Recursive types to D and beyond

Transcription

1 Recursive types to D and beyond Felice Cardone Dipartimento di Informatica, Sistemistica e Comunicazione, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, I Milano, Italy Abstract The paper presents an analysis of the interpretation of recursive types as logical relations in the framework of iterative algebraic theories. This allows a smoother presentation of the underlying technique, and easier proofs of properties of the interpretation. Key words: Recursive types, Logical relations, Models of the lambda-calculus, Iterative algebraic theories 1 Introduction In this paper I present a rational reconstruction of the interpretation of simple recursive types introduced by Coppo in [1] within the framework of the iterative algebraic theories of Elgot and his school [2], regarded as a canonical theory of recursive structures (see e.g. [3]). The interpretation of [1] shares with the analysis of recursive types contained in [4] the same semantical structure, namely the collection of non-empty, Scottclosed subsets (viz., ideals) of a domain that solves a suitable equation (whose shape depends on the constructors of the underlying untyped language). Ideals are interesting for several reasons. First, constructions on ideals can be generalized easily to relational structures, in particular to partial equivalence relations. Second, because these formalize directly the intuition of types as sets of values, they model subtyping rules, [5], even in the presence of polymorphic type constructors (and, of course, type recursion; see for example [6]). address: cardone@disco.unimib.it (Felice Cardone). Preprint submitted to Theoretical Computer Science 27 October 2006

2 However, there are differences as to the specific strategy followed in the two approaches in finding solutions of recursive type equations. The strategy of [4] exploits a structure of complete metric space on the collection of ideals, and then solves recursive type equations by appealing to Banach s theorem after observing that the allowed patterns of recursion define contractive mappings. On the other hand, the path followed in [1] amounts essentially to performing the systematic construction, for each type expression, of a logical relation over the underlying domain, by exploiting both the syntactic details of the type system and the fine structure of the domain in a way that is reminiscent of early applications of logical relations in [7 9]. This involves a considerable amount of details that tend to obscure the overall intuition on which the technique is based. The algebraic perspective that I describe in this paper reconciles the approach based on metric spaces with the latter construction, by subsuming both under a common framework. It also shows how natural it is to find solutions of recursive type equations by interpreting directly the infinite regular trees obtained by pushing into infinity occurrences of recursively defined type symbols (and, with hindsight, it offers further support to some choices made in [10]). The story that I will tell here (a similar one is in [11]) introduces in Section 2 the presentation of recursive types by means of equations, then highlights in Section 3 the basic ingredients used in [1] for the interpretation of recursive simple types ( 3.1, 3.2) and shows how types can be interpreted in a wide class of iterative algebraic theories ( 3.3). The final Section 4 outlines a few easy or well-known applications of the relational interpretation of types, suggesting some directions for futher research. 2 Defining types recursively Recursive types are commonly defined by means of a recursion operator added to other type constructors of some extended system of λ-terms; a type of the form µx.a is thought of as a notation for the solution of the type equation X = A [4,10,12,13]. I shall use here another presentation, which allows the simultaneous definition of finite sequences of recursive type expressions by means of systems of equations, called simultaneous recursions following [14]. This is perhaps closer to the way actual programming languages introduce recursive datatype declarations, and is in fact the original presentation used in [1]. However, the two presentations are equivalent, and indeed both can naturally be compiled into the set of combinators built in the semantical framework used in this paper, Elgot s iterative algebraic theories. Definition 1 (Types and simultaneous recursions) (a) Let ξ = ξ 1,..., ξ n, for some n 0, be a sequence of distinct indeterminates, and X = X 1,..., X p 2

