:
4745:
4691:
4489:
716:
261:
221:
7338:
4628:
4545:
4261:
4179:
61:
34:
7630:
7545:
7512:
7428:
7401:
7397:
7048:
5462:
4695:
4595:
4512:
225:
6828:. We construct r' inductively in the following way. Let the first state of r' be same as
7135:
Proc. International
Congress on Logic, Method, and Philosophy of Science, Stanford, 1960
27:
7641:
7423:
7405:
7349:
7245:
4714:
4670:
4485:
433:
272:
253:
7168:
7067:
Proceedings of the 29th Annual
Symposium on Foundations of Computer Science (FOCS '88)
4731:
Check whether there is a non-trivial SCC that is reachable and contains a final state.
7744:
7726:
7207:
7188:
7092:
7282:
6988:
into a Büchi automaton. This construction is doubly exponential in terms of size of
7222:
7062:
5484:
This construction is particularly hard relative to the constructions for the other
5478:
445:
7009:
Büchi, J.R. (1962). "On a
Decision Method in Restricted Second Order Arithmetic".
7302:
7018:
6057:
to all the states in Q' and does not have a run that passes through any state in
4779:
Transforming from other models of description to non-deterministic Büchi automata
7693:
7615:
7540:
7298:
7249:
7184:
7322:
Thomas, Wolfgang (1990). "Automata on infinite objects". In Van
Leeuwen (ed.).
4484:
An ω-language is recognizable by a deterministic Büchi automaton if it is the
318:
17:
4690:
provide algorithms that can translate a Büchi automaton into a deterministic
7074:
4842:∆' = { ( (q,i), a, (q',j) ) | (q,a,q') ∈ ∆ and if q ∈ F
7152:(1966), "Testing and generating infinite sequences by a finite automaton",
2673:{\displaystyle \scriptstyle A'=(Q_{C}\cup Q_{A},\Sigma ,\Delta ',I',F_{A})}
7267:
180:
denotes the infinite repetition of a string. It rejects the infinite word
5739:
The following three theorems provide a construction of the complement of
787:
708:{\displaystyle \scriptstyle C=(Q_{C},\Sigma ,\Delta _{C},I_{C},{F}_{C})}
620:{\displaystyle \scriptstyle B=(Q_{B},\Sigma ,\Delta _{B},I_{B},{F}_{B})}
532:{\displaystyle \scriptstyle A=(Q_{A},\Sigma ,\Delta _{A},I_{A},{F}_{A})}
7133:(1962), "On a decision method in restricted second order arithmetic",
6363:) occurs. By the infinite Ramsey theorem, we can find an infinite set
5723:
is a map definable in this way, we denote the (unique) class defining
4709:
An effective algorithm that can check emptiness of a Büchi automaton:
6784:
must be an infinite set. Similarly, we can split the word w'. Let w'
4817:
are sets of accepting states then the equivalent Büchi automaton is
5081:. This construction closely follows the first part of the proof of
3176:
but there are no incoming transitions to initial states of A'. So,
7656:
7112:
5525:
be an equivalence relation over elements of Σ such that for each
2100:{\displaystyle \scriptstyle F'=\{(q_{A},q_{B},2)|q_{B}\in F_{B}\}}
26:
5009:
as follows: At the beginning of the input word, the execution of
1336:{\displaystyle \scriptstyle \Delta '=\Delta _{1}\cup \Delta _{2}}
7370:
4910:
are sets of accepting states. An equivalent Büchi automaton is
1282:{\displaystyle \scriptstyle Q'=Q_{A}\times Q_{B}\times \{1,2\}}
7069:, Washington DC, US: IEEE Computer Society, pp. 319–327,
6428:
be a defining map of an equivalence class such that for each
5141:
is a relation between two states also interpreted as an edge,
6090:∈ Q'}, which is the set of non-empty words that cannot move
5452:
about the other variables that could be present in the model.
5158:
The Büchi automaton will have the following characteristics:
5021:
decides of jump into newly added states via a transition in ∆
2397:{\displaystyle \scriptstyle r_{B}=q_{B}^{0},q_{B}^{1},\dots }
2322:{\displaystyle \scriptstyle r_{A}=q_{A}^{0},q_{A}^{1},\dots }
2000:{\displaystyle \scriptstyle I'=I_{A}\times I_{B}\times \{1\}}
7725:
Each category of languages, except those marked by a , is a
7252:(July 2001), "Weak alternating automata are not that weak",
7352:"An automata-theoretic approach to linear temporal logic".
1160:{\displaystyle \scriptstyle A'=(Q',\Sigma ,\Delta ',I',F')}
4728:) to find which SCCs are reachable from the initial state.
7013:. Stanford: Stanford University Press. pp. 425–435.
6049:∈ Q'}, which is the set of non-empty words that can move
5761:
has finitely many equivalent classes and each class is a
6911:
is infinite. Therefore, r' is an accepting run and w' ∈
5784:
has finitely many equivalence classes. Now we show that
5200:{\displaystyle Q_{\text{final}}=Q\cup \{{\text{init}}\}}
3840:
There is a Büchi automaton that recognizes the language
2477:
There is a Büchi automaton that recognizes the language
1027:
There is a Büchi automaton that recognizes the language
728:
There is a Büchi automaton that recognizes the language
4950: ={ ( q, a, (i,q',∅) ) | (q, a, q') ∈ ∆ and q' ∈ F
4961:={ ( (i,q,R), a, (i,q',R') ) | (q,a,q')∈∆ and q,q' ∈ F
3847:
3691:
3571:
3564:
by adding a loop back to the initial state of A'. So,
3533:
3502:
3451:
3300:
3183:
3152:
3114:
3065:
3034:
2911:
2735:
2687:
2587:
2545:
2484:
2444:
2415:
2340:
2265:
2118:
2015:
1949:
1649:
1351:
1297:
1225:
1174:
1091:
1034:
886:
840:
798:
735:
634:
546:
458:
6404:
be a defining map of an equivalence class such that
6335:
of size 2. We assign the colors as follows. For each
5400:
5327:
5286:
5251:
5214:
5166:
4631:
4598:
4548:
4515:
4425:
4348:
4293:
4264:
4215:
4182:
4151:
4112:
4051:
3994:
3938:
3922:
Deterministic versus non-deterministic Büchi automata
3846:
3690:
3570:
3532:
3501:
3450:
3299:
3182:
3151:
3113:
3064:
3033:
2910:
2734:
2686:
2586:
2544:
2483:
2443:
2414:
2339:
2264:
2117:
2014:
1948:
1648:
1350:
1296:
1224:
1173:
1090:
1033:
885:
839:
797:
734:
633:
545:
457:
186:
159:
123:
91:
64:
37:
7137:, Stanford: Stanford University Press, pp. 1–12
3930:
A non-deterministic Büchi automaton that recognizes
3882:{\displaystyle \scriptstyle \Sigma ^{\omega }/L(A).}
7225:(October 1988), "On the complexity of ω-automata",
5001:and adds extra states on them. The Büchi automaton
4413:is generated which causes A to visit some state in
4406:{\displaystyle 0^{i_{0}}10^{i_{1}}10^{i_{2}}\dots }
4043:is a deterministic Büchi automaton that recognizes
5413:
5352:
5312:
5271:
5236:
5199:
5149:are the set of atomic propositions that form
4651:
4613:
4568:
4530:
4503:can be viewed as a deterministic finite automaton
4463:
4405:
4330:
4277:
4248:
4195:
4164:
4125:
4086:
4029:
3973:
3881:
3823:
3676:
3556:
3518:
3487:
3436:
3285:
3168:
3137:
3091:
3050:
3008:
2895:
2718:
2672:
2572:
2518:
2458:
2429:
2396:
2321:
2246:
2099:
1999:
1931:
1633:
1335:
1281:
1208:
1159:
1068:
1010:
871:
825:
769:
707:
619:
531:
208:
172:
145:
109:
77:
50:
6880:, we can extend the run along the word segment w'
275:as an automata-theoretic version of a formula in
5990:∈ Q'}, which is the set of words that can move
5353:{\displaystyle \delta =q{\overrightarrow {a}}p}
4342:. Continuing with this construction the ω-word
264:. They are named after the Swiss mathematician
6852:. By induction hypothesis, we have a run on w'
6585:), otherwise the theorem holds trivially. Let
117:. As an example, it accepts the infinite word
7382:
5609:→ 2 × 2 in the following way: for each state
310:) that consists of the following components:
8:
7229:Symposium on Foundations of Computer Science
5307:
5299:
5266:
5258:
5194:
5186:
3814:
3714:
3667:
3654:
3648:
3635:
3612:
3599:
3427:
3325:
3260:
3247:
3224:
3211:
3092:{\displaystyle \scriptstyle L(C)^{\omega }.}
2889:
2773:
2573:{\displaystyle \scriptstyle Q_{C}\cap Q_{A}}
2519:{\displaystyle \scriptstyle L(C)\cdot L(A).}
2093:
2027:
1993:
1987:
1925:
1663:
1627:
1365:
1275:
1263:
826:{\displaystyle \scriptstyle Q_{A}\cap Q_{B}}
104:
92:
7704:Counter-free (with aperiodic finite monoid)
7065:(1988), "On the complexity of ω-automata",
5998:to all the states in Q' via some state in
3557:{\displaystyle \scriptstyle L(C)^{\omega }}
3138:{\displaystyle \scriptstyle L(C)^{\omega }}
2719:{\displaystyle \scriptstyle L(C)\cdot L(A)}
1209:{\displaystyle \scriptstyle L(A)\cap L(B),}
1069:{\displaystyle \scriptstyle L(A)\cap L(B).}
770:{\displaystyle \scriptstyle L(A)\cup L(B).}
7414:
7389:
7375:
7367:
7339:"Finite-state Automata on Infinite Inputs"
6922:By the above theorems, we can represent Σ-
5062:states again and again. So, if the states
872:{\displaystyle \scriptstyle L(A)\cup L(B)}
432:For more comprehensive formalism see also
7357:
7307:. Springer Science & Business Media.
