## A Proof Theory for Generic Judgments (2003)

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@MISC{Miller03aproof,

author = {Dale Miller and Alwen Tiu},

title = {A Proof Theory for Generic Judgments},

year = {2003}

}

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### Abstract

this paper, we do this by adding the #-quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important

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Citation Context ...stractions over names that are used for such things as nonces in security protocols [Cervesato et al. 1999], locations for reference cells [Chirimar 1995; Miller 1996], or new communication channels [=-=Milner et al. 1992-=-]. One declarative way to capture these features in the inference rule setting is to employ scoped (eigen)variables. Given the logic F Oλ ∇ , we now have the ability to scope variables within sequents... |

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Citation Context ...o. N, December 2004.s4 · D. Miller and A. Tiu with local scope. The syntax of the formula ∇τ x.B is like that for the universal and existential quantifiers. Following Church’s Simple Theory of Types [=-=Church 1940-=-], formulas are given the type o, and for all types τ not containing o, ∇τ is a constant of type (τ → o) → o. The expression ∇τ λx.B is usually abbreviated as simply ∇τ x.B or as ∇x.B if the type info... |

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Citation Context ...tered higher-order unification problems and higher-order substitutions, the unification problems generated from this particular example fall within Lλ-unification or higher-order pattern unification [=-=Miller 1991-=-; Nipkow 1993]. This subset of the unification of simply typed λ-terms has complexity similar to that of first-order unification, in that it is decidable and has most general unifiers when unifiers ex... |

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Citation Context ...uent calculus proofs, such new variables are called eigenvariables, and they are used to prove universally quantified formulas generically. In Gentzen’s original presentation of the sequent calculus [=-=Gentzen 1969-=-], eigenvariables are immutable during proof search: once an eigenvariable is introduced (reading proofs bottom-up), it is not used as a site for substitution. In other words, eigenvariables did not v... |

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Citation Context ...t an attempt at a notion of name “freshness” or a semantics for “name generation”. It is natural to ask about possible connections between the ∇-quantifier and the new quantifier of Pitts and Gabbay [=-=Gabbay and Pitts 2001-=-; Pitts 2003]. Both are self dual and both have similar sets of applications in mind. The focus on ∇ has been proof theoretic while the work on Pitts and Gabbay has been model theoretic. More concrete... |

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Citation Context ...Focusing on their intensional nature and guarantee of newness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus [Miller 1993], nonces in security protocols [=-=Cervesato et al. 1999-=-], reference locations in imperative programming [Pfenning and Rohwedder 1992; Chirimar 1995; Cervesato and Pfenning 1996; Miller 1996], new assumptions in encodings of natural deduction or sequent ca... |

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Citation Context ...ogic specification involves instantiations of eigenvariables. Similarly, focusing on their extensional nature guaranteed by cut-elimination, enrichments to the sequent calculus have been proposed by [=-=Hallnäs and Schroeder-Heister 1991-=-; SchroederHeister 1992; Girard 1992; McDowell and Miller 2000] in which eigenvariables are intended as variables to be substituted during proof search. This enrichment to proof theory (discussed here... |

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Citation Context ...imar 1995; Cervesato and Pfenning 1996; Miller 1996], new assumptions in encodings of natural deduction or sequent calculi [Felty and Miller 1988], and constructors hidden within abstract data-types [=-=Miller 1989-=-]. Eigenvariables also provide an essential aspect of recursive programming with data encoded using λ-tree syntax [Miller 2000]: to move recursively through syntax that is an outermost binder, instant... |

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Citation Context ...a or abstraction over a formula. In (6), x is not free in B. inference rules of Figure 2, we write (hσ) to denote (hx1 . . . xn). For the sake of consistency with a naming convention from the papers [=-=McDowell 1997-=-; McDowell and Miller 2000], we shall refer to the inference system defined with just the rules in Figure 2 as F Oλ (mnemonic for a “first-order logic for λexpressions”). The proof system resulting fr... |

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Citation Context ...order unification problems and higher-order substitutions, the unification problems generated from this particular example fall within Lλ-unification or higher-order pattern unification [Miller 1991; =-=Nipkow 1993-=-]. This subset of the unification of simply typed λ-terms has complexity similar to that of first-order unification, in that it is decidable and has most general unifiers when unifiers exist. Certain ... |

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Citation Context ...een used to encode name restrictions in the π-calculus [Miller 1993], nonces in security protocols [Cervesato et al. 1999], reference locations in imperative programming [Pfenning and Rohwedder 1992; =-=Chirimar 1995-=-; Cervesato and Pfenning 1996; Miller 1996], new assumptions in encodings of natural deduction or sequent calculi [Felty and Miller 1988], and constructors hidden within abstract data-types [Miller 19... |

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Citation Context ...ause. Then the following inference rules can be shown interadmissible with defL: {Σθ; Bθ, Γθ −→ Cθ | θ ∈ CSU(A, H) for some clause ∀¯ h[H △ = B]} Σ; A, Γ −→ C defLcsu. This rule is originally due to [=-=Eriksson 1991-=-] and is also used in [McDowell and Miller 2000]. The proof of its interadmissibility with defL follows the same outline as the one in [McDowell and Miller 2000]. The meta-theoretic analysis of defini... |

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Citation Context ...EFINITIONS Introduction rules are, generally, restricted to logical connectives and quantifiers. The recent development of a proof theoretic notion of definitions [Hallnäs and Schroeder-Heister 1991; =-=Schroeder-Heister 1992-=-; Girard 1992; McDowell and Miller 2000] provides left and right introduction rules also for non-logical predicate symbols, provided that they are “defined” appropriately. Given certain restrictions o... |

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Citation Context ...d the existing free names are represented in the same way. This approach is also interesting in that it relates certain aspects of names to the way they are quantified, in particular, it is shown in [=-=Tiu and Miller 2004-=-] (where precise connections between this style specification and open and late bisimulations are given) that different ways of quantifying free names result in different bisimulation relations. Note ... |

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Citation Context ... ↓X −−⇀ N P | Q τ close −−→ νn.(Mn | Nn) P ↑XY −−→ P ′ Q ↓X −−⇀ N P | Q τ −−→ P ′ | (NY ) Consider encoding π-calculus [Milner et al. 1992] using λ-tree syntax following [Miller and Palamidessi 1999; =-=Miller and Tiu 2002-=-]. Since we are focused here on abstractions in syntax, we shall deal with only finite π-calculus expression; that is, expressions without ! or defined constants. Extending this work to infinite proce... |

1 |
A proof theory for generic judgments · 33
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- 2004
(Show Context)
Citation Context ...ntailed by ∀ and entails ∃. In Pitts and Gabbay, the domain of quantification is fixed to a certain denumerably infinite set of names, while the ∇ quantifier works at any type. In their recent paper [=-=Gabbay and Cheney 2004-=-], Gabbay and Cheney provide some initial connections between these two quantifiers. ACM Transactions on Computational Logic, Vol. V, No. N, December 2004.s32 · D. Miller and A. Tiu Pursuing such a co... |

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