02-13-actual.ml Fri Feb 14 11:50: (* Lecture 6: 02/13/2014 *) (* Substitution example:

Transcription

1 02-13-actual.ml Fri Feb 14 11:50: Lecture 6: 02/13/2014 Substitution example: let x = 5 in (let y = 12 in x + y) * (let x = x + 1 in x - 3) (let y = 12 in 5 + y) * (let x = in x - 3) (5 + 12) * (let x = in x - 3) 17 * (let x = in x - 3) 17 * (let x = 6 in x - 3) 17 * (6-3) 17 * 3 51 if0 e1 then e2 else e3 if0 0 then e2 else e3 e2 if0 6 then e2 else e3 e3 if0 (fun x -> x) then e2 else e3 --- ERROR open Core.Std * Adding conditionals and functions * type op type var = Plus Minus Times = string type expr = ValE of value OpE of expr * op * expr LetE of var * expr * expr VarE of var If0E of expr * expr * expr AppE of expr * expr and value = IntV of int FunV of var * expr let rec p_expr (e : expr) = match e with ValE v ->... p_value v...

2 02-13-actual.ml Fri Feb 14 11:50: OpE(e1, op, e2) ->... p_expr e1... p_expr e2... LetE(x, e1, e2) ->... p_expr e1... p_expr e2... VarE x ->... If0E(e1, e2, e3) ->... p_expr e1... p_expr e2... p_expr e3... AppE(e1, e2) ->... p_expr e1... p_expr e2... and p_value (v : value) = match v with IntV z ->... FunV(x, e) ->... p_expr e... let inte (z : int) : expr = ValE (IntV z) 3 let three = inte let four = OpE(intE 3, Plus, inte 1) let two = 2 in two + (two + 1) let five = LetE("two", inte 2, OpE(VarE "two", Plus, OpE(VarE "two", Plus, inte 1))) let square = fun x -> x * x in square (square 2) let sixteen = LetE("square", ValE(FunV("x", OpE(VarE "x", Times, VarE "x"))), AppE(VarE "square", AppE(VarE "square", inte 2))) let add = fun x -> fun y -> x + y in let inc = add 1 in let dec = add (-1) in dec (inc 51) let fiftyone = LetE("add", ValE(FunV("x", ValE(FunV("y", OpE(VarE "x", Plus, VarE "y"))))), LetE("inc", AppE(VarE "add", inte 1), LetE("dec", AppE(VarE "add", inte (-1)), AppE(VarE "dec", AppE(VarE "inc", inte 51))))) let add = fun x -> fun y -> x + y in let inc = fun z -> add 1 z in let dec = add (-1) in dec (inc 51) let fiftyone = LetE("add", ValE(FunV("x", ValE(FunV("y", OpE(VarE "x", Plus, VarE "y"))))), LetE("inc", ValE(FunV("z", AppE(AppE(VarE "add", inte 1), VarE "z"))), LetE("dec", AppE(VarE "add", inte (-1)), AppE(VarE "dec", AppE(VarE "inc", inte 51))))) 3 + x -- Error! let error_expr = OpE(intE 3, Plus, VarE "x") exception UnboundVariable of var exception TypeError Evaluate an arithemetic operation on two values let rec eval_op (v1 : value) (op : op) (v2 : value) : value = match op, v1, v2 with Plus, IntV z1, IntV z2 -> IntV (z1 + z2) Minus, IntV z1, IntV z2 -> IntV (z1 - z2) Times, IntV z1, IntV z2 -> IntV (z1 * z2) _ -> raise TypeError Substitute a value for a variable in an expression

