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Lexical Scoping and Closures

Lexical Scoping and Closures. COS 441 Princeton University Fall 2004. Lexical Scoping. The correct environment semantics for  -calculus gives rise to lexical scoping Evaluates body of function in the environment where the function was created Requires introducing notion of a closure

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Lexical Scoping and Closures

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  1. Lexical Scoping and Closures COS 441 Princeton University Fall 2004

  2. Lexical Scoping • The correct environment semantics for -calculus gives rise to lexical scoping • Evaluates body of function in the environment where the function was created • Requires introducing notion of a closure • Does not allow for unbound variables to occur in body of an expression • Lexical scope is the modern trend • Scheme, SML, Java Inner Classes, Python 2.1, Mathmetica “Module” and R (GPL S-Plus clone)

  3. Lexical Scoping Values are closures An expression and a environment that that holds bindings for all the free variables of the expression

  4. (Env,E1) lam(X’.E’)[Env’] (Env,E2) V’ (Env’[X’V’], E’)  V eval-X eval-L (Env,lam(X.E))  lam(X.E)[Env] (Env,X) Env(X) eval-A (Env,apply(E1,E2)) V Lexical Scoping

  5. (Env,E1) lam(X’.E’)[Env’] (Env,E2)  V’ (Env’[X’  V’], E’)  V eval-X eval-L (Env,lam(X.E))  lam(X.E)[Env] (Env,X)  Env(X) eval-A (Env,apply(E1,E2)) V Lexical Scoping Capture current environment

  6. (Env,E1) lam(X’.E’)[Env’] (Env,E2)  V’ (Env’[X’  V’], E’)  V eval-X eval-L (Env,lam(X.E))  lam(X.E)[Env] (Env,X)  Env(X) eval-A (Env,apply(E1,E2)) V Lexical Scoping Environment captured by closure

  7. Lexical Scoping • Can modify rule eval-L to only capture minimum set of variables needed to evaluate body • Compilers can pre-compute this set during semantic analysis X1,…, Xn2 FN(lam(X.E)) Envmin = {X1 Env(X1),…,Xn Env(Xn)} eval-L (Env,lam(X.E))  lam(X.E)[Envmin]

  8. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  V ({},(x.(y.x+y)) 1))  ?? ({},2)  2 (????)  V

  9. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  V ({},(x.(y.x+y)) 1))  ?? ({},2)  2 (????)  V

  10. Example: Lexical Scope ({},(x.(y.x+y)) 1))  ?? ({},(x.(y.x+y))  (x.(y.x+y))[{}] ({},1)  1 ({}[x1],(y.x+y))  ??

  11. Example: Lexical Scope ({},(x.(y.x+y)) 1))  (y.x+y)[{x1}] ({},(x.(y.x+y))  (x.(y.x+y))[{}] ({},1)  1 ({x1},(y.x+y))  (y.x+y)[{x1}]

  12. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  V ({},(x.(y.x+y)) 1))  ?? ({},2)  2 (????)  V

  13. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  V ({},(x.(y.x+y)) 1))  (y.x+y)[{x2}] ({},2)  2 ({x1}[y2],x+y)  V

  14. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  V ({},(x.(y.x+y)) 1))  (y.x+y)[{x2}] ({},2)  2 ({x1,y2},x+y)  V

  15. Example: Lexical Scope ({},(((x.(y.x+y)) 1) 2))  3 ({},(x.(y.x+y)) 1))  (y.x+y)[{x2}] ({},2)  2 ({x1,y2},3)  3

  16. E-Machine • E-machine is small-step semantics with environment semantics • Stacks now must contain references to environment capture when the stack frame was push on the stack

  17. Example: lwith Numbers

  18. Example: lwith Numbers • Machine frames contain machine values • Not all machine frames capture an environment Why?

  19. E-Machine Rules

  20. E-Machine Rules

  21. E-Machine Rules

  22. E-Machine Rules

  23. E-Machine Rules

  24. E-Machine Rules

  25. Example: Evaluation (²,{},(((x.(y.+(x,y))) 1) 2)) E((¤ 2)[{}] B²,{},((x.(y.+(x,y))) 1)) E((¤ 1)[{}] B(¤ 2)[{}] B²,{},(x.(y.+(x,y)))) E((x.(y.+(x,y)))[{}],(¤ 1)[{}] B(¤ 2)[{}] B²) E(((x.(y.+(x,y)))[{}] ¤)[{}] B(¤ 2)[{}] B²,{},1) E(1,((x.(y.+(x,y)))[{}] ¤)[{}] B(¤ 2)[{}] B²) E((¤ 2)[{}] B²,{x1},(y.+(x,y))) E((y.+(x,y))[{x1}],(¤ 2)[{}] B²) E(((y.+(x,y))[{x1}] ¤)[{}] B²,{},2) E (2,((y.+(x,y))[{x1}] ¤)[{}] B²) E (²,{x1,y2},+(x,y)) … E(3,²)

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