Consider the language L = {w |w ∈ {0, 1} ∗ , 2 × #0 (w) = #1 (w)} , here #0(w) and #1(w) means the numbers of 0s and 1s in w respectively. That is, this is the language consisting of all strings with exactly twice as many 1s as 0s: for example, 101 ∈ L, 010 ∈/ L. In this problem we will show how to construct a context-free grammar for it. (a) (2 points) For a length n word x, we define a function f (i) = (2 × number of 0s in locations [1 . . . i]) − (number of 1s in locations [1 . . . i]). Show that we have f(i) = f(j) for some i < j if and only if the substring in indices [i + 1, . . . j] has exactly twice as many 1s as 0s (i. e. xi+1,i+2,...,j ∈ L).

Before you enter an intersection on a green light make sure

Word vocabulary words: print, proofread, status line, graphics, font effects, left margin, justification, line spacing, copy/paste, data. review words: font point, bold, save, center, error. fill in the correct word for the definition and then transfer the letters to the appropriate spot by number. some numbers will be found multiple times. you will end up with a quotation about…… what else? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k 16 17 18 19 20 21 22 23 24 25 8 27 28 29 w 31 32 k 34 35 36 w h 39 40 41 42 8 44 45 46 47 48 49 50 51 52 53 54 55 .1. a software function that records keystrokes on a disk or drive so information can be 5 4 52 9 later retrieved. p n 2. to produce a paper copy of information. 10 7 12u n 3. a display that shows the location of the cursor, pages, etc. 45 46 18 27 36 20 42p4. pictures or images, located in clip art or other files. 6 24 44 28 34 49 555. any mis-stroke of a key. 47 41 48 2 10 n6. allows major changes to the font such as shadow, emboss, etc. 21 25 46 35 23 21 29 14 22 17 n7. a feature that centers lines of text horizontally. 49 53 46 9 51 p8. size of the font 31 16 22 b l 9. a feature that prints designated text darker than the rest to add emphasis. 32 3 . p10. to compare copy on a display screen or printout to the original 24 39 25 23 54 9 50 3 and correct errors. j un 11. a feature that allows text to be aligned at the left 11 12 7 21 16 49 40 46 34 2 and right margins. leftn 12. amount of blank space on the left side of the paper. 8 18 41 6 34 linen 13. number of blank lines between lines of text. 17 4 49 13 1914. any information inputted into the computer. 3 4 46 44 p /p15. feature that duplicates text from one location and places it in another.

You are consulting for a beverage distributor who is interested in determining the benefits it could achieve from implementing new information systems. what will you advise as the first step?