3 02-13-actual.ml Fri Feb 14 11:50: let substitute (v : value) (x : var) (e : expr) : expr = let rec subst e = match e with ValE v -> ValE (subst_v v) OpE(e1, op, e2) -> OpE(subst e1, op, subst e2) LetE(y, e1, e2) -> let e1 = subst e1 in let e2 = if x = y then e2 else subst e2 in LetE(y, e1, e2 ) VarE y -> if x = y then ValE v else VarE y If0E(e1, e2, e3) -> If0E(subst e1, subst e2, subst e3) AppE(e1, e2) -> AppE(subst e1, subst e2) and subst_v v = match v with IntV z -> IntV z FunV(y, e) -> if x = y then FunV(y, e) else FunV(y, subst e) in subst e Evaluate an expression let rec eval : expr -> value = function ValE v -> v OpE(e1, op, e2) -> eval_op (eval e1) op (eval e2) LetE(x, e1, e2) -> eval (substitute (eval e1) x e2) VarE x -> raise (UnboundVariable x) If0E(e1, e2, e3) -> (match eval e1 with IntV z -> if z = 0 then eval e2 else eval e3 FunV(_, _) -> raise TypeError) AppE(e1, e2) -> (match eval e1 with IntV _ -> raise TypeError FunV(x, e3) -> eval (substitute (eval e2) x e3)) assert( eval three = IntV 3 ) assert( eval four = IntV 4 ) assert( eval five = IntV 5 ) assert( eval sixteen = IntV 16 ) assert( eval fiftyone = IntV 51 ) assert( eval fiftyone = IntV 51 ) assert( eval (LetE("x", inte 3, LetE("x", inte 4, VarE "x"))) = IntV 4 ) assert( eval (LetE("x", inte 3, LetE("y", inte 4, VarE "x"))) = IntV 3 ) type a tree = Leaf Branch of a tree * a * a tree let tree0 = Leaf let tree1 = Branch(Leaf, 1, Leaf) let tree2 = Branch(Branch(Leaf, 1, Leaf), 3, Branch(Leaf, 5, Branch(Leaf, 7, Leaf)))

4 02-13-actual.ml Fri Feb 14 11:50: let rec p_tree (t :... tree) = Leaf ->... Branch(tl, x, tr) ->... p_tree tl... p_tree tr... To sum the ints in an int tree let rec sum_tree (t : int tree) : int = Leaf -> 0 Branch(tl, x, tr) -> x + sum_tree tl + sum_tree tr assert( sum_tree tree0 = 0 ) assert( sum_tree tree1 = 1 ) assert( sum_tree tree2 = 16 ) To find the maximum int in an int tree let rec max_tree (t : int tree) : int option = let max_opt o1 o2 = match o1, o2 with Some z1, Some z2 -> Some (max z1 z2) None, Some z2 -> Some z2 Some z1, None -> Some z1 None, None -> None in Leaf -> None Branch(tl, x, tr) -> max_opt (Some x) (max_opt (max_tree tl) (max_tree tr)) assert( max_tree tree0 = None ) assert( max_tree tree1 = Some 1 ) assert( max_tree tree2 = Some 7 ) let rec in_order_tree (t : a tree) : a list = Leaf -> [] Branch(tl, x, tr) -> in_order_tree x :: in_order_tree tr let rec fold_tree (t : a tree) (init : b) (f : b -> a -> b -> b) : b = Leaf -> init Branch(tl, x, tr) -> f (fold_tree tl init:init f:f) x (fold_tree tr init:init f:f) let sum_tree (t : int tree) : int = fold_tree t init:0 f:(fun zl z zr -> zl + z + zr) assert( sum_tree tree0 = 0 ) assert( sum_tree tree1 = 1 ) assert( sum_tree tree2 = 16 ) let in_order_tree : a tree -> a list = fold_tree init:[] f:(fun lstl x lstr -> x :: lstr) assert( in_order_tree tree2 = [1; 3; 5; 7] ) let max_tree : int tree -> int option = let max_opt o1 o2 = match o1, o2 with Some z1, Some z2 -> Some (max z1 z2) None, Some z2 -> Some z2 Some z1, None -> Some z1 None, None -> None in

5 02-13-actual.ml Fri Feb 14 11:50: fold_tree init:none f:(fun ml z mr -> max_opt ml (max_opt (Some z) mr)) assert( max_tree tree2 = Some 7 ) assert( max_tree tree0 = None )