3 a sequence of distinct type variables (the parameters). These will be called collectively the atoms. The set T( ξ; X) of simple types over ξ, X is defined inductively by the productions: T( ξ; X) T ::= ξ i for each i = 1... n X j for each j = 1... p T T (b) A simultaneous recursion (over indeterminates ξ and parameters X) is a finite system of equations of the form E = {ξ 1 = T 1,..., ξ n = T n n 0} where ξ 1,..., ξ n are distinct indeterminates and T i T( ξ; X) is not an atom, for all i = 1,..., n. In general, recursive types are considered modulo a type congruence, viz. an equivalence relation. = E over T( ξ; X) such that In particular, we have: ξ i = T i E = ξ i. =E T i (. = E -unfold/fold) A. = E B, A. =E B = A A. =E B B (. = E - ) Definition 2 (Weak equivalence from simultaneous recursions) If E is a simultaneous recursion, types A, B T( ξ; X) are weakly equivalent modulo E, written A E B, if they are identified modulo the smallest type congruence. Another interesting type congruence arises when recursive types are viewed as notations for the regular trees obtained by pushing into infinity occurrences of recursively defined symbols, using (. = E -unfold/fold). This stronger notion of equivalence will be defined later (Definition 22), after setting up the semantical framework for the interpretation of recursive types. Both congruences are studied in [10,15]. Systems for building recursively typed λ-terms can now be introduced in a straightforward way:. Definition 3 (Recursively typed terms) Let = E, for a simultaneous recursion E, be any type congruence over T( ξ; X). For any A T( ξ; X), the sets Λ E (A) of λ-terms of type A are defined inductively by the following rules, assuming that for each type A there is a denumerable set of variables of type A, ranged over by x A, y A, z A,...: x A Λ E (A) M Λ E (A B), N Λ E (A) = (MN) Λ E (B) M Λ E (B) = (λx A.M) Λ E (A B) M Λ E (A), A =. E B = M Λ E (B) (var) (app) (abs) (equiv) 3

4 As an example, if X is any type variable and E = {ξ = ξ X}, where X is a type variable, the following is the construction tree of the term (λx ξ.xx) Λ E (ξ X): x ξ Λ E (ξ) ξ =. E ξ X (equiv) x Λ E (ξ X) x ξ Λ E (ξ) xx Λ E (X) λx ξ.xx Λ E (ξ X) (abs) (app) Then the construction tree of (λx ξ.xx)(λx ξ.xx) Λ E (X) is:. λx ξ.xx Λ E (ξ X). λx ξ.xx Λ E (ξ X) (λx ξ.xx)(λx ξ.xx) Λ E (X) λx ξ.xx Λ E (ξ) ξ. = E ξ X (app) (equiv) Observe that any pure λ-term M, possibly containing free variables x 1,..., x n, has a reflexively typed version M Λ E (A), where A E A A, assuming that each x i is of type A. 3 Interpreting recursively defined types 3.1 Approximating λ-models Let D be a reflexive cpo [16, 5.4], namely a cpo D with continuous functions F : D [D D] and G : [D D] D such that F G = I [D D]. One basic ingredient in the interpretation of recursive types of [1] is the assumption of the existence of a denumerable family of approximating mappings over D that behave well with respect to application: Definition 4 (Notion of approximation) Given a reflexive cpo D, F, G, a family of continuous functions {( ) n : D D n N} is a notion of approximation over D if it satisfies the following conditions: (i) For all d D, and all n, m N: 0 = (1) n m d n d m (2) (d n ) m = d min(n,m) = (d m ) n (3) 4

5 (ii) For all d, e D and n N: d = d n. (4) n N d 0 e = d 0 = (d ) 0 (5) d n+1 e n = d n+1 e = (d e n ) n (6) d e = (d n+1 e n ). (7) n N These properties hold for the D λ-models built by the classical construction of Scott [16, Lemma , Proposition ], although some of these may fail for the same contruction as modified by Park [16, Exercise ]. Slightly beyond D, they also apply to models of the λ-calculus not explicitly obtained by means of an inverse limit construction, like Engeler s models D A [18] or (quite appropriately in view of the occasion of this paper) the filter λ-model F introduced in [19]: Proposition 5 F is a reflexive cpo with a notion of approximation. PROOF. The proof that F is a reflexive cpo is implicit in [19]. The existence of a notion of approximations is proved by observing that intersection types can given a height σ as follows: ω = 0, ϕ i = 1, for all type variables ϕ i, σ τ = max{ σ, τ }, σ τ = 1 + max{ σ, τ }. Given a filter d F, it can be cut at level n (n N) by defining d[n] = {σ d σ n} but this is not a filter in general: for example, ω ω d[0] even though ω d[0] and ω ω ω. So, define d n as d[n], where X is the filter generated by the set X, namely the set {σ τ 1,..., τ n X.τ 1... τ n σ}. Then the mappings d d n : F F endow F with a notion of approximation. The details are tedious but straightforward, with the exception perhaps of Equation (6), which need a closer analysis of the preorder on intersection types. Claim 6 (Invertibility) If τ ω and σ τ σ τ, then σ σ and τ τ. Proof of Claim 6 An immediate consequence of [19, Lemma 2.4(ii)]. 6) (Claim 5

