Problem 1. (20pts) Parser Technology a) Look again at our grammar for well- matched bracket sequences:

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1 Long Quiz3 45mins Name: Person Number: Problem 1. (20pts) Parser Technology a) Look again at our grammar for well- matched bracket sequences: S epslon TS T (S) Please write down First set of S and T, and Follow Set of S and T (considering LL(1) parser) First(S) = { (, because an S can begin with ( or be empty First(T ) = { ( because a T must begin with ( Follow(S) = { ), $ because within a complete phrase, an S can be followed by ) or appear at the end Follow(T ) = { (, ), $ because a T can be followed by ( or ) or appear at the end b) Say you have an xml document <?xml version='1.0' encoding='us-ascii'?> <slideshow title="sample Slide Show" date="date of publication" author="yours Truly" > <slide type="all"> <title>wake up to WonderWidgets!</title> </slide> <slide type="all"> <title>overview</title> <item>why <em>wonderwidgets</em> are great</item> <item/> <item>who <em>buys</em> WonderWidgets</item> </slide> </slideshow> Please convert it to Context Free Grammar for parsing.

2 Xml - > SlideShow SlideShow - > SlideList SlideList - > Slide SlideList Slide - > Title Title ItemList ItemList - > Item ItemList epslon Problem 2. (20pts) Analysis of Definition The Fibonacci function F(n) defined for natural numbers n is as follows. F (n) = 0 if n = 0 = 1 if n = 1 = F(n- 1) + F(n- 2) otherwise Define an SML- function cntcalls that computes the number of calls to F generated for various input values. (E.g., cntcalls(0) = 1, cntcalls(3) = 5, etc.) What are the values of F(5) and cntcalls(5)? fun cntcall 0 = 1 cntcall 1 = 2 cntcall n = cntcall(n- 1) + cntcall(n- 2) ;

3 Problem 3. (20pts) Understanding Function Definition fun f [] = [] f (x::xs) = let val s = f xs in (map (fn y => y@[x]) s) end; a) What is the signature of f? f = fn : 'a list - > 'a list list b) Informally describe the list function computed by f. Give the value and type returned for f ["a"]? - f["a"]; val it = [] : string list list c) Now formalize what f does using the Induction Principle. There are 2 rules here: 1) x : x' xs : xs' f :! y@ys :[y', ys']. f (ys)@y x@xs :[x', xs'] f (x@xs :[x', xs']) : [a' list, x'] Where a is type variable mapping f(xs ) - > a

4 2) x : x' xs : Nil f :! y : Nil. Nil x@xs : x' f (x@nil) : x' Problem4. (20 pts) Writing Function Definition The following SML- definition specifies two concrete datatypes etype and expr. datatype etype = Int Real; datatype expr = I J A B Plus of (expr * expr) Mul of (expr * expr); The type of I and J is Int. The type of A and B is Real. The type of a Plus- expression is Int, if the subexpressions have Int type; otherwise it is an error. (E.g., if both arguments are Real, it is an error!) The type of a Mul- expression is always Real as long as the subexpressions are well- typed; otherwise it is an error. (Note, a subexpression is well- typed if its type either Int or Real.) Write an SML- function typeinfer that infers the type of an expression whereever feasible (according to the specified notion of type inference) and throws an exception called Error otherwise. For example, - typeinfer I; val it = Int : etype - typeinfer (Mul (A,Plus(J,I))); val it = Real : etype - typeinfer (Mul (A,Plus(B,J)));

5 fun typeinfer e = case e of I => Int Plus(e1, e2) => Real Mul(e1, e2) => Real ; Problem 5. (20pts) Please write down result of following program. #include <iostream> using namespace std; enum note { middlec, Csharp, Eflat ; // Etc. class Instrument { public: void play(note) const { cout << "Instrument::play" << endl; ; // Wind objects are Instruments // because they have the same interface: class Wind : public Instrument { public: // Redefine interface function: void play(note) const { cout << "Wind::play" << endl; ; void tune(instrument& i) { //... i.play(middlec); int main() { Wind flute; tune(flute); // Upcasting ///:~ The function Instrument::play