## A Simpler Proof Theory for Nominal Logic (2005)

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Venue: | In FOSSACS 2005, number 3441 in LNCS |

Citations: | 20 - 10 self |

### BibTeX

@INPROCEEDINGS{Cheney05asimpler,

author = {James Cheney},

title = {A Simpler Proof Theory for Nominal Logic},

booktitle = {In FOSSACS 2005, number 3441 in LNCS},

year = {2005},

pages = {379--394},

publisher = {Springer-Verlag}

}

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### Abstract

Nominal logic is a variant of first-order logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the new-quantifier, but we argue that none of these attempts is completely satisfactory. In this paper we develop a new sequent calculus for nominal logic in which the rules for the newquantifier are much simpler than in previous attempts. We also prove several structural and metatheoretic properties, including cut-elimination, consistency, and conservativity with respect to Pitts' axiomatization of nominal logic; these proofs are considerably simpler for our system. 1

### Citations

208 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
- 2003
(Show Context)
Citation Context ...st-order and reasonably well-behaved fragment of Fraenkel-Mostowski set theory, the setting for Gabbay and Pitts’ earlier foundational work on formalizing names, freshness, and binding using swapping =-=[6]-=-. One of the most interesting features of nominal logic is the presence of a novel form of quantification: quantification over fresh names. The formulaN a.ϕ means, intuitively, “for fresh names a, ϕ h... |

167 | Nominal logic: A first order theory of names and binding
- Pitts
(Show Context)
Citation Context ..., including cut-elimination, consistency, and conservativity with respect to Pitts’ axiomatization of nominal logic; these proofs are considerably simpler for our system. 1 Introduction Nominal logic =-=[8]-=- is a variant of first-order logic with additional constructs for dealing with names and binding (or name-abstraction) based on the primitive notions of bijective renaming (swapping) and nameindepende... |

53 | Nominal unification
- Urban, Pitts, et al.
- 2004
(Show Context)
Citation Context ...αProlog. In this paper we present a simplified sequent calculus for nominal logic, called NL⇒ , in which slices are not needed in the rules forN (or anywhere else). Following Urban, Pitts, and Gabbay =-=[11, 4]-=-, we employ a new syntactic class of name-symbols a, b, . . .. Like variables, such name-symbols may be bound (byN ), but unlike variables, two distinct name-symbols are always regarded as denoting di... |

41 | A proof theory for generic judgments: An extended abstract
- Miller, Tiu
(Show Context)
Citation Context ...s much more complicated than experience with αProlog suggests. Gabbay and Cheney also gave a translation from F Oλ∇ , a logic introduced by Miller and Tiu that also includes a self-dual quantifier, ∇ =-=[9]-=- into F LSeq. This translation was sound (mapped derivable sequents to derivable sequents), but incompletesΓ, a # x ⇒ ϕ, ∆ (†) Γ ⇒N a.ϕ, ∆ Γ ⊢ u # t Γ ⊢ ϕ[u/a] (∗) Γ ⊢N a.ϕ Γ, u # t ⇒ ϕ[u/a] (∗) Γ, u ... |

37 | Alpha-prolog: a logic programming language with names, binding and alpha-equivalence, in: Bart Demoen, Vladimir Lifschitz (Eds
- Cheney, Urban
- 2004
(Show Context)
Citation Context ... )). These rules seem simpler and more deterministic; however, they still involve slices. Experience gained in the process of implementing αProlog, a logic programming language based on nominal logic =-=[1]-=-, suggests a much simpler reading ofN as a proof-search operation than that implied by the F L-style rules. In αProlog, when aN -quantifier is encountered (either in a goal or program clause), proof s... |

35 | Meta-programming with names and necessity
- Nanevski
- 2002
(Show Context)
Citation Context ... constraints are not formulas of their logic. These rules are similar in spirit to (and partly inspired) the slice-based rules of F L and F LSeq. Another related system is the type system of Nanevski =-=[10]-=-, which includes rules similar to those of F L forN -quantified types. A third closely related system is Schöpp and Stark’s dependent type theory for names and binding [13], in which a bunched context... |

29 |
Structural Proof Theory
- Negri, Plato
- 2001
(Show Context)
Citation Context ...l rules correspond to first-order universal axioms of nominal logic (Figure 7), which may be incorporated into sequent systems in a uniform fashion using the Ax rule without affecting cut-elimination =-=[7]-=-. The remaining nonlogical rules are as follows. Rule A2 expresses an invertibility property for abstractions: two abstractions are equal only if they are structurally equal or equal by virtue of A1. ... |

23 | Nominal Logic Programming
- Cheney
(Show Context)
Citation Context ... decidability of the equality and typechecking judgments. In this report we present a new and simpler sequent calculus for nominal logic (developed in the course of the author’s dissertation research =-=[3]-=-). Its main novelty is the use of a freshness context to manage freshness information needed in reasoning aboutN -quantified formulas, rather than the technically more cumbersome slices used in F L an... |

17 |
Basic Proof Theory,” Number 43
- Troelstra, Schwichtenberg
- 1996
(Show Context)
Citation Context ...s in which Π and Π ′ both start with a rule in which ϕ is principal. All cases involving first-order rules exclusively are standard, and are shown in any standard proof of cutelimination (e.g. [7] or =-=[10]-=-). In addition, Negri and von Plato [7] showed that nonlogical rules of the form we consider can be added to sequent systems like G3c or G3im without damaging cut-elimination. Hence, it will suffice t... |

15 | A Dependent Type Theory with Names and Binding
- Schöpp, Stark
- 2004
(Show Context)
Citation Context ... and Cheney [5] presented F LSeq, a sequent calculus version of Fresh Logic. Schöpp and Stark have developed a dependent type theory of names and binding that contains nominal logic as a special case =-=[9]-=-. However, none of these formalizations is ideal. Hilbert systems have well-known deficiencies for computer science applications. F L and F LSeq rely on a complicated technical device called slices fo... |

2 |
Fresh logic: A logic of FM
- Gabbay
- 2003
(Show Context)
Citation Context ...or all fresh names; in either case, we say that N a.ϕ holds. Several formalizations of nominal logic have been investigated. Pitts introduced nominal logic as a Hilbert-style axiomatic system. Gabbay =-=[4]-=- proposed Fresh Logic (F L), an intuitionistic Gentzen-style natural deduction system. Gabbay and Cheney [5] presented F LSeq, a sequent calculus version of Fresh Logic. Schöpp and Stark have develope... |

2 |
A spatial logic for concurrency–II
- Caires, Cardelli
- 2004
(Show Context)
Citation Context ...malizations of nominal logic by Pitts, Gabbay, and Cheney (surveyed in Section 1.1), several other logics and type systems have considered rules for -quantified formulas or types. Caires and Cardelli =-=[1]-=- investigated a logic incorporating proof rules forN -quantified formulas based on maintaining a set of side-conditions involving freshness constraints. However, the freshness constraints are not form... |

1 |
A sound and complete translation of generic judgments into nominal logic
- Cheney
- 2005
(Show Context)
Citation Context ...dness and completeness. We also sketch a proof that the original translation is complete with respect to F Oλ∇ with ∇-weakening and exchange. Full proofs will be given in a companion technical report =-=[4]-=-. Our translation T departs from TGC in two ways. First, TGC translated c-formulas such as x⊲ϕ∧ψ by first usingN -quantifiers for the local context, then translating ϕ∧ψ, and finally substituting n(a)... |