7206:
7167:
7125:
7123:
7111:
6841:, we can choose a run segment on word w'
5401:
5399:
5337:
5326:
5313:{\displaystyle F=Q\cup \{{\text{init}}\}}
5302:
5285:
5261:
5250:
5225:
5213:
5189:
5171:
5165:
4630:
4597:
4547:
4514:
4464:{\displaystyle (0\cup 1)^{*}0^{\omega }.}
4452:
4442:
4424:
4392:
4387:
4375:
4370:
4358:
4353:
4347:
4320:
4315:
4303:
4298:
4292:
4269:
4263:
4237:
4225:
4220:
4214:
4187:
4181:
4156:
4150:
4117:
4111:
4078:
4068:
4050:
4021:
4011:
3993:
3965:
3955:
3937:
3858:
3852:
3845:
3768:
3745:
3736:
3689:
3661:
3642:
3606:
3590:
3569:
3547:
3531:
3500:
3449:
3421:
3379:
3361:
3335:
3316:
3298:
3270:
3254:
3218:
3202:
3181:
3150:
3128:
3112:
3079:
3063:
3032:
2999:
2986:
2966:
2960:
2940:
2934:
2921:
2912:
2909:
2883:
2846:
2828:
2822:
2802:
2764:
2751:
2733:
2685:
2660:
2619:
2606:
2585:
2563:
2550:
2543:
2482:
2449:
2442:
2420:
2413:
2381:
2376:
2363:
2358:
2345:
2338:
2306:
2301:
2288:
2283:
2270:
2263:
2228:
2215:
2210:
2197:
2192:
2173:
2160:
2155:
2142:
2137:
2116:
2087:
2074:
2065:
2050:
2037:
2013:
1978:
1965:
1947:
1911:
1897:
1891:
1878:
1869:
1863:
1844:
1825:
1813:
1807:
1788:
1769:
1757:
1736:
1720:
1689:
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1654:
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1613:
1599:
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1580:
1571:
1565:
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1527:
1515:
1509:
1490:
1471:
1459:
1438:
1422:
1391:
1378:
1356:
1349:
1326:
1313:
1295:
1254:
1241:
1223:
1172:
1089:
1032:
995:
990:
980:
975:
965:
952:
939:
926:
907:
894:
884:
838:
816:
803:
796:
733:
695:
690:
680:
667:
648:
632:
607:
602:
592:
579:
560:
544:
519:
514:
504:
491:
472:
456:
200:
185:
164:
158:
137:
122:
90:
69:
63:
42:
36:
7324:Handbook of Theoretical Computer Science
5077:accepts corresponding input and so does
4748:, which can be done in polynomial time.
4417:infinitely often and the word is not in
4087:{\displaystyle (0\cup 1)^{*}0^{\omega }}
4030:{\displaystyle (0\cup 1)^{*}0^{\omega }}
3974:{\displaystyle (0\cup 1)^{*}0^{\omega }}
3925:
7255:ACM Transactions on Computational Logic
7011:The Collected Works of J. Richard Büchi
7001:
5485:
5051:then it is reset to the empty set and ∆
4338:the automaton will visit some state in
3488:{\displaystyle \scriptstyle L(C)=L(A')}
2254:is a run of automaton A' on input word
7596:Linear context-free rewriting language
7044:
7034:
6597:. We need to show that each word w' ∈
6321:. Consider the set of natural numbers
4249:{\displaystyle 0^{i_{0}}10^{\omega }.}
252:to infinite inputs. Each are types of
7521:Linear context-free rewriting systems
6631:occurs in r' infinitely often. Since
6375:of size 2 has same color. Let 0 <
5591:if and only if this is possible over
5567:if and only if this is possible over
5414:{\displaystyle {\overrightarrow {a}}}
879:is recognized by the Büchi automaton
7:
7304:Automata Theory and its Applications
6351:} be the equivalence class in which
5486:closure properties of Büchi automata
5467:complementation of a Büchi automaton
4143:after reading some finite prefix of
3916:construction of a ω-regular language
3107:The Büchi automaton that recognizes
6984:) is an ω-regular language. We can
6627:over input w' such that a state in
5272:{\displaystyle I=\{{\text{init}}\}}
4331:{\displaystyle 0^{i_{0}}10^{i_{1}}}
31:A Büchi automaton with two states,
7729:of the category directly above it.
6887:such that the extension reaches s
6753:if and only if the run segment in
6623:), i.e., there exists a run r' of
5743:using the equivalence classes of ~
5215:
5073:are visited infinitely often then
4997:keeps original set of states from
4768:Semi-deterministic Büchi automaton
4499:Any deterministic Büchi automaton
3849:
3807:
3750:
3704:
3693:
3625:
3618:
3418:
3366:
3313:
3302:
3237:
3230:
2880:
2833:
2761:
2748:
2737:
2635:
2628:
2580:is empty then the Büchi automaton
2459:{\displaystyle \scriptstyle r_{B}}
2430:{\displaystyle \scriptstyle r_{A}}
1860:
1804:
1651:
1562:
1506:
1353:
1323:
1310:
1299:
1124:
1117:
936:
923:
916:
664:
657:
576:
569:
488:
481:
25:
5852:nondeterministic finite automaton
4902:}) is a Muller automaton, where F
3519:{\displaystyle \scriptstyle L(C)}
3169:{\displaystyle \scriptstyle L(C)}
3051:{\displaystyle \scriptstyle L(C)}
271:Büchi automata are often used in
238:non-deterministic Büchi automaton
7284:Büchi Complementation Made Tight
250:nondeterministic finite automata
5145:is the label for the state and
5055:try to visit all the states of
5029:try to visit all the states of
5005:simulates the Muller automaton
4969:then R'= ∅ otherwise R'=R∪{q} }
4846:then j=((i+1) mod n) else j=i }
260:, the infinite word version of
256:. Büchi automata recognize the
6745:be a set of indices such that
6325:. Let equivalence classes of ~
5770:Since there are finitely many
5237:{\displaystyle \Sigma =2^{AP}}
5137:is the set of initial states,
4646:
4635:
4608:
4602:
4563:
4552:
4525:
4519:
4439:
4426:
4065:
4052:
4008:
3995:
3952:
3939:
3872:
3866:
3800:
3777:
3746:
3742:
3717:
3670:
3583:
3544:
3537:
3512:
3506:
3481:
3470:
3461:
3455:
3411:
3388:
3362:
3358:
3328:
3276:
3195:
3162:
3156:
3125:
3118:
3076:
3069:
3044:
3038:
2873:
2855:
2803:
2799:
2776:
2712:
2706:
2697:
2691:
2666:
2599:
2509:
2503:
2494:
2488:
2234:
2185:
2179:
2130:
2066:
2062:
2030:
1853:
1818:
1797:
1762:
1758:
1754:
1751:
1713:
1701:
1669:
1666:
1555:
1520:
1499:
1464:
1460:
1456:
1453:
1415:
1403:
1371:
1368:
1199:
1193:
1184:
1178:
1153:
1103:
1059:
1053:
1044:
1038:
1001:
887:
865:
859:
850:
844:
760:
754:
745:
739:
701:
641:
613:
553:
525:
465:
240:, later referred to just as a
209:{\displaystyle aaab^{\omega }}
146:{\displaystyle (ab)^{\omega }}
134:
124:
1:
7326:. Elsevier. pp. 133–164.
7169:10.1016/s0019-9958(66)80013-x
6119:is regular. By definitions,
4719:strongly connected components
3910:Büchi automata recognize the
444:The set of Büchi automata is
336:Σ is a finite set called the
289:deterministic Büchi automaton
268:, who invented them in 1962.
246:deterministic finite automata
230:deterministic Büchi automaton
7208:10.1016/0304-3975(87)90008-9
7194:Theoretical Computer Science
7019:10.1007/978-1-4613-8928-6_23
6970:are equivalence classes of ~
6510:be equivalence classes of ~
6331:be the colors of subsets of
5491:
5025:. Then, the transitions in ∆
4713:Consider the automaton as a
6273:to prove this theorem. Let
5793:is a regular language. For
4773:Generalized Büchi automaton
4578:. An ω-word is accepted by
4165:{\displaystyle 0^{\omega }}
4126:{\displaystyle 0^{\omega }}
351: × Σ →
173:{\displaystyle x^{\omega }}
7772:
7611:Deterministic context-free
7536:Deterministic context-free
4821: = (Q', Σ, ∆',q'
4786:generalized Büchi automata
448:the following operations.
355:is a function, called the
7722:
7684:Nondeterministic pushdown
7412:
6371:such that each subset of
5013:follows the execution of
4914: = (Q', Σ, ∆',Q
4540:is the limit language of
4139:will visit some state in
2942: is empty then
7231:, White Plains, New York
6555:Suppose there is a word
6534:) is either a subset of
4724:Run a search (e.g., the
4209:also accepts the ω-word
7155:Information and Control
7075:10.1109/SFCS.1988.21948
6589:be an accepting run of
6271:infinite Ramsey theorem
5778:→ 2 × 2, the relation ~
5543:if and only if for all
4623:is a limit language of
3918:for a Büchi automaton.
3897:The proof is presented
110:{\displaystyle \{a,b\}}
7689:Deterministic pushdown
7565:Recursively enumerable
7297:Bakhadyr Khoussainov;
6986:translate the language
6891:and visits a state in
6776:contains a state from
5415:
5354:
5314:
5273:
5238:
5201:
5133:is the set of states,
5090:From Kripke structures
4717:and decompose it into
4653:
4615:
4570:
4532:
4465:
4407:
4332:
4279:
4250:
4197:
4166:
4127:
4088:
4031:
3983:
3975:
3906:Recognizable languages
3883:
3825:
3678:
3558:
3520:
3489:
3438:
3287:
3170:
3139:
3093:
3052:
3010:
2897:
2720:
2674:
2574:
2520:
2460:
2431:
2398:
2323:
2248:
2101:
2001:
1933:
1635:
1337:
1283:
1210:
1161:
1070:
1012:
873:
827:
771:
709:
627:be Büchi automata and
621:
533:
217:
210:
174:
147:
111:
79:
52:
7281:Schewe, Sven (2009).
