Administration • Homework 2 due on Monday • Scribes needed CS 611 • Winskel×2, Gunter available on reserve in Advanced Programming Languages Engineering library Andrew Myers Cornell University Lecture 13 Domain Constructions 22 Sep 00 CS 611 Fall '00 -- Andrew Myers, Cornell University 2 Fixed points Meta-language • Denotational semantics for IMP rely on • Idea: define restricted language for taking fixed point to define �� while � expressing mathematical functions • Fixed points occur in most language • All functions expressible in this language definitions: needed to deal with loops are continuous –control flow loops: while • Looks like a programming language (ML) –data loops: recursive functions, recursive data –not executed: just mathematical notation structures, recursive types –can talk about non-termination! • Only know how to find least fixed pts for –“evaluation” is lazy (vs. eager in ML) continuous functions f • Need easy way to ensure continuity CS 611 Fall '00 -- Andrew Myers, Cornell University 3 CS 611 Fall '00 -- Andrew Myers, Cornell University 4 “Types” for Meta-language Lifting • Meta-language contains domain • If D is a domain (for now: cpo), can “lift” declarations indicating the set of values by adding new bottom element to form meta-variables can take on, e.g. pointed cpo D � λ f ∈Σ ⊥ →Σ ⊥ . λσ∈Σ ⊥ . if ¬ �� b � σ then σ else f ( �� c � ) • cpo defined by underlying set plus complete ordering relation � • Domains will function as types for meta- language • Elements of D � are � d i � , � where d i ∈ D –but with precisely defined meaning, ordering • Ordering relation: relation, etc. � d i ��� d � i � d i � d i ⇔ D –T 1 * T 2 is not necessarily modeled by T 1 ×T 2 ! ��� d i � D � • Meta-language consists of domains and • Complete? associated operations � CS 611 Fall '00 -- Andrew Myers, Cornell University 5 CS 611 Fall '00 -- Andrew Myers, Cornell University 6 1
Discrete cpos let • Various discrete cpos: booleans ( T ), natural • Useful syntax: given d ∈ D ⊥ numbers ( ω ), integers ( Z ), … let x = d in e ( λ x ∈ D . e ) * d ≡ • Corresponding functions over discrete cpos exist: + : Z → Z , ∧ : T → T • Expresses evaluation of e that is strict in d • Often want to lift discrete cpos to take fixed • Example: �� while � points; helpful to extend fcns to pointed cpos • If f ∈ D → E, then f ⊥ ∈ D ⊥ → E ⊥ , f * ∈ D ⊥ → E are f ⊥ = λ d ∈ D ⊥ . if d= ⊥ then ⊥ else f ( d ) = fix λ f ∈Σ ⊥ →Σ ⊥ . λσ � ∈Σ ⊥ . f * = λ d ∈ D ⊥ . if d= ⊥ then ⊥ else f ( d ) (if E pointed) let σ=σ � in if ¬ �� b � σ then σ else f ( �� c � ) • 2 + ⊥ 2 = 4, 3 + ⊥ ⊥ = ⊥ , ⊥ ∧ ⊥ true = true • If f continuous, are f ⊥ , f * ? CS 611 Fall '00 -- Andrew Myers, Cornell University 7 CS 611 Fall '00 -- Andrew Myers, Cornell University 8 Unit Products • Simplest cpo: empty set ( ∅ ) • If D 1 , D 2 are domains, then D 1 × D 2 is a product domain • Next simplest: unit domain ( U ) Hasse diagram • Underlying set: pairs � d 1 , d 2 � where d i ∈ D i –single element: u . u –ordering relation: reflexive • Ordering: –complete: only directed set is { u } � d 1 , d 2 � � � d � 1 , d � 2 � iff • Used to represent computations that d 1 � d � 1 & d 2 � d � 2 terminate but do not produce a value, D 1 D 2 • Extends to n -tuples D 1 × D 2 argument for functions that need no • Operations: argument –tupling: � d 1 ,…, d m � • Also building block for other domains –projection: π i � d 1 ,…, d m � = d i CS 611 Fall '00 -- Andrew Myers, Cornell University 9 CS 611 Fall '00 -- Andrew Myers, Cornell University 10 CPO? Sums • Is product domain a cpo if D 1 , D 2 are? • Sometimes want to allow values of one kind or another: D 1 + D 2 • Any chain � d 0 , d � 0 � �� d 1 , d � 1 � �� d 2 , d � 2 � � … D 1 D 2 • Elements of domain are must have LUB in D 1 × D 2 elements of D 1 or D 2 tagged • Definition of � : d 0 � d 