## A proof theory for generic judgments: An extended abstract (2003)

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Venue: | In LICS 2003 |

Citations: | 41 - 15 self |

### BibTeX

@INPROCEEDINGS{Miller03aproof,

author = {Dale Miller and Alwen Tiu},

title = {A proof theory for generic judgments: An extended abstract},

booktitle = {In LICS 2003},

year = {2003},

pages = {118--127},

publisher = {IEEE}

}

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### Abstract

A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the π-calculus and the encoding of objectlevel provability.

### Citations

995 | A calculus of mobile processes
- Milner, Parrow, et al.
- 1992
(Show Context)
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847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
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302 |
Higher-order abstract syntax
- Pfenning, Elliot
- 1988
(Show Context)
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287 | A logic programming language with lambda-abstraction, function variables, and simple unification
- Miller
- 1991
(Show Context)
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259 |
Investigations into logical deduction
- Gentzen
- 1969
(Show Context)
Citation Context ...of, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are called eigenvariables. In Gentzen’s original presentation of the sequent calculus =-=[5]-=-, eigenvariables were immutable: reading proofs bottom-up, once an eigenvariable is introduced it is not used as a site for substitution. In other words, Gentzen’s eigenvariables did not vary in proof... |

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- Gabbay, Pitts
- 2002
(Show Context)
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- Pitts
(Show Context)
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- Cervesato, Durgin, et al.
- 1999
(Show Context)
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124 | Unification under a mixed prefix
- Miller
- 1992
(Show Context)
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101 | The π-calculus as a theory in linear logic: Preliminary results
- Miller
- 1993
(Show Context)
Citation Context ...bles as fresh, scoped constants Focusing on their intensional nature and guarantee of newness or freshness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus =-=[15]-=-, nonces in security protocols [1], reference locations in imperative programming [2, 16], and constructors hidden within abstract data-types [12]. Eigenvariables also provide an essentialΣ : (σ, y :... |

90 | Reasoning with higher-order abstract syntax in a logical framework - McDowell, Miller |

68 |
A proof-theoretic approach to logic programming. II. Programs as definitions
- Hallnäs, Schroeder-Heister
- 1991
(Show Context)
Citation Context ...logic specification involves instantiations of eigenvariables. Similarly, focusing on their extensional nature guaranteed by cut-elimination, enrichments to the sequent calculus have been proposed by =-=[7, 24, 6, 9]-=- in which eigenvariables are intended as variables to be substituted. This enrichment to proof theory (discussed here in Section 4) holds promise for providing proof systems for the direct reasoning o... |

65 | Lexical scoping as universal quantification
- Miller
- 1989
(Show Context)
Citation Context ...en used to encode name restrictions in the π-calculus [15], nonces in security protocols [1], reference locations in imperative programming [2, 16], and constructors hidden within abstract data-types =-=[12]-=-. Eigenvariables also provide an essentialΣ : (σ, y : γ) ⊲ B[y/x], Γ −→ C Σ : σ ⊲ ∇γx.B, Γ −→ C Σ : Γ −→ (σ, y : γ) ⊲ B[y/x] Σ : Γ −→ σ ⊲ ∇γx.B ∇L ∇R Figure 1. Rules for the ∇-quantifier. aspect of r... |

61 | Cut-elimination for a logic with definitions and induction
- McDowell, Miller
- 2000
(Show Context)
Citation Context ...f h will be γ1 → · · · → γn → γ0 instead of simply γ0. In the inference rules of Figure 2, we write (hσ) to denote (hx1 . . . xn). For the sake of consistency with a naming convention from the papers =-=[8, 9]-=-, 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 λ-expressions”). The proof system resulting from the addition of the rul... |

