Topology in Denotational Semantics

Transcription

1 Journées Topologie et Informatique Thomas Ehrhard Preuves, Programmes et Systèmes (PPS) Université Paris Diderot - Paris 7 and CNRS March 21, 2013

2 What is denotational semantics? The usual stateful approach to programming Most current programming languages are based on states. A program is a sequence of instructions (including while or goto instructions allowing for loops). An instruction induces a change of the state of the machine, and its behaviour depends on the state. Abstract models behind this approach: Turing machines, automata...

3 What is denotational semantics? The functional approach There is another approach, more in the spirit of recursion theory where programs are functions taking values to values. It is based on rewriting, does not require a machine (automaton) and has no notion of state.

4 What is denotational semantics? Pros conceptually simpler allows to write higher order programs (taking programs as arguments) direct connection with mathematical proof systems (Curry Howard) stateful extensions are possible, in an elegant way (monads). Cons computational complexity harder to evaluate some algorithms are more naturally expressed in a stateful model.

5 What is denotational semantics? A functional language: PCF The language Programming Computational Functions (PCF) has been introduce by Plotkin in It is Turing complete. One ground type ι of natural numbers. If σ and τ are types, then σ τ is a type. Example of types: ι, ι ι, ι (ι ι), (ι ι) ι. Any partial recursive function from N to N can be represented as a PCF term of type ι ι.

6 What is denotational semantics? Syntax and typing rules for PCF The language uses variables x,y,... Typing context: finite partial function Γ from variables to types. Typing judgement Γ M : σ means that the term M is of type σ when its variables have types specified by Γ. There is a set of deduction rules for deriving such judgements.

7 What is denotational semantics? For each natural number N, there is a constant which is a term of PCF: Each variable is a term Γ n : ι Γ,x : σ x : σ

8 What is denotational semantics? Successor, predecessor and conditional constructions Γ M : ι Γ succ(m) : ι Γ M : ι Γ pred(m) : ι Γ M : ι Γ P : σ Γ Q : σ Γ if(m,p,q) : σ

9 What is denotational semantics? PCF: the lambda-calculus constructs To define a function: Γ,x : σ M : τ Γ λx σ M : σ τ and to apply a function to an argument: Γ M : σ τ Γ N : σ Γ (M)N : τ

10 What is denotational semantics? PCF: the fixpoint operator The last construct is essential for Turing-completeness. Γ M : σ σ Γ fix(m) : σ

11 What is denotational semantics? PCF: operational semantics Until now, we have said nothing about the meaning of these terms, ie. on their computational behaviour. This operational semantics is specified by a rewriting system, defined by the following rules. succ(n) n+1 pred(0) 0 pred(n+1) n M M succ(m) succ(m ) M M pred(m) pred(m )

12 What is denotational semantics? The rewriting rules for conditionals: if(0,p,q) P if(n+1,p,q) Q M M if(m,p,q) if(m,p,q)

13 What is denotational semantics? The rewriting rules for the lambda-calculus fragment The main rule is β-reduction: (λx σ M)N M[N/x]

14 What is denotational semantics? Then we have to specify how to reduce in an abstraction or an application. M M λx σ M λx σ M P P and P is not of the shape λx σ M (P)N (P )N This last rule specifies that we have a call-by-name operational semantics.

15 What is denotational semantics? The last rule concerns fixpoints: fix(m) (M)fix(M) Fact For any term M, there is at most one rule which can apply for reducing it to a term M : the rewriting relation is a deterministic evaluation strategy on PCF terms. Easy to see by induction on M. The strategy can be implemented by means of an abstract machine (a stateful device).

16 What is denotational semantics? Subject reduction Reduction is compatible with typing. Fact If Γ M : σ and M M then Γ M : σ.

17 What is denotational semantics? Programming in PCF: an example We want to represent in PCF a total function f : N N N which computes the inf of two natural numbers: f(n,0) = n, f(0,m) = m and f(n+1,m +1) = 1+f(n,m). One defines a closed term F of type σ σ where σ = (ι (ι ι)) is the expected type for our function inf: F = λf σ λx ι λy ι if(x,y,if(y,x,((f)pred(x))pred(y))) and then inf = fix(f) is a closed term of type σ which has the expected behaviour.

