## A formal calculus for informal equality with binding (2007)

Venue: | In WoLLIC’07: 14th Workshop on Logic, Language, Information and Computation, volume 4576 of LNCS |

Citations: | 13 - 2 self |

### BibTeX

@INPROCEEDINGS{Gabbay07aformal,

author = {Murdoch J. Gabbay and Aad Mathijssen},

title = {A formal calculus for informal equality with binding},

booktitle = {In WoLLIC’07: 14th Workshop on Logic, Language, Information and Computation, volume 4576 of LNCS},

year = {2007},

pages = {162--176}

}

### OpenURL

### Abstract

Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing capture-avoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as first-order logic or the lambda-calculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1

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Citation Context ...particularly process calculi such as those derived from the π-calculus, which often feature quite complex binding side-conditions and for which algebraic reasoning principles are frequently developed =-=[36, 37, 38]-=-. It is possible to extend nominal algebra with abstractions of the form [t]u, and freshness judgements of the form t#u. Other extensions are possible and case studies will tell which of them are most... |

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Citation Context ... moderated unknown. We write Id·X just as X, for brevity. In π · X, X will get substituted for a term and then π will permute the atoms in that term; see Sect. 3. This notion is grounded in semantics =-=[8]-=- and permits a succinct treatment of α-renaming atoms (see CORE below and [9]). A signature Σ is some set of term-formers with their arities. For example: – {lam : 1, app : 2} is a signature for the λ... |

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Citation Context ...s which make them something other than ‘just equalities’. Ways have been developed to attain the simplicity and power of the theory of equality between terms. For example we can work with combinators =-=[1]-=- orcombinatory logic [2], cylindric algebra [3], higher-order algebra [4] or higherorder logic [5]. Roughly speaking: combinatory approaches reject object-level variables entirely; cylindric approach... |

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Citation Context ...(perm) a#X, b#X ⊢ (b a) · X = X Lemma 3.2 below shows that this axiom with the derivation rules of nominal algebra give the native notion of α-equivalence on nominal terms from previous work [9]. See =-=[10]-=- for an axiomatisation of first-order logic. Similar development for other systems with binding, such as System F [11] and the π-calculus [12], should also be possible. 3 A Derivation System Now we ne... |

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Citation Context ...ibes inductive programming and reasoning 6 The reader familiar with a theorem-prover such as Isabelle [16] might like to imagine that # maps to Prop and = maps to o.principles for trees-with-binding =-=[21]-=-. FreshML is a programming language... for trees-with-binding [22] in ML [23] style. Aside from applications to trees-with-binding, ideas from nominal techniques have been used in logic [24] (this was... |

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Citation Context ...h-binding [21]. FreshML is a programming language... for trees-with-binding [22] in ML [23] style. Aside from applications to trees-with-binding, ideas from nominal techniques have been used in logic =-=[24]-=- (this was still reasoning on trees, since the logic had a fixed syntactic model) and recently in semantics [25]. Nominal rewriting considers equalities between terms not directly to do with an underl... |

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Citation Context ...h a theorem-prover such as Isabelle [16] might like to imagine that # maps to Prop and = maps to o.principles for trees-with-binding [21]. FreshML is a programming language... for trees-with-binding =-=[22]-=- in ML [23] style. Aside from applications to trees-with-binding, ideas from nominal techniques have been used in logic [24] (this was still reasoning on trees, since the logic had a fixed syntactic m... |

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Citation Context ...plications to trees-with-binding, ideas from nominal techniques have been used in logic [24] (this was still reasoning on trees, since the logic had a fixed syntactic model) and recently in semantics =-=[25]-=-. Nominal rewriting considers equalities between terms not directly to do with an underlying model of trees [6]. α-prolog [26] takes a similar tack but in logic programming. Still, the emphasis is squ... |

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Citation Context ...ta-variables and reject object-level variables, preferring to encode their expressive power in the term-formers. Examples are lambda-abstraction algebras [31] for the λcalculus and cylindric algebras =-=[32, 33]-=- for first-order logic. Polyadic algebras have a slightly more general treatment, for a brief but clear discussion of the design of these and related systems is in [34, Appendix C]. Combinators [1] re... |

