1. Tim Sheard Oregon Graduate Institute Lecture 8: Operational Semantics of MetaML CSE 510 Section FSC Winter 2005 Winter 2005

2. 2Cse583 Winter 2002 Assignments Reading Assignment A Methodology for Generating Verified Combinatorial Circuits. A Methodology for Generating Verified Combinatorial Circuits By Kiseyov, Swadi, and Taha. Available on the readings page of the course web- site Homework # 5 assigned Tuesday Feb. 1, 2005 due in class Tuesday Feb. 8, 2005 Details on the assignment page of the course web- site.

3. 3Cse583 Winter 2002 Map of today’s lecture Thanks to Walid Taha. Todays lecture comes from chapter 4 of his thesis Simple interpreter for lambda calculus Addition of constants for  Operators  If-then-else over integer  Y combinator Add staging annotations Fix up environment mess

4. 4Cse583 Winter 2002 Terms and Values datatype exp = EI of int (* integers *) | EA of exp * exp (* applications *) | EL of string * exp (* lambda-abs *) | EV of string; (* variables *) datatype value = VI of int | VF of (value -> value);

5. 5Cse583 Winter 2002 Example terms val ID = EL("x",EV "x"); val COMPOSE = EL("f", EL("g", EL("x",EA(EV "f", EA(EV "g",EV "x"))))); fn x => x fn f => fn g => fn x => f(g x)

6. 6Cse583 Winter 2002 Example Values val id = VF(fn x => x); val compose = VF(fn (VF f) => VF (fn (VF g) => VF(fn x => f(g x)))); val square = VF (fn (VI x) => VI(x * x)); val bang = let fun fact n = if n=0 then 1 else n*(fact (n-1)) in VF (fn (VI x) => VI(fact x)) end; Note how “primitive” functions can be implemented

7. 7Cse583 Winter 2002 An interpreter The interpreter ev0 : (string -> value) -> exp -> value Environments type env =string -> value; exception NotBound of string val env0 = fn x => (print x; raise (NotBound x)); fun ext env x v = (fn y => if x=y then v else env y);

8. 8Cse583 Winter 2002 Semantics in 8 lines fun ev0 env e = case e of EI i => VI i | EA(e1,e2) => (case (ev0 env e1,ev0 env e2) of (VF f,v) => f v) | EL(x,e1) => VF(fn v => ev0 (ext env x v) e1) | EV x => env x; -| val ex1 = ev0 env0 (EA(ID,EI 5)); val ex1 = VI 5 : value

9. 9Cse583 Winter 2002 Printing Terms fun showe e = case e of EI n => show n | EA(f,x as EA(_,_)) => (showe f)^" ("^(showe x)^")" | EA(f,x) => (showe f)^" "^(showe x) | EL(x,e) => "(fn "^x^" => "^(showe e)^")" | EV x => x

10. 10Cse583 Winter 2002 Making it useful The interpreter is good for the lambda calculus with integer constants, but we have no operations on integers Need to add primitive operations Arithmetic Case analysis over integers Recursion over integers This is accomplished by adding an initial environment with values for some constants

11. 11Cse583 Winter 2002 Arithmetic val plus = VF(fn (VI x) => VF(fn (VI y) => VI(x+y))); val minus = VF(fn (VI x) => VF(fn (VI y) => VI(x-y))); val times = VF(fn (VI x) => VF(fn (VI y) => VI(x*y))); val env1 = ext env0 "*" times; val SQUARE = EL("x",EA(EA(EV "*",EV "x"),EV "x")); -| val ex2 = ev0 env1 (EA (SQUARE,EI 5)); val ex2 = VI 25 : value

12. 12Cse583 Winter 2002 If-zero function fun Z n tf ff = if n=0 then tf 0 else ff n; val Z : int -> (int -> 'a ) -> (int -> 'a ) -> 'a val ifzero = VF(fn (VI n) => VF(fn (VF tf) => VF(fn (VF ff) => if n=0 then tf(VI 0) else ff(VI n))));

13. 13Cse583 Winter 2002 Recursion fun Y f = f (fn v => (Y f) v); Y : (('b -> 'a ) -> 'b -> 'a ) -> 'b -> 'a val recur = let fun recur' (VF f) = f(VF (fn v=> (case (recur' (VF f)) of VF fp => fp v | w => error “not function“ ))) in VF recur' end;

