Name: Section: CS / CE 4337 Sample Final Exam Instructor: Brian W. DeVries May 2, 2013 Instructions: Read the following instructions before beginning the exam. • Before beginning, write your name at the top of each page of the exam. Also write your section number on this page. This will be worth points — if you cannot read a specification, how can you expect to implement it? • Answer each question in the space provided. Additional paper is available upon request. Please write your name at the top of these pages as well, and clearly indicate what question you are answering. • Please write legibly and clearly! No credit can be given for unreadable answers. • Don’t panic! You have more than enough time to complete the exam. If you feel yourself becoming overwhelmed, sit back for a minute and relax. Trying to throw down an answer in a hurry is an easy way to make mistakes and lose points. • By the same token, if you finish early, don’t rush out the door. Relax for a minute, then go back through and check your answers. In particular, be on the lookout for careless mistakes. • Unless explicitly stated, you should not need to make any assumptions in order to answer the questions. If you feel that a question is unclear, then please raise your hand. Name: CS / CE 4337 Midterm Exam March 6, 2013 Prolog and Logic Programming 1. Consider the following logic program: /** * edge( Source, Weight, Destination ) * * A collection of facts documenting the edge relation of a graph. Note that * the set of vertices is implicit. */ edge( a, 5, b ). edge( b, 3, a ). /** * path( Source, Weight, Destination ) * * The path relation for the above graph (i.e., the transitive closure of the * edge relation). */ path( X, W, Y ) :edge( X, W, Y ). path( X, W, Y ) :edge( X, W1, Z ), path( Z, W2, Y ), W #= W1 + W2. Recall our discussion of the semantics of logic programs: we take the set of things we know to be true (call it Sk ), substitute them in for the bodies of the clauses in our programs, and see what we can prove true at that point (call it Sk+1 ) based on whether the body of the clause is satisfied. We continue doing this until we reach a fixed point. With that in mind, answer the following questions: (a) Given that all of these predicates are inductive, where do we begin this process. In other words, what is S0 ? (b) What is S1 ? Page 1 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 (c) What is S2 ? (d) What is S3 ? (e) By repeating this process ad infinitum, will we arrive at a least fixed point or a greatest fixed point? In other words, is induction a least fixed point or a greatest fixed point semantics? Page 2 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 (f) Consider the query ?- path( a, W, Y ), W = 10. What would happen if this query were entered at the prompt? Why? (g) What effect would tabling path / 3 have on this behavior? Page 3 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 2. Consider the following logic program: p :- q. q :- r. r :- p. s :- t. t. Answer the following questions: (a) What propositions hold under an inductive interpretation? (b) What propositions hold under a coinductive interpretation? Page 4 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 Haskell Questions 3. Recall the following higher-order functions in Haskell: -- Function composition (.) :: ( b -> c ) -> ( a -> b ) -> a -> c f . g = \ x -> f ( g x ) map :: ( a -> b ) -> [ a ] -> [ b ] map fn [] = [] map fn ( x : xs ) = ( fn x ) : ( map fn xs ) foldl :: ( a -> b -> a ) -> a -> [ b ] -> a foldl fn x [] = x foldl fn x ( y : ys ) = foldl fn ( fn x y ) ys flip :: ( a -> b -> c) -> b -> a -> c flip fn x y = fn y x Suppose we have the following functions: foo :: Integer -> Integer foo x = 2 * x + 1 bar = ( foldl (+) 0 ) . ( map foo ) (a) What is the type of bar? (b) Give two example inputs to bar and their corresponding outputs. (c) What does this function do? Page 5 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 4. Take the following exponent function: (^) :: ( Num a, Integral b ) => a -> b -> a -- Examples: 4 ^ 2 -- = 16 5.3 ^ 4 -- = 789.0481 4 ^ 5.3 -- Type error Write a single-line Haskell function prodSq that uses the higher-order functions in the previous problem to compute the product of the squares of a list of numbers. In other words, you should not write a recursive function to do this. Provide the type signature of sumSq. Your type signature should be as general as possible; in particular, it does not contain the type Integer, Int, etc. Hint: Be careful of the ordering of the arguments of (^)! Page 6 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 T-Diagrams 5. Describe the following diagrams: (a) (b) (“Py” is short for Python) (c) Page 7 of 8 Name: CS / CE 4337 Midterm Exam March 6, 2013 6. Suppose I have the following compiler. (NOTE: You should know the name for this sort of compiler!) Suppose I have the following tools available: (Here, “Hs” is short for Haskell.) (a) What program could I write that would bootstrap this compiler on the x86 architecture? (There are two possible answers.) Draw its T-diagram. (b) Draw the T-diagram for bootstrapping our Imp compiler using your choice above. Page 8 of 8