## Predicate transformers and Linear Logic - yet another Denotational Model (2004)

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Venue: | In http://jumpstart.anr.mcnc.org |

Citations: | 7 - 6 self |

### BibTeX

@INPROCEEDINGS{Hyvernat04predicatetransformers,

author = {Pierre Hyvernat},

title = {Predicate transformers and Linear Logic - yet another Denotational Model},

booktitle = {In http://jumpstart.anr.mcnc.org},

year = {2004},

pages = {115--129},

publisher = {Springer-Verlag}

}

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### Abstract

In the refinement calculus, monotonic predicate transformers are used to model specifications for (imperative) programs. Together with a natural notion of simulation, they form a category enjoying many algebraic properties. We build on this structure to make predicate transformers into a denotational model of full linear logic: all the logical constructions have a natural interpretation in terms of predicate transformers (i.e. in terms of specifications). We then interpret proofs of a formula by a safety property for the corresponding specification.

### Citations

615 | Linear logic
- Girard
- 1987
(Show Context)
Citation Context ...tions). We then interpret proofs of a formula by a safety property for the corresponding specification. Introduction The first denotational model for linear logic was the category of coherent spaces (=-=[1]-=-). In this model, formulas are interpreted by graphs and proofs by cliques (complete subgraphs): this forms a special case of domain a la Scott. From a conceptual point of view, the construction of in... |

59 | Hypercoherences: a strongly stable model of linear logic
- Ehrhard
- 1993
(Show Context)
Citation Context ...ery crude model interprets formulas by sets; and proofs by subsets. It is degenerate in the sense that any formula is identified with its linear negation! Coherent spaces ([1]), hypercoherent spaces (=-=[5]-=-), finiteness spaces ([6]) remove (part of) this degeneracy by adding structure on top of the relational model. We follow the same approach: Definition 4. A interface X is given by a set |X | (called ... |

40 |
Refinement Calculus: a systematic introduction. Graduate texts in computer science
- Back, Wright
- 1998
(Show Context)
Citation Context ...e want to reach) a set of initial states (which guarantee that we will reach our goal). 3 For a complete introduction to the field of predicate transformers in relation to specifications, we refer to =-=[7]. In the c-=-oherence semantics, a "point" is a complete subgraph, 4 called a clique. Since the intuitions behind our objects are quite di#erent, we change the terminology. Definition 5. Let X be a inter... |

30 | Refinement Calculus
- Back, Wright
- 1998
(Show Context)
Citation Context ... (|X |s|Y |, PX# P Y ) where PX# P Y (r) is the predicate transformer r ## [ xy#r PX (x) P Y (y) . We write it X# Y . PX# P Y is the most natural transformer to construct on |X |s|Y |. It was used in =-=[8]-=- to model parallel execution of independent pieces of programs. The intuition is the following: a program satisfies PX# P Y if when you start it in the pair (a i , b i ) # PX# P Y (r) of initial state... |

20 | O.: An algebraic construction of predicate transformers
- Gardiner, Martin, et al.
- 1994
(Show Context)
Citation Context ...; -- Rel, where morphisms are (binary) relations; -- Pow, where morphisms are monotonic predicate transformers. One can go from Set to Rel and from Rel to Pow using the same categorical construction (=-=[4]-=-) which cannot be applied further. Definition 2. A predicate transformer from A to B is a function from P(A) to P(B). A predicate transformer P is monotonic if x # x # implies P (x) # P (x # ). From n... |

19 |
T.: On phase semantics and denotational semantics: the exponentials
- Bucciarelli, Ehrhard
- 2001
(Show Context)
Citation Context ...is a "non-uniform" model in the sense that the web of !X contains all finite multisets, not just those whose underlying set is a seed. It is thus closer to non-uniform (hyper)coherence seman=-=tics (see [10]-=- or [11]) than to the traditional (hyper)coherence semantics. 7 defined on the disjoint sum of the di#erent index sets 8 The interpretation of !, like that of# is a synchronous operation. Let's prove ... |

8 |
The differential lambda-calculus. Theoretical Computer Science, 309(1-3):1–41, 2003. 14 Thomas Ehrhard and Laurent Regnier. Böhm trees, Krivine machine and the Taylor expansion of ordinary lambda-terms
- Ehrhard, Regnier
(Show Context)
Citation Context ... or finiteness spaces. A promising direction for further research is to explore the links between the model presented below and non-determinism as it appears both in the differential lambda-calculus (=-=[2,3]-=-) and different kind of process calculi. We expect such a link because of the following remarks: this model comes from the semantics of imperative languages; it can be extended to a model of the diffe... |

4 |
Non-uniform hypercoherences
- Boudes
(Show Context)
Citation Context ...n-uniform" model in the sense that the web of !X contains all finite multisets, not just those whose underlying set is a seed. It is thus closer to non-uniform (hyper)coherence semantics (see [10=-=] or [11]-=-) than to the traditional (hyper)coherence semantics. 7 defined on the disjoint sum of the di#erent index sets 8 The interpretation of !, like that of# is a synchronous operation. Let's prove a simple... |

3 | R.: A specification structure for deadlockfreedom of synchronous processes. Theoretical Computer Science 222
- Abramsky, Gay, et al.
- 1999
(Show Context)
Citation Context ...is category is an enrichment of the usual category Rel. The construction can be summarized in the following way: Lemma 8. Int is obtained by lifting Rel through the following specification structure (=-=[9]-=-): -- if X is a set, PrX # P(X) # P(X); -- if r # X Y , P # PrX and Q # Pr Y , then P{r}Q i# #r#sP # Qs#r#. Let's now turn our attention to the structure of this category: Lemma 9. In Int, # is termin... |

2 |
L.: Differential interaction nets. unpublished note
- Ehrhard, Regnier
- 2004
(Show Context)
Citation Context ...e or finiteness spaces. A promising direction for further research is to explore the links between the model presented below and non-determinism as it appears both in the di#erential lambda-calculus (=-=[2, 3]-=-) and di#erent kind of process calculi. We expect such a link because of the following remarks: this model comes from the semantics of imperative languages; it can be extended to a model of the di#ere... |

2 |
Finiteness spaces. to appear
- Ehrhard
- 2004
(Show Context)
Citation Context ...s formulas by sets; and proofs by subsets. It is degenerate in the sense that any formula is identified with its linear negation! Coherent spaces ([1]), hypercoherent spaces ([5]), finiteness spaces (=-=[6]-=-) remove (part of) this degeneracy by adding structure on top of the relational model. We follow the same approach: Definition 4. A interface X is given by a set |X | (called the state space) and a pr... |

1 |
L.: The di#erential lambda calculus. Theoretical Computer Science 309
- Ehrhard, Regnier
- 2003
(Show Context)
Citation Context ...e or finiteness spaces. A promising direction for further research is to explore the links between the model presented below and non-determinism as it appears both in the di#erential lambda-calculus (=-=[2, 3]-=-) and di#erent kind of process calculi. We expect such a link because of the following remarks: this model comes from the semantics of imperative languages; it can be extended to a model of the di#ere... |