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Type Inference with Simple Selftypes is NPcomplete
, 1997
"... . The metavariable self is fundamental in objectoriented languages. Typing self in the presence of inheritance has been studied by Abadi and Cardelli, Bruce, and others. A key concept in these developments is the notion of selftype, which enables flexible type annotations that are impossible with ..."
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. The metavariable self is fundamental in objectoriented languages. Typing self in the presence of inheritance has been studied by Abadi and Cardelli, Bruce, and others. A key concept in these developments is the notion of selftype, which enables flexible type annotations that are impossible with recursive types and subtyping. Bruce et al. demonstrated that, for the language TOOPLE, type checking is decidable. Open until now is the problem of type inference with selftype. In this paper we present a simple type system with selftype, recursive types, and subtyping, and we prove that type inference is NPcomplete. With recursive types and subtyping alone, type inference is known to be Pcomplete. Our example language is the object calculus of Abadi and Cardelli. Both our type inference algorithm and our lower bound are the first such results for a type system with selftype. CR Classification: Categories and Subject Descriptors: D.3.2 [Programming Languages]: Language Classifications...
Extensions of... with Decidable Typing
 Centre
, 1994
"... Both subtyping and typing relations in the system F , the wellknown secondorder polymorphic typed  calculus with subtyping [CW85, BL90, BTCCS91, CG92, CMMS94] appeared to be undecidable [Pie92]. We demonstrate an infinite class F of extensions of the system F , where both relations are decidable ..."
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Both subtyping and typing relations in the system F , the wellknown secondorder polymorphic typed  calculus with subtyping [CW85, BL90, BTCCS91, CG92, CMMS94] appeared to be undecidable [Pie92]. We demonstrate an infinite class F of extensions of the system F , where both relations are decidable. Our extensions are based on the converging hierarchies of decidable extensions of the Fsubtyping relation introduced in [Vor94c]. Every system c F from the class F satisfies the following properties: ffl all subtyping \Gamma ` oe and typing \Gamma ` t : judgments provable in F are also c Fprovable; in particular, every F typable term is also c F typable, but not conversely: an F typable term may have additional types in c F , and there exist c F typable terms that are not F typable; ffl the c F canonical types, analogous to the Fminimum types [CG92], are effectively computable (as opposed to F ); there exists a decision procedure, which given a context \Gamma and a term t a...
Theory of Finite Trees Revisited: Application of ModelTheoretic Algebra
, 1994
"... . The theory of finite trees in finite signature \Sigma is axiomatized by the simple set of axioms E : 1. 8x x 6= t(x) for every nonvariable term t(x) containing x, 2. 8x 8y ( f(x) = f(y) , x = y ) for every f 2 \Sigma , 3. 8x 8y (f(x) 6= g(y)) for different f , g 2 \Sigma , plus the usual eq ..."
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Cited by 2 (1 self)
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. The theory of finite trees in finite signature \Sigma is axiomatized by the simple set of axioms E : 1. 8x x 6= t(x) for every nonvariable term t(x) containing x, 2. 8x 8y ( f(x) = f(y) , x = y ) for every f 2 \Sigma , 3. 8x 8y (f(x) 6= g(y)) for different f , g 2 \Sigma , plus the usual equality axioms, plus the following Domain Closure Axiom: 8x f2\Sigma 9z ( x = f(z) ) (DCA) postulating that every element of a model is in the range of some (perhaps 0ary) function, i.e., there are no isolated elements. The theory E [ (DCA) has numerous applications in Automated deduction, Constraint solving, Unification theory, Logic programming, Database theory. It was proved complete by Maher [Mah88] using the straightforward quantifier elimination, and also by Lescanne and Comon [CL89] by a direct transformational method. Earlier Kunen [Kun87] proved that E (without (DCA)) is complete in the case of infinite signatures with constants (this case is much more simple), again by...
Proof Normalization and Subject Reduction in Extensions of Fsub
, 1995
"... System F , the secondorder polymorphic typed calculus with subtyping appeared to be undecidable because of the undecidability of its subtyping component. The discovery of decidable extensions of the F subtyping relation put forward a challenging problem of incorporating these extensions into an F ..."
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System F , the secondorder polymorphic typed calculus with subtyping appeared to be undecidable because of the undecidability of its subtyping component. The discovery of decidable extensions of the F subtyping relation put forward a challenging problem of incorporating these extensions into an F like typing in a decidable and coherent manner. In this paper we describe a family of systems combining the standard F typing rules with converging hierarchies of decidable extensions of the F subtyping and give decidable criteria for successful proof normalization and subject reduction.
[Programming Languages]: Formal Definitions and Theory General Terms: Languages
"... Bounded quantification allows quantified types to specify subtyping bounds for the type variables they introduce. It has undecidable subtyping and type checking. This paper shows that subclassingbounded quantification—type variables have subclassing bounds—has decidable type checking. The main diff ..."
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Bounded quantification allows quantified types to specify subtyping bounds for the type variables they introduce. It has undecidable subtyping and type checking. This paper shows that subclassingbounded quantification—type variables have subclassing bounds—has decidable type checking. The main difficulty is that, type variables can have either upper bounds or lower bounds, which complicates the minimal type property.
Unified CardelliMitchell's Polymorphic Calculus with Subtyping (Preliminary Report)
"... . We join Cardelli's and Mitchell's approaches to polymorphic subtyping by introducing a new unified calculus G that subsumes both Cardelliinspired systems with rigid subtyping F , F bounded [CW85, CCH + 89, CG92, Pie92, Kat92, Ghe93, CMMS94] on the one hand, and the more liberal Mi ..."
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. We join Cardelli's and Mitchell's approaches to polymorphic subtyping by introducing a new unified calculus G that subsumes both Cardelliinspired systems with rigid subtyping F , F bounded [CW85, CCH + 89, CG92, Pie92, Kat92, Ghe93, CMMS94] on the one hand, and the more liberal Mitchell's containment calculus F` [Mit88] on the other. Up until now both approaches were considered incompatible, since F`provable containments between functional and universal types are unprovable in F and F bounded, but F` lacks the proper treatment of bounded quantification. We give a logical motivation for the new subtyping system G by interpreting it in the secondorder intuitionistic propositional logic. As our system turns out to be stronger (i.e., types more terms) than F , F , F bounded, we prove the strong normalization theorem by applying the recent evaluation semantics techniques due to McAllester, Kuchan, and Otth [MKO95]. 1 Introduction The burst of interest to the semantics of...
Fsub with Recursive Types: "TypesAsPropositions" Interpretation in M. Rabin's S2S
, 1995
"... Subtyping judgments of the polymorphic secondorder typed calculus F extended by recursive types and different known inference rules for these types could be interpreted in S2S, M.Rabin's monadic secondorder theory of two successor functions. On the one hand, this provides a comprehensible ..."
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Subtyping judgments of the polymorphic secondorder typed calculus F extended by recursive types and different known inference rules for these types could be interpreted in S2S, M.Rabin's monadic secondorder theory of two successor functions. On the one hand, this provides a comprehensible model of the parametric and inheritance polymorphisms over recursive types, on the other, proves that the corresponding subtyping theories are not essentially undecidable, i.e., possess consistent decidable extensions.