Type Checking and Inference for Polymorphic and Existential Types in Multiple-Quantifier and Type-Free Systems

A multiple quantifier is a quantifier having inference rules that introduce or eliminate arbitrary number of quantifiers by one inference. This paper introduces the lambda calculus with negation, conjunction, and multiple existential quantifiers, and the lambda calculus with implication and multiple universal quantifiers. Their type checking and type inference are proved to be undecidable. This paper also shows that the undecidability of type checking and type inference in the type-free-style lambda calculus with negation, conjunction, and existence is reduced to the undecidability of type checking and type inference in the type-free-style polymorphic lambda calculus.