18 What is denotational semantics? Computations at ground type ι A term M is normal if M M for no M (M cannot be reduced). And M is closed if it has no free variables (the notion of free variable is the same as in first order logic, considering λ as a binder). Fact If M is closed and normal and M : ι, then M = n for some n N.

19 What is denotational semantics? Turing completeness of PCF So if M : ι (M is closed), there are two possibilities: either M n (where n N is uniquely defined), notation M n or there is a uniquely defined sequence of terms (M i ) i N such that M 0 = M and M i M i+1 for each i, notation M.

20 What is denotational semantics? Theorem Let f : N N be a partial recursive function. There is a term M with M : ι ι such that, for any n,m N if f(n) = m then (M)n m and if f(n) is undefined then (M)n. The converse is obviously true (Church Thesis).

21 What is denotational semantics? Observational equivalence Let M and M be closed with M : σ and M : σ. We can say that M and M are equivalent if they produce the same results when used in the same environment. Definition M M if, for any closed term C with C : σ ι one has n N (C)M n (C)M n. This is a nice definition, but it is usually quite difficult to prove that two terms are equivalent. Denotational semantics is a convenient way of proving such equivalences.

22 Continuity and computability The Myhill Shepherdson Theorem Let P be the set of partial recursive functions N N. An effective operation is a partial function Φ : P N such that there is e N such that, for any n N The MS Theorem implies: Theorem Let Φ be an effective operation. Φ(ϕ n ) = ϕ e (n). If f g P, then Φ(f) = n Φ(g) = n. If Φ(f) = n then there is a finite function f 0 f such that Φ(f 0 ) = n.

23 Continuity and computability Let P be the set of all partial functions N N, so that P P. Any effective operation Φ : P N can be extended to a partial function Φ : P N setting n if there is f 0 f, f 0 finite Φ (f) = such that Φ(f 0 ) = n undefined otherwise

24 Continuity and computability Complete partial orders (cpos) A subset D of a partially ordered set (X, ) is directed if D is non empty and u 1,u 2 D u D u 1 u and u 2 u. Intuition: natural generalization of increasing sequences. NB: a finite directed set has a maximal element. (X, ) is a cpo if all directed subsets D of X have a least upper bound, denoted as D. (P, ) is a cpo, but (P, ) is not.

25 Continuity and computability Add to N a new element corresponding to the undefined computation: N = N { } endowed with the following order relation: u v if u = or u = v. N is (trivially) a cpo: directed sets in N have at most 2 elements! Then Φ is a total function P N which satisfies f g Φ (f) Φ (g) and if D P is directed then Φ ( D) = f D Φ (f).

26 Continuity and computability This observation is the starting point of denotational semantics (Dana Scott). Towards a Mathematical Semantics of Programming Languages, Dana Scott and Christopher Strachey, Even if they considered stateful programming languages, this paper settled the bases of the denotational semantics of functional languages. Stateful languages can be translated in functional languages (this syntactic transformation is also known as denotational semantics).

27 Continuity and computability Idea: interpret PCF types as cpos PCF terms as monotone and directed lub preserving functions. This does not work, because the category of ω-algebraic cpos and monotone and directed lubs preserving functions is not cartesian closed. However there are suitable subcategories which are cartesian closed. For instance, the category of Scott domains.

28 Continuity and computability A Scott domain is a cpo (X, ) which has a least element where each bounded subset has a lub where any element is the lub of its compact lower bounds (algebraicity) and which has at most countably many compact elements (ω-algebraicity). u X compact means that if D X is directed and u D, then v D u v. In some sense, a compact element is finite.

29 Continuity and computability Scott Topology Any cpo (X, ) has a natural topology for which U X is open if: u,v X (u v and u U) v U and for any D X directed, if D U then D U. Equivalently F X is closed if u,v X (u v and v F) u F and for any D X directed, if D F then D F.