7268:10.1145/377978.377993
6930:) as finite union of
6796:... = w' such that w'
5416:
5355:
5315:
5274:
5239:
5202:
4890: = (Q,Σ,∆,Q
4862:Linear temporal logic
4797: = (Q,Σ,∆,q
4654:
4652:{\displaystyle L(A')}
4616:
4571:
4569:{\displaystyle L(A')}
4533:
4466:
4408:
4333:
4280:
4278:{\displaystyle i_{1}}
4251:
4198:
4196:{\displaystyle i_{0}}
4167:
4128:
4096:with final state set
4089:
4032:
3976:
3929:
3884:
3826:
3679:
3559:
3521:
3490:
3439:
3288:
3171:
3140:
3094:
3053:
3011:
2968: otherwise
2898:
2721:
2675:
2575:
2521:
2461:
2432:
2399:
2324:
2249:
2102:
2002:
1934:
1636:
1338:
1284:
1211:
1162:
1071:
1013:
874:
828:
772:
710:
622:
534:
425:is replaced by a set
277:linear temporal logic
211:
175:
148:
112:
80:
78:{\displaystyle q_{1}}
53:
51:{\displaystyle q_{0}}
30:
7674:Tree stack automaton
6542:) or disjoint from
6343:, let the color of {
5492:Büchi's construction
5398:
5325:
5284:
5249:
5212:
5164:
5083:McNaughton's Theorem
4809:}) is a GBA, where F
4763:Weak Büchi automaton
4688:Safra's construction
4684:McNaughton's Theorem
4629:
4614:{\displaystyle L(A)}
4596:
4546:
4531:{\displaystyle L(A)}
4513:
4423:
4346:
4291:
4262:
4256:Therefore, for some
4213:
4180:
4149:
4110:
4049:
3992:
3936:
3844:
3688:
3568:
3530:
3499:
3448:
3297:
3180:
3149:
3111:
3062:
3031:
2908:
2732:
2684:
2584:
2542:
2481:
2441:
2412:
2337:
2262:
2115:
2012:
1946:
1646:
1348:
1294:
1222:
1171:
1088:
1084:The Büchi automaton
1031:
883:
837:
795:
732:
631:
543:
455:
396:acceptance condition
266:Julius Richard Büchi
184:
157:
121:
89:
62:
35:
7582:range concatenation
7507:range concatenation
7301:(6 December 2012).
6932:ω-regular languages
6871:. By definition of
6832:. By definition of
5689:can move automaton
5658:can move automaton
4935:∆'= ∆ ∪ ∆
4287:, after the prefix
3912:ω-regular languages
2386:
2368:
2311:
2293:
2220:
2202:
2165:
2147:
1852:
1796:
1744:
1728:
1554:
1498:
1446:
1430:
357:transition function
258:ω-regular languages
7233:, pp. 319–327
5957:) ∈ Δ }. Let Q' ⊆
5475:ω-regular language
5411:
5350:
5310:
5269:
5234:
5197:
4828:Q' = Q × {1,...,n}
4758:Co-Büchi automaton
4726:depth-first search
4702:Emptiness checking
4649:
4611:
4566:
4528:
4461:
4403:
4328:
4275:
4246:
4193:
4162:
4123:
4084:
4027:
3984:
3971:
3879:
3878:
3821:
3820:
3674:
3673:
3554:
3553:
3516:
3515:
3485:
3484:
3434:
3433:
3283:
3282:
3166:
3165:
3135:
3134:
3089:
3088:
3048:
3047:
3006:
3005:
2893:
2892:
2716:
2715:
2670:
2669:
2570:
2569:
2516:
2515:
2456:
2455:
2427:
2426:
2394:
2393:
2372:
2354:
2319:
2318:
2297:
2279:
2244:
2243:
2206:
2188:
2151:
2133:
2097:
2096:
1997:
1996:
1929:
1928:
1871: and if
1840:
1784:
1732:
1716:
1631:
1630:
1573: and if
1542:
1486:
1434:
1418:
1333:
1332:
1279:
1278:
1206:
1205:
1157:
1156:
1066:
1065:
1008:
1007:
869:
868:
823:
822:
767:
766:
705:
704:
617:
616:
529:
528:
440:Closure properties
321:. The elements of
218:
206:
170:
143:
107:
75:
48:
7738:
7737:
7717:
7716:
7679:Embedded pushdown
7575:Context-sensitive
7500:Context-sensitive
7434:Abstract machines
7419:Chomsky hierarchy
7314:978-1-4612-0171-7
7028:978-1-4613-8930-9
5595:. Each class of ~
5481:complementation.
5409:
5345:
5305:
5264:
5192:
5174:
5044:becomes equal to
5036:and keep growing
4981:{i} × F
4928:{i} × F
4694:or deterministic
4582:if it will force
3739:
3664:
3645:
3609:
3338:
3257:
3221:
2969:
2943:
2915:
2831:
2437:is accepting and
2111:By construction,
1914:
1900:
1872:
1816:
1616:
1602:
1574:
1518:
412:non-deterministic
372:is an element of
283:Formal definition
262:regular languages
16:(Redirected from
7763:
7733:
7730:
7694:Visibly pushdown
7668:Thread automaton
7616:Visibly pushdown
7584:
7541:Visibly pushdown
7509:
7496:(no common name)
7415:
7402:formal languages
7391:
7384:
7377:
7368:
7363:
7361:
7345:
7343:
7327:
7318:
7289:
7288:
7278:
7272:
7270:
7242:
7236:
7234:
7227:Proc. 29th IEEE
7219:
7213:
7211:
7210:
7201:(2–3): 217–237,
7180:
7174:
7172:
7171:
7146:
7140:
7138:
7127:
7118:
7117:
7115:
7103:
7097:
7095:
7059:
7053:
7052:
7046:
7042:
7040:
7032:
7006:
6862:that reaches to
6725:after consuming
6721:be the state in
6269:We will use the
6218:∈ Σ, there are ~
5886:) ∈ Δ} ∪ { ((1,
5830:, Σ, Δ
5763:regular language
5701:via a state in
5571:and furthermore
5445:
5431:
5420:
5418:
5417:
5412:
5410:
5402:
5371:
5359:
5357:
5356:
5351:
5346:
5338:
5319:
5317:
5316:
5311:
5306:
5303:
5278:
5276:
5275:
5270:
5265:
5262:
5243:
5241:
5240:
5235:
5233:
5232:
5206:
5204:
5203:
5198:
5193:
5190:
5176:
5175:
5172:
5136:
5128:
5127:
5097:Kripke structure
4746:DFA minimization
4692:Muller automaton
4660:
4658:
4656:
4655:
4650:
4645:
4622:
4620:
4618:
4617:
4612:
4589:
4585:
4581:
4577:
4575:
4573:
4572:
4567:
4562:
4539:
4537:
4535:
4534:
4529:
4506:
4502:
4490:regular language
4472:
4470:
4468:
4467:
4462:
4457:
4456:
4447:
4446:
4412:
4410:
4409:
4404:
4399:
4398:
4397:
4396:
4382:
4381:
4380:
4379:
4365:
4364:
4363:
4362:
4337:
4335:
4334:
4329:
4327:
4326:
4325:
4324:
4310:
4309:
4308:
4307:
4286:
4284:
4282:
4281:
4276:
4274:
4273:
4255:
4253:
4252:
4247:
4242:
4241:
4232:
4231:
4230:
4229:
4204:
4202:
4200:
4199:
4194:
4192:
4191:
4174:, say after the
4173:
4171:
4169:
4168:
4163:
4161:
4160:
4134:
4132:
4130:
4129:
4124:
4122:
4121:
4095:
4093:
4091:
4090:
4085:
4083:
4082:
4073:
4072:
4038:
4036:
4034:
4033:
4028:
4026:
4025:
4016:
4015:
3982:
3980:
3978:
3977:
3972:
3970:
3969:
3960:
3959:
3888:
3886:
3885:
3880:
3862:
3857:
3856:
3830:
3828:
3827:
3822:
3813:
3799:
3773:
3772:
3760:
3749:
3741:
3740:
3737:
3710:
3699:
3683:
3681:
3680:
3675:
3666:
3665:
3662:
3647:
3646:
3643:
3631:
3611:
3610:
3607:
3595:
3594:
3579:
3563:
3561:
3560:
3555:
3552:
3551:
3525:
3523:
3522:
3517:
3494:
3492:
3491:
3486:
3480:
3443:
3441:
3440:
3435:
3426:
3425:
3410:
3384:
3383:
3365:
3357:
3340:
3339:
3336:
3321:
3320:
3308:
3292:
3290:
3289:
3284:
3275:
3274:
3259:
3258:
3255:
3243:
3223:
3222:
3219:
3207:
3206:
3191:
3175:
3173:
3172:
3167:
3144:
3142:
3141:
3136:
3133:
3132:
3098:
3096:
3095:
3090:
3084:
3083:
3057:
3055:
3054:
3049:
3015:
3013:
3012:
3007:
3004:
3003:
2991:
2990:
2978:
2970:
2967:
2965:
2964:
2952:
2944:
2941:
2939:
2938:
2926:
2925:
2916:
2913:
2902:
2900:
2899:
2894:
2888:
2887:
2851:
2850:
2832:
2829:
2827:
2826:
2814:
2806:
2798:
2769:
2768:
2756:
2755:
2743:
2725:
2723:
2722:
2717:
2679:
2677:
2676:
2671:
2665:
2664:
2652:
2641:
2624:
2623:
2611:
2610:
2595:
2579:
2577:
2576:
2571:
2568:
2567:
2555:
2554:
2525:
2523:
2522:
2517:
2465:
2463:
2462:
2457:
2454:
2453:
2436:
2434:
2433:
2428:
2425:
2424:
2403:
2401:
2400:
2395:
2385:
2380:
2367:
2362:
2350:
2349:
2328:
2326:
2325:
2320:
2310:
2305:
2292:
2287:
2275:
2274:
2253:
2251:
2250:
2245:
2233:
2232:
2219:
2214:
2201:
2196:
2178:
2177:
2164:
2159:
2146:
2141:
2126:
2106:
2104:
2103:
2098:
2092:
2091:
2079:
2078:
2069:
2055:
2054:
2042:
2041:
2023:
2006:
2004:
2003:
1998:
1983:
1982:
1970:
1969:
1957:
1938:
1936:
1935:
1930:
1915:
1913: else
1912:
1901:
1899: then
1898:
1896:
1895:
1883:
1882:
1873:
1870:
1868:
1867:
1848:
1830:
1829:
1817:
1814:
1812:
1811:
1792:
1774:
1773:
1761:
1740:
1724:
1694:
1693:
1681:
1680:
1659:
1658:
1640:
1638:
1637:
1632:
1617:
1615: else
1614:
1603:
1601: then
1600:
1598:
1597:
1585:
1584:
1575:
1572:
1570:
1569:
1550:
1532:
1531:
1519:
1516:
1514:
1513:
1494:
1476:
1475:
1463:
1442:
1426:
1396:
1395:
1383:
1382:
1361:
1360:
1342:
1340:
1339:
1334:
1331:
1330:
1318:
1317:
1305:
1288:
1286:
1285:
1280:
1259:
1258:
1246:
1245:
1233:
1215:
1213:
1212:
1207:
1166:
1164:
1163:
1158:
1152:
1141:
1130:
1113:
1099:
1075:
1073:
1072:
1067:
1017:
1015:
1014:
1009:
1000:
999:
994:
985:
984:
979:
970:
969:
957:
956:
944:
943:
931:
930:
912:
911:
899:
898:
878:
876:
875:
870:
832:
830:
829:
824:
821:
820:
808:
807:
776:
774:
773:
768:
717:finite automaton
714:
712:
711:
706:
700:
699:
694:
685:
684:
672:
671:
653:
652:
626:
624:
623:
618:
612:
611:
606:
597:
596:
584:
583:
565:
564:
538:
536:
535:
530:
524:
523:
518:
509:
508:
496:
495:
477:
476:
222:computer science
215:
213:
212:
207:
205:
204:
179:
177:
176:
171:
169:
168:
152:
150:
149:
144:
142:
141:
116:
114:
113:
108:
84:
82:
81:
76:
74:
73:
57:
55:
54:
49:
47:
46:
21:
7771:
7770:
7766:
7765:
7764:
7762:
7761:
7760:
7756:Finite automata
7741:
7740:
7739:
7734:
7731:
7724:
7718:
7713:
7635:
7579:
7558:
7504:
7485:
7408:
7406:formal grammars
7398:Automata theory
7395:
7359:10.1.1.125.8126
7350:Vardi, Moshe Y.
