Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....tipo ad x. Altri tas importanti in letteratura sono il sistema di assegnazione di tipi del secondo ordine, meglio conosciuto come Polymorphic Type Assignment System (F2) Lei83] il sistema di assegnazione di tipi di ordine superiore (F ) GR88] il sistema di assegnazione di tipi ricorsivi ( CC90] ed il sistema di assegnazione di tipi intersezione, o Intersection Type Assignment System ( BCD83] 1.2 Sistemi Tipati per il I tipi possono essere usati direttamente per decorare i termini del calcolo. Il linguaggio decorato viene detto calcolo tipato [Chu41] Nel calcolo tipato, a ....

F. Cardone and M. Coppo. Type Inference with Recursive Types: Syntax and Semantics. Information and Computation, 1990.

....also be of interest to extend the semantic analysis to other type disciplines. The combination of the polymorphic type discipline with the intersection type discipline of Coppo et al. is of immediate interest, and one can also consider, for example, existential and union types and recursive types [1,6,7,24]. In a different direction, it would be interesting to extend the language to include other features such as, perhaps, a recursion combinator. Continuing in this way, it would be particularly interesting to see a treatment of a programming language equipped with an operational semantics (rather ....

Cardone, F. and Coppo, M. Type inference with recursive types: syntax and semantics. Information and Computation 92(1) pp. 48-80 (1991)

....of simple types is due to Hindley and Milner. This is an instance of a more general and very productive connection, encountered again later between other forms of unification and other forms of type inference. 2 with recursive types has been studied (and shown decidable) by many authors [MPS86, CC91] However, much less is known about the type inference problem in the presence of recursive types and universal types. The advantage of static vs. dynamic type checking is the ability to distinguish good terms from bad terms at compile time as opposed to at run time. Traditionally, bad terms ....

F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

....relation over TExp considering two type expressions of equivalent modulo if they have the same (possibly infinite) type expression obtained by unfolding recursive types X:A occurring in them to A[ X:A=X ] indefinitely. This equivalence relation is known to be decidable (see [8] and [3] Definition 3 (Typing contexts) A typing context, or a context for short, of is a finite mapping that assigns a type expression of to each individual variable of its domain. We use , 0 , to denote contexts, and fx 1 : A 1 ; xm : Am g to denote a context that ....

....straightforward to construct a non trivial realizability model of . The first non trivial model was developed by MacQueen, Plotkin and Sethi [22] by constructing a complete metric space of types and by interpreting recursive types as the fixed points of contractive type constructors (see also [2, 8]) Proposition 1 (Soundness of ) Let V ; V ; T ; T be a realizability model of . If fx 1 : A 1 ; xn : An g M : B is derivable, then [ M ] V 2 [ B] T for every and provided (x i ) 2 [ A i ] T (i = 1; 2; n) Nevertheless, ....

F. Cardone and M. Coppo. Type inference with recursive types: syntax and semantics. Information and Computation, 92(1):48--80, 1991.

.... can be found in Davey and Priestley s text [DP90] Properties of infinite and regular trees are surveyed by Courcelle [Cou83] Recursive types in computer science go back to (at least) Morris [Mor68] Basic syntactic and semantic properties (without subtyping) are collected in Cardone and Coppo [CC91]. Amadio and Cardelli [AC93] gave the first subtyping algorithm for recursive types. Their paper defines three relations: an inclusion relation between infinite trees, an algorithm that checks subtyping between types, and a reference subtyping relation between types defined as the least fixed ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, 1991.

....and its logical soundness under the Curry Howard interpretation, recursive types preserve and extend the well typed programs don t go wrong property of terms. To use recursive types it is necessary to add the rule A e : 0 A e : 0 (Equal) for simple typing with recursive types [CC91] or A e : 0 A e : 0 (Subtype) for simple subtyping with recursive types [AC91, AC93] The question, now, is when two recursive types are equal or in the subtyping relation. This is what we study in this paper. 1.1. Recursive Types Definition 1.1. The recursive types (in canonical ....

....recursive types and write fv( for the set of free type variables in . 1.2. Regular Trees We define Tree( to be the regular (possibly infinite) tree obtained by completely unfolding all occurrences of ff: to [ff: For a precise definition of Tree( regular trees and their properties see [Cou83, CC91, AC93]. We write T 1 T 2 for the tree T with root label , left subtree T 1 and right subtree T 2 . Henceforth we shall assume that all trees are over the ranked alphabet f 0 ; 2 ; 0 g [ fff 0 : ff 2 TVarg of labels, which are ordered by the reflexive transitive closure of ff ....