6 Claim 7 If χ σ τ and χ n + 1, then there exist σ, τ such that σ, τ n and χ σ τ σ τ. Proof of Claim 7 First define the relation on types by the same axioms and rules as, with the exception of transitivity. Then an easy induction on the length of the proof that σ τ shows that σ τ if and only if σ( ) + τ, that means that all applications of transitivity can be postponed. Assume that χ σ τ, then χ( ) + σ τ and therefore there is a finite sequence χ χ 0 χ 1 χ n σ τ (8) for some n > 0. If σ τ > n + 1 (otherwise there is nothing to prove), then take the leftmost χ i in (8) such that χ i < χ i+1, and consider the only two possibilities: (1) χ i ω, χ i+1 ω ω. Then take σ τ ω; (2) χ i (µ ν) and χ i+1 (µ ν ), where the rule applied at this step is µ µ ν ν µ ν µ ν and µ > n or ν > n Observe that µ ν n + 1, so take σ µ and τ ν. (Claim 7) Equation (6): I only show that d n+1 e = d n+1 e n. Let τ d n+1 e = def {τ σ e.(σ τ) d n+1 }, and assume τ ω (otherwise τ d n+1 e n needs no proof). Then there is χ d[n + 1] such that χ σ τ. By Claim 7 there is σ τ d[n+1] such that σ τ σ τ, so by Claim 6 σ σ and τ τ. Then σ e[n] because σ σ e and, as σ τ d[n + 1] d n+1 we have that τ d n+1 e n and finally also that τ d n+1 e n. Conversely, τ d n+1 e n implies σ τ d n+1 for some σ e n. Assume that τ ω, otherwise the proof is immediate. Then by Claim 7 there is σ τ d[n + 1] d n+1 such that σ τ σ τ and therefore, again by Claim 6, σ σ and τ τ, with σ, τ n. Now σ e n e and τ d n+1 e, so finally τ τ d n+1 e. The verification of the other equations is similar. (Proposition 5) 3.2 Types as logical relations Recursive types will be interpreted as certain logical relations over reflexive cpo s with a notion of approximation. Specializations of these to partial 6

7 equivalence relations have been used in [20,6] for the semantics of the system described in [5], which includes recursive types, subtyping and bounded quantification, but I shall argue later that the history of the relational interpretation of types is even longer. Definition 8 (Complete relations) Given reflexive cpo s with a notion of approximation D and E, a relation R D E is complete iff: D R E, if {d (j) j N}, {e (j) j N} are increasing chains in D and E, respectively, such that d (j) R e (j) for all j N, then d (j) R e (j) j N j N The relations must respect the notion of approximation, in the following sense: Definition 9 (Uniformity) Given reflexive cpo s with a notion of approximation D and E, let be a complete relation over D 0 E 0 such that D E. A relation R D E is uniform if the following properties hold where, for n N, R n = def { d n, e n d, e R}: R 0 =, R n R for all n N. Let CU (D, E) denote the set of complete and uniform relations over D, E with base. The following Proposition summarizes the main properties of CU (D, E); the (mostly straightforward) proofs are omitted. Proposition 10 (Basic properties of complete and uniform relations) For any R, S CU (D, E) and any n N: (1) R n CU (D, E); (2) R = S if and only if, for all n N, R n = S n ; (3) R n = (R n+1 ) n ; (4) Let R (i), for i N, be a sequence of complete and uniform relations such that, for all n N: R (n) = (R (n+1) ) n. (9) Then: For all k n R (n) R (k) ; For all k n (R (n) ) k = R (k) ; Define R ( ) as { d, e d n, e n R (n) }. Then R ( ) is the unique complete and uniform relation such that, for all n N, (R ( ) ) n = R (n). 7