61 |
Functional unification of higher-order patterns
- Nipkow
- 1993
(Show Context)
Citation Context ...rorder unification problems and higher-order substitutions, the unification problems generated from this particular example fall within the higher-order pattern unification or Lλ unification problems =-=[13, 21]-=-. This subset of the unification of simply typed λ-terms has complexity similar to that of first-order unification: it is decidable (in linear time) and has most general unifiers when unifiers exist. ... |

42 | Proof Theoretic Approach to Specification Languages
- Chirimar
- 1995
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Citation Context ...newness or freshness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus [15], nonces in security protocols [1], reference locations in imperative programming =-=[2, 16]-=-, and constructors hidden within abstract data-types [12]. Eigenvariables also provide an essentialΣ : (σ, y : γ) ⊲ B[y/x], Γ −→ C Σ : σ ⊲ ∇γx.B, Γ −→ C Σ : Γ −→ (σ, y : γ) ⊲ B[y/x] Σ : Γ −→ σ ⊲ ∇γx.... |

40 |
A fixpoint theorem in linear logic. An email posting to the mailing list linear@cs.stanford.edu
- Girard
- 1992
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Citation Context ...hment to proof theory (discussed here in Section 4) holds promise for providing proof systems for the direct reasoning of logic specifications (see, for example, the above mentioned papers as well as =-=[10, 11]-=-). These two approaches are, however, at odds with each other. Consider, for example, the problem of representing restriction of names or nonces using ∀ quantification. (The following example can be d... |

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- Schroeder-Heister
- 1992
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- Miller
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Citation Context ...newness or freshness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus [15], nonces in security protocols [1], reference locations in imperative programming =-=[2, 16]-=-, and constructors hidden within abstract data-types [12]. Eigenvariables also provide an essentialΣ : (σ, y : γ) ⊲ B[y/x], Γ −→ C Σ : σ ⊲ ∇γx.B, Γ −→ C Σ : Γ −→ (σ, y : γ) ⊲ B[y/x] Σ : Γ −→ σ ⊲ ∇γx.... |

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Citation Context ... −−→ νn.(Mn | Nn) ↑xy −−→ P ′ Q ↓x P | Q −−⇀ N τ −−→ P ′ | (Ny) close com Figure 4. The rules for the (late) π-calculus. Consider encoding π-calculus [19] using higher-order abstract syntax following =-=[17, 18]-=-. 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 proces... |

8 |
A hybrid encoding of Howe’s method for establishing congruence of bisimilarity
- Momigliano, Ambler, et al.
- 2002
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Citation Context ...s than those shown here, one needs an implementation of F Oλ∆∇ . The Isabelle theorem prover should provide a promising setting for building an 9interactive theorem prover given the work reported in =-=[20]-=-. A natural next step is to attempt adding directly to F Oλ ∆∇ induction and co-induction: induction should work much as it does in F Oλ ∆IN [9]. Some related work on co-induction appears in [20]. Ack... |

3 | Encoding generic judgments
- Miller, Tiu
- 2002
(Show Context)
Citation Context ... −−→ νn.(Mn | Nn) ↑xy −−→ P ′ Q ↓x P | Q −−⇀ N τ −−→ P ′ | (Ny) close com Figure 4. The rules for the (late) π-calculus. Consider encoding π-calculus [19] using higher-order abstract syntax following =-=[17, 18]-=-. 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 proces... |

3 | Cut elimination for a logic with generic judgments and induction, Tech. rep., CoRR, extended version of LFMTP’06 paper. Available from http://arxiv.org/abs/ 0801.3065
- Tiu
- 2008
(Show Context)
Citation Context ...lvl(∇x.B) = lvl(∃x.B) = lvl(B). We shall require that for every definitional clause ∀¯x[p ¯t △ = B], lvl(B) ≤ lvl(p). This requirement allows us to prove cut-elimination for F Oλ∆∇ (see Section 5 and =-=[9, 25]-=-). Introduction rules for defined atoms involve the use of substitutions. We recall some basic definitions related to substitutions. A substitution θ is a mapping (with application written in postfix ... |