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....calculus [Gir72, Rey74] that allow types with quantification over all types. The T type system is essentially predicative in another sense. The T dependent types are based on well founded recursion schemes in contrast to type systems 97 with recursive types that have no finite basis [MPS86, CC91] Within these dimensions of strong normalization versus general recursion, Curry versus Church philosophies, explicit versus implicit types, and predicative versus impredicative polymorphic constructions we can distinguish our system from others. Thus, for example, we differ from the Calculus ....

....Calculus of Constructions (COC) CH88] Martin Lof type theory [Mar75] and the Logical Framework (LF) HHP93, CH88] because our types are Currystyle meta constructions only implicitly associated with programs via type reconstruction. We differ from systems with recursively defined types [MPS86, CC91] because our dependent types are predicative; either finite or well founded recursions. On the other hand, we are similar to other implicitly typed and first order or predicative systems in the ML family [KTU93a, Hen93, Lei91, KT92, McC84] 4.2.2 Polymorphism and Dependent Types There are many ....

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics second order lambda calculus. Information and Computation, 92:48--80, 1991.

....b (Reference (pointer) t 1 , t 2 , t n ) b (Tuple) int b (Scalar) 4 a is a metavariable that ranges over an infinite set of type variables. Ignoring the b tags (described below) this is a completely standard polymorphic type system. We use the standard inference algorithm W [9, 2] to compute the types of all variables of a source program (with no initial type declarations) Roughly speaking, W works by assigning variables to types and unifying type expressions as the program structure requires, in the manner described informally above. W is defined for a simple functional ....

....[14] Similarly, comparing the tags on function types with the tags on declared functions gives us an analysis of higherorder control flow. 7 Actually in the polymorphic type system we must use a more complicated relation than just tag equality; see below. regular trees (see Cardone and Coppo [2] for details) We do not use polymorphic recursion; that is, let and letrec bindings are the only places where we perform polymorphic generalization. We use a value restriction on polymorphic lets to make side effects safe [16] Our representation tags correspond closely to the region variables of ....

F. Cardone and M. Coppo. Type inference with recursive types: syntax and semantics. Information and Computation, 1992, number 1, pp. 48-80.

....int. An alternative way of viewing the ffi rules is to consider them as type equivalences so that the algebra of types is no longer a free term algebra, but rather a quotient algebra factored over the compatible closure of the ffi rules. Such approaches are common in dealing with recursive types [6] and with record types [40] For specialization time integer expressions, all computation in the translated term takes place on the type level. However, we still need an expression of that type. This expression is never inspected, it is just there because we cannot have a type without an ....

Felica Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

....solving subtype satisfaction problems for types with quantifiers and intersections at unlimited depth. 9 Related work Principal typings are not a new concept. A number of existing type systems have principal typings, including the simply typed lambda calculus [31] the system of recursive types [5], the system of simple subtypes [25] and the system of intersection types [4] Our contribution is to highlight the practical uses of the principal typing property, and to distinguish it from the principal type property. A number of authors have published offhand claims that ML possesses the ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....we can now extract from the new execution tree a proof of a R b . Applying the completeness of the rules for type equivalence we conclude a= R a and b= R b . Finally we derive a R b by (eq R ) and (trans R ) M 6. A Per Model We sketch the main features of a model described in [1] see also [14] for a related work) based on complete uniform pers over a D l model [26] Per (partial equivalence relation) models provide an interpretation of subtyping as settheoretic containment of the relations [7] In addition, these structures have very interesting categorical properties (in particular ....

Cardone, F., and Coppo, M. Type inference with recursive types: syntax and semantics, Info.&Comp., 92, 1, pp 48-80.

.... functions and fixed points can be found in [1] and [9] Properties of infinite and regular trees are surveyed by Courcelle [7] Recursive types in computer science go back to (at least) Morris [17] Basic syntactic and semantic properties (without subtyping) are collected in Cardone and Coppo [6]. Amadio and Cardelli [3] gave the first subtyping algorithm for recursive types. Their paper defines three relations: an inclusion relation between infinite trees, an algorithm that checks subtyping between types, and a reference subtyping relation between types defined as the least fixed ....

