## Mechanizing Logical Relations (1994)

Venue: | Proc. Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science 802, Springer-Verlag |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Stoughton94mechanizinglogical,

author = {Allen Stoughton},

title = {Mechanizing Logical Relations},

booktitle = {Proc. Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science 802, Springer-Verlag},

year = {1994},

pages = {1--9}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We give an algorithm for deciding whether there exists a definable element of a finite model of an applied typed lambda calculus that passes certain tests, in the special case when all the constants and test arguments are of order at most one. When there is such an element, the algorithm outputs a term that passes the tests; otherwise, the algorithm outputs a logical relation that demonstrates the nonexistence of such an element. Several example applications of the C implementation of this algorithm are considered. 1 Introduction Given a model of an applied typed lambda calculus, it is natural to consider the problem of determining whether an element of that model is definable by a term, or, more generally, of determining whether there exists a definable element of the model that passes certain tests. One approach to settling such questions makes use of so-called "logical relations" [Plo80]. Building on recent work on logical relations by Sieber [Sie92], we give an algorithm for dec...

### Citations

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...hat a = [[M ]]. We say that A is finite iff A is finite, and that A is infinite otherwise. Our example model in the sequel will be the monotone function model of Finitary PCF : the restriction of PCF =-=[Plo77]-=- to the booleans. We write FPCF for the family of constants such that FPCF ' = f\Omega ; tt; ffg, FPCF ' 3 = If, and FPCF oe = ; for all other oe 2 T , and define a finite (FPCF)- model F as follows. ... |

239 | The lazy lambda calculus
- Abramsky
- 1990
(Show Context)
Citation Context ...; ffi is in the complement of the relation. Figure 3 shows that parallel or remains non-definable when parallel convergence is added to Finitary PCF (the original proof of this result can be found in =-=[Abr90]-=-). This time the C program produced by Lambda was run in verbose mode, with the consequence that the elements of the resulting relation are labeled with the stages at which they were found. The relati... |

141 |
Type systems for programming languages
- Mitchell
- 1990
(Show Context)
Citation Context ...pported by ESPRIT project CLICS-II and was performed while the author was on the faculty of the School of Cognitive and Computing Sciences of the University of Sussex. calculus can be found, e.g., in =-=[Mit90]. The set of-=- types T is least such that (i) ' 2 T , (ii) oe ! �� 2 T if oe 2 T and �� 2 T . We let ! associate to the right. The order ord oe 2 ! of a type oe 2 T is defined by ord ' = 0 and ord(oe!��... |

58 |
Lambda-definability in the full type hierarchy
- Plotkin
- 1980
(Show Context)
Citation Context ... or, more generally, of determining whether there exists a definable element of the model that passes certain tests. One approach to settling such questions makes use of so-called "logical relati=-=ons" [Plo80]-=-. Building on recent work on logical relations by Sieber [Sie92], we give an algorithm for deciding whether there exists a definable element of a finite model that passes certain tests, in the special... |

32 |
Reasoning about sequential functions via logical relations
- Sieber
- 1992
(Show Context)
Citation Context ...able element of the model that passes certain tests. One approach to settling such questions makes use of so-called "logical relations" [Plo80]. Building on recent work on logical relations =-=by Sieber [Sie92]-=-, we give an algorithm for deciding whether there exists a definable element of a finite model that passes certain tests, in the special case when all the constants and test arguments are of order at ... |

26 | Tiuryn, A New Characterization of Lambda Definability
- Jung, A
- 1993
(Show Context)
Citation Context ...ralization of Theorem 4.1 of [Sie92] shows that it does provide a complete method in the special case where the orders of C and the oe i 's are at most one (cf., Theorem 1 of [Plo80] and Theorem 5 of =-=[JT93]-=-). Definition 3.3 Suppose A is a (C )-model and \Delta = h \Delta i 2 A ff oe i j i 2 m i, for m 2 ! and an ordinal ff and where C and the oe i 's have order at most one. Then, R(\Delta) is the ff-ary... |

18 |
Studying the fully abstract model of PCF within its continuous function model
- Jung, Stoughton
- 1993
(Show Context)
Citation Context ... order-three elements of the full type hierarchy over a seven element set.) The algorithm was first implemented in Standard ML and used to find an interesting nondefinability proof (see Lemma 4.16 of =-=[JS93]-=-). An efficient implementation of the algorithm in ANSI C has now been written and applied to various definability problems, some examples of which are described below. A copy of this program, Lambda,... |

4 |
The undecidability of -definability
- Loader
- 1994
(Show Context)
Citation Context ...es the tests; otherwise, the algorithm outputs a logical relation that demonstrates the nonexistence of such an element. Loader's recent proof of the undecidability of the lambda definability problem =-=[Loa94]-=- shows that the restriction to constants and test arguments of order at most one is necessary. (Specifically, Loader shows the undecidability of the problem of determining the definability of order-th... |