Consider the following EBNF definition of a small language:
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1 xercise 1: Consider the following BNF definition of a small language: program ::= <decl>+ decl ::= <vardecl> <procdecl> vardecl ::= "var" <name> ":" <typename> "=" <expr> ";" procdecl ::= "procedure" <name> "(" <formalparamlist>? ")" "" <blockbody> "" formalparamlist ::= <formalparameter> ( "," <formalparameter> )* formalparameter ::= <name> ":" <typename> blockbody ::= <vardecl>* <statement>* statement ::= <printstatement> <callstatement> <assignstatment> <ifstatement> printstatement ::= "print" <expr> ";" callstatement ::= "call" <name> "(" <actualparamlist> ")" ";" assignstatment ::= <name> ":=" <expr> ";" ifstatement ::= "if" <expr> "" <blockbody> "" actualparamlist ::= <expr> ( "," <expr> )* expr ::= <name> <integer> "true" "false" In the productions above, nonterminals are enclosed within angled brackets (< and >), and terminals are quoted. Plus (+) is used for one or more repetitions, asterisk (*) for zero or more, and question mark (?) for zero or one. Regular parentheses are used for grouping symbols in the grammar. The nonterminals name, typename and integer are not specified in any further detail, but it will suffice to say that name and typename are strings of letters and numbers, and integer denotes ordinary integer numbers. 1a) Below is a small program in the language defined by the grammar above:
2 var x : int = 0; var y : int = 1; procedure M(x : int) var y : int = 42; x := 1; call N(x); procedure N(x : int) var z : int = 2; if true var z : int = 42; print x; print y; print z; procedure Main() call M(x); xplain what this program does; describe what will be printed, and why. Is there more than one reasonable interpretation? xplain your reasoning. Assume that the program execution starts with the execution of the procedure Main(), and that print <expr> will print the value of the expression to the console/terminal. Furthermore, assume that the program is legal and valid (it has no errors) in the language in question. Solution: in this exercise, the student should show some maturity, and be able to discuss some alternatives for semantics for this language. To gain full score, the student should mention, and show that he/she understands, the following: - Static scope - Dynamic scope - Variable hoisting.
3 Furthermore, there should be an explanation of what will be printed in each case. - Static scope: x = 1, y = 1, z = 2 - Dynamic scope: x = 1, y = 42, z = 2 - Variable hoisting, static function scope: x = 1, y = 1, z = 42 - Variable hoisting, dynamic function scope: x = 1, y = 42, z = 42 - Variable hoisting, program scope: x = 1, y = 42, z = 42 The double declaration of z should be mentioned. If not the above, one could also discuss static typing vs dynamic checking, but since the language has types specified explicitly, we better not take that too far. Also, different mechanism of variable passing (by value, by ref, by name, etc) could be discussed. 1b) Draw an abstract syntax tree for the procedure M from 1a) Solution: Variants of this, e.g. with a little less details, are OK. 1c)
4 Draw a runtime stack for the program from 1a) with activation records containing variables, parameters, access links and control links when the program has just executed the print z statement. xplain briefly the assumptions you make about the semantics of the language. Solution: Assuming static scoping, and no variable hoisting: For dynamic scoping, access link will be equal to the control link. Values will then be as stated in the solution for 1a. 1d)
5 xt the grammar provided at the top of this exercise to allow for procedures within procedures..g., the following should be allowed, where we have introduced new procedures P and Q: procedure N(x : int) var z : int = 2; procedure Q(v : int) print v; if true var z : int = 42; procedure P() print z; call P(); print x; print y; print z; call Q(y); You only need to write out the productions that you modify. Solution: We modify the blockbody production: blockbody ::= ( <vardecl> <statement> <procdecl> ) * Note that we cannot just add <procdecl> to the original setup, e.g. like this: blockbody ::= <vardecl>* <statement>* <procdecl>* as this involves an ordering of the decls and statements that does not fit with the program. Putting <procdecl>* before <statement>* in the above does work with the example, but is clearly not a very elegant solution. 1e)
6 Assume now, that in addition to procedures within procedures, we also allow procedures as parameters to other procedures. xplain briefly how this can be implemented in the runtime system. Solution: The students should know how this is done through the use of closures, where the closure is a pair pointing to the procedure s code and the procedure s environment (access link). A call to the parameter procedure will then utilize this closure in order to be able to find it s environment through its access link, regardless of where the call site actually is. 1f) Modify the procedure named N from 1d) so that this procedure now has a new formal parameter FP that is a procedure, and so that N performs a recursive call to itself with the procedure P as the actual parameter. Write down the new program, and the runtime stack right after the first recursive call to N. (As in 1a), we assume that program execution starts with the Main() procedure.) You only need to write down the parts of the program you modify. You do not need to write any modified version of the grammar of the language in this exercise, just assume a suitable syntax for this purpose. If the program s up with infinite recursion (non-termination), that is OK. Briefly state any further assumptions that you make about the language. Solution: First, we invent some syntax for formal and actual procedure parameters, and modify the procedure N accordingly, e.g. procedure N(x : int, proc FP(int)) var z : int = 2; procedure Q(v : int) print v; if true var z : int = 42; procedure P() print z; call P();
7 call N(z, P); print x; print y; print z; call Q(y); We also need to modify M, that calls N procedure M(x : int) var y : int = 42; x := 1; proc Dummy(d : int) call N(x, Dummy); Then, we need to draw the runtime stack Note how there are two closures for P, one for each call to N. One could also make an extra activation block for the if-statement in the procedure N, in between the last and the second-to-last block above.
