Theorem 3.8. in .NET Implement PDF-417 2d barcode in .NET Theorem 3.8.

Theorem 3.8. generate, create none none on none projects barcode pdf417 Given a fin none none ite automaton A = <!" S, so, 8, F>,. (Vs E S)(Vt E S)(Vi E N)(s E i+ 1A t SEiA t/\ (V none for none a E !")(8(s, a) EiA 8(t, a))). Proof. Let s E S, t E S. Then SEi+lA t~ (Vx E!"*. 11xl:s; i + 1)(8(s, x) E F~8(t,x) E F) ~ (Vx E!"* 11xl:s; i )[8(s, x) E F ~8(t,x) E F] /\ (Vy E!"* 11yl = i + 1)[8(s,y) EF~8(t,y) EF] ~ (Vx E!"* 11xl:s; i ) [8(s , x) E F~8(t,x) E F]/\. (Vy E!"* 11:s;lyl:S;i ~ (Vx E!"*. + 1)[8(s,Y)EF~8(t,Y)EF]. 11xl:s; i ) [8(s, x) E F ~8(t,x) E F]/\ (Va E !")(Vx E!, * 11xl :s; i )[8(s, ax) E F ~ 8(t, ax) E F] 11x I:s; i )(8(s, ax) E F ~ 8(t, ax) E F) 11x I:s; i )(8(8(s, a), x) E F ~ 8(8(t, a),x) E F). (Va E !")(Vx E!"*. ~SEiAt (Va E !")(Vx E!"*. ~ SEiA t /\ (Va E !" none for none )(8(s, a) EiA 8(t, a)). Note that T heorem 3.8 gives a far superior method for determining successive EjA relations. The definition required the examination of many (long) strings using the 8 function; Theorem 3.

8 allows us to simply check a few letters using the 8 function. Theorems 3.9, 3.

10, and 3.11 will assure us that EA will eventually be. Minimization of Finite Automata Chap. 3 found. The none for none following theorem guarantees that the relations, should they ever begin to look alike, will continue to look alike as successive relations are computed. V Theorem 3.

9. Given a finite automaton A = <I". s, so, 8, F>,. E mA ). (3m EN ;l EmA = E m +1A ). => (Vk E N)(Em+kA = Proof. By induction on k; see the exercises. The result none for none in Theorem 3.9 is essential to the proof of the next theorem, which guarantees that when successive relations look alike they are identical to EA. V Theorem 3.

10. Given a finite automaton A = <I". (3m EN ;l EmA = E m 1A ) + S, so, 8, F>,. => EmA = EA Proof. Assume 3m EN EmA = Em+1A and let q, rES:. 1. By Lemma 3.2, qEAr qEmAf. 2. Conversely, assume q EmA r. Then => (by a ssumption) q Em+1A r => (by Theorem 3.9). q EmA r (Vj none none 2:m)(qEjAr) Furthermore, by Lemma 3.1, (Vj:5 m)(qEjAr), and so (Vj E N)(qEjAr); but by definition or E A, this implies q EA f. We have just shown that qEmAr qEAf.

. 3. Combining (1) and (2), we have (Vq, r E S)(qEmAr :>qEAr), and so EmA = EA. The next th none none eorem guarantees that these relations will eventually look alike (and so by Theorem 3.10, we are assured that successive computations of EiA will yield an expression representing the relation EA)" V Theorem 3.11.

Given a finite automaton A = <I" S, so, 8, F>,. (3m EN ;l m :5IISII/\ EmA = E m 1A). + Proof. Assu none none me the conclusion is false; that is, that E OA , E 1A , ..

. ,EllsllA are all distinct. Since EllsllA C .

.. C EIA C E OA , the only way for two successive relations to be different is for the number of equivalence classes to increase.

Thus,. o< rk (E OA) < rk (E1A) < rk (E2A) < ...

< rk (EllsIIA),. which means none for none that rk(Ells1IA) > IISII, which is a contradiction (why ). Therefore,. Sec. 3.2 Minimization Algorithms not all the se relations can be distinct, and so there is some index m for which E mA = E m +1A". V Corollary 3.5. computing EA. Given a DFA A = <I, S, so, S, F>, there is an algorithm for Proof. EA c none for none an be found by using Lemma 3.2 to find E OA , and computing successive EiA relations using Theorem 3.

8 until EiA = E i+1A ; this EiA will equal E A, and this will all happen before i reaches liS II, the number of states in S. Tl).e procedure is therefore guaranteed to halt.

. Since EA wa s the key to producing a reduced machine, we now have an algorithm for taking a DFA and finding an equivalent DFA that is reduced. The other necessary step needed to find the minimal machine was to produce a connected DFA from a given automaton. This construction hinged on the calculation of SC, the set of connected states.

The algorithm suggested by the definition of SC is by no means the most efficient; it involves checking long strings with the 8" function and hence massive duplication of effort. Furthermore, the definition seems to imply that all the strings in I * must be checked, which certainly cannot be completed if it is done one string at a time. Theorem 2.

7 can be used to justify that it is unnecessary to check any strings longer than II SII (see the exercises). Thus SC = {8"(so, x) Ilxl < IISII}. While this set, being based on a finite number of words, justifies that there is an algorithm for findingSC (and hence there exists an algorithm for constructing N), it is still a very inefficient way to calculate the set of accessible states.

As with the calculation of E A, there is a way to avoid using 8" to process long strings when computing SC. In this case, a better strategy is to begin with So and find all the new states that can be reached from So with just one transition. Note that this can be done by simply examining the row of the state transition table corresponding to so, and hence the computation can be accomplished quite fast.

Each of these new states should then be examined in the same fashion to see if they lead to still more states, and this process can continue until all connected states are found. A sequence of state sets is thereby constructed, in a similar manner to the way successive partial state equivalence relations EiA were built. This approach is reflected in Definition 3.

10. V Definition 3.10.

Given a finite automaton A = <I, S, so, S, F>, the ith partial state set Ci is defined by the following rules: Let Co = {so} and recursively define.
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