7348:
7341:
7337:
7334:
7321:
7315:
7296:
7293:
7292:
7280:
7279:
7275:
7244:
7243:
7239:
7221:
7220:
7216:
7183:Sistla, A. P.;
7182:
7181:
7177:
7148:
7147:
7143:
7129:
7128:
7121:
7105:
7104:
7100:
7085:
7061:
7060:
7056:
7043:
7033:
7029:
7008:
7007:
7003:
6998:
6976:. Therefore, Σ-
6975:
6969:
6960:
6951:
6942:
6890:
6886:
6879:
6870:
6861:
6855:
6851:
6844:
6840:
6827:
6818:
6808:
6799:
6795:
6791:
6787:
6775:
6765:
6740:
6731:
6720:
6711:
6702:
6689:
6680:
6669:
6663:
6659:
6652:
6643:
6614:
6605:
6584:
6575:
6551:
6533:
6524:
6515:
6509:
6500:
6485:
6476:
6463:
6454:
6445:
6423:
6414:
6395:
6388:
6381:
6330:
6320:
6310:
6289:
6283:
6265:
6263:
6254:
6241:
6232:
6223:
6191:
6184:
6177:
6176:
6160:
6159:
6152:
6140:
6139:
6127:
6118:
6111:
6104:
6085:
6067:
6044:
6026:
6008:
5985:
5967:
5956:
5949:
5938:
5931:
5924:
5917:
5913:
5906:
5899:
5892:
5885:
5878:
5871:
5864:
5857:
5850:)} ) be a
5837:
5833:
5826:= ( {0,1}×
5825:
5792:
5783:
5766:
5760:
5748:
5735:
5718:
5711:
5676:
5645:
5630:
5623:
5600:
5555:has a run from
5538:
5524:
5517:Büchi automaton
5510:
5494:
5469:is the task of
5463:automata theory
5459:
5457:Complementation
5433:
5429:
5396:
5395:
5361:
5323:
5322:
5282:
5281:
5247:
5246:
5221:
5210:
5209:
5167:
5162:
5161:
5134:
5105:
5100:
5092:
5071:
5060:
5054:
5049:
5034:
5028:
5024:
4988:
4985: × {F
4984:
4980:
4977:
4968:
4964:
4960:
4953:
4949:
4942:
4938:
4931:
4927:
4924:
4921:Q' = Q ∪
4917:
4909:
4905:
4901:
4897:
4893:
4883:
4880:Muller automata
4866:
4852:
4845:
4838:
4834:
4824:
4816:
4812:
4808:
4804:
4800:
4790:
4781:
4754:
4696:Rabin automaton
4678:Determinization
4668:
4638:
4627:
4626:
4624:
4594:
4593:
4591:
4587:
4583:
4579:
4555:
4544:
4543:
4541:
4511:
4510:
4508:
4504:
4500:
4473:Contradiction.
4448:
4438:
4421:
4420:
4418:
4388:
4383:
4371:
4366:
4354:
4349:
4344:
4343:
4316:
4311:
4299:
4294:
4289:
4288:
4265:
4260:
4259:
4257:
4233:
4221:
4216:
4211:
4210:
4183:
4178:
4177:
4175:
4152:
4147:
4146:
4144:
4113:
4108:
4107:
4105:
4074:
4064:
4047:
4046:
4044:
4017:
4007:
3990:
3989:
3987:
3961:
3951:
3934:
3933:
3931:
3924:
3908:
3848:
3842:
3841:
3836:Complementation
3806:
3792:
3764:
3753:
3732:
3703:
3692:
3686:
3685:
3657:
3638:
3624:
3602:
3586:
3572:
3566:
3565:
3543:
3528:
3527:
3497:
3496:
3473:
3446:
3445:
3417:
3403:
3375:
3350:
3331:
3312:
3301:
3295:
3294:
3266:
3250:
3236:
3214:
3198:
3184:
3178:
3177:
3147:
3146:
3124:
3109:
3108:
3075:
3060:
3059:
3029:
3028:
2995:
2982:
2971:
2956:
2945:
2930:
2917:
2906:
2905:
2879:
2842:
2830: and
2818:
2807:
2791:
2760:
2747:
2736:
2730:
2729:
2682:
2681:
2656:
2645:
2634:
2615:
2602:
2588:
2582:
2581:
2559:
2546:
2540:
2539:
2479:
2478:
2445:
2439:
2438:
2416:
2410:
2409:
2404:is run of B on
2341:
2335:
2334:
2329:is run of A on
2266:
2260:
2259:
2224:
2169:
2119:
2113:
2112:
2083:
2070:
2046:
2033:
2016:
2010:
2009:
1974:
1961:
1950:
1944:
1943:
1887:
1874:
1859:
1821:
1815: and
1803:
1765:
1685:
1672:
1650:
1644:
1643:
1589:
1576:
1561:
1523:
1517: and
1505:
1467:
1387:
1374:
1352:
1346:
1345:
1322:
1309:
1298:
1292:
1291:
1250:
1237:
1226:
1220:
1219:
1169:
1168:
1145:
1134:
1123:
1106:
1092:
1086:
1085:
1029:
1028:
989:
974:
961:
948:
935:
922:
903:
890:
881:
880:
835:
834:
812:
799:
793:
792:
730:
729:
689:
676:
663:
644:
629:
628:
601:
588:
575:
556:
541:
540:
513:
500:
487:
468:
453:
452:
442:
424:
416:Büchi automaton
371:
325:are called the
305:
285:
242:Büchi automaton
226:automata theory
196:
182:
181:
160:
155:
154:
133:
119:
118:
87:
86:
65:
60:
59:
38:
33:
32:
23:
22:
15:
12:
11:
5:
7769:
7767:
7759:
7758:
7753:
7751:Model checking
7743:
7742:
7736:
7735:
7723:
7720:
7719:
7715:
7714:
7712:
7711:
7709:Acyclic finite
7706:
7701:
7696:
7691:
7686:
7681:
7676:
7670:
7665:
7660:
7659:Turing Machine
7654:
7652:Linear-bounded
7649:
7644:
7642:Turing machine
7638:
7636:
7634:
7633:
7628:
7623:
7618:
7613:
7608:
7603:
7601:Tree-adjoining
7598:
7593:
7590:
7585:
7577:
7572:
7567:
7561:
7559:
7557:
7556:
7551:
7548:
7543:
7538:
7533:
7528:
7526:Tree-adjoining
7523:
7518:
7515:
7510:
7502:
7497:
7494:
7488:
7486:
7484:
7483:
7480:
7477:
7474:
7471:
7468:
7465:
7462:
7459:
7456:
7453:
7450:
7447:
7444:
7440:
7437:
7436:
7431:
7426:
7421:
7413:
7410:
7409:
7396:
7394:
7393:
7386:
7379:
7371:
7365:
7364:
7346:
7333:
7332:External links
7330:
7329:
7328:
7319:
7313:
7291:
7290:
7273:
7262:(3): 408–429,
7237:
7214:
7175:
7162:(5): 521–530,
7150:McNaughton, R.
