Extensional Models of Untyped Lambda-mu Calculus

This paper proposes new mathematical models of the untyped Lambda-mu calculus. One is called the stream model, which is an extension of the lambda model, in which each term is interpreted as a function from streams to individual data. The other is called the stream combinatory algebra, which is an extension of the combinatory algebra, and it is proved that the extensional equality of the Lambda-mu calculus is equivalent to equality in stream combinatory algebras. In order to define the stream combinatory algebra, we introduce a combinatory calculus SCL, which is an abstraction-free system corresponding to the Lambda-mu calculus. Moreover, it is shown that stream models are algebraically characterized as a particular class of stream combinatory algebras.