## External and internal syntax of the λ-calculus

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Venue: | In: Buchberger, Ida, Kutsia (Eds.), Proc. of the Austrian-Japanese Workshop on Symbolic Computation in Software Science, SCSS 2008. No. 08–08 in RISC-Linz Report Series |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Sato_externaland,

author = {Masahiko Sato and Randy Pollack},

title = {External and internal syntax of the λ-calculus},

booktitle = {In: Buchberger, Ida, Kutsia (Eds.), Proc. of the Austrian-Japanese Workshop on Symbolic Computation in Software Science, SCSS 2008. No. 08–08 in RISC-Linz Report Series},

year = {},

pages = {176--195}

}

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### Abstract

There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic and computer science, we cannot avoid teaching the

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Citation Context ...1) and Church (1941), perhaps because of the influence of Russell and Whitehead (1910). McCarthy contributed to semantical understanding of syntactic objects by introducing Lisp symbolic expressions (=-=McCarthy, 1960-=-), and the concept abstract syntax (McCarthy, 1963), providing functions to analyze and synthesize syntactic objects while hiding the concrete representation of these objects. This approach works well... |

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Citation Context ...uk (Randy Pollack). 1 Pollack is partially supported by EPSRC Platform Grant EPE/005713/1. Preprint submitted to Elsevier 1 October 2009We think that Frege (1879), McCarthy (1960, 1963), Martin-Löf (=-=Nordström et al., 1990-=-, Chapter 3) and Gabbay and Pitts (1999, 2002) contributed much to the semantical study of syntax. The work which we report here is influenced by these works and in particular by the works of Frege an... |

206 | A new approach to abstract syntax with variable binding
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(Show Context)
Citation Context ...) and we have the same result for [(lam Y (lam X (app Y X)))]. We now verify the adequacy (see Harper et al. (1993)) of our definition of α-equivalence against the definition of α-equivalence due to (=-=Gabbay and Pitts, 2002-=-; Pitts, 2003). Their definition, in our notation, is as follows. M : Λ M =α M M =α P N =α Q (app M N) =α (app P Q) [X /Z]M =α [Y /Z]N Z ̸∈ GV(M) ∪ GV(N) (lam X M) =α (lam Y N) The adequacy is establi... |

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Citation Context ... α-equivalence on languages with binders using explicit variable names (as opposed to nameless variables, e.g. de Bruijn indices). Nominal techniques have sinsce been extened to more standard logics (=-=Pitts, 2003-=-; Urban, 2008). In this paper we work in standard mathematics and develop our theory by introducing a new notion of B-algebra (‘B’ is for ‘binding’) which is an algebra equipped with the mechanism of ... |

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Citation Context ...uijn indices rather than natural numbers serving as names. This approach goes back to the original paper on indices by de Bruijn (1972), and its formalization in a computer proof tool is detailed in (=-=Aydemir et al., 2008-=-). This locally nameless representation works very well, but the syntactic representation of terms does not have a name at binding points, which differs from informal practice. Further, some of the an... |

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(Show Context)
Citation Context ...e on languages with binders using explicit variable names (as opposed to nameless variables, e.g. de Bruijn indices). Nominal techniques have sinsce been extened to more standard logics (Pitts, 2003; =-=Urban, 2008-=-). In this paper we work in standard mathematics and develop our theory by introducing a new notion of B-algebra (‘B’ is for ‘binding’) which is an algebra equipped with the mechanism of variable bind... |

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Citation Context ...lam [x ][x/X]M) : L (∗). In this “forward” rule, X and M vary together. To see where case (3) comes from notice rule (∗) could equivalently be stated as a “backwards” rule, analogous to that used in (=-=McKinna and Pollack, 1993-=-, 1999; Aydemir et al., 2008) X ̸∈ GV(M) [X/x]M : L x = HX([X/x]M) (lam [x ]M) : L (∗∗). 12In this form X varies independently of M, and it is clear that any sufficiently fresh X will do in the premi... |

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Citation Context ...tion. Theorem 1 (Substitution). If P, Q : L then [Q/Y ]P : L. 13The proof of theorem 1 encounters a problem that is well known and discussed in the literature, e.g. (McKinna and Pollack, 1993, 1999; =-=Urban et al., 2007-=-; Pitts, 2003). We want to do induction on the derivation of P : L. In case P is an abstraction, the induction hypothesis mentions some particular X. To make the argument go through we need to swap X ... |

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7 |
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Citation Context ...ally. To answer this question, we have to study syntax semantically. Our contribution in this paper is the result of such a study. We have already contributed in this study in (Sato and Hagiya, 1981; =-=Sato, 1983-=-, 1985, 1991, 2002, 2008b) by investigating the mathematical structure of symbolic expressions. ⋆ This paper is an extensively revised version of Sato (Sato, 2008a) which was presented at SCSS 2008. E... |

5 |
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(Show Context)
Citation Context ...nt of view. This is because in order to apply this rule and obtain a new λ-term as the result of the application, we must somehow know the very λ-term we wish to construct. As we already stressed in (=-=Sato, 2002-=-), we believe that every mathematical object, including of 20course every λ-term, must be constructed by applying a constructor function to already created objects. This rule does not follow this ont... |

4 | Elements of mathematics. Theory of sets, Addison-Wesley Publ - Bourbaki - 1968 |

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3 | A canonical locally named representation of binding
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Citation Context ...tion of λ-terms, such as the nominal Isabelle representation (Urban, 2008), or the locally nameless representation (Aydemir et al., 2008). You can see where we stand at time of writing by looking at (=-=Pollack and Sato, 2009-=-). Beyond theory, there is application. We claim our representation is more elegant than the closest comparable work, (Aydemir et al., 2008). But is it more practical to use? We must do more interesti... |

3 | Inverting Inductively Defined Relations in - McBride - 1996 |

2 | A framework for checking proofs naturally - Sato, 2008b |

1 | A basis for a mathematical theory of computation - unknown authors - 1963 |

1 |
Hyperlisp
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- 1981
(Show Context)
Citation Context ...ating symbols syntactically. To answer this question, we have to study syntax semantically. Our contribution in this paper is the result of such a study. We have already contributed in this study in (=-=Sato and Hagiya, 1981-=-; Sato, 1983, 1985, 1991, 2002, 2008b) by investigating the mathematical structure of symbolic expressions. ⋆ This paper is an extensively revised version of Sato (Sato, 2008a) which was presented at ... |