7141:
7119:
7098:
7083:
7054:
7045:|journal=
7027:
7000:
6999:
6997:
6994:
6971:
6965:
6956:
6947:
6938:
6888:
6881:
6875:
6866:
6857:
6853:
6849:
6842:
6836:
6823:
6814:
6804:
6797:
6793:
6789:
6785:
6770:
6761:
6736:
6729:
6716:
6707:
6698:
6685:
6678:
6667:
6661:
6657:
6648:
6639:
6610:
6601:
6580:
6571:
6529:
6520:
6511:
6505:
6496:
6481:
6472:
6459:
6450:
6440:
6419:
6412:
6393:
6386:
6379:
6326:
6315:
6306:
6287:
6281:
6259:
6250:
6237:
6228:
6219:
6189:
6182:
6174:
6162:
6157:
6150:
6142:
6137:
6129:
6123:
6113:
6106:
6099:
6076:
6062:
6035:
6017:
6003:
5976:
5962:
5954:
5947:
5936:
5929:
5922:
5915:
5911:
5904:
5897:
5890:
5883:
5876:
5869:
5862:
5855:
5835:
5831:
5813:
5809:∈ {0,1}, let
5788:
5779:
5756:
5744:
5731:
5716:
5709:
5674:
5643:
5628:
5621:
5601:defines a map
5596:
5575:has a run via
5534:
5520:
5508:
5500: = (
5493:
5490:
5458:
5455:
5454:
5453:
5448:
5447:
5408:
5405:
5391:
5390:
5389:
5388:
5349:
5344:
5341:
5336:
5333:
5330:
5320:
5309:
5301:
5298:
5295:
5292:
5289:
5279:
5268:
5260:
5257:
5254:
5244:
5231:
5228:
5224:
5220:
5217:
5207:
5196:
5188:
5185:
5182:
5179:
5170:
5155:
5154:
5099:be defined by
5095:Let the given
5091:
5088:
5087:
5086:
5069:
5058:
5052:
5047:
5032:
5026:
5022:
4992:
4991:
4990:
4986:
4982:
4978:
4975:
4972:
4971:
4970:
4966:
4962:
4958:
4955:
4951:
4947:
4940:
4939: ∪ ∆
4936:
4933:
4932: × 2
4929:
4925:
4922:
4915:
4907:
4903:
4899:
4895:
4891:
4882:
4876:
4875:
4874:
4865:
4858:
4857:
4856:
4855:
4854:
4850:
4847:
4843:
4840:
4836:
4832:
4829:
4822:
4814:
4810:
4806:
4802:
4798:
4789:
4782:
4780:
4777:
4776:
4775:
4770:
4765:
4760:
4753:
4750:
4741:
4740:
4733:
4732:
4729:
4722:
4715:directed graph
4704:
4703:
4680:
4679:
4671:Model checking
4667:
4664:
4663:
4662:
4648:
4644:
4641:
4637:
4634:
4610:
4607:
4604:
4601:
4565:
4561:
4558:
4554:
4551:
4527:
4524:
4521:
4518:
4494:
4486:limit language
4460:
4455:
4451:
4445:
4441:
4437:
4434:
4431:
4428:
4402:
4395:
4391:
4386:
4378:
4374:
4369:
4361:
4357:
4352:
4323:
4319:
4314:
4306:
4302:
4297:
4272:
4268:
4245:
4240:
4236:
4228:
4224:
4219:
4190:
4186:
4159:
4155:
4120:
4116:
4081:
4077:
4071:
4067:
4063:
4060:
4057:
4054:
4024:
4020:
4014:
4010:
4006:
4003:
4000:
3997:
3968:
3964:
3958:
3954:
3950:
3947:
3944:
3941:
3923:
3920:
3907:
3904:
3903:
3902:
3891:
3890:
3877:
3874:
3871:
3868:
3865:
3861:
3855:
3851:
3832:
3831:
3819:
3816:
3812:
3809:
3805:
3802:
3798:
3795:
3791:
3788:
3785:
3782:
3779:
3776:
3771:
3767:
3763:
3759:
3756:
3752:
3748:
3744:
3735:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3709:
3706:
3702:
3698:
3695:
3672:
3669:
3660:
3656:
3653:
3650:
3641:
3637:
3634:
3630:
3627:
3623:
3620:
3617:
3614:
3605:
3601:
3598:
3593:
3589:
3585:
3582:
3578:
3575:
3550:
3546:
3542:
3539:
3536:
3514:
3511:
3508:
3505:
3483:
3479:
3476:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3432:
3429:
3424:
3420:
3416:
3413:
3409:
3406:
3402:
3399:
3396:
3393:
3390:
3387:
3382:
3378:
3374:
3371:
3368:
3364:
3360:
3356:
3353:
3349:
3346:
3343:
3334:
3330:
3327:
3324:
3319:
3315:
3311:
3307:
3304:
3281:
3278:
3273:
3269:
3265:
3262:
3253:
3249:
3246:
3242:
3239:
3235:
3232:
3229:
3226:
3217:
3213:
3210:
3205:
3201:
3197:
3194:
3190:
3187:
3164:
3161:
3158:
3155:
3131:
3127:
3123:
3120:
3117:
3101:
3100:
3087:
3082:
3078:
3074:
3071:
3068:
3046:
3043:
3040:
3037:
3023:ω-closure
3019:
3018:
3017:
3016:
3002:
2998:
2994:
2989:
2985:
2981:
2977:
2974:
2963:
2959:
2955:
2951:
2948:
2937:
2933:
2929:
2924:
2920:
2914: if
2903:
2891:
2886:
2882:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2849:
2845:
2841:
2838:
2835:
2825:
2821:
2817:
2813:
2810:
2805:
2801:
2797:
2794:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2767:
2763:
2759:
2754:
2750:
2746:
2742:
2739:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2668:
2663:
2659:
2655:
2651:
2648:
2644:
2640:
2637:
2633:
2630:
2627:
2622:
2618:
2614:
2609:
2605:
2601:
2598:
2594:
2591:
2566:
2562:
2558:
2553:
2549:
2534:If we assume,
2528:
2527:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2469:
2468:
2452:
2448:
2423:
2419:
2392:
2389:
2384:
2379:
2375:
2371:
2366:
2361:
2357:
2353:
2348:
2344:
2317:
2314:
2309:
2304:
2300:
2296:
2291:
2286:
2282:
2278:
2273:
2269:
2242:
2239:
2236:
2231:
2227:
2223:
2218:
2213:
2209:
2205:
2200:
2195:
2191:
2187:
2184:
2181:
2176:
2172:
2168:
2163:
2158:
2154:
2150:
2145:
2140:
2136:
2132:
2129:
2125:
2122:
2109:
2108:
2107:
2095:
2090:
2086:
2082:
2077:
2073:
2068:
2064:
2061:
2058:
2053:
2049:
2045:
2040:
2036:
2032:
2029:
2026:
2022:
2019:
2007:
1995:
1992:
1989:
1986:
1981:
1977:
1973:
1968:
1964:
1960:
1956:
1953:
1941:
1940:
1939:
1927:
1924:
1921:
1918:
1910:
1907:
1904:
1894:
1890:
1886:
1881:
1877:
1866:
1862:
1858:
1855:
1851:
1847:
1843:
1839:
1836:
1833:
1828:
1824:
1820:
1810:
1806:
1802:
1799:
1795:
1791:
1787:
1783:
1780:
1777:
1772:
1768:
1764:
1760:
1756:
1753:
1750:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1692:
1688:
1684:
1679:
1675:
1671:
1668:
1665:
1662:
1657:
1653:
1641:
1629:
1626:
1623:
1620:
1612:
1609:
1606:
1596:
1592:
1588:
1583:
1579:
1568:
1564:
1560:
1557:
1553:
1549:
1545:
1541:
1538:
1535:
1530:
1526:
1522:
1512:
1508:
1504:
1501:
1497:
1493:
1489:
1485:
1482:
1479:
1474:
1470:
1466:
1462:
1458:
1455:
1452:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1394:
1390:
1386:
1381:
1377:
1373:
1370:
1367:
1364:
1359:
1355:
1329:
1325:
1321:
1316:
1312:
1308:
1304:
1301:
1289:
1277:
1274:
1271:
1268:
1265:
1262:
1257:
1253:
1249:
1244:
1240:
1236:
1232:
1229:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1155:
1151:
1148:
1144:
1140:
1137:
1133:
1129:
1126:
1122:
1119:
1116:
1112:
1109:
1105:
1102:
1098:
1095:
1078:
1077:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1019:
1018:
1006:
1003:
998:
993:
988:
983:
978:
973:
968:
964:
960:
955:
951:
947:
942:
938:
934:
929:
925:
921:
918:
915:
910:
906:
902:
897:
893:
889:
867:
864:
861:
858:
855:
852:
849:
846:
843:
833:is empty then
819:
815:
811:
806:
802:
785:If we assume,
779:
778:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
703:
698:
693:
688:
683:
679:
675:
670:
666:
662:
659:
656:
651:
647:
643:
640:
637:
615:
610:
605:
600:
595:
591:
587:
582:
578:
574:
571:
568:
563:
559:
555:
552:
549:
527:
522:
517:
512:
507:
503:
499:
494:
490:
486:
483:
480:
475:
471:
467:
464:
461:
441:
438:
422:
408:
407:
385:
369:
364:
345:
334:
303:
295: = (
284:
281:
273:model checking
203:
199:
195:
192:
189:
167:
163:
140:
136:
132:
129:
126:
106:
103:
100:
97:
94:
72:
68:
45:
41:
24:
18:Büchi automata
14:
13:
10:
9:
6:
4:
3:
2:
7768:
7757:
7754:
7752:
7749:
7748:
7746:
7728:
7727:proper subset
7721:
7710:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7671:
7669:
7666:
7664:
7661:
7658:
7655:
7653:
7650:
7648:
7645:
7643:
7640:
7639:
7637:
7632:
7629:
7627:
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7599:
7597:
7594:
7591:
7589:
7586:
7583:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7562:
7560:
7555:
7554:Non-recursive
7552:
7549:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7524:
7522:
7519:
7516:
7514:
7511:
7508:
7503:
7501:
7498:
7495:
7493:
7490:
7489:
7487:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7460:
7457:
7454:
7451:
7448:
7445:
7442:
7441:
7439:
7438:
7435:
7432:
7430:
7427:
7425:
7422:
7420:
7417:
7416:
7411:
7407:
7403:
7399:
7392:
7387:
7385:
7380:
7378:
7373:
7372:
7369:
7360:
7355:
7351:
7347:
7340:
7336:
7335:
7331:
7325:
7320:
7316:
7310:
7306:
7305:
7300:
7295:
7294:
7286:
7285:
7277:
7274:
7269:
7265:
7261:
7257:
7256:
7251:
7247:
7246:Kupferman, O.
