## Computational Adequacy of the FIX-Logic (1995)

### BibTeX

@MISC{Crole95computationaladequacy,

author = {Roy L. Crole},

title = {Computational Adequacy of the FIX-Logic},

year = {1995}

}

### OpenURL

### Abstract

This paper presents computational adequacy results for the FIX logical system introduced by Crole and Pitts in LICS '90. More precisely, we take two simple PCF style languages (whose dynamic semantics follow a call-by-value and call-by-name regime) give translations of the languages into suitable judgements in the FIX-logic and prove that the translations are adequate for the static and dynamic semantics. This shows that the FIX-logic can be regarded as a programming metalogic which will uniformly interpret both call-by-value and call-by-name languages. The proofs of dynamic adequacy make use of a logical relations technique which is based on the methods of Plotkin and Tait. We also show that there is some choice in the translation of recursion; certain translations make use of an existence property of the FIX-logic to prove computational adequacy.

### Citations

1285 | A structural approach to operational semantics
- Plotkin
- 2004
(Show Context)
Citation Context ...| λx: σ.m, 6and we write Canσ ⊆ Exp σ for the set of canonical expressions assigned the type σ. We now give the syntax of QL a call by value dynamic semantics via an evaluation relation (see [7] and =-=[12]-=-) which will take the form m ⇓ c, where m and c are closed QL expressions and c is canonical. The rules for generating the evaluation relation are given below: Canonical Forms c canonical c ⇓ c Condit... |

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ... conditional at ground types and fixpoint operators. This syntax is then equipped with a call-by-name operational semantics, giving rise to the language PCF which was first investigated by Plotkin in =-=[11]-=-. The two languages we investigate here, which we call QL and HPCF, resemble PCF in that their syntax consists essentially of simply typed lambda calculus with extra arithmetical, procedural and fixpo... |

301 | Natural Semantics - Kahn - 1987 |

267 |
Semantics of Programming Languages: Structures and Techniques. Foundations of Computing
- Gunter
- 1992
(Show Context)
Citation Context ... semantics in the same way: for details of the original specification of Plotkin’s PCF in this style of semantics see [11]. Note that good textbook accounts of such operational semantics are [14] and =-=[6]-=-. We omit the details, but remark that the reflexive, transitive closure of such a single step reduction relation will yield the natural semantics style reduction relation: Proposition 5.2 Let m and c... |

134 | Elements of Intuitionism - Dummett - 1977 |

117 | Observable Properties of Higher Order Functions that Dynamically Create Local Names, or What’s new
- Pitts, Stark
- 1993
(Show Context)
Citation Context ... to see just how useful logics such as FIX are for reasoning about realistic languages. Recent work of Pitts and Stark has highlighted the problems associated with local store in the language ML; see =-=[9]-=-. Work in progress is considering “realistic” fragments of ML (not involving local state) and developing programming logics which are based on both FIX and also Evaluation Logic [8]. We hope to implem... |

88 | The HOL logic
- Pitts
- 1993
(Show Context)
Citation Context ...existence property of the FIX-logic to prove computational adequacy. 11 Introduction This paper makes use of the FIX logical system (which was introduced in [3]) as a programming metalogic (see also =-=[8]-=-) into which programming languages can be translated and reasoned with. In this introduction we give a summary of the crucial elements of the FIX logical system and then give a brief heuristic outline... |

69 |
The Formal Semantics of Programming Languages. Foundation of Computing Series
- Winskel
- 1993
(Show Context)
Citation Context ...erational semantics in the same way: for details of the original specification of Plotkin’s PCF in this style of semantics see [11]. Note that good textbook accounts of such operational semantics are =-=[14]-=- and [6]. We omit the details, but remark that the reflexive, transitive closure of such a single step reduction relation will yield the natural semantics style reduction relation: Proposition 5.2 Let... |

43 |
New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic
- Crole, Pitts
- 1990
(Show Context)
Citation Context ...n give a brief heuristic outline of two simple programming languages which will be translated into the FIX-logic. A Summary of the FIX Logical System NB: A complete description of FIX can be found in =-=[4]-=-; we shall give only an outline of the format of FIX, and readers are urged to consult the latter paper in detail if they are not familiar with the FIX logical system. FIX is an intuitionistic predica... |

33 |
Denotational semantics with partial functions
- Plotkin
- 1985
(Show Context)
Citation Context ..., we have xi: σi, f: σ ′ ⇒ σ, x: σ ′ ⊢ m: σ and xi: σi ⊢ n: σ ′ , from which xi: σi ⊢ R(f.x.m,n): σ is immediate. ✷ Dynamic Adequacy of QL We shall prove a theorem based on Plotkin’s methods given in =-=[13]-=-. We write D(−) for the composition [−]ωCPO ◦ ⃗u.[−] v (⃗x): QL → FIX → ωCPO where [−]ωCPO is the standard domain theoretic semantics of FIX. We define a relation ⊳σ between elements d ∈ D(σ) and cano... |

18 |
Categories for Types. Cambridge Mathematical Textbooks
- Crole
- 1993
(Show Context)
Citation Context ... existence of a term denoting that value.) 3A Categorical Semantics for FIX We outline the essence of a categorical semantics for a FIX-theory—for a general introduction to categorical semantics see =-=[2]-=-. Let C be a category. A FIX-hyperdoctrine is a C-indexed poset C: C op → POSet which has enough structure to model FIX. The types are modelled by objects in C, and FIX-terms by morphisms. Each FIXpro... |

12 | Programming Metalogics with a Fixpoint Type
- Crole
- 1992
(Show Context)
Citation Context ...nd the sequents-in-context are interpreted by inequalities (inclusions) in such posets. As regards notation in this paper, we just note that ωCPO is an example of a cartesian closed let-category (see =-=[1]-=-) for which the operation of lifting gives rise to the let-category structure. Let D, D ′ and D ′′ be three ω-cpos. We write D × D ′ for the binary product of D and D ′ , D ⇒ D ′ for the exponential o... |

12 | New Foundations for Fixpoint Computations
- Crole, Pitts
- 1990
(Show Context)
Citation Context ...sion; certain translations make use of an existence property of the FIX-logic to prove computational adequacy. 11 Introduction This paper makes use of the FIX logical system (which was introduced in =-=[3]-=-) as a programming metalogic (see also [8]) into which programming languages can be translated and reasoned with. In this introduction we give a summary of the crucial elements of the FIX logical syst... |

7 | Call by name, call by value and the -calculus - Plotkin - 1975 |

5 |
Call by name, call by value and the λ calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...tional semantics is call-by-value. HPCF has a call-by-name operational semantics and apart from conditionals at higher types is identical to PCF. A discussion of evaluation strategies can be found in =-=[10]-=-. 2 The Language QL We define the language QL by specifying the basic syntax of types and raw expressions; this syntax will then be given a static and dynamic semantics. 5The Types and Expressions of... |

2 | New foundations for computations - Crole, Pitts - 1990 |