30 Continuity and computability Scott domains are topological spaces Fact If (X, ) is a Scott domain, the Scott topology on X is T 0 and is the specialization order of the Scott topology. T 0 is the weakest form of separation: if u v there is an open set such that u U and v / U, or such that v U and u / U. The specialization order of a T 0 space is defined by: u v if any open set which contains u also contains v.

31 Continuity and computability Fact If (X, ) and (Y, ) are Scott domains and f : X Y, the two following conditions are equivalent f is continuous for the Scott topologies of X and Y f is monotone and preserves the lubs of all directed sets. This shows that the category of Scott domains and monotone and directed lubs preserving maps is a full subcategory of the category of topological spaces and continuous maps.

32 Continuity and computability Fact The category of Scott domains and Scott continuous functions is cartesian closed. If (X, ) and (Y, ) are Scott domains, then X Y equiped with the product order is a Scott domain and the projections are continuous the set X Y of Scott continuous function X Y, equiped with the pointwise order (f g if a X f(a) g(a)) is a Scott domain the evaluation map (X Y) X Y is Scott continuous.

33 Continuity and computability This fact is surprising because the category of topological spaces and continuous functions is not cartesian closed. It is probably related to the following very specific feature of Scott topology. Fact If (X, ), (Y, ) and (Z, ) are Scott domains, a function f : X Y Z is continuous as soon as it is separately continuous. NB: the Scott topology of X Y is the product topology of the Scott topologies of X and Y.

34 Continuity and computability Any Scott continuous function f : X X has a least fixpoint, namely n N f n ( ). (Indeed f( ) f 2 ( ) ). By cartesian closeness, the function (X X) X f n N f n ( ) is itself Scott continuous. This provides a direct interpretation of the fix(m) construction of PCF.

35 Continuity and computability With each type σ of PCF we can associate a Scott domain [σ]: [ι] = N [σ τ] = [σ] [τ]. and with any term M such that x 1 : σ 1,...,x n : σ n M : τ we can associate a continuous function [M] : [σ 1 ] [σ n ] [τ]. Theorem Soundness: if M M then [M] = [M ]. This interpretation is an invariant of the reduction.

36 Continuity and computability So in particular, if M : ι and M n, then [M] = n N = [ι]. The converse is also true. Theorem Adequacy: if M : ι and [M] = n, then M n. The proof uses a very powerful tool in the semantics of lambda-calculus: logical relations, introduced by Tait (reducibility).

37 Continuity and computability A logical relation The following idea is due to Plotkin. By induction on types one defines for each type σ a relation R σ between elements of [σ] and closed terms of type σ. n R ι M if n M n. f R σ τ M if, for all u [σ] and P such that P : σ, if u R σ P then f(u) R τ (M)P. By induction on terms one proves that if M : σ, then [M] R σ M. (One needs actually to prove a more complicated statement involving non closed terms). Adequacy follows. Main lemma: {u [σ] u R σ M} is Scott closed.

38 Continuity and computability A consequence of adequacy Theorem If M and M are closed of type σ, then [M] = [M ] M M. Indeed, let C : σ ι. If (C)M n then [(C)M] = n by soundness. But [(C)M] = [C]([M]) = [C]([M ]) = [(C)M ]. So [(C)M ] = n and hence (C)M n by adequacy.

39 Beyond Scott semantics Full abstraction is the converse property: M M [M] = [M ]. This property does not hold in Scott domains but is highly desirable because it means that the model provide an abstract and complete way of reasoning on programs.

40 Beyond Scott semantics The reason is that it contains non deterministic morphisms such as por : N N N { 0 if n = 0 or m = 0 (n,m) otherwise This function is Scott continuous.

41 Beyond Scott semantics Let Ω = fix(λx ι x), it is the everlooping program of type N (Ω (λx ι x)ω Ω...). One has [Ω] =. Consider λh ι (ι ι) Ω, also denoted as Ω, and port = λh ι (ι ι) if(((h)0)ω,if(((h)ω)0,0,ω),ω). then we have [Ω] [port] since [Ω](por) = and [port](por) = 0. But port Ω as we shall see. port: por taster.