7241:
7238:
7232:
7230:
7224:
7218:
7215:
7209:
7204:
7200:
7196:
7195:
7190:
7186:
7179:
7176:
7170:
7165:
7161:
7157:
7156:
7151:
7145:
7142:
7136:
7132:
7126:
7124:
7120:
7114:
7109:
7102:
7099:
7094:
7090:
7086:
7084:0-8186-0877-3
7080:
7076:
7072:
7068:
7064:
7058:
7055:
7050:
7038:
7030:
7024:
7020:
7016:
7012:
7005:
7002:
6995:
6993:
6991:
6987:
6983:
6979:
6974:
6968:
6964:
6959:
6955:
6950:
6946:
6941:
6937:
6933:
6929:
6925:
6920:
6918:
6914:
6910:
6906:
6902:
6898:
6894:
6884:
6878:
6874:
6869:
6865:
6860:
6848:
6839:
6835:
6831:
6826:
6822:
6817:
6812:
6809:and for each
6807:
6803:
6783:
6779:
6773:
6769:
6764:
6760:
6756:
6752:
6748:
6744:
6739:
6735:
6728:
6724:
6719:
6715:
6710:
6706:
6701:
6697:
6693:
6690:and for each
6688:
6684:
6677:
6673:
6666:
6656:
6651:
6647:
6642:
6638:
6634:
6630:
6626:
6622:
6618:
6615:) is also in
6613:
6609:
6604:
6600:
6596:
6592:
6588:
6583:
6579:
6574:
6570:
6566:
6562:
6558:
6554:
6549:
6545:
6541:
6537:
6532:
6528:
6523:
6519:
6514:
6508:
6504:
6499:
6495:
6491:
6487:
6484:
6480:
6475:
6471:
6467:
6462:
6458:
6453:
6449:
6443:
6439:
6435:
6431:
6427:
6422:
6418:
6411:
6407:
6403:
6399:
6392:
6385:
6378:
6374:
6370:
6366:
6362:
6358:
6354:
6350:
6346:
6342:
6338:
6334:
6329:
6324:
6318:
6314:
6309:
6305:
6301:
6297:
6293:
6286:
6280:
6276:
6272:
6268:
6262:
6258:
6253:
6249:
6245:
6240:
6236:
6231:
6227:
6222:
6217:
6213:
6209:
6207:
6203:
6199:
6195:
6188:
6181:
6173:
6169:
6165:
6156:
6149:
6145:
6136:
6132:
6126:
6122:
6116:
6109:
6102:
6097:
6093:
6089:
6084:
6080:
6075:
6071:
6065:
6060:
6056:
6052:
6048:
6043:
6039:
6034:
6030:
6025:
6021:
6016:
6012:
6006:
6001:
5997:
5993:
5989:
5984:
5980:
5975:
5971:
5965:
5960:
5953:
5946:
5942:
5935:
5928:
5921:
5914:) ∈ Δ}, and Δ
5910:
5903:
5896:
5889:
5882:
5875:
5868:
5861:
5853:
5849:
5845:
5841:
5829:
5824:
5820:
5816:
5812:
5808:
5804:
5800:
5796:
5791:
5787:
5782:
5777:
5773:
5769:
5764:
5759:
5754:
5750:
5747:
5742:
5737:
5734:
5730:
5726:
5722:
5715:
5708:
5705:}. Note that
5704:
5700:
5696:
5692:
5688:
5684:
5680:
5673:
5669:
5665:
5661:
5657:
5653:
5649:
5642:
5638:
5634:
5627:
5620:
5616:
5612:
5608:
5604:
5599:
5594:
5590:
5586:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5542:
5537:
5532:
5528:
5523:
5518:
5514:
5507:
5503:
5499:
5489:
5487:
5482:
5480:
5476:
5472:
5471:complementing
5468:
5464:
5456:
5450:
5449:
5444:
5440:
5436:
5427:
5423:
5406:
5403:
5393:
5392:
5387:
5383:
5379:
5375:
5369:
5365:
5347:
5342:
5339:
5334:
5331:
5328:
5321:
5296:
5293:
5290:
5287:
5280:
5255:
5252:
5245:
5229:
5226:
5222:
5218:
5208:
5183:
5180:
5177:
5168:
5160:
5159:
5157:
5156:
5152:
5148:
5144:
5140:
5132:
5125:
5121:
5117:
5113:
5109:
5103:
5098:
5094:
5093:
5089:
5084:
5080:
5076:
5072:
5065:
5061:
5050:
5043:
5039:
5035:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4993:
4973:
4956:
4945:
4944:
4934:
4920:
4919:
4913:
4889:
4885:
4884:
4881:
4877:
4872:
4868:
4867:
4863:
4859:
4848:
4841:
4830:
4827:
4826:
4820:
4796:
4792:
4791:
4787:
4783:
4778:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4755:
4751:
4749:
4747:
4738:
4737:
4736:
4730:
4727:
4723:
4720:
4716:
4712:
4711:
4710:
4707:
4701:
4700:
4699:
4697:
4693:
4689:
4685:
4677:
4676:
4675:
4672:
4665:
4642:
4639:
4632:
4605:
4599:
4559:
4556:
4549:
4522:
4516:
4498:
4495:
4493:
4491:
4487:
4482:
4479:
4478:
4477:
4474:
4458:
4453:
4449:
4443:
4435:
4432:
4429:
4416:
4400:
4393:
4389:
4384:
4376:
4372:
4367:
4359:
4355:
4350:
4341:
4321:
4317:
4312:
4304:
4300:
4295:
4270:
4266:
4243:
4238:
4234:
4226:
4222:
4217:
4208:
4188:
4184:
4157:
4153:
4142:
4138:
4118:
4114:
4103:
4099:
4079:
4075:
4069:
4061:
4058:
4055:
4042:
4022:
4018:
4012:
4004:
4001:
3998:
3966:
3962:
3956:
3948:
3945:
3942:
3928:
3921:
3919:
3917:
3913:
3905:
3900:
3896:
3893:
3892:
3889:
3875:
3869:
3863:
3859:
3853:
3837:
3834:
3833:
3817:
3810:
3803:
3796:
3793:
3789:
3786:
3783:
3780:
3774:
3769:
3765:
3761:
3757:
3754:
3733:
3729:
3726:
3723:
3720:
3711:
3707:
3700:
3696:
3658:
3651:
3639:
3632:
3628:
3621:
3615:
3603:
3596:
3591:
3587:
3580:
3576:
3573:
3548:
3540:
3534:
3509:
3503:
3477:
3474:
3467:
3464:
3458:
3452:
3430:
3422:
3414:
3407:
3404:
3400:
3397:
3394:
3391:
3385:
3380:
3376:
3372:
3369:
3354:
3351:
3347:
3344:
3341:
3332:
3322:
3317:
3309:
3305:
3279:
3271:
3267:
3263:
3251:
3244:
3240:
3233:
3227:
3215:
3208:
3203:
3199:
3192:
3188:
3185:
3159:
3153:
3129:
3121:
3115:
3106:
3103:
3102:
3099:
3085:
3080:
3072:
3066:
3041:
3035:
3024:
3021:
3020:
3000:
2996:
2992:
2987:
2983:
2979:
2975:
2972:
2961:
2957:
2953:
2949:
2946:
2935:
2931:
2927:
2922:
2918:
2904:
2884:
2876:
2870:
2867:
2864:
2861:
2858:
2852:
2847:
2843:
2839:
2836:
2823:
2819:
2815:
2811:
2808:
2795:
2792:
2788:
2785:
2782:
2779:
2770:
2765:
2757:
2752:
2744:
2740:
2728:
2727:
2709:
2703:
2700:
2694:
2688:
2661:
2657:
2653:
2649:
2646:
2642:
2638:
2631:
2625:
2620:
2616:
2612:
2607:
2603:
2596:
2592:
2589:
2564:
2560:
2556:
2551:
2547:
2537:
2533:
2530:
2529:
2526:
2512:
2506:
2500:
2497:
2491:
2485:
2474:
2473:Concatenation
2471:
2470:
2450:
2446:
2421:
2417:
2407:
2390:
2387:
2382:
2377:
2373:
2369:
2364:
2359:
2355:
2351:
2346:
2342:
2332:
2315:
2312:
2307:
2302:
2298:
2294:
2289:
2284:
2280:
2276:
2271:
2267:
2257:
2240:
2237:
2229:
2225:
2221:
2216:
2211:
2207:
2203:
2198:
2193:
2189:
2182:
2174:
2170:
2166:
2161:
2156:
2152:
2148:
2143:
2138:
2134:
2127:
2123:
2120:
2110:
2088:
2084:
2080:
2075:
2071:
2059:
2056:
2051:
2047:
2043:
2038:
2034:
2024:
2020:
2017:
2008:
1990:
1984:
1979:
1975:
1971:
1966:
1962:
1958:
1954:
1951:
1942:
1922:
1919:
1916:
1908:
1905:
1902:
1892:
1888:
1884:
1879:
1875:
1864:
1856:
1849:
1845:
1841:
1837:
1834:
1831:
1826:
1822:
1808:
1800:
1793:
1789:
1785:
1781:
1778:
1775:
1770:
1766:
1748:
1745:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1710:
1707:
1704:
1698:
1695:
1690:
1686:
1682:
1677:
1673:
1660:
1655:
1642:
1624:
1621:
1618:
1610:
1607:
1604:
1594:
1590:
1586:
1581:
1577:
1566:
1558:
1551:
1547:
1543:
1539:
1536:
1533:
1528:
1524:
1510:
1502:
1495:
1491:
1487:
1483:
1480:
1477:
1472:
1468:
1450:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1412:
1409:
1406:
1400:
1397:
1392:
1388:
1384:
1379:
1375:
1362:
1357:
1344:
1343:
1327:
1319:
1314:
1306:
1302:
1290:
1272:
1269:
1266:
1260:
1255:
1251:
1247:
1242:
1238:
1234:
1230:
1227:
1218:
1217:
1202:
1196:
1190:
1187:
1181:
1175:
1149:
1146:
1142:
1138:
1135:
1131:
1127:
1120:
1114:
1110:
1107:
1100:
1096:
1093:
1083:
1080:
1079:
1076:
1062:
1056:
1050:
1047:
1041:
1035:
1024:
1021:
1020:
1004:
996:
991:
986:
981:
976:
971:
966:
962:
958:
953:
949:
945:
940:
932:
927:
919:
913:
908:
904:
900:
895:
891:
862:
856:
853:
847:
841:
817:
813:
809:
804:
800:
790:
789:
784:
781:
780:
777:
763:
757:
751:
748:
742:
736:
725:
722:
721:
720:
718:
696:
691:
686:
681:
677:
673:
668:
660:
654:
649:
645:
638:
635:
608:
603:
598:
593:
589:
585:
580:
572:
566:
561:
557:
550:
547:
520:
515:
510:
505:
501:
497:
492:
484:
478:
473:
469:
462:
459:
449:
447:
439:
437:
435:
430:
428:
421:
417:
413:
405:
401:
397:
393:
389:
386:
383:
379:
378:initial state
376:, called the
375:
368:
365:
362:
358:
354:
350:
346:
343:
339:
335:
332:
328:
324:
320:
316:
313:
312:
311:
309:
302:
298:
294:
290:
282:
280:
278:
274:
269:
267:
263:
259:
255:
251:
247:
243:
239:
234:
231:
227:
223:
201:
197:
193:
190:
187:
165:
161:
138:
130:
127:
101:
98:
95:
70:
66:
43:
39:
29:
19:
7663:Nested stack
7606:Context-free
7531:Context-free
7492:Unrestricted
7323:
7303:
7283:
7276:
7259:
7253:
7250:Vardi, M. Y.