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Citation Context ... (between terms in the signature of T) by the rules in Fig. 2. Here (fr) is subject to a condition that a ̸∈ t, u, ∆ and the square brackets denote discharge of assumptions in natural deduction style =-=[13]-=-. Write ∆ ⊢ t = u when we may derive t = u from ∆, using the signature from T theory T and admitting only the axioms it contains. We write ∆ ⊢ A as a T convenient shorthand for ∆ ⊢ t = u when A is t =... |

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Citation Context ...’ algebraic techniques exist. These embrace meta-variables and reject object-level variables, preferring to encode their expressive power in the term-formers. Examples are lambda-abstraction algebras =-=[31]-=- for the λcalculus and cylindric algebras [32, 33] for first-order logic. Polyadic algebras have a slightly more general treatment, for a brief but clear discussion of the design of these and related ... |

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Citation Context ...particularly process calculi such as those derived from the π-calculus, which often feature quite complex binding side-conditions and for which algebraic reasoning principles are frequently developed =-=[36, 37, 38]-=-. It is possible to extend nominal algebra with abstractions of the form [t]u, and freshness judgements of the form t#u. Other extensions are possible and case studies will tell which of them are most... |

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Citation Context ...-prover such as Isabelle [16] might like to imagine that # maps to Prop and = maps to o.principles for trees-with-binding [21]. FreshML is a programming language... for trees-with-binding [22] in ML =-=[23]-=- style. Aside from applications to trees-with-binding, ideas from nominal techniques have been used in logic [24] (this was still reasoning on trees, since the logic had a fixed syntactic model) and r... |

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Citation Context ...particularly process calculi such as those derived from the π-calculus, which often feature quite complex binding side-conditions and for which algebraic reasoning principles are frequently developed =-=[36, 37, 38]-=-. It is possible to extend nominal algebra with abstractions of the form [t]u, and freshness judgements of the form t#u. Other extensions are possible and case studies will tell which of them are most... |

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Citation Context ...orting system so it will be possible to write ‘silly’ terms. There is no problem in principle with extending the system with a sort or type system if convenient, perhaps along the lines of other work =-=[6, 7]-=-. Let π range over (finitely supported) permutations. So π bijects atoms with themselves and there is a finite set of atoms S such that π(a) = a for all atoms not in S. Write Id for the identity permu... |

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Citation Context ... design of these and related systems is in [34, Appendix C]. Combinators [1] rejectobject-level variables altogether. Algebras over (untyped) combinators can then express first-order predicate logic =-=[35]-=-. These systems are effective for their applications but there are things that nominal algebra allows us to say particularly naturally, because of the way it isolates abstraction, has explicit meta-va... |

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Citation Context ...ssed in this paper by the theory CORE are from this semantics. The name ‘nominal algebra’ acknowledges this debt. Nominal unification considers the computational aspects of unifying treeswith-binding =-=[9]-=-. Nominal logic describes inductive programming and reasoning 6 The reader familiar with a theorem-prover such as Isabelle [16] might like to imagine that # maps to Prop and = maps to o.principles fo... |

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Citation Context ...k for applications in which higher-order logic can be used [27] remains to be made. However we can make some remarks: Unification of nominal terms is decidable whereas higher-order unification is not =-=[28]-=-. (First-order) logic can be axiomatised [10] and the treatment of quantification is very smooth. In particular it closely models the informal specification of quantification; the ∀-intro rule in Isab... |

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Citation Context ...orting system so it will be possible to write ‘silly’ terms. There is no problem in principle with extending the system with a sort or type system if convenient, perhaps along the lines of other work =-=[6, 7]-=-. Let π range over (finitely supported) permutations. So π bijects atoms with themselves and there is a finite set of atoms S such that π(a) = a for all atoms not in S. Write Id for the identity permu... |

1 | Quine’s NF, 60 years on
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Citation Context ...nominal algebra offers two levels of variable, why not extend this to allow an infinite hierarchy of variables, by analogy with type hierarchies in the λ-calculus [1],or stratification in set theory =-=[39]-=-? The first author has considered this extension of nominal terms in other publications [40, 41] but introducing such a hierarchy in a logical context poses unique challenges, and in particular, we ha... |