42 Beyond Scott semantics The trouble with por: por(, ) = por(0,0) = 0 so that por needs its arguments to produce the result 0 from (0,0) but since por(0, ) = 0 and por(,0) = 0 it is impossible to say which are the arguments really used by the function to compute the result.

43 Beyond Scott semantics Say that f : X N is stable if f is Scott continuous if f(u) = n there is u 0 u such that f(u 0 ) = n and for any v u, if f(v) = n then u 0 v. u 0 is the part of u that f has used to produce the result n. por is not stable: take u = (0,0), there is no u 0 with the required property.

44 Beyond Scott semantics Coherence spaces One can build nice categories with stable functions as morphisms. However there is no topology for which continuity would be equivalent to stability. The simplest of these categories is that of coherence spaces and stable functions. A coherence space is a structure X = ( X, X) where X is a countable set and X is a binary reflexive and symmetric relation on X.

45 Beyond Scott semantics A clique of X is a subset u of X such that a,a x a X a. Ordered by, the cliques of X form a (very well-behaved) Scott domain. Let Cl(X) be this Scott domain. A function f : Cl(X) Cl(Y) is stable if it is Scott continuous and moreover u 1,u 2 Cl(X) u 1 u 2 Cl(X) f(u 1 u 2 ) = f(u 1 ) f(u 2 ).

46 Beyond Scott semantics Fact A function f : Cl(X) Cl(Y) is stable iff for all u Cl(X) and all b f(u) there exists u 0 u finite such that b f(u 0 ) and, for all u 1 u, if b f(u 1 ) then u 0 u 1. Theorem The category of coherence spaces and stable functions is cartesian closed and is a model of PCF. ι interpreted by the coherence space N such that N = N and n N m if n = m. Cl(N) is clearly isomorphic to N.

47 Beyond Scott semantics Moreover, this model satisfies the adequacy property. In this model [port] = = [Ω]. This shows that port Ω. The stable model is not fully abstract (there is a stable function N 3 N which is not definable in PCF and one can define a taster for this function).

48 Linear logic as an underlying structure Linear decomposition of stable functions A function f : Cl(X) Cl(Y) is linear if it is stable and commutes with all unions: u 1,u 2 Cl(X) u 1 u 2 Cl(X) f(u 1 u 2 ) = f(u 1 ) f(u 2 ).

49 Linear logic as an underlying structure Fact There is a canonical bijection between linear functions Cl(X) Cl(Y) and cliques of the coherence space X Y where X Y = X Y and (a,b) X Y (a,b ) if a X a b Y b b Y b a X a where a X a if a = a, or a X a does not hold.

50 Linear logic as an underlying structure If t Cl(X Y), the associated linear map f : Cl(X) Cl(Y) is defined by f(u) = {b Y a u and (a,b) t}. This suggests to define X by X = X and a 1 X a 2 if a 1 X a 2 so that X = X. Then X (X ) where is the one point coherence space. If one think of as the field (1-dimensional vector space), then X is the analogue of a linear dual of X. X Y by X Y = X Y and (a 1,b 1 ) X Y (a 2,b 2 ) if a 1 X a 2 and b 1 Y b 2. Then X Y = (X Y ).

51 Linear logic as an underlying structure We have a structure completely similar to that of the category of finite dimensional vector space and linear map, but less degenerated because here, it is not true that (X Y) X Y. Coherence spaces and stable functions form a cartesian closed category: we have a coherence space X Y with a canonical bijection between Cl(X Y) and stable functions Cl(X) Cl(Y).

52 Linear logic as an underlying structure Decomposition of implication Girard s observation: X Y =!X Y where!x is the set of all finite (multi)cliques of X and u 1!X u 2 if u 1 u 2 Cl(X). This led him to the introduction of a refinement of intuitionistic and classical logic: linear logic.