7240:
7226:
7217:
7198:
7192:
7185:Vardi, M. Y.
7178:
7159:
7153:
7144:
7134:
7131:Büchi, J. R.
7101:
7066:
7057:
7010:
7004:
6989:
6981:
6977:
6972:
6966:
6962:
6957:
6953:
6948:
6944:
6939:
6935:
6934:of the form
6927:
6923:
6921:
6916:
6912:
6908:
6904:
6900:
6896:
6892:
6882:
6876:
6872:
6867:
6863:
6858:
6846:
6837:
6833:
6829:
6824:
6820:
6815:
6810:
6805:
6801:
6781:
6777:
6771:
6767:
6762:
6758:
6754:
6750:
6746:
6742:
6737:
6733:
6726:
6722:
6717:
6713:
6708:
6704:
6699:
6695:
6691:
6686:
6682:
6675:
6671:
6664:
6654:
6649:
6645:
6640:
6636:
6632:
6628:
6624:
6620:
6616:
6611:
6607:
6602:
6598:
6594:
6590:
6586:
6581:
6577:
6572:
6568:
6564:
6560:
6556:
6552:
6547:
6543:
6539:
6535:
6530:
6526:
6521:
6517:
6512:
6506:
6502:
6497:
6493:
6489:
6488:
6482:
6478:
6473:
6469:
6465:
6460:
6456:
6451:
6447:
6441:
6437:
6433:
6429:
6425:
6420:
6416:
6409:
6405:
6401:
6397:
6390:
6383:
6376:
6372:
6368:
6364:
6360:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6327:
6322:
6316:
6312:
6307:
6303:
6299:
6295:
6291:
6284:
6278:
6274:
6266:
6260:
6256:
6251:
6247:
6243:
6238:
6234:
6229:
6225:
6220:
6215:
6211:
6210:
6205:
6201:
6197:
6193:
6186:
6179:
6171:
6167:
6163:
6154:
6147:
6143:
6134:
6130:
6124:
6120:
6114:
6107:
6100:
6095:
6091:
6087:
6082:
6078:
6073:
6069:
6063:
6058:
6054:
6050:
6046:
6041:
6037:
6032:
6028:
6023:
6019:
6014:
6010:
6004:
5999:
5995:
5991:
5987:
5982:
5978:
5973:
5969:
5963:
5958:
5951:
5944:
5940:
5933:
5926:
5919:
5908:
5901:
5894:
5887:
5880:
5873:
5866:
5859:
5847:
5843:
5839:
5827:
5822:
5818:
5814:
5810:
5806:
5802:
5798:
5794:
5789:
5785:
5780:
5775:
5771:
5767:
5757:
5752:
5751:
5745:
5740:
5738:
5732:
5728:
5724:
5720:
5713:
5706:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5671:
5667:
5663:
5659:
5655:
5651:
5647:
5640:
5636:
5632:
5625:
5618:
5614:
5610:
5606:
5602:
5597:
5592:
5588:
5584:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5544:
5540:
5535:
5530:
5526:
5521:
5516:
5512:
5505:
5501:
5497:
5495:
5483:
5479:closed under
5466:
5460:
5442:
5438:
5434:
5425:
5421:
5385:
5381:
5377:
5373:
5367:
5363:
5150:
5146:
5142:
5138:
5130:
5123:
5119:
5115:
5111:
5107:
5101:
5078:
5074:
5067:
5063:
5056:
5045:
5041:
5037:
5030:
5018:
5014:
5010:
5006:
5002:
4998:
4994:
4918:,F'), where
4911:
4887:
4825:,F'), where
4818:
4794:
4742:
4739:Minimization
4734:
4708:
4705:
4681:
4669:
4496:
4483:
4480:
4475:
4414:
4339:
4206:
4140:
4136:
4101:
4097:
4040:
3985:
3909:
3894:
3839:
3835:
3104:
3026:
3022:
2535:
2531:
2476:
2472:
2405:
2330:
2255:
1081:
1026:
1023:Intersection
1022:
786:
782:
727:
723:
450:
446:closed under
443:
431:
426:
419:
415:
411:
409:
403:
399:
395:
391:
387:
381:
377:
373:
366:
360:
356:
352:
348:
341:
337:
330:
326:
322:
314:
307:
300:
296:
292:
288:
287:Formally, a
286:
270:
241:
237:
235:
229:
219:
7672:restricted
7299:Anil Nerode
6593:over input
5842:)}, {(
5838:, {(0,
5617:, we have (
5428:belongs to
5372:belongs to
4205:th letter.
2680:recognizes
1167:recognizes
434:ω-automaton
291:is a tuple
7745:Categories
7189:Wolper, P.
6996:References
6813:> 0, w'
6674:such that
6490:Theorem 3:
6242:such that
6212:Theorem 2:
5854:, where Δ
5753:Theorem 1:
4965:and if R=F
4666:Algorithms
3444:Note that
319:finite set
254:ω-automata
7626:Star-free
7580:Positive
7570:Decidable
7505:Positive
7429:Languages
7354:CiteSeerX
7223:Safra, S.
7113:1007.1333
7093:206559211
7063:Safra, S.
7047:ignored (
7037:cite book
6952:), where
6845:to reach
6214:For each
5918:= { ((0,
5639:), where
5407:→
5394:and init
5343:→
5329:δ
5297:∪
5216:Σ
5184:∪
4590:. Hence,
4454:ω
4444:∗
4433:∪
4401:…
4239:ω
4158:ω
4119:ω
4080:ω
4070:∗
4059:∪
4023:ω
4013:∗
4002:∪
3967:ω
3957:∗
3946:∪
3854:ω
3850:Σ
3808:Δ
3804:∈
3762:∈
3751:∃
3712:∪
3705:Δ
3694:Δ
3626:Δ
3619:Σ
3597:∪
3549:ω
3419:Δ
3415:∈
3373:∈
3367:∃
3323:∪
3314:Δ
3303:Δ
3238:Δ
3231:Σ
3209:∪
3130:ω
3081:ω
2993:∪
2928:∩
2881:Δ
2877:∈
2840:∈
2834:∃
2816:∈
2771:∪
2762:Δ
2758:∪
2749:Δ
2738:Δ
2701:⋅
2636:Δ
2629:Σ
2613:∪
2557:∩
2498:⋅
2391:…
2316:…
2241:…
2081:∈
1985:×
1972:×
1885:∈
1861:Δ
1857:∈
1805:Δ
1801:∈
1652:Δ
1587:∈
1563:Δ
1559:∈
1507:Δ
1503:∈
1354:Δ
1324:Δ
1320:∪
1311:Δ
1300:Δ
1261:×
1248:×
1188:∩
1125:Δ
1118:Σ
1048:∩
987:∪
959:∪
937:Δ
933:∪
924:Δ
917:Σ
901:∪
854:∪
810:∩
749:∪
665:Δ
658:Σ
577:Δ
570:Σ
489:Δ
482:Σ
202:ω
166:ω
139:ω
7424:Grammars
7287:. STACS.
6907:, since
6694:> 0,
6290:... and
6224:classes
6045:)-{ε} |
5858:= { ((0,
5126:⟩
5106:⟨
4943:, where
4752:Variants
4643:′
4560:′
4488:of some
4104:accepts
3811:′
3797:′
3758:′
3708:′
3697:″
3684:, where
3629:″
3577:″
3495:because
3478:′
3408:′
3355:′
3306:′
3241:′
3189:′
2976:′
2950:′
2812:′
2796:′
2741:′
2726:, where
2650:′
2639:′
2593:′
2536:w.l.o.g.
2124:′
2021:′
1955:′
1850:′
1794:′
1742:′
1726:′
1552:′
1496:′
1444:′
1428:′
1303:′
1231:′
1150:′
1139:′
1128:′
1111:′
1097:′
788:w.l.o.g.
347:δ:
338:alphabet
153:, where
7647:Decider
7621:Regular
7588:Indexed
7546:Regular
7513:Indexed
6653:), let
6516:. Then
6464:. Then
6396:.... ∈
6112:, and γ
6068:= ∩{ Σ-
6061:. Let γ
6002:. Let β
5961:. Let α
5533: ~
5519:. Let ~
5515:) be a
5040:. Once
4864:formula
4721:(SCCs).
