Administration Homework 2 due in one week (25 th ) Gunter, - PDF document

Administration • Homework 2 due in one week (25 th ) • Gunter, Mitchell, Stoy placed on reserve in CS 611 Engineering library Advanced Programming Languages Andrew Myers Cornell University Lecture 11 Fixed Points and CPOs 18 Sep 00 CS 611 Fall '00 -- Andrew Myers, Cornell University 2 Last time Problem: while • Denotational semantics: meaning function �� while b do c � σ = if ¬ �� b � σ then σ �� e � maps language syntax e into domain else �� while b do c � ( �� c � σ) with ‘intrinsic’ meaning • Defined semantics for IMP through � b , f b � � while b do c , f w � � c , f c � meaning functions �� a � , �� a � , �� c � � � � � � � � � � � � � � � � � � � � while b do c , λσ . if f b σ then σ else f w ( f c σ) � • Meaning functions defined as syntax- � � � � � � directed translation • No finite proof tree for � � while b do c , f � � ! � � � � � a 0 , f 0 � � a 1 , f 1 � • Definitions of denotations will be total if �� a 0 +a 1 � = λσ . �� a 0 � σ + �� a 1 � σ � a 0 +a 1 , λσ . f 0 σ + f 1 σ � based on structural induction CS 611 Fall '00 -- Andrew Myers, Cornell University 3 CS 611 Fall '00 -- Andrew Myers, Cornell University 4 Denotation of while Approximations �� while b do c � is solution to • Consider sequence of approximations to denotation of while : x = Γ ( x ) with 1. λσ .if ¬ �� b � σ then σ else … (works for 0 iterations) Γ = λ f ∈Σ ⊥ →Σ ⊥ . if ¬ �� b � σ then σ else f ( � ) 2. λσ. if ¬ �� b � σ then σ else if ¬ �� b ��� c � σ then �� c � σ else … Idea: �� while b do c � σ = (0 or 1 iterations) fix ( Γ ) 3. λσ. if ¬ �� b � σ then σ else if ¬ �� b ��� c � σ then �� c � σ else = fix ( λ f ∈Σ ⊥ →Σ ⊥ . if ¬ �� b � σ then σ else f ( �� c � σ)) if ¬ �� b ��� c ��� c � σ then �� c ��� c � σ else … (0-2 iterations) • What fixed point do we want (least)? • How do we define least fixed point operator fix ? • “limit” of this sequence is denotation of while CS 611 Fall '00 -- Andrew Myers, Cornell University 5 CS 611 Fall '00 -- Andrew Myers, Cornell University 6 1

Orderings Partial orders • A partial-order is • Fixed points of denotation of while differ –a set of elements S only in case of non-termination –an relation x � y that is • We want �� while true do skip � σ = ⊥ • reflexive: x � x • Idea: define ordering on fixed points of Γ • transitive: ( x � y ∧ y � z ) � x � z such that least fixed point is the one we • anti-symmetric: ( x � y ∧ y � x ) � x = y want –two elements may be incomparable • Compare to inductive definitions • Examples ( S , � ) –ordering was ⊆ (Z, ≤ ) (Z,=)? (Z, <)? –doesn’t work here: how to order elements of ( 2 S , ⊆ ) ( 2 S , � ) Σ ⊥ →Σ ⊥ ? ( S , � ) CS 611 Fall '00 -- Andrew Myers, Cornell University 7 CS 611 Fall '00 -- Andrew Myers, Cornell University 8 Hasse diagram: 2 {a,b,c} , ⊆ LUBs and Chains ⊆ ⊆ ⊆ • Given a subset B ⊆ S , y is an upper bound y of B if ∀ x ∈ B . x � y {a,b,c} x � y • y is a least upper bound ( � B) if y � z for {b,c} x {a,c} all upper bounds z … • A chain is a sequence of elements {c} x 2 x 0 , x 1 , x 2 , … such that x 0 � x 1 � x 2 � … {a,b} x 1 � = ⊆ • For any finite chain x 0 ,…, x n , x n is LUB x 0 {b} {a} • What about infinite chains? { } CS 611 Fall '00 -- Andrew Myers, Cornell University 9 CS 611 Fall '00 -- Andrew Myers, Cornell University 10 Complete partial orders Information content • We consider one domain element to be • A complete partial order (cpo) is a partial less than another if it gives less order in which every chain has a least information upper bound • Non-termination gives less information • Examples ( S , � ) than any store ( � � x ) ( 2 S , ⊆ ) • Stores σ are incomparable unless equal ( ω ∪ { ∞ }, ≤ ) … σ 1 σ 2 σ 3 σ 4 … ([0,1], ≤ ) Σ � : cpo? ( S ,=)? ( S , � )? � • Recall: trying to find least fixed point in • cpo may have least element � : pointed Σ � →Σ � ; how to order functions ? CS 611 Fall '00 -- Andrew Myers, Cornell University 11 CS 611 Fall '00 -- Andrew Myers, Cornell University 12 2

Pointwise ordering Back to while • Functions are ordered on their results • Γ = λ f ∈Σ ⊥ →Σ ⊥ . if ¬ �� b � σ then σ else f ( �� c � σ) • Approximations to denotation of while : • Given f ∈ D → E, g ∈ D → E, E cpo, 1. Γ(⊥) = λσ. if ¬ �� b � σ then σ else ⊥ = f � D → E g � ∀ x ∈ D . f ( x ) � E g ( x ) 2. Γ(Γ(⊥)) = λσ. if ¬ �� b � σ then σ else • Example ( Z → Z ⊥ ) if ¬ �� b ��� c � σ then �� c � σ else ⊥ λ x ∈ Z . if x = 0 then ⊥ else x 3. Γ(Γ(Γ(⊥))) = λσ. if ¬ �� b � σ then σ else � if ¬ �� b ��� c � σ then �� c � σ else λ x ∈ Z . x if ¬ �� b ��� c ��� c � σ then �� c ��� c � σ else ⊥ • D → E is pointed cpo if E is pointed: • Denotation of while is � n ∈ω Γ n ( ⊥ ) ⊥ D → E = λ x ∈ D . ⊥ E • Gives �� while true do skip � = ⊥ Σ ⊥ →Σ ⊥ = λ σ ∈Σ ⊥ . ⊥ CS 611 Fall '00 -- Andrew Myers, Cornell University 13 CS 611 Fall '00 -- Andrew Myers, Cornell University 14 Fixed points • Want to show � n ∈ω Γ n (⊥) is solution to equation x = Γ ( x ) • Not true for arbitrary Γ! • Consider: Γ( x ) = if x = � then 1 0 1 else if x = 1 then � � else if x = 0 then 0 Need: monotonicity ( Γ n (⊥) not a chain!) • Consider a monotonic function: R � {- � , � } Γ( x ) = if x ≤ 0 then tan -1 (x) else 1 Need: continuity ( Γ(0) ≠ 0 ) CS 611 Fall '00 -- Andrew Myers, Cornell University 15 3