F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, 1991.

....its logical soundness under the Curry Howard interpretation, recursive types preserve and extend the welltyped programs don t go wrong property of terms. To use recursive types it is necessary to add the rule A e : 0 A e : 0 (Equal) for simple typing with recursive types [CC91] or A e : 0 A e : 0 (Subtype) for simple subtyping with recursive types [AC91, AC93] The question, now, is when two recursive types are equal or in the subtyping relation. This is what we study in this paper. M.Brandt and F.Henglein Coinductive Axiomatization of Recursive ....

....and sum, but for clarity we shall not treat these extensions here. 1.2. Regular Trees We define Tree( to be the regular (possibly infinite) tree obtained by completely unfolding all occurrences of ff: to [ ff: For a precise definition of Tree( regular trees and their properties see [Cou83, CC91, AC93]. We write T 1 T 2 for the tree T with root label , left subtree T 1 and right subtree T 2 . Henceforth we shall assume that all trees are over the ranked alphabet f 0 ; 2 ; 0 g [ fff 0 : ff 2 TVarg of labels, which are ordered by the reflexive transitive closure of ff ....

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

.... : F; F 0 ; 0 ) Lemma 3.2 Let t be a term and (F ; a K type system such that Gamma t : oe and Gamma t : are judgements. If Gamma contains all bounded variables of t then oe holds. Proof: By induction on the structure of t. 2 F. Cardone and M. Coppo showed in [CC91] the principal typing property for the calculus with recursive types. Here is the analogous result for K type systems: Theorem 3.3 (Principal Typing Theorem) For each term t exists a principal K type system. Proof: We assume that no variable in t occurs freely and bounded or bounded more than ....

F. Cardone and M. Coppo. Type inference with recursive types: syntax and semantics. Information and Computation, 92:48--80, 1991.

....of interesting program properties, such as finding abstract datatypes, determining if certain variables or functions are never used, and detecting memory leaks. Lackwit uses a polymorphic type system to describe representation constraints and perform a variant of the M type inference algorithm [Mil78,CC92]. Types are merged as it finds the representation of variables and expressions in the program to be constrained to be the same. Type inferencing provides a good method for determining comparability because it is fully automatic, the basic algorithm is straightforward, it can be done statically, ....

....as it finds the representation of variables and expressions in the program to be constrained to be the same. Type inferencing provides a good method for determining comparability because it is fully automatic, the basic algorithm is straightforward, it can be done statically, and it is efficient [Mil78,CC92]. Jakewit I applied applied the experiences from Lackwit and M algorithm to code an invariant type inference module for our tool. I use invariant type to describe the type inferred by our algorithm since it describes what invariants may be checked for over the associated variable. The ....

F. Cardone and M. Coppo. Type inference with recursive types: syntax and semantics. Information and Computation, 1:48-80, 1992.

....dual to those encountered for Sigma 2 , and we leave to the reader to construct an appropriate structure. The result is the first type inference algorithm for this system. 6 Related work Huet [16] gave the first unification algorithm for recursive types; see also the papers by Cardone and Coppo [7, 8]. Mitchell [23, 24] gave the first inference algorithm for atomic subtyping, without recursive types. With no further assumptions about the partial order, this problem is PSPACE complete [35, 15, 12] and if the partial order is a disjoint union of lattices or trees, then type inference is in ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....type quantifier prefixes. In the presence of recursive types, it seems preferable that in the definition of the match function obove we read j not as syntactical identity of type expressions, but rather as identity of the rational trees obtained by infinite unfolding of the operator (c.f. [4]) The fold and unfold rules for recursive types should then be replaced by the stronger rule of equality for recursive types, see [4] 4 Open problems By providing an interpretation in the ideal model, it has been shown that dynamic types can be combined with recursive types and explicit or ....

....we read j not as syntactical identity of type expressions, but rather as identity of the rational trees obtained by infinite unfolding of the operator (c.f. 4] The fold and unfold rules for recursive types should then be replaced by the stronger rule of equality for recursive types, see [4]. 4 Open problems By providing an interpretation in the ideal model, it has been shown that dynamic types can be combined with recursive types and explicit or implicit polymorphism in a sound way. This gives a partial answer to questions of Abadi e.a. 1] and adds semantical support to ....