4659:
4625:
4621:
4592:
4576:
4542:
4538:
4509:
4471:
4419:
4285:
4258:
4203:
4176:
4172:
4145:
4133:
4106:
4094:
4045:
4037:
3988:
3981:
3932:
394:is the
7699:Finite
7631:Finite
7476:Type-3
7467:Type-2
7449:Type-1
7443:Type-0
7356:
7311:
7091:
7081:
7025:
6741:. Let
6712:. Let
6670:... =
6553:Proof:
6432:>0,
6424:. Let
6400:. Let
6267:Proof:
6128:= ∩{ α
5900:)) | (
5872:)) | (
5768:Proof:
5670:} and
5539:
5129:where
4906:,...,F
4898:,...,F
4835:= ( q
4813:,...,F
4805:,...,F
4497:Proof:
4481:Lemma:
4135:. So,
3895:Proof:
3293:where
3105:Proof:
2532:Proof:
1216:where
1082:Proof:
783:Proof:
410:In a (
327:states
7657:PTIME
7342:(PDF)
7108:arXiv
7089:S2CID
6856:...w'
6757:from
6389:<
6382:<
6339:<
6094:from
6053:from
6009:= ∩{
5994:from
5968:= ∩{
5932:)) |
5925:),(1,
5893:),(1,
5865:),(0,
5719:. If
5693:from
5662:from
5587:over
5579:from
5563:over
5529:∈ Σ,
5504:,Σ,Δ,
5173:final
4878:From
4860:From
4853:× {1}
4788:(GBA)
4784:From
724:Union
715:be a
317:is a
299:,Σ,δ,
7404:and
7309:ISBN
7079:ISBN
7049:help
7023:ISBN
6961:and
6567:) ∩
6501:and
6492:Let
6455:) ∈
6415:) ∈
6302:) =
6233:and
6200:) ∧
6086:) |
5986:) |
5805:and
5441:) =
5432:and
5384:) =
5376:and
5304:init
5263:init
5191:init
4871:here
4849:F'=F
4839:,1 )
4686:and
3899:here
2333:and
539:and
451:Let
248:and
228:, a
224:and
58:and
7264:doi
7203:doi
7164:doi
7071:doi
7015:doi
6919:).
6895:if
6889:i+1
6766:to
6732:...
6550:).
6486:).
6408:(0,
6311:...
6264:).
6208:}.
6178:| (
6161:∩ γ
6141:∩ β
6117:,Q'
6110:,Q'
6105:, β
6103:,Q'
6066:,Q'
6007:,Q'
5966:,Q'
5943:∧ (
5727:by
5697:to
5677:= {
5666:to
5646:= {
5631:)=
5583:to
5559:to
5551:,
5545:p,q
5527:v,w
5461:In
5424:if
5360:if
5075:A'
5019:A'
5003:A'
4995:A'
4979:i=0
4974:F'=
4926:i=0
4894:,{F
4801:,{F
3738:new
3663:new
3644:new
3608:new
3337:new
3256:new
3220:new
3027:If
2467:A'.
2258:if
380:of
359:of
340:of
329:of
220:In
7747::
7400::
7258:,
7248:;
7199:49
7197:,
7187:;
7158:,
7122:^
7087:,
7077:,
7041::
7039:}}
7035:{{
7021:.
6992:.
6899:∈
6885:+1
6819:∈
6800:∈
6792:w'
6788:w'
6780:.
6774:+1
6749:∈
6703:∈
6681:∈
6635:∈
6559:∈
6468:∈
6444:-1
6367:⊆
6319:-1
6246:∈
6204:∈
6192:)=
6077:0,
6036:1,
6027:)-
6018:0,
5977:1,
5939:∈
5834:∪Δ
5801:∈
5749:.
5736:.
5712:⊆
5685:|
5681:∈
5654:|
5650:∈
5613:∈
5547:∈
5465:,
5366:,
5147:AP
5124:AP
5122:,
5118:,
5114:,
5110:,
5104:=
5011:A'
4912:A'
4831:q'
4819:A'
4698:.
4588:A'
4505:A'
4385:10
4368:10
4313:10
4235:10
4100:.
3025::
2538:,
2475::
2408:.
1025::
791:,
726::
719:.
436:.
414:)
398:.
279:.
236:A
7592:—
7550:—
7517:—
7482:—
7479:—
7473:—
7470:—
7464:—
7461:—
7458:—
7455:—
7452:—
7446:—
7390:e
7383:t
7376:v
7362:.
7344:.
7317:.
7271:.
7266::
7260:2
7235:.
7212:.
7205::
7173:.
7166::
7160:9
7139:.
7116:.
7110::
7096:.
7073::
7051:)
7031:.
7017::
6990:A
6982:A
6980:(
6978:L
6973:A
6967:g
6963:L
6958:f
6954:L
6949:g
6945:L
6943:(
6940:f
6936:L
6928:A
6926:(
6924:L
6917:A
6915:(
6913:L
6909:I
6905:F
6901:I
6897:i
6893:F
6883:i
6877:g
6873:L
6868:i
6864:s
6859:i
6854:0
6850:0
6847:s
6843:0
6838:f
6834:L
6830:r
6825:g
6821:L
6816:i
6811:i
6806:f
6802:L
6798:0
6794:2
6790:1
6786:0
6782:I
6778:F
6772:i
6768:s
6763:i
6759:s
6755:r
6751:I
6747:i
6743:I
6738:i
6734:w
6730:0
6727:w
6723:r
6718:i
6714:s
6709:g
6705:L
6700:i
6696:w
6692:i
6687:f
6683:L
6679:0
6676:w
6672:w
6668:2
6665:w
6662:1
6660:w
6658:0
6655:w
6650:g
6646:L
6644:(
6641:f
6637:L
6633:w
6629:F
6625:A
6621:A
6619:(
6617:L
6612:g
6608:L
6606:(
6603:f
6599:L
6595:w
6591:A
6587:r
6582:g
6578:L
6576:(
6573:f
6569:L
6565:A
6563:(
6561:L
6557:w
6548:A
6546:(
6544:L
6540:A
6538:(
6536:L
6531:g
6527:L
6525:(
6522:f
6518:L
6513:A
6507:g
6503:L
6498:f
6494:L
6483:g
6479:L
6477:(
6474:f
6470:L
6466:w
6461:g
6457:L
6452:j
6448:i
6446:,
6442:j
6438:i
6436:(
6434:w
6430:j
6426:g
6421:f
6417:L
6413:0
6410:i
6406:w
6402:f
6398:X
6394:2
6391:i
6387:1
6384:i
6380:0
6377:i
6373:X
6369:N
6365:X
6361:j
6359:,
6357:i
6355:(
6353:w
6349:j
6347:,
6345:i
6341:j
6337:i
6333:N
6328:A
6323:N
6317:j
6313:a
6308:i
6304:a
6300:j
6298:,
6296:i
6294:(
6292:w
6288:1
6285:a
6282:0
6279:a
6277:=
6275:w
6261:g
6257:L
6255:(
6252:f
6248:L
6244:w
6239:g
6235:L
6230:f
6226:L
6221:A
6216:w
6206:Q
6202:p
6198:p
6196:(
6194:f
6190:2
6187:Q
6185:,
6183:1
6180:Q
6175:1
6172:Q
6170:-
6168:Q
6166:,
6164:p
6158:2
6155:Q
6153:-
6151:1
6148:Q
6146:,
6144:p
6138:2
6135:Q
6133:,
6131:p
6125:f
6121:L
6115:p
6108:p
6101:p
6096:p
6092:A
6088:q
6083:q
6081:,
6079:p
6074:A
6072:(
6070:L
6064:p
6059:F
6055:p
6051:A
6047:q
6042:q
6040:,
6038:p
6033:A
6031:(
6029:L
6024:q
6022:,
6020:p
6015:A
6013:(
6011:L
6005:p
6000:F
5996:p
5992:A
5988:q
5983:q
5981:,
5979:p
5974:A
5972:(
5970:L
5964:p
5959:Q
5955:2
5952:q
5950:,
5948:1
5945:q
5941:F
5937:1
5934:q
5930:2
5927:q
5923:1
5920:q
5916:2
5912:2
5909:q
5907:,
5905:1
5902:q
5898:2
5895:q
5891:1
5888:q
5884:2
5881:q
5879:,
5877:1
5874:q
5870:2
5867:q
5863:1
5860:q
5856:1
5848:q
5846:,
5844:i
5840:p
5836:2
5832:1
5828:Q
5823:q
5821:,
5819:p
5817:,
5815:i
5811:A
5807:i
5803:Q
5799:q
5797:,
5795:p
5790:f
5786:L
5781:A
5776:Q
5774::
5772:f
5765:.
5758:A
5755:~
5746:A
5741:A
5733:f
5729:L
5725:f
5721:f
5717:1
5714:Q
5710:2
5707:Q
5703:F
5699:q
5695:p
5691:A
5687:w
5683:Q
5679:q
5675:2
5672:Q
5668:q
5664:p
5660:A
5656:w
5652:Q
5648:q
5644:1
5641:Q
5637:p
5635:(
5633:f
5629:2
5626:Q
5624:,
5622:1
5619:Q
5615:Q
5611:p
5607:Q
5605::
5603:f
5598:A
5593:w
5589:v
5585:q
5581:p
5577:F
5573:A
5569:w
5565:v
5561:q
5557:p
5553:A
5549:Q
5541:w
5536:A
5531:v
5522:A
5513:F
5511:,
5509:0
5506:Q
5502:Q
5498:A
5446:.
5443:a
5439:q
5437:(
5435:L
5430:I
5426:q
5422:q
5404:a
5386:a
5382:p
5380:(
5378:L
5374:R
5370:)
5368:p
5364:q
5362:(
5348:p
5340:a
5335:q
5332:=
5308:}
5300:{
5294:Q
5291:=
5288:F
5267:}
5259:{
5256:=
5253:I
5230:P
5227:A
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