## Logical Foundations of Eval/Quote Mechanisms, and the Modal Logic S4 (1997)

Venue: | IN PRESS S1570-8683(05)00043-1/FLA AID:71 Vol.•••(•••) [DTD5] P.12 (1-12) JAL:m1a v 1.40 Prn:15/07/2005; 8:08 jal71 by:SL p. 12 12 N. Alechina, D. Shkatov / Journal of Applied Logic |

Citations: | 5 - 0 self |

### BibTeX

@TECHREPORT{Goubault-larrecq97logicalfoundations,

author = {Jean Goubault-larrecq},

title = {Logical Foundations of Eval/Quote Mechanisms, and the Modal Logic S4},

institution = {IN PRESS S1570-8683(05)00043-1/FLA AID:71 Vol.•••(•••) [DTD5] P.12 (1-12) JAL:m1a v 1.40 Prn:15/07/2005; 8:08 jal71 by:SL p. 12 12 N. Alechina, D. Shkatov / Journal of Applied Logic},

year = {1997}

}

### OpenURL

### Abstract

Starting from the idea that cut elimination is the precise meaning of program execution, we design two languages of constructions for the minimal logic S4, yielding -calculi with idealized versions of Lisp's eval and quote. The first, the S4 -calculus, is based on Bierman and De Paiva's proposal, and has all desirable logical properties, except for its non-operational flavor. The second, the evQ-calculus, is more complicated, but has a clear operational meaning: it is a tower of interpreters in the style of Lisp's reflexive tower. Remarkably, this language was developed from purely logical principles, but nonetheless provides some operational insight into eval/quote mechanisms. 1 Introduction Let's consider two dual questions. The first is: is there a proofs-as-programs, formulasas -types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and - calculi [How80], and also for classical logic [Gri90] or linear logic [Abr93], so why not S4? A...

### Citations

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(Show Context)
Citation Context ...consider two dual questions. The first is: is there a proofs-as-programs, formulasas -types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and - calculi =-=[How80]-=-, and also for classical logic [Gri90] or linear logic [Abr93], so why not S4? As already noted by Davies and Pfenning [DP95], the answer is a kind of -calculus augmented with polished versions of Lis... |

391 | Explicit substitutions
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Citation Context ...s have been omitted for lack of space. They can be found in the full papers [GL96a, GL96b, GL96c]. We assume the reader to be familiar with the CurryHoward correspondence [GLT89], and with oe-calculi =-=[ACCL90]-=-. 2 Related Works The search for functional interpretations of modal logics began with Gabbay and De Queiroz [GdQ90], who encode Kripke models directly inside the proof structure (pp.38--39). They int... |

281 | Computational interpretations of linear logic
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Citation Context ...s-programs, formulasas -types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and - calculi [How80], and also for classical logic [Gri90] or linear logic =-=[Abr93]-=-, so why not S4? As already noted by Davies and Pfenning [DP95], the answer is a kind of -calculus augmented with polished versions of Lisp's eval and quote primitives. The second question is: what ty... |

234 | A formulae-as-types notion of control
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Citation Context ... is: is there a proofs-as-programs, formulasas -types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and - calculi [How80], and also for classical logic =-=[Gri90]-=- or linear logic [Abr93], so why not S4? As already noted by Davies and Pfenning [DP95], the answer is a kind of -calculus augmented with polished versions of Lisp's eval and quote primitives. The sec... |

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(Show Context)
Citation Context ... S4? There is one between minimal and intuitionistic logics and - calculi [How80], and also for classical logic [Gri90] or linear logic [Abr93], so why not S4? As already noted by Davies and Pfenning =-=[DP95]-=-, the answer is a kind of -calculus augmented with polished versions of Lisp's eval and quote primitives. The second question is: what type system can we graft on eval and quote that would make them u... |