53 A vector space model of LL Since LL has a strong intuitive background of linear algebra, it was tempting to define models where formulae are interpreted as vector spaces. However, the construction!x will create infinite dimensional vector spaces, and therefore topology will come in naturally. It will be quite different from Scott topology.

54 A vector space model of LL A starting point: observe that if u Cl(X) and u Cl(X ) then u u has at most one element. Interpret u and u as {0,1}-valued vectors: the sum Σ a X u a u a has at most one non-zero term. Let us relax this condition by requiring u u to be finite so that the corresponding sum will only have a finite number of non-zero terms.

55 A vector space model of LL A typical LL definition Given a set I and a subset F or P(I), set Observe that F G G F F F. hence F = F. F = {u I u F u u finite}. This means that any set G which has been defined as F for some F will satisfy G = G.

56 A vector space model of LL A finiteness space is a structure X = ( X,F(X)) where X is a countable set and F(X) P( X ) satisfies F(X) = F(X). Similarity with coherence spaces: F(X) looks like a set of cliques. For instance: u 1 u 2 F(X) u 1 F(X). But there are big differences: u 1,u 2 F(X) u 1 u 2 F(X), whereas the union of two cliques is not a clique in general any finite subset of X belongs to F(X) F(X) is not closed under directed unions, whereas cliques are.

57 A vector space model of LL Let k be a field (no topological assumptions). Let k X be the set of all functions x : X k such that supp(x) = {a X x(a) 0} is an element of F(X). Since supp(x 1 +x 2 ) supp(x 1 ) supp(x 2 ) and supp(αx) supp(x) (for α k), Fact k X is a k-vector space.

58 A vector space model of LL Topology Let u F(X). Let V(u ) = {x k X supp(x) u = }. This is a linear subspace of k X. Observe that V(u 1 u 2 ) = V(u 1 ) V(u 2 ). Say that U k X is open if x U u F(X) x +V(u ) U. This defines a topology on k X which is T 2.

59 A vector space model of LL Fact For each u F(X), the space V(u ) is both open and closed. So k X is zero-dimensional (the x +V(u ) are a basis of clopen sets). Fact This topology is compatible with the linear structure (addition and scalar multiplication are continuous). We give k the discrete topology. Remark: when X is finite, k X = k X is finite dimensional and its topology is discrete.

60 A vector space model of LL Linearly topologized vector spaces The corresponding general notion of topological vector space has been introduced by S. Lefschetz in 1942: linearly topologized vector spaces (ltvs). They are a purely algebraic analogue of locally convex topological vector spaces (based on the topology of R or C).

61 A vector space model of LL Cauchy completeness A net is a family (x i ) i I of elements of k X where I is a directed ordered set. It converges to x k X if for all neighborhood U of 0 there exists i I such that x j x U for all j i. It is Cauchy if for all neighborhood U of 0 there exists i I such that x j x j U for all j,j i.

62 A vector space model of LL Theorem Any Cauchy net in k X converges: k X is Cauchy complete. The proof uses the fact that F(X) = F(X) in an essential way. Basic examples: N = (N,P fin (N)); indeed P fin (N) = P(N) and P(N) = P fin (N). Then k N = k (N), with the discrete topology. And k N = k N with the product topology (remember that k is discrete).

63 A vector space model of LL Matrix of a linear map Given a X define e a k X by (e a ) a = Let f : k X k Y be linear and continuous. { 1 if a = a 0 otherwise. Define the matrix of f as M(f) k X Y such that M(f) a,b = f(e a ) b.

64 A vector space model of LL Define X Y by X Y = X Y and F(X Y) = {u v u F(X) and v F(Y) } Theorem The operation f M(f) is a linear isomorphism between the space of linear and continuous maps k X k Y and k X Y.

65 A vector space model of LL Linear boundedness A linear subspace B of k X is linearly bounded if, for any open (and therefore closed) linear subspace U of k X, the image of B by the canonical projection k X k X /U is finite dimensional. Fact B is linearly bounded iff x B supp(x) F(X).