8 The operation on the class of complete and uniform relations that provides the interpretations of function types can now be defined. Definition 11 For R, S CU (D, E), define for d D and e E: d (R S) e iff a D, b E (a R b = d a S e b) Proposition 12 If R, S CU (D, E) then R S CU (D, E). PROOF. I first show that = (R S) 0. Assume d, e, and let a, b R. Then d a, e b = d 0 a, e 0 b (as D 0 E 0 ) = d, e (by Definition 4(5)) therefore d a, e b = S 0 S by uniformity. So R S, and (R S) 0 as = 0. Conversely, let d 0, e 0 (R S) 0. Then d 0, e 0 = (d D ) 0, (e E ) 0 (by Definition 4(5)) but D, E = R 0 R, hence d D, e D S, because d, e R S, so (d D ) 0, (e E ) 0 S 0 = and finally d 0, e 0. Completeness and uniformity are easy. The following property of the interpretation of function types is the key to the whole construction of the interpretation of recursive types in [1]: Proposition 13 For any R, S CU (D, E) and n N, (R S) n+1 = (R n S n ) n+1. PROOF. Assume that d n+1, e n+1 (R S) n+1 a n, b n R n R. Then d n+1 a n, e n+1 b n S. But R S and that d n+1 a n, e n+1 b n = (d n+1 a n ) n, (e n+1 b n ) n S n by Definition 4(6), so d n+1, e n+1 R n S n and this entails d n+1, e n+1 (R n S n ) n+1. Conversely, let d n+1, e n+1 (R n S n ) n+1, and assume that a, b R. Then a n, b n R n. Observe now that d n+1 a, e n+1 b = d n+1 a n, e n+1 b n S n S using again Definition 4(6) and uniformity of S, so d n+1, e n+1 R S and finally d n+1, e n+1 (R S) n+1. 8

9 It is possible now to outline the idea behind the construction of type interpretations in [1]. By way of example, consider the simultaneous recursion E = {ξ = ξ ξ}. The interpretation Ξ of the type symbol ξ as a complete and uniform relation is defined locally, in denumerably many stages Ξ (0), Ξ (1), Ξ (2),.... The 0-th stage is Ξ (0) =. Then, whatever Ξ will be eventually, Ξ 0 = Ξ (0). For later stages, set up the recurrence relation Ξ (n+1) = (Ξ (n) Ξ (n) ) n+1. If we can show that for all n N ( Ξ (n+1) ) n = Ξ(n), (10) then Proposition 10(4) suggests taking Ξ = Ξ ( ), and then Ξ n = Ξ (n) for all n N. The proof of (10) is the core of the technique (cf. [8, Lemma 1.1]), and appeals in an essential way to Proposition 13. The fact that Ξ solves E, that is, Ξ = Ξ Ξ, is then a direct consequence of Proposition 10(2). Of course, this process should be carried out in parallel for all type expressions, not only for the unknowns ξ of the simultaneous recursion E. It is at this point that syntactic details creep into the technique, and it becomes useful to adopt the more abstract viewpoint to which I turn next. 3.3 Complete and uniform relations as an iterative algebraic theory Fix arbitrary reflexive cpo s with a notion of approximation D, E and D 0 E 0, and let CU abbreviate CU (D, E). It is interesting to frame the solution of recursive equations over complete and uniform relations in the context of a standard theory of recursion: to this end, I will show that (not too surprisingly) the iterative algebraic theories of Elgot provide such a context. I will review the basic notions of iterative theories along with their interpretations in an iterative theory of complete and uniform relations, following the comprehensive monograph [2] with slight and easily recognizable changes in notation when appropriate. The category CU has the natural numbers as objects, and as morphisms CU (n, p) the functions f : CU p CU n, where CU p is the p-fold power of CU, with CU 0 = { }, such that, for every i N and R (1),..., R (p), if f(r (1),..., R (p) ) = S (1),..., S (n) then: ( f((r (1) ) i,..., (R (p) ) i ) ) i = (S(1) ) i,..., (S (n) ) i (11) Morphisms in CU (1, p) are called scalar morphisms. In the category CU, for every n N, there are n distinguished morphisms i n : 1 n mapping R (1),..., R (n) to R (i), for i = 1,..., n. The distinguished morphisms have the following universal property: for any p 0 and any family f 1,..., f n : 1 p of scalar morphisms there is a unique morphism f 1,..., f n : n p (the 9

