Questions and Answers of Implementing Programming Languages

=+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
=+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}
=+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
=+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
=+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
=+1. Let M = (Q, X, ft) be the ffsm, where Q = {ql, q2}, X = {a}, and ft is defined by ft(q l ,a, qi) = s, ft(ql ,a, q2) = 0, ft(g2,a, qi) = 0, and
=+14. Prove Theorem 5 .18 .14 with > in {x E X* I A(x) > c} replaced by=, >, and < .
=+13. Prove Theorem 5 .18.4 for the case k = 2.
=+12. Prove Theorem 5.18.1 for the cases k = 2,3.
=+11. For k = 2, 3, state and prove a result similar to Theorem 5 .16.7 replacing Ax with P) .
=+10. Prove Theorem 5 .16.7 for the cases k = 2,3.
=+9. Prove that every stochastic language under maximal interpretation is w .3 .m.
=+8. Determine whether or not the family of w.i .m. languages is a subset of the family of w.i .s . languages.
=+7. Determine whether pushdown automata can approximate pa's.
=+6. Determine the most powerful class of np devices that suffices for approximating the pa's.
=+(b) pushdown automata,(c) sequential automata.
=+5. Characterize the fsf's (the pa's) that can be approximated by(a) linear bounded automata,
=+a definite automaton, prove that A is quasi-definite .
=+4. Let A be a probabilistic automaton. If A can be E-approximated by
=+3. Prove Proposition 5.2 .6 .1 if A(x) > c2 0 otherwise.
=+2. Prove that p(xy) = 7r(x)9F (y), where x, y E X* .
=+1. Let A = (Q, 7r, {A(u) }, F) be a probabilistic automaton. Prove that 7r(xy) = 7r(x)A(y), where x, y E X* .
=+10. Prove for an F-TA that results corresponding to Lemma 4 .10.4 and Theorems 4.10.5 and 4 .10.6 hold.
=+9. Show that G is closed under union, intersection with regular sets, concatenation with CFL, substitution by CFL, homomorphism, inverse homomorphism, reversal, and a-transducer.
=+8. Use Theorem 4.5 .13 and the proof of Theorem 4.5 .11 to show that G is not closed under intersection .
=+7. Prove that Theorems 4.5 .4 and 4.5 .5 hold if > is replaced by > .
=+6. Show that Theorems 4.4 .11 and 4.4 .12 do not hold without the assumption A is finitary.
=+5. Use induction on Ixl to finish the proof of Theorem 4.4 .4 .
=+CHAD,AFB,A~a,B~b,D~b}.Show that AG(aaab)=( .9) ( .8) ( .6) ( .6) ( .4) .
=+4. Consider the grammar G of Example 4.4 .2, where P = {so -' 9-> aAC,
=+yields a grammar in Chomsky normal form that is equivalent to G,[96, Example 4.9, p. 93].
=+Ana, B~CbS, B~CaD2, Bib, D1 ~AA, D2~BB, Caa, Ca b,
=+B ----> aBB, B bS, B ----> b} . Show that S~Cb A, S~Ca B, A~Ca S, A~Cb Dj ,
=+3. Consider the grammar G = (N, T, P, S), where N = (S, A, B), T ={a, b}, and P = {S ----> ab, S ----> aB, A ----> bAA, A ----> aS, A ----> a,
=+Definition 1.8 .7 in that, given G, there exists a G' such that L(G) _ L(G) and vice versa.
=+2. Show that the definition of a type 2 grammar G in this chapter (with c = 1) is equivalent to the definition of a type 2 grammar G' of
=+1. Show that the operation of concatenation of fuzzy languages is associative.
=+8. Prove that Theorems 3.6 .5 to 3.6 .8 hold if > is replaced by > or =?
=+7. Show that Theorems 3.5 .1 and 3.5 .2 do not hold in general for weak regular fuzzy languages.
=+6. Complete the proof of Theorem 3.4 .16.
=+5. Let X, Y E 'P(V) be such that X C Y. If every element of Y can be expressed uniquely in the form VT_ l ai Tx2 , a2 E [0,1], x2 E X, prove that X is a set of vertices of Y.
=+Y, prove that X C X'. Conclude that a set of vertices is unique .
=+4. Let X,X',Y E P(V) . If X is a set of vertices of Y and X' generates
=+set of vertices of Y. Prove that X is a set of vertices of Y if and only if Y = C(X) and Vx E X, x ~ C(XV X}) .
=+3. Let X,Y E P(V) be such that X C_ Y. X is said to generate Y if Y = C(X), where C is defined in Exercise 2. If X generates Y and there does not exists X' C X which generates Y, then X is called a
=+(c) VX E P(V), C(C(X)) = C(X) .
=+(b) VX,Y E P(V), X C Y implies C(X) C C(Y) ;
=+(a) VX E P(V), X C C(X) ;
=+Prove that the following assertions hold:
=+C(X) = {Vml aZTx2 I ai E [0, 1], x2 E X, 2 = 1, . . . , n2; m E N}.
=+For all a E [0,1] and Vx, y E V, let aTx denote (aTal . . . . , aTan), where x = (a,, .an) and let x V y denote (al V bl . . . . , an V bn), where y = (bl, . . . , bn ) . Define the function C : 'P
=+2. Let V = {(al, ., an) I a2 E [0,1], i = 1, . . . , n}, where n E N. Let T be a t-norm on [0,1] . Assume that Va, b E [0,1], a < 1 implies aTb < b.
=+1. Let T be a norm on [0,1] . Prove that Va E A, aTa
=+8. Prove that all the assertions of Theorem 2 .10.4 are valid if B" is replaced by A' .
=+7. Show that I and 12 of Example 2 .9 .19 are equivalent if r7(gi) =97(g2) _ 972 (s o ) and 97 (q3) =972(ss)-
=+6. Let V be a vector space over a field F. Define s : P(V) ~ P(V)by b'X E P(V), s(X) = the intersection of all subspaces of V that contain X. Show that s satisfies the Exchange Property.
=+3. Prove Theorem 2.4 .5 .
=+5. Let V be a vector space over a field F. Prove that the intersection of any collection of subspaces of V is a subspace of V.
=+4. Show that T defined in Theorem 2.5 .5 has the desired properties.
=+2. Let X = {(1, 0), (1,1)} . Show that X is not a basis of C(X).
=+either x1 < 1 or x2 < 1-
=+1. Let X = {(x1 ,0 , (O,x2)} . Show that X is not a basis of C(X) if
=+language . Prove that 3n E hY such that b'z E L, ~zj >_ n implies that 3u, v, w, x, y such that z = uvwxy, vx 1 >_ 1, vwx j
=+Prove that 3n E hY such that b'z E L, I zI >_ n implies that 3u, v, w such that z = uvw, ~uvj _ 1, and Vi E hY U {0}, uv'w E L.Moreover, n is not larger than the number of states of the smallest
=+22. (Pumping lemma for regular languages.) Let L be a regular language.
=+21. Determine the possible move sequence of M ,Z of Example 1 .11.4 for the input string aabbaa.
=+20. Determine the validity of the following statement: If L is a regular language, then so is L' = {un I u E L, n = 1, 2. . . . } .
=+19. Show, by example, that there are context-free languages Ll and L2 such that Ll n L2 is not context-free.

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