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....liberal annotations (earlier binding time values for expressions) This can be done without changing the rules in Fig. 7. It is sufficient to extend the equality on bindingtime types in such a way that all types that have the same regular type are equal (according to the translation described in [CC91]) 4.3 Insertion of Multi Level Specialization Points We chose dynamic conditionals and dynamic abstractions for inserting specialization points because this strategy is straightforward, surprisingly effective in practice, and easily generalized to the multi level case. We will only explain this ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

.... the type reconstruction problem to be undecidable [13] The importance of recursive types and positive recursive types in programming language theory has been recognized for many years; a sample of recent results, restricted to aspects of type checking and typereconstruction, can be found in [1, 3, 17]. The report is organized as follows. We first give a precise definition of recursive and positive recursive types (Section 2) and introduce the systems ML fix ) and ML fix pos (Section 3) We call the two system S and S for short. These two systems are in fact pared down versions which are ....

....types. For a type oe of the form ff: the unfolding of oe for one step results in the type [ff : ff: Every recursive type oe represents an underlying regular type obtained by unfolding oe infinitely many times. More formally there is a map ( T T reg . We refer the reader to [3] for an exact definition of ( It is also true that every type in T reg has a notation (not unique) in T . We refer the reader to [2] for the proof of this fact, the reference also contains a detailed discussion of infinite and regular types. This means that, whenever appropriate, we can ....

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

....(n) ranges over type tags of arity n. We treat the procedure tag as a binary constructor, and we write 0 for procedure( 0 ) also, we write 0 for pair( 0 ) Recursive type abstractions of the form ff: are finite notations for possibly infinite, regular trees, as in [CC91]. Types of the form P tc tc( are discriminative sums, where the top level type constructors of the summands are required to be distinct (see [Reh95] and also [Fag90] for more details. In this paper, we generally assume that tc ranges over the whole type constructor alphabet (denoted T ) in ....

....with rules for recursive types, coercion application, coercion abstraction, coercion instantiation, and definitions, respectively. In the recursion rule (ninth rule from top) the relation 0 holds, if and only if the (possibly) infinite regular unfoldings of and 0 are identical. See [CC91] for further information. In the rule for definitions (last rule) the expression C ) denotes type s1 ) sn ) where C is the set of coercion signatures s1 ; sn . Note that any pure expression which is well typed is a completion (of itself. In the presence of recursive types, ....

F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....solving subtype satisfaction problems for types with quantifiers and intersections at arbitrary depth. 8 Related work Principal typings are not a new concept. A number of existing type systems have principal typings, including the simply typed lambda calculus [22] the system of recursive types [4], the system of simple subtypes [16] and the system of intersection types [3] Our contribution is to highlight the practical uses of the principal typing property, and to distinguish it from the principal type property. A number of authors have published offhand claims that ML possesses the ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....do not arise in another. In particular, we will see that having and actually makes type inference easier, and we give the first algorithms for the system with but not , and the system with but not . Well known languages with but not include F : 6] and O 1 [1] Cardone and Coppo [7, 8] have studied the system with but not , and claim type inference is solved, but their method is flawed, as we will explain. Our algorithm uses well known ideas, starting with the equivalence between type inference and finding solutions to sets of constraints. A constraint set is simply a ....

....natg, is sometimes called the system of partial types, after the work of Thatte [33, 34, 35] who studied a nonrecursive version. Other well known calculi with but not include F : 6] and O 1 [1] ffl The system Sigma 3 , where Sigma 3 = Tv [ f ; natg, was studied by Cardone and Coppo [7, 8]. Although Cardone and Coppo claim that type inference for this system is solved, we believe we have the first type inference algorithm for the system. ffl The system Sigma 0 , where Sigma 0 = Tv [ f ; natg, is the system of recursive types without subtyping ( is just syntactic equality of ....

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

.... Int. The substitution 1 which maps fi into RECfi: fi; Int; Int) as well as the substitution 2 which maps fi into RECfi: fi; Int) will satisfy this constraint; but using the axiomatisation in Fig. 1 it seems hard to find a sense in which 1 and 2 are comparable. A remedy, inspired by [CC91] where a similar observation is made concerning recursive types, might be to consider RECfi:b equivalent to its infinite unfolding (cf. rule R1 in Fig. 1 which states that RECfi:b is equivalent to its finite unfoldings) It is not yet clear whether such a change in the axiomatisation will be ....