10 tupling of f 1,..., f n ) such that i n f 1,..., f n = f i : 1 p: just set, for R = R (1),..., R (p), f 1,..., f n ( R) = def f 1 ( R),..., f n ( R) (where, on the right hand side, corners delimit an element of CU n ). Then every morphism f CU (n, p) has the form f 1,..., f n where f i : 1 p (take f i = i n f). Tupling can be applied more generally to morphisms f : n p and g : m p by setting f, g : n + m p = def 1 n f,..., n n f, 1 m g,..., m m g. Furthermore, for any n N there is one morphism 0 n : 0 n. Observe also that 1 1 = I 1 and, in general, the identity morphisms I n have the form 1 n,..., n n, from the universal property of tupling. Therefore CU is an algebraic theory [2, 2.3], but it does not yet support recursion over complete and uniform relations, because there are too many morphisms. However, we can single out a class of morphisms of this theory in such a way that each of these has a unique fixed point: Definition 14 (Ideal morphisms) A scalar morphism f : 1 p of CU is ideal if, for every i N and R (1),..., R (p) : ( f(r (1),..., R (p) ) ) = ( f((r (1) ) i+1 i,..., (R (p) ) i ) ) (12) i+1 Morphisms and ideal morphisms of CU correspond to the rank-preserving, resp. rank-increasing functions of [21], but these are also clearly related to nonexpansive, resp. contractive mappings on CU regarded as a complete metric space as in [20]. Proposition 15 (Scalar iteration) If f : p is ideal, then there is a unique morphism f : 1 p that solves the scalar iteration equation for f: f = f f, I p (13) PROOF. That f is ideal means that for all R, R (1),..., R (p) : ( f(r, R (1),..., R (p) ) ) i+1 = ( f(r i, (R (1) ) i,..., (R (p) ) i ) ) i+1 = ( f(r i, R (1),..., R (p) ) ) i+1 for all i N. Now, define inductively the sequence: Φ (0) = Φ (n+1) = ( f(φ (n), R) ) n+1 10

11 Then, for all n N, Φ (n) = ( Φ (n+1)). In fact, n Φ(0) = = ( Φ (1)). Assuming 0 that Φ (n 1) = ( Φ (n)) for n > 0, we have n 1 ( ) Φ (n+1) = ( (f(φ (n), ) R)) n n+1 (by definition) n = ( f(φ (n), R) ) (by uniformity) n = ( f ( (Φ (n) ) n 1, R )) (f is ideal) n = ( f(φ (n 1), R) ) (induction hypothesis) n = Φ (n) (by definition) Using Proposition 10(4), define f ( R) as Φ ( ), and observe that ( f(φ ( ), R) ) n = ( f((φ ( ) ) n 1, R) ) n (as f is ideal) = ( f(φ (n 1), R) ) n = Φ (n) (by definition) hence f(f ( R), R) = f(φ ( ), R) = Φ ( ) = f ( R). Uniqueness of f follows by a straightforward induction, using Proposition 10(2). Proposition 13 can then be rephrased by saying that : 1 2 is an ideal morphism in CU. Observe that the scalar ideal morphisms form a scalar ideal in CU, viz. if f : 1 p is ideal then also f g : 1 q is a scalar ideal morphism, for any g CU (p, q). In fact, let g(r (1),..., R (q) ) = S (1),..., S (p), then ( f(g(r (1),..., R (q) )) ) i+1 = f(s(1),..., S (p) ) i+1 = f ( (S (1) ) i,..., (S (p) ) i ) i+1 = f ( g((r (1) ) i,..., (R (q) ) i ) ) i+1 (as f is ideal) (as g CU (p, q)) = ( (f g)((r (1) ) i,..., (R (q) ) i ) ) i+1 Define now the subtheory CU of CU whose objects are the same as those of CU, and where f : n p if and only if, for every i = 1,..., n, i n f either is a distinguished morphism or is ideal. General results on iterative theories [2, Proposition , Theorem 5.2.2], together with Proposition 13 yield: Proposition 16 CU is an iterative theory whose ideal morphisms have the form f 1,..., f n where each f i is a scalar ideal morphism. In particular, for any ideal morphism f : n n + p, there is a unique ideal morphism f : n p solving the iteration equation for f, i.e., f = f f, I p. 11