8 Question 2. ML (weight 40%) 2a valuate the following ML expressions: a) List.foldr (-) 4 [3,2,1] -2 b) List.foldr (fn (y,x) => 2*x + y) 4 [1,2,3] 49 2b Assume the standard definition of foldl: fun foldl (f: 'a*'b->'b) (acc: 'b) (l: 'a list): 'b = case l of [] => acc x::xs => foldl f (f(x,acc)) xs With the help of foldl, define the function val myreverse = fn : 'a list -> 'a list which returns the input list in reverse order. 2c fun myrev zs = List.foldl (fn (x,xs) => x::xs) [] zs; 1) Define a function that sums up all values in a list of integers (and consider also question (4) below): val sum = fn : int list -> int 2) Define a function that finds the largest element in a list of integers: val max = fn : int list -> int 3) Define the function val summax = fn : int list list -> int which takes a list of lists of integers and sums up the largest values from each list. xample: summax [[1,2],[4,2],[3,6,5,9]] = 15 4) Give function sum from the question (2c1) above as a tail-recursive function (if you have not done so immediately you do not have to answer both questions separately if you directly give the tail-recursive version)! If you cannot answer question (2c1), explain the difference between a tail-recursive and a naïve implementation of a recursive function in a few sentences.
9 NO SOLUTION b/c easy J 2d 1) Define a datatype for binary trees, 'a tree, with a constructor mpty, and a constructor Node with a value of type 'a and two subtrees. 2) Define a function val mirror = fn : 'a tree -> 'a tree that returns a mirrored version of the tree passed as an argument. That is, for a node, each subtree must be mirrored. xamples ( for mpty): ) Define a function val sym = fn : 'a tree -> 'a tree -> bool which returns true, if the two trees are symmetric (that is, one is a mirror of the other), or false otherwise. datatype 'a tree = m Br of ('a * 'a tree * 'a tree); fun mir m = m mir (Br (a, l, r)) = Br (a, (mir r), (mir l)); fun sym m m = true sym (Br (a, l1, r1)) (Br (b, l2, r2)) = (a = b) andalso (mir l1 = r2) andalso (mir r1 = l2) sym = false; -- or: fun sym2 t1 t2 = (mir t1) = t2; 2e Calculate the type for the following expression according to the ML type inference algorithm: fn x => fn y => x (y x). Use the provided type variables in the parse graph below. Derive the corresponding equations for the parse graph, and solve the resulting equation system to obtain the type of the root node R.
10 λ:r x:x R = X -> A A=Y->B X=C->B Y=X->C ~ Y = (C->B)->C ~ R = (C->B) -> ((C->B)->C) -> B
11 Question 3. Prolog (weight 20%) 3a Give PROLOG s answer (that is, the substitutions for all variables in a query, if the terms can be unified) for each of the queries below, or write no if no solution exists and give the reason. Be careful with the last question, the == is no typo! 1. g(a,x,y) = g(z,c,f(z)). 2. g(a,x,y) = g(y,c,f(x)). 3. f(x,y,d) == f(y,h(x),z).?- g(a,x,y) = g(z,c,f(z)). X = c Y = f(a) Z = a?- g(a,x,y) = g(y,c,f(x)). no?- f(x,y,d) == f(y,h(x),z). no 3b Define the predicate rotate(l,n,r) which rotates cyclically a list L with N positions to the left, i.e. the following queries succeed: rotate([a,b,c,d], 1, [b,c,d,a]). rotate([a,b,c,d], 6, [c,d,a,b]). rotate([a,b,c,d], 3, [d,a,b,c]). rotate([],[],_). rotate(l,l,0). rotate([x H],Z,N) :- N is 1, app(h,[x],z). rotate(y,z,n) :- N > 1, X is N-1, rotate(y,z1,1), rotate(z1,z,x).
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