Fuzzy Automata And Languages Theory And Applications 1st Edition John N Mordeson ,Davender S Malik - Solutions

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This chapter discussed several trends that will influence the future of training. Based on future social, economic, political, or technological factors, identify one or two additional trends that you think will influence training. Write a two- to three-page paper summarizing your ideas. Make sure
Interview a manager. Ask him to evaluate his company’s training department in terms of training delivery, service, expertise, and contribution to the business. Ask him to explain the rationale for his evaluation. Summarize this information. Based on the information you gathered, make
What are the implications of virtual work arrangements for training?
What is organization development? Describe the interventions used to create change.
Explain what you believe are the advantages and disadvantages of creating a training consortium or partnership with other companies.
What misconceptions do managers have about training? How could you change those misconceptions?
Discuss the steps necessary to introduce a new training practice from a change model perspective.
What is process reengineering? Why is it relevant to training?
What is benchmarking? Explain the process you would use to benchmark a company’s safety training programs.
What is rapid instructional design? How does it differ from the traditional training design process discussed in Chapter 1? (See Figure 1.1.)
Discuss how new technologies are likely to impact training in the future.
What is the relationship between talent management and employee engagement? What role can training and development practices play in keeping employee engagement high during poor economic times? Explain.
Explain how training relates to attracting new employees, employee retention, and motivation.
How has new technology improved training and development? What are some of the limitations of using iPods or PDAs for training?
What are the implications of the aging work force? What strategies should companies consider from a training and development perspective to best utilize older employees and prepare for their retirement?
Describe the training courses that you have taken. How have they helped you? Provide recommendations for improving the courses.
Which of the training professionals’ roles do you believe is most difficult to learn? Which is easiest?
How is Starbucks using training to benefit the company during difficult economic times?
=+14. Prove that the condition in Definition 10.17.5 implies that no state can be entered and left on the same input, else S would not be single
=+13. In Example 10.15 .6, show that SQ U Sx1 T°Sx C T° for x E {xl, X21-
=+. Then determine T19 from
=+T14 and T15 .1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 1
=+12. In Example 10.14.6, show that T7 and T9 yield T14 = 1 11 and Tg and T9 yield T15 =
=+11. Prove (1) of Theorem 10.14 .3 .1
=+10. In Definition 10.13 .4, prove that T is the least tolerance such that all injections tj are fuzmaps, and TIT2 is the largest tolerance such that all projections pry are fuzmaps.
=+9. If g : Y ----> (X, T) is a set-theoretic map, prove that g*T is the biggest tolerance on Y such that g : (Y,g*T) ----> (X, T) is a fuzmap.
=+set-theoretic maps g : Y ~ (Z, Q), f - g : (X, T) ~ (Z, Q) if and only if g : (Y,f. T) ~ (Z,a) is a fuzmap.
=+is the unique tolerance on Y such that b' tolerances Q on Z and b'
=+8. For f in Definition 10.13.2, prove that f*T is the least tolerance on Y such that f : (X, T) ~ (Y, f*T) is a fuzmap. Prove also that it
=+7. Show that (No , -), where No = hY U {0} and ({0,1, . . . , n}, i), of Example 10.13.1 are tolerance spaces.
=+6. Prove (2) of Proposition 10.11.7 .
=+5. Let M be the median operator of (10 .5) . Show that M is associative, not strictly increasing, and contractive .
=+4. Prove Lemma 10.11 .2 .
=+3. Prove that y(anbm) = 'A' in Example 10.8 .2 .
=+2. Prove that the matrix in Example 10.6 .2 has period equal to 3.
=+1,2 . . . . , m} I j = 1, 2, . . . , n}) V {a} = A{V{aj2 Aa I i = 1, 2, . . . , ml I j = 1, 2, . . . , n}.
=+1. Leta, aii E II8 for i = 1, 2, . . . , m and j = 1, 2, . . . , n. Prove that(V{A{aj2 i = 1,2, . . . , m} I j = 1,2, . . . , n}) n {a} = V{n{aj2 lea I i = 1,2, . . . , m} I j = 1,2 . . . . , n} and that (A{V{aj2 I i =
=+6. (149 Show that the generative power of type 3 (regular) B-fuzzy grammars is equal to that of the ordinary type 3 grammars.
