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implementing programming languages
Implementing Programming Languages An Introduction To Compilers And Interpreters 1st Edition Department Of Philosophy Aarne Ranta - Solutions
=+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 (V) ~ 'P(V) by VX E 'P (V),
=+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 finite-state automaton accepting L . 23. (Pumping lemma for context-free languages.) Let L be a
=+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.
=+18. Show that if Ll and L2 are regular languages over X and S is the set of all strings over X, then each of S\L1, Ll U L2 , L+, and LIL2 is a regular language.
=+17. Show that the set L = {xl . . . xn I xl . . . xn = xn . . . xl } of strings over {a, b} is not a regular language.
=+16. Prove Theorem 1.9 .10.
=+15. Show that the string x = abba is not accepted by the nondeterministic finite state automaton of Figure 1 .12.
=+14. Locate the path in representing the string x = aabaabbb in Example 1.9 .9 .
=+13. Write a regular grammar that generates the strings of Exercise 12.
=+12. Design a nondeterministic finite-state automaton that accepts strings over {a, b} having the property of having each b preceded and followed by an a.
=+11. Show that the language L = {anbnck I n, k = 1, 2 . . . . } is a contextfree language .
=+10. Let G be a grammar and let A denote the empty string. Show that if every production is of the form A~xorA~xBorA~A, where A, B E N, x E T*\{A}, then there is a regular grammar G'with L(G) = L(G') .
=+Show that AJA) = Ll U L2-
=+9. Let LZ be a finite set of strings accepted by the finite-state automaton AZ = (QZ, X, f2, AZ, s2), i = 1, 2. Let A = (Qi x Q2, X,f, A, s), where f((gi,q2),x) = (fi(gi,x)j2(q2,x)),= {(qi, q2) I qi E A1 or q2 E ,A2}, s = (81,82) .
=+Show that A,(A) = Ll rl L2 .
=+8. Let LZ be a finite set of strings accepted by the finite-state automaton AZ = (QZ, X, f2, AZ , s2 ), i = 1, 2. Let A = (Qi x Q2, X,f, A, s), where f((qi, q2), x) = (fi(gi, x), f2(g2, x)),= {(qi, q2) I qi E A 1 and q2 E ,A2}, s = (81,82) .
=+7. Let L be a finite set of strings over {0,1} . Show that there is a finitestate automaton that accepts L.
=+6. Show that there is no finite state machine that receives a bit string and outputs 1 whenever the number of 1's input equals the number of 0's input and outputs 0 otherwise .
=+5. Design a finite state machine that outputs 1 when it sees the first 0 and until it sees another 0; thereafter, outputs 0; in all other cases outputs 0.
=+4. Design a finite state machine that outputs 1 when it sees 101 and thereafter; otherwise outputs 0 .
=+3. Design a finite state machine that outputs 1 if an even number of 1's have been input ; otherwise outputs 0.
=+2. Prove Theorem 1.5 .4 .
=+1. Prove Theorem 1.5 .1 .
12. What is a rapid needs assessment? How would you conduct a rapid needs assessment so that it is valuable and accurately identifies training needs?
11. How is competency modeling similar to traditional needs assessment? How does it differ?
10. Discuss the types of evidence that you would look for in order to determine whether a needs analysis has been improperly conducted.
9. Review the accompanying sample tasks and task ratings for the electronic technician’s job. What tasks do you believe should be emphasized in the training program? Why? Task Importance Frequency Difficulty 1. Replaces components 1 2 1 2. Repairs equipment 2 5 5 3. Interprets instrument
8. Assume you have to prepare older employees with little computer experience to attend a training course on how to use the World Wide Web. How will you ensure that they have high levels of readiness for training? How will you determine their readiness for training?
7. What conditions would suggest that a company should buy a training program from an outside vendor? Which would suggest that the firm should develop the program itself?
6. Explain how you would determine if employees had the reading level necessary to succeed in a training program. How would you determine if employees had the necessary computer skills needed to use a Web-based training program?
5. Why should upper-level managers be included in the needs assessment process?
4. Needs assessment involves organization, person, and task analyses. Which one of these analyses do you believe is most important? Which is least important? Why?
3. If you were going to use online technology to identify training needs for customer service representatives for a Web-based clothing company, what steps would you take to ensure that the technology was not threatening to employees?
2. If you had to conduct a needs assessment for a new job at a new plant, describe the method you would use.
1. Which of the factors that influence performance and learning do you think is most important? Which is least important?
Complete the self-assessment exercise in Table 11.4. What changes would you make in the exercise to improve it?
Go to online.onetcenter.org. Click on Skills Search. Complete the skills search, and click Go. What occupations match your skills? How might Skills Search be useful for career management?
The World Wide Web is increasingly being used by companies to list job openings and by individuals to find jobs. Using the Web sites listed here (or sites you find yourself by surfing the Web), find two job openings that you may be qualified for. The Web sites include www.collegerecruiter.com
Go to www.ncsu.edu/careerkey, the Web site for Career Key, an assessment tool that can be used for self-assessment and career management. Roll the cursor over Your Personality, and click on Holland’s Theory of Career Choice and You. According to Holland’s theory, when will people be most
Go to www.monster.com. Roll over to Career Tools. Review Career Snapshot, Career Benchmark, and Career Mapping. How are each of these tools helpful for career management?
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