12 One important consequence of this Proposition is that CU satisfies all equations valid in iterative theories [2, 5.3]. Among these we have, for example, a general form of Bekič s theorem: f, g = f h, I p, h (14) where f : n n+m+p, g : m n+m+p are ideal and h = g f, I m+p : m m + p. Another equation that is worth recording is (f (I n g)) = f g (15) for any ideal f : n n+p and all g : p q. The notation h k : n+m p + q for h : n p, k : m q, is defined as the tuple h κ, k λ, where κ : p p + q, λ : q p + q are given by i p κ = i p+q i q λ = p + i p+q for i = 1,..., p for i = 1,..., q Regarding theories as categories, it is appropriate to define homomorphisms of theories as a certain kind of functors that are the identity on objects: Definition 17 (Homomorphisms of iterative theories) If T and T are iterative theories, then a homomorphism is a family of mappings F = F n,p : T(n, p) T (n, p) for n, p 0 such that for f : n p and g : p q in T, F n,q (f g) = F n,p (f) F p,q (g) in T (n, q), F 1,p (i p ) = i p for any i = 1,..., p, if F 1,p (f) is distinguished in T, then f is distinguished in T or, equivalently, for every ideal f : n p, F n,p (f) is ideal in T. Then it can be shown [2, Proposition 5.2.3] that, if F : T T is an iterative theory morphism, then it preserves tupling and iteration, more precisely F n,p (f ) = (F n,n+p (f)) for every ideal morphism f : n n + p in T. Let R be the iterative algebraic theory whose morphisms in R(n, p) are n- tuples or regular trees over the signature with a binary operation symbol, in which may occur variables in the list X = X 1,..., X p [22, 4]. Distinguished morphisms of R(1, p) are i p = X i, and composition of t : n p and s = s 1,..., s p : p q is t s = def t 1,..., t n, where t i is given by substitution, t i = t i [X 1 := s 1,..., X p := s p ]. This is the initial iterative algebraic theory [2, Corollary ]: for any iterative theory T and any interpretation of as an ideal morphism T(1, 2), there is a unique homomorphism of iterative theories Ψ T : R T. 12

13 3.4 Interpretations of types The iterative algebraic theory of complete and uniform relations provides a uniform ambient for interpreting recursive types in any presentation. The discussion below focuses on simultaneous recursions, but it will be clear that the results apply to recursive types defined by means of the recursion operator as well. Actually, types and simultaneous recursions shall be interpreted quite generally in any iterative algebraic theory T with an ideal T(1, 2), as morphisms of the appropriate kinds. As particular cases, we shall obtain interpretations in the iterative theories R and CU. The role of a simultaneous recursion E = {ξ 1 = T 1,..., ξ n = T n } is that of defining (recursively and simultaneously) the type symbols ξ = ξ 1,..., ξ n to be used as constants, together with the parameters X = X 1,..., X p, in the constructions of the elements of T( ξ; X). Accordingly, the interpretation of that simultaneous recursion in T, given a type environment η = η (1),..., η (p) : p 0 that assigns meaning η (i) T(1, 0) to the parameter X i, must return a tuple Ξ = Ξ (1),..., Ξ (n) : n 0, such that Ξ (i) T(1, 0) is the interpretation of the type symbol ξ i. Types in T( ξ; X) are then assigned meanings by means of a straightforward inductive extension of η and Ξ. Throughout this section, to avoid notational clutter, the sequences of indeterminates ξ = ξ 1,..., ξ n and parameters X = X 1,..., X p, for some n, p 0, as well as the simultaneous recursion E = {ξ 1 = T 1,..., ξ n = T n }, are arbitrary but fixed. I shall also assume throughout that T is an iterative theory with an ideal morphism : 1 2. Definition 18 (Interpreting simultaneous recursions) For any A T( ξ; X), define a morphism T[A] : 1 n + p of T as follows, by induction on the structure of A: T[ξ i ] = i n+p, T[X j ] = (n + j) n+p, T[A A ] = T[A ], T[A ]. Observe that T[T 1 ],..., T[T n ] is an ideal morphism and define T[E ] = T[T 1 ],..., T[T n ] : n p. Definition 19 (Interpreting recursive types) Given a type A T( ξ; X), define the interpretation of A under E, T E [A] T(1, p), as T[A] T[E ], I p. The standard identities hold for this interpretation of types, for example T E [A A ] turns out to be the same as T E [A ], T E [A ]. Furthermore, weak equivalence of recursive types (Definition 2) is respected by the 13