14. 14Cse583 Winter 2002 New initial environment val env1 = ext(ext(ext(ext(ext env0 "+" plus) "-" minus) "*" times) "Z" ifzero) "Y" recur;

15. 15Cse583 Winter 2002 Using the constants Start with a recursive function in ML Transform it removing ML features Example function fun h1 n z = if n=0 then z else (fn x => (h1 (n-1) (x+z))) n;

16. 16Cse583 Winter 2002 Steps Remove recursion by using Y val h1 = Y(fn h1' => fn n => fn z => if n=0 then z else (fn x => (h1' (n-1) (x+z))) n); Remove if val h1 = Y(fn h1' => fn n => fn z => Z n (fn _ => z) (fn n => (fn x => (h1' (n-1) (x+z))) n));

17. 17Cse583 Winter 2002 Make a term fun minus x y = EA(EA(EV "-",x),y); fun plus x y = EA(EA(EV "+",x),y); val H1 = EA(EV "Y",EL("h1",EL("n",EL("z", EA(EA(EA(EV "Z",EV "n"),EL("w",EV "z")),EL("n",EA(EL("x",EA(EA(EV "h1", minus (EV "n") (EI 1)),plus (EV "x") (EV "z"))),EV "n")))))));

18. 18Cse583 Winter 2002 What does it look like? -| print(showe H1); Y (fn h1 => (fn n => (fn z => Z n (fn w => z) (fn n => (fn x => h1 (- n 1)(+ x z)) n))))

19. 19Cse583 Winter 2002 Running it -| val IT = EA(EA(H1,EI 3),EI 4); -| val answer = ev0 env1 IT; val answer = VI 10 : value

20. 20Cse583 Winter 2002 Adding Staging What are the new kinds of values Code Represented as embedding exp in value What are the new kinds of terms Brackets Escapes Run Cross stage persistent constants

21. 21Cse583 Winter 2002 Terms and values datatype value = VI of int | VF of (value -> value) | VC of exp and exp = EI of int (* integers *) | EA of exp * exp (* applications *) | EL of string * exp (* lambda-abstractions *) | EV of string (* variables *) | EB of exp (* brackets *) | ES of exp (* escape *) | ER of exp (* run *) | EC of string * value; (* cross-stage constants *)

22. 22Cse583 Winter 2002 Generalizing environments type env = string -> exp; fun env0 x = EV x; val env1 = ext(ext(ext(ext(ext env0 "+" (EC("+",plus))) "-" (EC("-",minus))) "*" (EC("*",times))) "Z" (EC("if",ifzero))) "Y" (EC("Y",recur)); Note the initial values in the environment are cross-stage persistent Expressions. Other values will also be injected using EC

23. 23Cse583 Winter 2002 Printing terms fun showe e = case e of EI n => show n | EA(f,x as EA(_,_)) => (showe f)^" ("^(showe x)^")" | EA(f,x) => (showe f)^" "^(showe x) | EL(x,e) => "(fn "^x^" => "^(showe e)^")" | EV x => x | EB e => " " | ES(EV x) => "~"^x | ES e => "~("^(showe e)^")" | ER e => "run("^(showe e)^")" | EC(s,v) => "%"^s;

24. 24Cse583 Winter 2002 Fresh Variables val ctr = ref 0; fun NextVar s = let val _ = ctr := (!ctr) + 1 in s^(toString (! ctr)) end; -| NextVar "x"; val it = "x1" : string -| NextVar "x"; val it = "x2" : string

25. 25Cse583 Winter 2002 The interpreter fun ev1 env e = case e of EI i => VI i | EA(e1,e2) => (case (ev1 env e1,ev1 env e2) of (VF f,v) => f v) | EL(x,e1) => VF(fn v => ev1 (ext env x (EC(x,v))) e1) | EV x => (case env x of EC(_,v) => v | w => VC w) | EB e1 => VC(eb1 1 env e1) | ER e1 => (case ev1 env e1 of VC e2 => ev1 env0 e2) | EC(_,v) => v | ES e1 => error "escape at level 0"