....constitute an unwarranted restriction upon the language. Future work will show how to overcome these problems. One avenue of research is to extend our axiomatization of w such that principal solutions exist; perhaps by identifying recursive behaviours with their infinite unfolding just as done in [CC91] for recursive types. Another avenue is to accept that there definitely will be applications of effect systems in theories that are known not to have principal solutions; in this case we may try to reduce our problems by incorporating a more general rule for subtyping into the inference ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

....results in updating the syntax and static semantics of the language. Recursive Effects The example given in section 7 cannot be handled with our current type and effect system since it would require recursively defined effects in the static semantics to typecheck. Following the approach of [2] for recursive types, we could allow recursively defined effects :oe, with the added equivalence rule: equiv) E e : oe 0 oe oe 0 E e : 0 ; oe 0 The addition of the rule (equiv) to the static semantics would allow equivalent types and effects for a given expression ....

Cardone, F., and Coppo, M. Type Inference with Recursive Types: Syntax and Semantics. In Information and Computation, Vol. 92, pages 4880. Academic Press 1991.

.... x: oe ) oe x: oe ) oe x: oe ) oe x: oe (E) x: oe ) oe x: oe ) oe x: oe ) oe x: oe (E) x: oe ) oe xx: E) x:xx: oe ) oe ( I) A completely different approach would consist in admitting recursive type definitions [22], and thus simply equating the two types: oe = oe . Adding the universal type not only trivially allows the typing of any term, even the term Omega , i.e. x:xx) x:xx) which cannot be typed in any other way, but owing to that allows non trivial (in a sense to be explained later) ....

F. Cardone, M. Coppo, "Type Inference with Recursive Types. Syntax and Semantics ", Information and Computation, 92(1), 1991, 48--80.

....logical soundness under the Curry Howard interpretation, recursive types preserve and extend the well typed programs don t go wrong interpretation of terms. To use recursive types it is necessary to add the rule A e : 0 A e : 0 (Equal) for simple typing with recursive types [CC91] or A e : 0 A e : 0 (Subtype) for simple subtyping with recursive types [AC91, AC93] The question, now, is when two recursive types are equal or in the subtyping relation. This is what we study in this paper. 1.1 Recursive Types Definition 1. The recursive types (in canonical ....

....types and write fv( for the set of free type variables in . 1. 2 Regular Trees We define Tree( to be the regular (possibly infinite) tree obtained by completely unfolding all occurrences of ff: to [ ff: For a precise definition of Tree( regular trees and their properties see [Cou83, CC91, AC93]. Henceforth we shall assume that all trees are over the ranked alphabet f 0 ; 2 ; 0 g [ fff 0 : ff 2 TVarg of labels, which are ordered by the reflexivetransitive closure of ff . We can define depth k lower and upper approximations T j k and T j k of a tree T as ....

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991.

....is downwards closed wrt. C if the following property holds for all ##b # C: if # # F then fv(b) # F . # Definition 4.7 Let F be a set of variables and let C be a set of (simple) Cconstraints. We say that F is upwards closed wrt. C if the following property to its finite unfoldings, and cf. [CC91] where a similar change in axiomatisation is made concerning recursive types) 2 We use # to range over type variables and behaviour variables collectively and use g to range over types and behaviours collectively. holds for all ##b # C: if fv(b) # F #= # then # # F . # We define the ....

Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

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F. Cardone, M. Coppo, "Type Inference with Recursive Types. Syntax and Semantics ", Information and Computation, 92(1), 1991, 48--80.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

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F. Cardone and M. Coppo. Type inference with recursivetypes: Syntax and semantics. Information and Computation, 92:48--80, 1991.

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Cardone, F. and Coppo, M. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92(1):48--80, May 1991. Cited on page 37.

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Cardone, F., Coppo, M.: Type inference with recursive types: syntax and semantics. Information and Computation 92 (1991) 48--80

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Cardone, F., Coppo, M. Type Inference with Recursive Types: Syntax and Semantics. In Information and Computation, Vol. 92, pages 48-80. Academic Press 1991.

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput., 92:48--80, 1991.

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F. Cardone and M. Coppo. Type inference with recursive types: Syntax and semantics. Inf. & Comput. , 92:48--80, 1991.

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Felice Cardone and Mario Coppo. Type inference with recursive types: Syntax and semantics. Information and Computation, 92:48--80, 1991.

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