Semantics of Higher-Order Functional Programming

Transcription

1 Semantics of Higher-Order Functional Programming Petros Barbagiannis µ λ July 14, 2014 Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

2 Introduction Higher-order functions are functions that either accept other functions as arguments or they return functions (ie, functions are first-class objects) Higher-order functions have been used widely in: Mathematics, (,, etc) Computability Theory (Kleene s T -Predicate, Church s λ-calculus, etc) Programming Languages (Lisp, Haskell, C, etc) Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

3 A Declarative Language Types τ ::= bool nat θ ::= val[τ] θ θ (θ) data types phrase types A type of the form θ θ is a functional type A functional type θ θ is called higher-order if θ is a functional type or θ is higher-order Example: (val[nat] val[bool]) val[nat] Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

4 Syntax Rules Bracketing X : θ (X) : θ Zero 0 : val[nat] Succ N : val[nat] succ N : val[nat] Cond B : val[bool] X 0 : θ X 1 : θ if B then X 0 else X 1 : θ Truth true : val[bool] Negation B : val[bool] notb : val[bool] Conjunction B 0 : val[bool] B 1 : val[bool] B 0 and B 1 : val[bool] Application P : θ θ Q : θ P Q : θ Local Definition [ι : θ] P : θ Q : θ let ι be P in Q : θ Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

5 Semantics Definition An environment u is a function which assigns meanings (values) to the identifiers in a phrase A function π is a phrase-type assignment which maps identifiers to types [[val[τ]]] = [[τ]] where [[τ]] is the set of values for data type τ [[θ θ ]] = [[θ]] [[θ ]] [[(θ)]] = [[θ]] Example: [[π(ι)]] is the set of values ι takes on Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

6 Semantic Equations For an identifier ι, [[ι]](u) = u(ι) { true, if [[B 0 ]]u = true and [[B 1 ]]u = true [[B 0 and B 1 ]]u = false, otherwise [[if B then X 0 else X 1 ]](u) = { [[X 0 ]](u) [[X 1 ]](u) if [[B]]u = true if [[B]]u = false [[P Q]]u = ([[P]]u)([[Q]]u) Note: Since [[P]]u is a function and [[Q]]u dom[[p]]u we can rewrite this equation as [[P]]u([[Q]]u) [[let ι be P in Q]]u = [[Q]](u ι [[P]]u) Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

7 Function Definition Function Definition [ι 0 : θ 0 ] [ι : θ] P : θ Q : θ let ι(ι 0 : θ) = P in Q : θ Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

8 Function Definition Function Definition [ι 0 : θ 0 ] [ι : θ] P : θ Q : θ let ι(ι 0 : θ) = P in Q : θ The above construct has the following semantic equation: [[let ι(ι 0 : θ 0 ) = P in Q]]u = [[Q]](u ι f ) where f : [[θ]] [[θ ]] is the function defined by for all a θ 0 f (a) = [[P]](u ι 0 a) Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

9 Lambda Expressions Abstraction Example: [ι : θ] P : θ λι : θp : θ θ (λn : val[nat]n + m) (3) The semantic equation for the lambda expression is: [[λι : θp]](u) = f [[θ]] [[θ ]] where f (a) = [[P]](u ι a) for all a [[θ]] Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

10 An Example Prelude> let myfunction f = f 10 in 5 * myfunction add2 [[let myfunction f = f 10 in 5 myfunction add2]]u = [[5 myfunction add2]](u myfunction [[f 10]](u f add2)) = [[5 myfunction add2]](u myfunction [[add2 10]]u) = [[5 12]]u = 5 12 = 60 Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

11 Computational Domains Definition (Definitions) A pair (D, ) consisting of a set D and a partial order is called partially-ordered set (poset) A poset D is complete iff for every chain d D ω, the least upper bound i ω d i exists in D A domain is any poset that is complete If A and D are domains, then the set A D is a domain where (a, d) A D (a, d ) if a A a and d D d If D is a domain, then so is D, ie, the set with a least element added to D if A is a set and D is a domain, then A D is a domain where f A D f if f (a) D f (a) for every a A Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

12 Continuous Functions Definition Let A and D be domains A function f : A D is called: monotonic if f (a) D f (a ) when a A a continuous if, for every chain d D ω, f ( d i ) = f (d i ) i ω i ω Definition If A and D are domains with least elements A and D, a function f : A D is called strict if f ( A ) = D Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

13 Continuous Functions Definition Let D be a domain d D ω ω is called a double chain if d i0 d i1 d i2 for every i ω and d 0j d 1j d 2j for every j ω Lemma If d is a double chain in D, the following limits are well-defined and equivalent: i ω j ω j ω i ω d ij d ij Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

14 Continuous Functions Theorem Let f be a chain of continuous functions from A to D Then i ω f i is continuous Proof Consider d A f i ( d j ) = f i (d j ) = f i (d j ) = ( f i )(d j ) i ω j ω i ω j ω j ω i ω j ω i ω Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

15 Fixed Points Theorem Let D be a domain with a least element and f : D D be a continuous function Then i ω f i ( ) is the least fixed point of f Proof f ( f i ( )) i ω = f (f i ( )) i ω = f i+1 ( ) i ω = f i ( ) i ω Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

16 Domain-Theoretic Semantics Recursion [ι : θ] [ι : θ] P : θ Q : θ letrec ι : θ be P in Q : θ Example: letrec double : val[nat] val[nat] be λn : val[nat]if n = 0 then 0 else 2 + double(n 1) in The meaning defined for ι by the letrec construct is the solution to the following equation: p = [[P]](u ι p) Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

17 Domain-Theoretic Semantics We redefine sets to be domains as follows: [[val[τ]]] is the flat domain [[τ]] [[θ θ ]] = [[θ]] [[θ ]] [[u]] is the set of all environments We must also redefine valuation functions so that all phrases P are mapped to continuous functions For all primitive operations (not, =, etc) we can take their strict extension Proposition For all phrases P, [[P]] is continuous and when π(p) = θ θ, then [[P]](u) is continuous Proof By structural induction on phrases Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

18 Recursion Let u [[u]] and define a function f : θ θ as follows: f (p) = [[P]](u i p) f is continuous if P is continuous [[letrec ι : θ be P in Q]] = [[Q]](u ι f i ( )) Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18

19 References R D Tennent, Semantics of Programming Languages, Prentice-Hall, 1991 C A Gunter, P D Mosses, D S Scott, Semantic Domains and Denotational Semantics, 1989 Petros Barbagiannis Semantics of Higher-Order Functional Programming July 14, / 18