The Environment Model of Evaluation Jim Royer CIS 352 March 22, - PowerPoint PPT Presentation

The Environment Model of Evaluation Jim Royer CIS 352 March 22, 2019 CIS 352 The Environment Model of Evaluation 1 / 31

References Structure and Interpretation of Computer Programs, 2/e, § 3.2: The Environment Model of Evaluation , by Harold Abelson and Gerald Sussman, MIT Press, 1996. https://mitpress.mit.edu/sicp/full-text/book/book.html William Cook, Anatomy of Programming Languages , Chapters 3 and 4, http://www.cs.utexas.edu/~wcook/anatomy/anatomy.htm CIS 352 The Environment Model of Evaluation 2 / 31

LFP = LC + λ + function application + variables LFP Expressions E :: = n | b | ℓ | E iop E | E cop E | if E then E else E | ! E | E : = E | skip | E ; E | while E do E | x | λ x . E | E E | let x = E in E � �� � the λ -calculus where x ∈ V , an infinite set of variables n ∈ Z (integers), b ∈ B (booleans), ℓ ∈ L (locations) iop ∈ (integer-valued binary operations) cop ∈ (boolean-valued binary comparisons) We focus on the ( λ -calculus + let ) part of LFP. CIS 352 The Environment Model of Evaluation 3 / 31

Application via substitution and its problems Call by name 1 , s ′ � 1 [ E 2 / x ] , s ′ � ⇓ � V , s ′′ � ⇓ -cbn: � E 1 , s � ⇓ � λ x . E ′ � E ′ � ( E 1 E 2 ) , s � ⇓ � V , s ′′ � Call by value 1 , s ′ � � E 2 , s ′ � ⇓ � V 2 , s ′′ � 1 [ V 2 / x ] , s ′′ � ⇓ � V , s ′′′ � ⇓ -cbv: � E 1 , s � ⇓ � λ x . E ′ � E ′ � ( E 1 E 2 ) , s � ⇓ � V , s ′′′ � Call-by-name and call-by-value are defined above via substitution . Substitution is: dandy for nailing down sensible meanings of application. stinko for everyday implementations. E.g., An implementation via substitution constantly needs to modify a program’s source code. Idea: In place of substituting a value v for a variable x : Keep a dictionary of variables & their values. When you need the value of x , look it up. CIS 352 The Environment Model of Evaluation 4 / 31

Environments ( Warning: Scary Greek letters) Definition An environment is just a table of variables and associated values . Consider an expression e = if z then x else y + 2. With environment { x �→ 3, y �→ 4, z �→ tt } , e evaluates to 3. With environment { x �→ 8, y �→ 5, z �→ ff } , e evaluates to 7. Etc. lookup ( ρ , x ) returns the value (if any) of x in environment ρ . update ( ρ , x , v ) returns a new environment ρ [ x �→ v ] ( ρ [ x �→ v ] is just like ρ except x has value v.) Evaluating variable x in environment ρ ≡ lookup ( ρ , x ) . CIS 352 The Environment Model of Evaluation 5 / 31

Revising call-by-value big-step semantics, 1 Definition ρ ⊢ � e , s � ⇓ V � v , s ′ � means that expression e with environment ρ and state s evaluates to value v and state s ′ . Var: ρ ⊢ � x , s � ⇓ V � v , s � ( v = lookup ( ρ , x )) Let: ρ ⊢ � e 1 , s � ⇓ V � v 1 , s ′ � ρ [ x �→ v 1 ] ⊢ � e 2 , s ′ � ⇓ V � v 2 , s ′′ � ρ ⊢ � let x = e 1 in e 2 , s � ⇓ V � v 2 , s ′′ � Examples/Exercises: Let ρ = { x �→ 7, y �→ 3 } . ρ ⊢ � x + y , s � ⇓ V ?? ρ ⊢ � let x = 1 in x + y , s � ⇓ V ?? ρ ⊢ � let x = 1 in ( let z = 11 in x + y + z ) , s � ⇓ V ?? CIS 352 The Environment Model of Evaluation 6 / 31

Revising call-by-value big-step semantics, 2 Preliminary versions of these rules: 1 , s ′ � � λ x . e ′ ρ ⊢ � e 1 , s � ⇓ V ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V 1 , s ′′ � � v , s ′′′ � ρ [ x �→ v 2 ] ⊢ � e ′ ⇓ V App: Fun: ρ ⊢ � λ x . e , s � ⇓ V � λ x . e , s � � v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V Examples/Exercises: Let ρ = { x �→ 7, y �→ 3 } . ρ ⊢ � let f = λ x . ( x + y ) in ( f 10 ) , s � ⇓ V ?? !!! ρ ⊢ � let f = λ x . ( x + y ) in ( let y = 100 in ( f 10 )) , s � ⇓ V ?? CIS 352 The Environment Model of Evaluation 7 / 31

Scoping Definition (Variable Scope) The scope of a variable binding/declaration is the region of a program where the binding is valid, i.e., when you use the variable, it uses that declaration for the binding (meaning) of the name. A Java example (static/lexical scoping) { int i = 23; the outer i ’s scope for (int i = 1; i<11; i++) { ... } the inner i ’s scope System.out.println(i); ... } CIS 352 The Environment Model of Evaluation 8 / 31

