## Higher-order Representation of Substructural Logics (2009)

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Citations: | 4 - 0 self |

### BibTeX

@MISC{Crary09higher-orderrepresentation,

author = {Karl Crary},

title = {Higher-order Representation of Substructural Logics},

year = {2009}

}

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### Abstract

We present a technique for higher-order representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, metatheoretic proofs about such logics can easily be developed in the Twelf proof assistant.

### Citations

696 | A Framework for Defining Logics
- Harper, Honsel, et al.
- 1993
(Show Context)
Citation Context ... without any linear or modal extensions. Using this encoding, metatheoretic proofs about such logics can easily be developed in the Twelf proof assistant. 1 Introduction The Logical Framework (or LF) =-=[6]-=- provides a powerful and flexible framework for encoding deductive systems such as programming languages and logics. LF employs an elegant account of binding structure by identifying object-language v... |

302 |
Higher-order abstract syntax
- Pfenning, Elliot
- 1988
(Show Context)
Citation Context ...es, object-language contexts with (fragments of) the LF context, and object-language binding occurrences with LF lambda abstraction. This account of binding, often called higher-order abstract syntax =-=[12]-=-, automatically handles most operations that pertain to binding, including alpha-equivalence, substitution, and variablefreshness conventions [3]. Since the object-language context is maintained impli... |

215 | Logical frameworks
- Pfenning
- 2001
(Show Context)
Citation Context ..., linear logics (by design) satisfy neither weakening nor contraction, so it would seem that they cannot be encoded in LF. One solution to this problem is to extend LF with linear features. Linear LF =-=[4]-=- extends LF with linear assumptions and connectives. This provides the ability to encode linear logics. However, linearity has yet to be implemented in Twelf [13], the proof assistant that implements ... |

190 | The Logic of Bunched Implications
- O’Hearn, Pym
- 1999
(Show Context)
Citation Context ...es as linear logic. We conjecture that contextual modal logic [9] is encodable along similar lines as judgemental modal logic. This is a good avenue for future work. The logic of bunched implications =-=[10]-=- is another. On the other hand, since our method relies on enforcing “substructuralness” on an assumption-by-assumption basis, there are some substructural logics it does not support, such as ordered ... |

159 | A judgmental reconstruction of modal logic
- Pfenning, Davies
(Show Context)
Citation Context ...d exactly once) in Mx.” In this paper, we illustrate the use of a substructural judgement (such as linear) in three settings: linear logic, dependently typed linear logic, and judgemental modal logic =-=[11]-=-. Many other substructural logics including affine logic and strict logic can be handled analogously. Some others, such as ordered logic [15, 14], cannot, because the rules of the logic make it imposs... |

100 | The Proof Theory and Semantics of Intuitionistic Modal Logic
- Simpson
- 1994
(Show Context)
Citation Context ...Lemma 3.4, it is easy to verify that �−� and �−� are inverses. □ 4 Modal Logic There are (at least) two ways to specify modal logic. One is using an explicit notion of Kripke worlds and accessibility =-=[16]-=-. Such a formulation does not behave as a substructural logic and can be encoded in LF without difficulty [8]. A second, which we consider here, is judgemental modal logic [11]. Judgemental modal logi... |

83 | Using typed lambda calculus to implement formal systems on a machine
- Avron, Honsell, et al.
- 1987
(Show Context)
Citation Context ...use of assumptions, we need look only at proof terms; there is no need to examine typing derivations. The idea of linearity as a judgement over proof terms dates to the early days of LF. Avron et al. =-=[2, 1]-=- suggested that linearity can be expressed by imposing a lattice structure on proof terms and defining linear proof terms as those that are strict and distributive, when viewed as a function of their ... |

83 | Engineering formal metatheory
- Aydemir, Charguéraud, et al.
- 2008
(Show Context)
Citation Context ...of binding, often called higher-order abstract syntax [12], automatically handles most operations that pertain to binding, including alpha-equivalence, substitution, and variablefreshness conventions =-=[3]-=-. Since the object-language context is maintained implicitly, as part of the built-in LF context, the structural properties of LF contexts (such as weakening and contraction) automatically apply to th... |

83 | On equivalence and canonical forms in the LF type theory - Harper, Pfenning |

56 |
Contextual modal type theory
- Nanevski, Pfenning, et al.
(Show Context)
Citation Context ...ategy applies to other substructural logics as well. For example, affine logic and strict logic can each be encoded along very similar lines as linear logic. We conjecture that contextual modal logic =-=[9]-=- is encodable along similar lines as judgemental modal logic. This is a good avenue for future work. The logic of bunched implications [10] is another. On the other hand, since our method relies on en... |

