. The metavariable self is fundamental in object-oriented 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 NP-complete. With recursive types and subtyping alone, type inference is known to be P-complete. 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---...

Both subtyping and typing relations in the system F , the well-known second-order 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 F-subtyping 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 F-provable; 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 F-minimum types [CG92], are effectively computable (as opposed to F ); there exists a decision procedure, which given a context \Gamma and a term t a...

. 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 non-variable 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 0-ary) 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...

System F , the second-order 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. Proof Normalization and Subject Reduction in Extensions of F Sergei Vorobyov Max-Planck-Institut fur Informatik Im Stadtwald, D-66123, Saarbrucken, Germany (e-mail: sv@mpi-sb.mpg.de, Phone: (49) 681-302-5391, Fax: (49) 681-302-5401) January 16, 1995 Abstract System F , the second-order polymorphic typed -calculus with subtyping [CW85, BL90, BTCCS91, CG92, CMMS94], appeared to be undecidable because ...

Subtyping judgments of the polymorphic second-order typed -calculus F extended by recursive types and different known inference rules for these types could be interpreted in S2S, M.Rabin's monadic second-order 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.