Dynamic Scoping, 1 Re: λ -expressions, functions, procedures, etc., there are two sorts of environments you have to worry about: The environment in force when the function was created . 1 The environment in force when the function is applied . 2 1 , s ′ � � λ x . e ′ ρ ⊢ � e 1 , s � ⇓ V ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V 1 , s ′′ � � v , s ′′′ � ☞ ρ [ x �→ v 2 ] ⊢ � e ′ ⇓ V Dynamic-App: � v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V Example: Let ρ = { x �→ 7, y �→ 3 } and consider ρ ⊢ � let f = λ x . x + y in let g = λ y . f ( y + 100 ) in (( f 10 ) + ( g 0 )) , s � ⇓ V ?? CIS 352 The Environment Model of Evaluation 9 / 31

Dynamic Scoping, 2 1 , s ′ � � λ x . e ′ ρ ⊢ � e 1 , s � ⇓ V ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V 1 , s ′′ � � v , , s ′′′ � ☞ ρ [ x �→ v 2 ] ⊢ � e ′ ⇓ V Dynamic-App: � v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V Under dynamic scoping, when you apply a function in environment (( λ x . e ′ 1 ) e 2 ) in environment ρ you evaluate e ′ 1 in environment ρ [ x �→ v 2 ] . Question: Is this a bug or a feature? CIS 352 The Environment Model of Evaluation 10 / 31

Dynamic Scoping, 3 1 , s ′ � � λ x . e ′ ρ ⊢ � e 1 , s � ⇓ V ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V ☞ ρ [ x �→ v 2 ] ⊢ � e ′ 1 , s ′′ � � v , s ′′′ � ⇓ V Dynamic-App: � ( v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V What goes right under dynamic scoping? let f = λ n . if n ≤ 0 then 1 else n ∗ ( f ( n − 1 )) in ( f 3 ) History Discovered and formalized in early ( ≈ 1960s) Lisp implementations. CIS 352 The Environment Model of Evaluation 11 / 31

Lexical Scoping, 1 Re: λ -expressions, functions, procedures, etc., there are two sorts of environments you have to worry about: The environment in force when the function is created . 1 The environment in force when the function is applied . 2 In human language, statements need to be understood in context: Such a fact is probable, but undoubtedly false. — Edward Gibbon in “Decline and Fall of the Roman Empire” When Gibbon was writing “probable” meant “well-recommended”. So in reading Gibbon we have to use a 1700’s English dictionary. We pull a similar trick for functions. CIS 352 The Environment Model of Evaluation 12 / 31

Lexical Scoping, 2 Definition A closure, e ρ , is an expression e with an environment ρ such that fv ( e ) ⊆ domain ( ρ ) , i.e., all of e ’s free variables are in ρ ’s dictionary. Ideas: A λ -expression evaluates to a closure. When we create a λ -expression, we “close” it with its definition-time environment. Lexical-Fun: ρ ⊢ � λ x . e , s � ⇓ V � ( λ x . e ) ρ , s � When we apply a function (i.e., closure ( λ x . e ′ ) ρ ′ ), we evaluate e ′ in ρ ′ [ x �→ v ] , where v is the value of the argument. 1 , s ′ � � ( λ x . e ′ 1 ) ρ ′ ρ ⊢ � e 1 , s � ⇓ V ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V 1 , s ′′ � � v , s ′′′ � ρ ′ 1 [ x �→ v 2 ] ⊢ � e ′ ⇓ V Lexical-App: � ( v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V CIS 352 The Environment Model of Evaluation 13 / 31

Lexical Scoping, 3 Lexical-Fun: ρ ⊢ � λ x . e , s � ⇓ V � ( λ x . e ) ρ , s � � �� � a closure a closure � �� � 1 , s ′ � ( λ x . e ′ 1 ) ρ ′ ρ ⊢ � e 1 , s � ⇓ V � ρ ⊢ � e 2 , s ′ � � v 2 , s ′′ � ⇓ V 1 , s ′′ � � v , s ′′′ � ρ ′ 1 [ x �→ v 2 ] ⊢ � e ′ ⇓ V Lexical-App: � ( v , s ′′′ � ρ ⊢ � ( e 1 e 2 ) , s � ⇓ V Examples/Exercises: Let ρ = { x �→ 7, y �→ 3 } . ρ ⊢ � let f = λ x . ( x + y ) in ( f 10 ) , s � ⇓ V ?? ρ ⊢ � let f = λ x . ( x + y ) in ( let y = 100 in ( f 10 )) , s � ⇓ V ?? ρ ⊢ � let f = λ n . if n ≤ 0 then 1 else n ∗ ( f ( n − 1 )) in ( f 3 ) , s � ⇓ V ?? CIS 352 The Environment Model of Evaluation 14 / 31

Puzzle 1 ρ 1 =[ a �→ 1, b �→ 2 ] e 1 = let q = λ a . ( a + b ) in let a = 5 ∗ b in let b = a ∗ b in ( q 100 ) What the value of e 1 in environment ρ 1 under call-by-value with lexical scoping? (a) dynamic scoping? (b) CIS 352 The Environment Model of Evaluation 15 / 31

Puzzle 1(a): Call-by-value, lexical scoping tag Environment Expression a �→ 1 ρ 1 : let q = . . . b �→ 2 ρ 1 =[ a �→ 1, b �→ 2 ] ↑ e 1 = let q = λ a . ( a + b ) in ρ 2 : q �→ ( λ a . ( a + b )) ρ 1 let a = . . . let a = 5 ∗ b in ↑ let b = a ∗ b in a �→ 10 let b = . . . ρ 3 : ↑ ( q 100 ) ρ 4 : b �→ 20 ( q 100 ) a �→ 100 → ρ 1 ( a + b ) ρ 5 : value of e 1 ρ 1 : 102 CIS 352 The Environment Model of Evaluation 16 / 31