1 � d 2 � … is chain in with origin: { in i ( d i ) | d i ∈ D i } D 1 , d � 0 � d � 1 � d � 2 � … is chain in D 2 D 1 + D 2 • If d � ∈ D 1 , d � � ∈ D 2 are respective LUBs, • Form of in i is irrelevant (could be λ d . � i , d � ) � d � , d � � � ∈ D 1 × D 2 is LUB of chain of pairs • Preserves ordering of individual domains: • Operations continuous? in i ( d i ) � in j ( d j ) iff i=j, d i � d j • Injection function in i is continuous π i � n ∈ω x n = � π i x n = � d in • Extends naturally to multi-domain sum � � x 1 n , …, x mn � = � � d 1 n , …, � d mn � • CPO, but not pointed CS 611 Fall '00 -- Andrew Myers, Cornell University 11 CS 611 Fall '00 -- Andrew Myers, Cornell University 12 2
Sums, cont’d Continuous functions • Why tag? Distinguishes identical domains • Given cpos D , E , define D → E as domain of – T = U + U, true = in 1 ( u ), false = in 2 ( u ) continuous functions mapping D to E • Sums unpacked with case construction: (subset of E D ) case e of x 1 . e 1 | x 2 . e 2 ≡ case e of D 1 ( x 1 ). e 1 | D 2 ( x 2 ). e 2 • Pointwise ordering: f � g iff f ( d ) � g ( d ) • Given e = in i ( d i ), has value f i ( d i ) ∈ E where f i ∈ D i → E = ( λ x i ∈ D i . e i ) • Complete? • Continuous function of e if all f i continuous: � n ∈ω f n = λ d ∈ D . � n ∈ω f n ( d ) continuous? � case e n of … = case � e n of … ? � f i ( d in ) = f i ( � d in ) ( λ d ∈ D . � n ∈ω f n ( d )) ( � m ∈ω d m ) = • Also continuous function of each f i � m ∈ω ( λ d ∈ D . � n ∈ω f n ( d )) ( d m ) ? � case e of f 1 n | f 2 = case e of � f 1 n | f 2 = � f 1n ( d 1 ) CS 611 Fall '00 -- Andrew Myers, Cornell University 13 CS 611 Fall '00 -- Andrew Myers, Cornell University 14 Proof of Continuity Lemma ( λ d ∈ D . � n ∈ω f n ( d )) ( � m ∈ω d m ) = � n � m f n ( d m ) = � n f n ( d n ) = � m � n f n ( d m ) Let e nm = f n ( d m ) � m ∈ω ( λ d ∈ D . � n ∈ω f n ( d )) ( d m )? = � n ∈ω f n ( � m ∈ω d m ) n ≤ n � , m ≤ m � � e nm � e n � m � e 02 e 20 = � n ∈ω � m ∈ω f n ( d m ) e 01 e 10 n m e 00 = � n ∈ω f n ( d n ) e nm � e n � n � for n � = max( m , n ), so � n,m e nm � � n e nn = � m ∈ω � n ∈ω f n ( d m ) e nn � � m e nm , so � n e nn � � n � m e nm , � m � n e nm = � m ∈ω ( λ d ∈ D . � n ∈ω f n ( d )) ( d m ) � m e nm � � n,m e nm , so � n � m e nm � � n,m e nm CS 611 Fall '00 -- Andrew Myers, Cornell University 15 CS 611 Fall '00 -- Andrew Myers, Cornell University 16 Operations on functions Meta-Language • apply ∈ ( D → E ) × D → E = λ p .( π 1 p )( π 2 p ) • Have defined various constructs that we can use to define continuous functions • curry ∈ (( D × E ) → F ) → ( D → E → F ) • Constructs are a syntax for a meta-language in = λ f ∈ ( D × E ) → F . λ d ∈ D. λ e ∈ E.f � d,e � which only continuous functions can be defined • compose = · � · ∈ ( D → E )×( E → F ) → ( D → F ) • How do we know when expression λ x . e is = λ � f,g � . λ d ∈ D . f ( g ( d )) continuous? • Idea: use structural induction on form of e so • fix ∈ ( D → D ) → D ( D pointed) every syntacally valid e can be abstracted over any variable to produce continuous function = λ g ∈ D → D . � n g n ( � ) • Problem: structural induction � need to = � n λ g ∈ D → D . g n ( � ) (LUB of cont.fcns!) consider open terms e CS 611 Fall '00 -- Andrew Myers, Cornell University 17 CS 611 Fall '00 -- Andrew Myers, Cornell University 18 3
Continuity in variables • Idea: consider a meta-language expression e to be implicitly function of its free variables • e is continuous in variable x if λ x . e is continuous for arbitrary values of other (non- x ) free variables in e • e is continuous in variables not free in e • structural induction: for each syntactic form, show that term is continuous in variables assuming sub-terms are CS 611 Fall '00 -- Andrew Myers, Cornell University 19 4
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