=+5. (149 Show that type 2 (context-free B-fuzzy grammars can generate type 1 (context-sensitive) languages although type 2 fuzzy grammars cannot generate type 1 languages [145] .
=+4. If P1 , P2, P3 are L-fuzzy relations on a set X, prove that (It, P2)P3 =It l(1t2ft3)
=+in T2, then ft n v is fuzzy recognized by a machine in T2 .
=+3. Show that if a L[o 1]-fuzzy language ft is fuzzy recognized by a machine in T2 and a L[o 1]-fuzzy language v is fuzzy recognized by a machine
=+2. Give the proof of Theorem 9.8 .2 for i = 0 and 3.
=+1. For Theorem 9.4 .1, prove the cases, where (2) or (3) holds.
=+8. Consider the system AoX = B of Example 8.7 .7 . Use the algorithms .8 .8 .7 is a maximal solution and .5 .5 0 and 0 are minimal solutions .
=+7. Compare minimality requirements of Mand M' under various equivalence assumptions .
=+6. If Mand M' are weakly equivalent, prove that V(q, x, y, q') E QxX* x Y* x Q, S* (q, x, y, q') > 0 if and only if ft*(q, x, q') Aw* (q, x, y) > 0.
=+.5
=+of Section 8.7 to show that.8 d 0 are minimal solutions .
=+6* (q, x, y, q~) = ft * (q, x, q~) n w* (q, x, y) .
=+5. If Mand M' are equivalent, prove that V(q, x, y, q') E Q xX* xY*
=+4. If Mand M' are strongly equivalent, prove that V(q, x, y, q') E Q x X* x Y* x Q, S* (q, x, y, q') = ft* (q, x, q') =w*(q, x, y) .
=+3. Prove (3) of Proposition 8.2 .1 .
=+(3) weakly equivalent if V(q, x, y, q') E Qx XxY xQ, S(q, x, y, q') > 0 if and only if ft (q, x, q') A w(q, x, y) > 0.
=+(2) equivalent if V(q, x, y, q') E QxXxYxQ, S(q, x, y, q') = ft(q, x, q')A w(q, x, y) ;
=+(1) strongly equivalent if V(q, x, y, q') E QxXxY x Q, S(q, x, y, q') _ ft (q, x, q~) =w(q, x, y) ;
=+M' = (Q, X, Y, ft, w) be a fuzzy finite state automaton (Section 8.5) . Call Mand M'
=+Let M = (Q, X, Y, S) be a fuzzy automaton (Definition 8.1 .5) and let
=+2. Prove Proposition 8.1 .2 .
=+1. Prove Proposition 8.1 .1 .
=+ft(q,a, ql) =4 is not possible.
=+2 and
=+while ft(q,a, ql) =z and ft(q,a, q2) =z is possible. Show also that S(q,a, 2) =ql and S(q,a, 4) = ql is possible, while ft(q,a, qi) =
=+13. Compare the fuzzy state transition map S : Q x X x [0,1] ----> Q in Definition 7.11 .2 with a function ft : Q x X x Q ----> [0,1] . Show for example that S(q,a, 2) = ql and S(q,a, 2) = q2 is not possible,
=+12. Prove that x - y(FL) implies x = y in Example 7 .10.12 .
=+11. Prove Theorem 7.8 .4 .
=+10. Prove (2) of Theorem 7.7.5 .
=+9. Prove (2) and (4) of Theorem 7.6 .18.
=+8. Prove Theorem 7.6 .6 .
=+7. Prove (2), (4), and (6) of Theorem 7.5 .4 .
=+6. Prove Theorems 7.5 .1 and 7.5 .2 .
=+is the identity map. If, in addition, Supp(t) z,4 0, prove that .M is reduced if and only if .M = .M A.
=+implies ql = q2 . Let A = L(M) . Prove that .M is reduced if and only if f is injective, where (f, g) : .M ----> MA is a homomorphism and g
=+5. (121) Let M= (M, t, T) be a complete accessible fuzzy X-recognizer of a ffsm M= (Q, X, [t) . Then .M is called reduced if ql1 oT = q21 OT
=+there exists a homomorphism of MA into .M, where MA is the fuzzy X-recognizer of MA . Show that if Supp(t) z,4 0, then MA is a homomorphic image of .M .