66 A vector space model of LL Theorem Let U be a set of linear and continuous maps k X k Y. The two following conditions are equivalent: {M(f) f U} is open in k X Y for each f U there exists a linearly bounded subspace B of k X and an open subspace V of k Y such that, for any linear and continuous g : k X k Y, if g(b) V then f +g U. This is the bounded-open topology.

67 A vector space model of LL Hypocontinuous bilinear functions A bilinear function k X 1 k X 2 k Y is hypocontinuous if: for any linearly bounded subspace B 1 of k X 1 and any neighborhood V of 0 in k Y, there is a neighborhood U 2 of 0 in k X 2 such that f(b 1 U 2 ) V and symmetrically. Warning: such a map is not necessarily continuous.

68 A vector space model of LL Tensor product X 1 X 2 is defined as X 1 X 2 = X 1 X 2 and F(X 1 X 2 ) = {u 1 u 2 u i F(X i ) for i = 1,2} Given x i k X i for i = 1,2, x 1 x 2 defined by (x 1 x 2 ) (a1,a 2 ) = (x 1 ) a1 (x 2 ) a2 belongs to k X 1 X 2. Theorem The map (x 1,x 2 ) x 1 x 2 is bilinear and hypocontinuous and is universal: for any Y and any hypocontinuous bilinear f : k X 1 k X 2 k Y, there is exactly one linear and continuous h : k X 1 X 2 k Y such that f(x 1,x 2 ) = h(x 1 x 2 ).

69 A vector space model of LL Remark: tensor and co-tensor do not coincide, (X Y) and X Y are not isomorphic in general.

70 A vector space model of LL Polynomial and entire functions A function f : k X k Y is polynomial if there are hypocontinuous multilinear maps f 0,f 1,...,f k with f i : k X i k Y such that f(x) = k f i (x,...,x) }{{} i i=0 They form a k-vector space that we can complete under the bounded-open topology to get a space (an ltvs actually) of entire functions k X k Y.

71 A vector space model of LL Theorem Assume k infinite. The ltvs of entire functions k X k Y equiped with the bounded-open topology is linearly homeomorphic to k (!X) Y where!x = M fin ( X ) and F(!X) = {M fin (u) u F(X)}.

72 A vector space model of LL Each p M fin ( X ) is a muti-exponent: given x k X, we define x p = a X xp(a) a k. Given M k (!X) Y we define an entire map k X k Y by for each b Y. f(x) b = p M fin ( X ) M p,b x p For each b, this sum is finite, but its number of non-zero terms is not bounded (when b varies).

73 A vector space model of LL Differentiation of entire functions An entire function f : k X k Y is not continuous in general. It can nevertheless be differentiated, one can define f : k X k (X Y) as an entire function. Iterating we have a nth derivative f (n) : k X k X n Y. The Taylor formula holds: f(x +y) = 1 n! f (n) (x)(y,...,y). }{{} n=0 This sum converges in the topology of k Y. n

74 A vector space model of LL The! construction introduces complicated topologies Theorem k X is metrizable iff there is a countable subset F of F(X) such that u F(X) v F u v. Fact Let 1 be the finiteness space with 1 = { }, so that k 1 = k. The space k (!(!1 1) 1) is not metrizable.

75 A vector space model of LL k!1 1 is the space of polynomials, with the discrete topology (isomorphic to k (N) ). k!(!1 1) 1 contains infinite objects such as the operations which maps a polynomial P to P(P(0)): n α i X i i=0 n α i α i 0 i=0

76 A vector space model of LL Conclusion The category of entire functions is cartesian closed and has some basic differential operations, providing solid foundations to calculi with resources introduced earlier (Boudol), and recasting them in a wider LL setting. Contrarily to Scott or stable semantics, it does not accomodate general fixpoints. It has suggested to introduce a new way of approximating lambda-terms by Taylor expansion: Tasson s talk this afternoon. One of the main applications of denotational semantics is to suggest sound extensions of syntaxes.