## Cut-elimination for a logic with definitions and induction (1997)

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Venue: | Theoretical Computer Science |

Citations: | 61 - 19 self |

### BibTeX

@ARTICLE{Mcdowell97cut-eliminationfor,

author = {Raymond Mcdowell and Dale Miller},

title = {Cut-elimination for a logic with definitions and induction},

journal = {Theoretical Computer Science},

year = {1997},

volume = {232},

pages = {2000}

}

### Years of Citing Articles

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### Abstract

In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic F Oλ ∆IN that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cut-elimination theorem for F Oλ ∆IN, adapting a technique due to Tait and Martin-Löf. This cut-elimination proof is technically interesting and significantly extends previous results of this kind. 1

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Citation Context ... only definitions with a finite number of clauses and restrict our uses of the def L rule to those formulas A such that for every definitional clause there is a finite, complete set of unifiers (CSU) =-=[11] of -=-A and the head of the clause. In this case we can implement Eriksson's rule [5] fB`; \Gamma` \Gamma! C` j ` 2 CSU(A;A 0 ) for some clause 8��x[A 0 4 = B]g A; \Gamma \Gamma! C def LCSU ; where the ... |

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Citation Context ...definitions to have only a finite number of clauses and to restrict the use of def L rule to those formulas A such that for every definitional clause there is a finite, complete set of unifiers (CSU) =-=[11]-=- of A and the head of the clause. Consider the following inference rule due to Eriksson [5] {Bθ, Γθ −→ Cθ | θ ∈ CSU(A, A ′ ) for some clause ∀¯x[A ′ △ = B]} A, Γ −→ C def LCSU , where the variables ¯x... |

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Citation Context ...tain completeness of uniform proofs, restrictions on logical formulas need to maintained. For example, completeness of uniform proofs can be achieved in classical logic by restricting to Horn clauses =-=[19]-=-; in intuitionistic logic by restricting to hereditary Harrop formulas [18]; and in linear logic by choosing the proper logical connectives [1, 17]. There are numerous examples of specifying computati... |

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Citation Context ...of of the cut-elimination theorem for F Oλ ∆IN uses a technique introduced by Tait [29] to prove normal form theorems. Martin-Löf extended the method to apply beyond terms to natural deduction proofs =-=[12]-=-, and we use it here in a sequent calculus setting. Rather than associate an induction measure with derivations, we use the derivations themselves as a measure by defining well-founded orderings on de... |

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27 | A finitary version of the calculus of partial inductive definitions
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Citation Context ...on natural numbers and inference rules for treating logic specifications as definitions instead of as theories. Our approach to definitions follows lines developed by Schroeder-Heister [25], Eriksson =-=[5]-=-, Girard [9], and Stärk [28]. Our needs for reasoning about specifications, however, forced us to develop a single extension to intuitionistic logic, called F Oλ ∆IN (pronounced “fold-n”), that goes b... |

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Citation Context ...s of the clauses. Definitions are employed in F Oλ∆IN via left and right introduction rules for atomic formulas. If we impose no restrictions on definitions, the cut-elimination theorem does not hold =-=[24]-=-. Two different approaches have been taken to retain the admissibility of cut. First, if the structural rule of contraction is removed or restricted (as it is in linear logic, for example), cut-elimin... |

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Citation Context ...fficulties can be overcome within F Oλ ∆IN . See [13, 14] for more on how F Oλ ∆IN can be used as a meta-logic for an intuitionistic and linear logical framework. The Pi derivation editor of Eriksson =-=[6]-=- was designed for the finitary calculus of partial inductive definitions [5]. Because of F Oλ ∆IN ’s close relationship with the finitary calculus of partial inductive definitions, the Pi editor can b... |

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Citation Context ...umbers and inference rules for treating logic specifications as definitions instead of as theories. Our approach to definitions follows lines developed by Schroeder-Heister [25], Eriksson [5], Girard =-=[9]-=-, and Stärk [28]. Our needs for reasoning about specifications, however, forced us to develop a single extension to intuitionistic logic, called F Oλ ∆IN (pronounced “fold-n”), that goes beyond the lo... |