=+4. (121) Suppose that M = (M,t,T) is a complete accessible fuzzy Xrecognizer of the ffsm M = (Q, X, ft) . Let A = L(M) . Show that
=+3. (121) Suppose that (f, g) is a homomorphism of .M into .M' and(h, k) is a homomorphism of .M' into .M" . Prove that (h of, k o g) is a homomorphism of .M into .M" .
=+and u2 E X, i = 1, 2, . . . , n.
=+(f, g) is a homomorphism of .M into .M' prove that b'p, q E Q and x E X* , ft * (p, x, q) < ft* (f(p),9(ui) . . .g(u,),f(q)), where x = ul . . .un
=+Let f : Q ----> Q' and g : X ----> X'. Then (f, g) is called a homomorphism of .M into AT if b'p, q E Q and Vu E X, (1) ft(p, u, q)
=+2. (121) Let .M = (M, t, T) be a fuzzy X-recognizer of M= (Q, X, /t)and let .M' = (M', t', T') be a fuzzy X'-recognizer of M' = (Q', X', ft') .
=+1. Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M= (Q, X, [t) . Let A = L(M) . If q E Q and t(q) n ft(p, x, q) > 0 for some p E Q and x E X*, then show that x-1 A = X{e} o T, where X{e} is the characteristic function of {q} .
=+x E X+ for every T-generalized state machine (Q, X, T) . Prove that there exists a T-generalized transformation semigroup inducible tnorm T.
=+t-norm T is said to be T-generalized transformation semigroup inducible if S(M) is finite and EgEQ T+ (p, x, q)
=+14. (113) A T-generalized state machine (Q, S, p) is called a T-generalized transformation semigroup if S is a finite semigroup such that(a) p(p, uv, q) = V{p(p, u, r) T p(r, v, q) I r E Q} Vp, q E Q and Vu, v E S and (b) Vu, v E S, p(p, u, q) = p(p, v, q) Vp, q E Q implies u = v . A
=+13. (113) Construct an example of a T-generalized state machine (Q, X, T)such that EgEQ T+ (p, x, q) > 1 and x E X+ .
=+12. (113) Let M = (Q, X, T) be a T-generalized state machine. Prove that since Q is finite, S(M) is finite if and only if Im(T+) is finite .
=+not be finite, where S(M) is defined in Exercise 10.
=+11. (113) Let M= (Q, X, T) be a T-generalized state machine, where T is the ordinary product on [0,1] . Show by example that S(M) need
=+10. (113) Let M= (Q, X, T) be a T-generalized state machine. Let --be a congruence relation defined in Exercise 6. Let [x] = {y E X+x - y} Vx E X+ and S(M) = {[x] I x E X+}. Prove that S(M)is a semigroup, where the binary operation on S(M) is defined by[x] [y] = [xy] V[x], [y] E S(M) .
=+that - is a congruence relation on X+ .
=+9. (113) Let (Q, X, T) be a T-generalized state machine . Define - on X+ by Vx, y E X+, x - y if T+ (p, x, q) =T+ (p, y, q) Vp, q E Q. Prove
=+T + (p, xy, q) =V{T+ (p, x, r) T T+(r, y, q) I r E Q}, for all p, q E Q and x, y E X+.
=+8. (113) Let (Q, X, T) be a T-generalized state machine. Prove that
=+7. Prove Theorems 6 .18 .12 and 6 .18 .13.
=+5. Prove (2), (3), and (4) of Theorem 6.17.1 .
=+6. Let A and ft be fuzzy subgroups of G such that A is a fuzzy normal subgroup of ft . Prove that ft/A is a fuzzy subgroup of Supp(ft)/Supp(A) .
=+4. In Example 6.6 .4, determine E(MI), E(M2), E(MI xM2), and E(MIA M2) when cl :A c2 and/or dl :?~ d2 .
=+3. Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and ft is defined by ft(ql ,a, qi) = 0, ft(ql ,a, q2) = 3, ft (q2 ,a, qi) = 3, and
=+2. Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and ft is defined by ft(q l ,a, gi) = 0, ft(gia, g2) = 3, ft(g2,a, qi) = 0, and

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Fuzzy Automata And Languages Theory And Applications 1st Edition John N Mordeson ,Davender S Malik - Solutions

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