## Domains and Lambda-CalculiThis book describes the mathematical aspects of the semantics of programming languages. The main goals are to provide formal tools to assess the meaning of programming constructs in both a language-independent and a machine-independent way and to prove properties about programs, such as whether they terminate, or whether their result is a solution of the problem they are supposed to solve. In order to achieve this the authors first present, in an elementary and unified way, the theory of certain topological spaces that have proved of use in the modeling of various families of typed lambda calculi considered as core programming languages and as meta-languages for denotational semantics. This theory is now known as Domain Theory, and was founded as a subject by Scott and Plotkin. One of the main concerns is to establish links between mathematical structures and more syntactic approaches to semantics, often referred to as operational semantics, which is also described. This dual approach has the double advantage of motivating computer scientists to do some mathematics and of interesting mathematicians in unfamiliar application areas from computer science. |

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### Contents

Notation ху | 1 |

Syntactic theory of the lcalculus | 22 |

Doo models and intersection types | 43 |

Interpretation of lcalculi in CCCs | 71 |

The Language PCF | 124 |

Domain equations | 144 |

Values and computations | 168 |

Powerdomains | 200 |

Stability | 270 |

Towards linear logic | 301 |

Sequentiality | 341 |

Domains and realizability | 388 |

Functions and processes | 421 |

Summary of recursion theory | 449 |

469 | |

480 | |

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### Common terms and phrases

abstract algebraic algorithms applied associated basic bound calculus called cartesian chapter claim closed coherence collection compact complete computation condition consider consists constant construction context continuous coprime correspondence cpo's dcpo defined definition denote derivation directed domain elements equations equivalence evaluation example Exercise exists extended fact figure finite function functor given Hence holds implies induction interpretation introduce isomorphic l-calculus labelled language least lemma linear locale logic means monotonic morphism names natural normal notion object observe obtained operator output pair partial order particular preorder present projection PROOF proposition prove recursive reduction relation Remark respectively result retraction rules satisfies Scott semantics sequential space stable structure subset Suppose theorem theory unique valof variables write

### Popular passages

Page 470 - Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183-220, 1992.