Smalltalk-80: the language and its implementation
Smalltalk-80: the language and its implementation
On understanding types, data abstraction, and polymorphism
ACM Computing Surveys (CSUR) - The MIT Press scientific computation series
Polymorphic type inference and containment
Logical foundations of functional programming
An extension of system F with subtyping
Information and Computation - Special conference issue: international conference on theoretical aspects of computer software
A Sequent Calculus for Subtyping Polymorphic Types
MFCS '96 Proceedings of the 21st International Symposium on Mathematical Foundations of Computer Science
ECOOP '95 Proceedings of the 9th European Conference on Object-Oriented Programming
Equational Axiomatization of Bicoercibility for Polymorphic Types
Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science
LICS '95 Proceedings of the 10th Annual IEEE Symposium on Logic in Computer Science
The Subtyping Problem for Second-Order Types is Undecidable
LICS '96 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science
Type systems for object-oriented programming languages
Type systems for object-oriented programming languages
A Subtyping for Extensible, Incomplete Objects
Fundamenta Informaticae
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This paper is devoted to a comprehensive study of polymorphic subtypes with products. We first present a sound and complete Hilbert style axiomatization of the relation of being a subtype in presence of →, × type constructors and the ∀ quantifier, and we show that such axiomatization is not encodable in the system with →,∀ only. In order to give a logical semantics to such a subtyping relation, we propose a new form of a sequent which plays a key role in a natural deduction and a Gentzen style calculi. Interestingly enough, the sequent must have the form E⊢T, where E is a non-commutative, non-empty sequence of typing assumptions and T is a finite binary tree of typing judgements, each of them behaving like a pushdown store. We study basic metamathematical properties of the two logical systems, such as subject reduction and cut elimination. Some decidability/undecidability issues related to the presented subtyping relation are also explored: as expected, the subtyping over →,×,∀ is undecidable, being already undecidable for the →,∀ fragment (as proved in [15]), but for the ×,∀ fragment it turns out to be decidable.