14 interpretation: Proposition 20 If A E B, then T E [A] = T E [B ]. The following properties exploit the special role played by the initial iterative theory R (see [22, 4.8]): Lemma 21 (i) For any type A T( ξ; X), (ii) For any simultaneous recursion E, (iii) For any type A T( ξ; X) T[A] = Ψ T (R[A]) : 1 n + p T[E ] = Ψ T (R[E ]) : n p T E [A] = Ψ T ( R E [A] ) : 1 p We finally can define a stronger equivalence on recursive types, as the type congruence that identifies types whose infinite unfoldings give rise to identical infinite (regular) trees. It follows directly from Lemma 21 that the type interpretation respects this stronger equivalence too: Definition 22 (Strong equivalence from simultaneous recursions) Two types A, B T( ξ; X) are strongly equivalent, written A E B, when R E [A] = R E [B ] : 1 p. Corollary 23 If A E B, then T E [A] = T E [B ]. 4 Concluding remarks There are several directions that are worth pursuing in order to understand the properties of the interpretation outlined in this paper. While [1,10] have focussed mainly on completeness properties of type inference or of strong equivalence over recursively defined types, I outline here the proofs of two properties that seem to point to a different area of applications (and also justify the study of a relatively weak system like that used throughout this paper). They are both consequences of a very important property of logical relations, that can be proved by induction on the construction of recursively typed terms (Definition 3), using the standard interpretation of terms in a λ-model [16, Ch. 5]: 14

15 Lemma 24 (Fundamental property of logical relations) Let M Λ E (A) where A T( ξ; X) and M contains possibly the free variables x A 1 1,..., x A k k. Let M be the pure untyped λ-term obtained by erasing all types in M, and let η = η (1),..., η (p) be a type environment. Then given (term) environments ρ 1, ρ 2 such that ρ 1 (x i ) ( CU E [A i ]η ) ρ 2 (x i ), for all i = 1,..., k where x i = (x A i i ), then [M ]ρ 1 ( CU E [A]η ) [M ]ρ 2. Now, as remarked earlier, every pure closed λ-term M has a reflexively typed version M Λ E (ξ), where ξ = ξ ξ E. Therefore [M ] Ξ [M ] where Ξ D D is CU E [ξ ] with base = { D, D } (a sort of inner model ) and D is any of the standard extensional models obtained by Scott s original construction [16, 18.2]. Then we take almost verbatim the proof of Plotkin in [8] (whose details are now hidden in the proofs of Section 3 above) to show Proposition 25 The only definable element of D 0 D is. In fact, if d D 0 is definable, then d = [M ] for some closed M Λ E (ξ). Therefore d Ξ d by Lemma 24, and then d 0 Ξ 0 d 0, so d d as d = d 0 and Ξ 0 =, because Ξ CU (D, D ). Hence d =. Another observation is that closed terms in Λ E (X) for a type variable X have a trivial computational behavior. Indeed: Proposition 26 If M Λ E (X) is closed, then M is unsolvable. Take F and set = { F, F }. Then [M ] R [M ] for R CU (F, F), in particular [M ] [M ], showing that [M ] = F, hence M : σ iff ω σ iff M is unsolvable, by [19, Theorem 4.13(i)]. The initial model L = [L L] of the lazy λ-calculus also satisfies a slight generalization of Definition 4 [17, 4.3]. This allows to show that the assumption of Proposition 26 even implies that M is an unsolvable term of order 0. It remains to be seen to what extent these remarks can be extended to a systematic approach to proving properties of pure, type-free λ-terms by means of recursisvely defined logical relations. A parallel direction for further investigations is the extension of the interpretation discussed in this paper from λ-models to structures that are closer to operational semantics including as a limiting case the syntax itself in the form of a labeled λ-calculus similar to that of [23], where the notion of approximation is built in the definition of labeled reduction. This extension should take into account the recent attempts towards the operational semantics of recursive types of [24,25] and provide a 15

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