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Finite-State Transducers use none none implement toembed none on none ISBN Chap. 7 the exerci ses, the hypothesis that Ml and Mz are reduced and connected is crucial to the proof of this part of the theorem.. Note that none for none Theorem 7.5 implies that, as long as we are dealing with reduced and connected machines, fMl = fM2 iff Ml = Mz. The conclusions discussed earlier now follow immediately from Theorem 7.

5. V Corollary 7.4.

Given a FST M, a necessary and sufficient condition for M to be minimal is that M is both reduced and connected.. Proof. See the exercises. V Corollary 7.5. Given a FST M, Me/EM" is minimal. Proof. Let M be a FST and let A be a minimal machine that is equivalent to M. By Corollaries 7.

2 and 7.3, A must be both reduced and connected. By Theorems 7.

3 and 7.4, Me/EM" is also reduced, connected, and equivalent to M (and hence to A). Theorem 7.

5 would then guarantee that A and Me/EM" are isomorphic, and therefore they have the same number of states. Since A was assumed to have the minimum possible number of states, Me/EM" also has that property and is thus minimal..

The minima none for none l machine can therefore be found as long as Me/EM" can be computed. Finding se (and from that Me) is accomplished in exactly the same manner as described in 3. The strategy for generating EM is likewise quite similar, and again uses the ith state equivalence relation, as outlined below.

. V Definiti on 7.15. Given a transducer M = <~,r,s,so,S,w> and a nonnegative integer i, define a relation between the states of M called EiM" the ith state equivalence relation on M, by.

(Vs, tES)(sEiMt (Vx E~*. 31xl :5 i) none none (w(s, x) = w(t, x))). Thus EiM r elates states that cannot be distinguished by strings of length i or less, whereas EM relates states that cannot be distinguished by any string of any length. All the properties attributable to the analogous relations for finite automata (EiA) carry over, with essentially the same proofs, to the relations for finite-state transducers (EiM)" V Lemma 7.3.

Given a transducer M = <~, r, s, so, S, w>:. a. Em+lM i none for none s a refinement of E mM ; that is, (Vs, t E S)(s Em+1M t ~ s EmM t)..

Sec. 7.3 Moore Sequential Machines s EmM t); hence,. b. EM is a refinement of E mM ; that is, (\fs, t E S)(s EM t EM~EmM". c. (3m E N none for none ~ EmM = E m IM ) ~ (\fk E N)(Em+kM = EmM)" + d. (3m E N ~ m :5IISII A EmM = E m IM ).

+ e. (3m E N ~ EmM = E m IM ) ~ EmM = EM. +.

Proof. The proof is similar to the proofs given in 3 for EiA (see the exercises). A V Lemma 7.4. Given a FST M = <I, r, S, sO, 8, w>:. a. EOM has none none just one equivalence classes, which consists of all of S. b.

ElM is defined by s ElM t ~ (\fa E I)(w(s, a) = w(t, a)). c. For i 2: 1, E i+1M can be computed from EiM as follows:.

(\fs E S)(\ft E S)(\fi 1)(s Ei+IM t ~ S EiM tA (\fa E I)(8(s, a) EiM 8(t, a))).. Proof. The proof is similar to the proofs given in 3 for EiA (see the exercises). A V Corollary 7.6. . Given a FST M computing EM. = <I, r, S, sO, 8, w>, there is an algorithm for Proof. Use none for none Lemma 7.4 to compute successive EiM relations from ElM until EiM = Ei+lM; by Lemma 7.

3, this EiM will equal EM, and this will all happen before i reaches IISII, the number of states in S. Thus the procedure is guaranteed to halt. A.

V Corollar y 7.7. Given a FST M = <I, r, S, sO, 8, w>, there is an algorithm for computing the minimal machine equivalent to M.

. Proof. Usi none for none ng the algorithm for computing the set of connected states, MC can be found. The output function is used to find E1Mc, and the state transition function is then used to calculate successive relations until EMc is found.

MC/EMC can then be defined and will be the minimal machine equivalent to M. A. 7.3 MOORE SEQUENTIAL MACHINES Moore mach none for none ines form another class of transducer that is equivalent in power to Mealy machines. They use a less complex output function, but often require more states than an equivalent Mealy machine to perform the same translation. An illustration of the convenience and utility of Moore machines can be found in.

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