39 | Structural Cut Elimination in Linear Logic
- Pfenning
- 1994
(Show Context)
Citation Context ...ssumption basis, without reference to contexts. Thus, wherever an assumption is introduced, as part of the typing rule that introduced it, we can check that that assumption is used linearly. Pfenning =-=[13]-=- proposed enforcing linearity using a metajudgement that traced the use of an assumption throughout a typing derivation. Avron, et al. [3] later used a similar approach for modal logic. Unfortunately,... |

35 | Ordered linear logic and applications
- Polakow
- 2001
(Show Context)
Citation Context ...ependently typed linear logic, and judgemental modal logic [11]. Many other substructural logics including affine logic and strict logic can be handled analogously. Some others, such as ordered logic =-=[15, 14]-=-, cannot, because the rules of the logic make it impossible to handle assumptions independently. We briefly discuss the latter in Section 5. The full Twelf development can be found on-line at: www.cs.... |

34 | Higher-order rewriting with dependent types
- Virga
- 1999
(Show Context)
Citation Context ...n contribute to t. More precisely, a type family s is subordinate to an type family t if there exist types S and T belonging to s and t such that objects of type S can appear within objects of type T =-=[17]-=-. If s is not subordinate to t, then assumptions whose types belong to s can be ignored while considering t. tp : type. A ::= ... · · · atomic : atom -> tp. | a const : constant -> term -> tp. | c(M) ... |

33 | Natural deduction for intuitionistic non-commutative linear logic
- Polakow, Pfenning
- 1999
(Show Context)
Citation Context ...ependently typed linear logic, and judgemental modal logic [11]. Many other substructural logics including affine logic and strict logic can be handled analogously. Some others, such as ordered logic =-=[15, 14]-=-, cannot, because the rules of the logic make it impossible to handle assumptions independently. We briefly discuss the latter in Section 5. The full Twelf development can be found on-line at: www.cs.... |

19 | Modal Types for Mobile Code
- Murphy
- 2008
(Show Context)
Citation Context ...o specify modal logic. One is using an explicit notion of Kripke worlds and accessibility [16]. Such a formulation does not behave as a substructural logic and can be encoded in LF without difficulty =-=[8]-=-. A second, which we consider here, is judgemental modal logic [11]. Judgemental modal logic distinguishes between two sorts of assumption, truth and validity. Although judgemental modal logic has no ... |

14 | Encoding modal logics in logical frameworks
- Avron, Honsell, et al.
- 1998
(Show Context)
Citation Context ...we can check that that assumption is used linearly. Pfenning [13] proposed enforcing linearity using a metajudgement that traced the use of an assumption throughout a typing derivation. Avron, et al. =-=[3]-=- later used a similar approach for modal logic. Unfortunately, the meta-judgement approach is very awkward to use in practice. Also both were able to prove adequacy for their encodings, neither (so fa... |

11 |
Twelf User’s Guide: Version 1.4, 2002. Avaiable from http://www.cs.cmu.edu/ ∼ twelf
- Pfenning, Schúrmann
(Show Context)
Citation Context ...end LF with linear features. Linear LF [4] extends LF with linear assumptions and connectives. This provides the ability to encode linear logics. However, linearity has yet to be implemented in Twelf =-=[13]-=-, the proof assistant that implements LF, in part due to unresolved complications that linearity creates in its metalogical apparatus. Consequently, Linear LF is not currently an option for those enga... |

9 | An overview of the edinburgh logical framework
- Avron, Honsell, et al.
- 1989
(Show Context)
Citation Context ...use of assumptions, we need look only at proof terms; there is no need to examine typing derivations. The idea of linearity as a judgement over proof terms dates to the early days of LF. Avron et al. =-=[2, 1]-=- suggested that linearity can be expressed by imposing a lattice structure on proof terms and defining linear proof terms as those that are strict and distributive, when viewed as a function of their ... |

6 | Explicit contexts in LF
- Crary
- 2008
(Show Context)
Citation Context ... not give us any assistance with other substructural logics, such as affine, strict, or modal logic. Another option is to break with standard LF practice and model object-language contexts explicitly =-=[5]-=-. Explicit 1 Substructural logics may be defined in various different ways. For our purposes, we define substructural logic to mean any logic in which it is not the case that every bound variable can ... |