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Citation Context ...n the premise of the rule. Our goal is to find correct constructions , u (eval u or ; u in Lisp) and u ` (quote, or rather `u) for the modal rules (2L) and (2R). Cut elimination works mainly as usual =-=[Sch77]-=-: the computational content of cut elimination arises from cuts between a left and a right rule, where the cut formula is the active formula in both rules. For instance, a cut between ()L) and ()R) yi... |

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(Show Context)
Citation Context ...ing new operators ? 2 ,s2 , ffi 2 , : : : We end up having the same problem, so we create new operatorss? 3 ,s3 , ffi 3 , and so on. Polishing the rough edges, we get the following calculus, based on =-=[HL89]-=- to avoid confluence problems: Definition 5.1 (evQ) The evQ-terms u, v, w, : : : are elements of the language T [ S, where: T ::= V j V \Delta T j TT j 1S js` T j T ? ` T j T ffi ` T S j 1 ` j ev ` T ... |

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(Show Context)
Citation Context ...by a compiler front-end. It is remarkable that the need for closures in, say, Bierman and De Paiva's language, parallels that for syntactic closures in the implementation of macro packages for Scheme =-=[BR88]-=-. We also feel that S4 provides an appropriate foundation for typing mobile code, provided that we understand quoting (wrapping up) as preparing a piece of program for sending (over a network), and ev... |

34 |
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Citation Context ... typed evQcalculus is confluent, then it is a conservative extension of the typeds�� S4 -calculus, i.e. for every typedsS4 -terms u and v, u /! S4 v if and only if G(u) /! evQG(v). This is similar=-= to [Har89]-=-. However, the evQ-calculus is not a conservative m-extension (in the sense of op.cit.): G(u) \Gamma! G(v) in evQ does not imply u \Gamma! v insS4 . (Take u = unbox (xy) ` , v = (unbox x ` )(unbox y `... |

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(Show Context)
Citation Context ...er to be familiar with the CurryHoward correspondence [GLT89], and with oe-calculi [ACCL90]. 2 Related Works The search for functional interpretations of modal logics began with Gabbay and De Queiroz =-=[GdQ90]-=-, who encode Kripke models directly inside the proof structure (pp.38--39). They introduce world variables w, write 8w \Delta \Phi(w) instead of 2\Phi; proof rules are then copied from the usual proof... |

12 |
On a modal -calculus for S4
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(Show Context)
Citation Context ...on they propose is however incomplete, insofar as there may remain cuts in proofs corresponding to normal terms, i.e. execution of programs may deadlock (see Section 4). Conversely, Pfenning and Wong =-=[PW95]-=- have tried to understand how S4 works, in the hope of understanding more about linear logic. They define thes!2 - calculus of constructions for S4. The careful reader will notice that their sequents ... |

8 | On computational interpretations of the modal logic S4 I. Cut elimination, Interner Bericht 1996-35 - Goubault-Larrecq - 1996 |

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- Goubault-Larrecq
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(Show Context)
Citation Context ... u \Gamma! v insS4 . (Take u = unbox (xy) ` , v = (unbox x ` )(unbox y ` ).) 1 This is a long and rather difficult proof, which we have not had time yet to write up in full; this should be ready soon =-=[GL96c]-=-. 7 Conclusion We have layed down rigorous logical foundations of idealized versions of Lisp's eval/quote mechanism. ThesS4 - calculus justifies the notion of quotation closures, generalizing the synt... |

7 |
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(Show Context)
Citation Context ...ons of S4 to staged computations and run-time code generation [DP95], to macrolanguages, to mobile code perhaps (see Section 2). Other exciting perspectives are to investigate Godel-like translations =-=[Fef84]-=- from intuitionistic logic to S4, which seem to be related to macro-expansions; a full study of this needs to understand how classical S4 behaves [GL96d]. The relationships between S4 and linear logic... |

6 | On computational interpretations of the modal logic S4 II. The evQ-calculus - Goubault-Larrecq - 1996 |

5 | Confluence and preservation of strong normalization in an explicit substitutions calculus - Hurtado - 1996 |

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1 | On computational interpretations of the modal logic S4 IV. Classical S4 - Goubault-Larrecq - 1996 |