26. 26Cse583 Winter 2002 Rebuilding terms and eb1 n env e = case e of EI i => EI i | EA(e1,e2) => EA(eb1 n env e1,eb1 n env e2) | EL(x,e1) => let val x' = NextVar x in EL(x',eb1 n (ext env x (EV x')) e1) end | EV y => env y | EB e1 => EB(eb1 (n+1) env e1) | ES e1 => if n=1 then (case ev1 env e1 of VC e => e) else ES(eb1 (n-1) env e1) | ER e1 => ER(eb1 n env e1) | EC(s,v) => EC(s,v)

27. 27Cse583 Winter 2002 Staging H1 val H2 = let fun minus x y = EA(EA(EV "-",x),y) fun plus x y = EA(EA(EV "+",x),y) in EA(EV "Y",EL("h1",EL("n",EL("z", EA(EA(EA(EV "Z",EV "n"),EL("w",EV "z")),EL("n",EB(EA(EL("x",ES(EA(EA(EV "h1", minus (EV "n") (EI 1)),EB(plus (EV "x") (ES (EV "z")))))),EV "n")))))))) end; -| print(showe H2); Y (fn h1 => (fn n => (fn z => Z n (fn w => z) (fn n => ~(h1 (- n 1) )) n>))))

28. 28Cse583 Winter 2002 Generating code val VC answer = ev1 env1 IT; -| val answer = EA (EL ("x7", EA (EL ("x8", EA (EL ("x9", EA (EA (EC ("+",VF fn ),EV "x9"), EA (EA (EC ("+",VF fn ),EV "x8"), EA (EA (EC ("+",VF fn ),EV "x7"),EI 4)))), EC ("n",VI 1))), EC ("n",VI 2))), EC ("n",VI 3)) -| showe answer; val it = "(fn x7 => (fn x8 => (fn x9 => %+ x9 (%+ x8 (%+ x7 4))) %n) %n) %n"

29. 29Cse583 Winter 2002 A problem -| val puzzle = run ((run ~( (fn x => ) (fn w => ) ) 5>) 3); -| val puzzle = 3 : int

30. 30Cse583 Winter 2002 In our language (* (fn x => ) *) val t1 = EL("x",EB(EV "x")); (* (fn w => ) *) val t2 = EL("w",EB(EV "a")); (* ~( (fn x => ) (fn w => ) ) 5> *) val t3 = EB(EL("a",EA(ES(EA(t1,t2)),EI 5))); val puzzle = ER(EA(ER t3,EI 3)); -| showe puzzle; val it = "run(run( ~((fn x => ) (fn w => )) 5)>) 3)“

31. 31Cse583 Winter 2002 Running it run(run( ~((fn x => ) (fn w => )) 5)>) 3)“ -| ev1 env1 puzzle; val it = VC EV "a14" : value What’s gone wrong?

32. 32Cse583 Winter 2002 Covers fun coverE env e = case e of EI i => EI i | EA(e1,e2) => EA(coverE env e1,coverE env e2) | EL(x,e1) => EL(x,coverE env e1) | EV y => env y | EB e1 => EB(coverE env e1) | ES e1 => ES(coverE env e1) | ER e1 => ER(coverE env e1) | EC(s,v) => EC(s,coverV env v) and coverV env v = case v of VI i => VI i | VF f => VF((coverV env) o f) | VC e => VC(coverE env e);

33. 33Cse583 Winter 2002 Final interpreter fun ev1 env e = case e of EI i => VI i | EA(e1,e2) => (case (ev1 env e1,ev1 env e2) of (VF f,v) => f v) | EL(x,e1) => VF(fn v => ev1 (ext env x (EC(x,v))) e1) | EV x => (case env x of EC(_,v) => v | w => VC w) | EB e1 => VC(eb1 1 env e1) | ER e1 => (case ev1 env e1 of VC e2 => ev1 env0 e2) | EC(s,v) => coverV env v

34. 34Cse583 Winter 2002 Final rebuilder and eb1 n env e = case e of EI i => EI i | EA(e1,e2) => EA(eb1 n env e1,eb1 n env e2) | EL(x,e1) => let val x' = NextVar x in EL(x',eb1 n (ext env x (EV x')) e1) end | EV y => env y | EB e1 => EB(eb1 (n+1) env e1) | ES e1 => if n=1 then (case ev1 env e1 of VC e => e) else ES(eb1 (n-1) env e1) | ER e1 => ER(eb1 n env e1) | EC(s,v) => EC(s,coverV env v)

35. 35Cse583 Winter 2002 Now it works -| ev1 env1 puzzle; val it = VI 3 : value