## On The Algebraic Models Of Lambda Calculus (1997)

Venue: | Theoretical Computer Science |

Citations: | 20 - 11 self |

### BibTeX

@ARTICLE{Salibra97onthe,

author = {Antonino Salibra},

title = {On The Algebraic Models Of Lambda Calculus},

journal = {Theoretical Computer Science},

year = {1997},

volume = {249},

pages = {197--240}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a first-order algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus...

### Citations

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Citation Context ...ized for some time among computer scientists because it leads among other things to more natural term rewriting systems, which are useful in the analysis of processes of computations. See for example =-=[1]-=-. In the transformation algebras and substitution algebras of LeBlanc [27] and Pinter [38] substitution is primitive and abstract quantification is defined in terms of it. A pure theory of abstract su... |

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Citation Context ...eferences will be [35] and [37] for lambda abstraction algebras and Barendregt's book [3] for lambda calculus. 4 ANTONINO SALIBRA Lambda calculus. The untyped lambda calculus was introduced by Church =-=[9, 10] as a foun-=-dation for logic. Although the appearance of paradoxes caused the program to fail, a consistent part of the theory turned out to be successful as a theory of "functions as rules" (formalized... |

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Citation Context ...1.23]). We follow Barendregt [3; pag. 217] and identify all Bohm-like trees that differ only in the names of bound -variables. The best way to do this is to use the notation of de Bruijn explained in =-=[7]-=- and in Appendix C of [3]. To keep matters readable we will write Bohm-like trees in the naive way. Consider the following algebra in the similarity type of LAA I 's BT = hB; \Delta BT ; x BT ; x BT i... |

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Citation Context ...algebras have a simple purely equational characterization. Curry also specified (by a considerably less natural set of axioms) a purely equational subclass of combinatory algebras, the -algebras (see =-=[3]-=-, 5.2.5), that he viewed as algebraic models of the lambda calculus. Although lambda calculus has been the subject of research by logicians since the early 1930's, its model theory developed only much... |

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Citation Context ...eferences will be [35] and [37] for lambda abstraction algebras and Barendregt's book [3] for lambda calculus. 4 ANTONINO SALIBRA Lambda calculus. The untyped lambda calculus was introduced by Church =-=[9, 10] as a foun-=-dation for logic. Although the appearance of paradoxes caused the program to fail, a consistent part of the theory turned out to be successful as a theory of "functions as rules" (formalized... |

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Citation Context ... the Cartesian product A n . 6. Conclusion and Related Work The way in which lambda abstraction theory arises from the lambda calculus almost exactly parallels the way cylindric and polyadic algebras =-=[22, 21]-=- are obtained from first-order logic. The axioms of first-order logic are like those of lambda calculus in that the formula variables cannot be substituted without restriction. In both cases the sourc... |

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Citation Context ...to reformulate the lambda calculus as a purely algebraic theory within the context of category theory: Obtu/lowicz and Wieger [31] via the algebraic theories of Lawvere; Adachi [2] via monads; Curien =-=[11]-=- via categorical combinators. There have also been several works that present an algebraic theory of the lambda calculus very close to lambda calculus in spirit. Locally finite functional LAA's are ve... |

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Citation Context ... the Cartesian product A n . 6. Conclusion and Related Work The way in which lambda abstraction theory arises from the lambda calculus almost exactly parallels the way cylindric and polyadic algebras =-=[22, 21]-=- are obtained from first-order logic. The axioms of first-order logic are like those of lambda calculus in that the formula variables cannot be substituted without restriction. In both cases the sourc... |

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Citation Context ...s. This abstraction of substitution is a characteristic feature of algebraic logic. Lambda abstraction theory has been extensively developed by Goldblatt, Pigozzi and the author in a series of papers =-=[17, 18, 32, 33, 34, 35, 37, 40]-=-, and, as in the theory of cylindric and polyadic algebras, the emphasis is on representation results. The most natural LAAs are functional algebras (FLAs) consisting of suitable functions obtained fr... |

6 |
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Citation Context ...s. This abstraction of substitution is a characteristic feature of algebraic logic. Lambda abstraction theory has been extensively developed by Goldblatt, Pigozzi and the author in a series of papers =-=[17, 18, 32, 33, 34, 35, 37, 40]-=-, and, as in the theory of cylindric and polyadic algebras, the emphasis is on representation results. The most natural LAAs are functional algebras (FLAs) consisting of suitable functions obtained fr... |

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Citation Context ...re arbitrary infinitary -terms. (ff') x:A = y:A[x := y], for any -variable y that does not occur free in A; (fi') (x:A)B = A[x := B]; (1) A = A; (2) A = B implies B = A; (3) A = B, B = C imply A = C; =-=(4)-=- A = B, C = D imply AC = BD; (5) A = B implies x:A = x:B. An infinitary lambda theory is any set of equations between infinitary -terms that is closed under (ff')- and (fi')-conversion and the five eq... |

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Citation Context ...roperties of a lambda theory by means of the variety of LAA's generated by its term algebra. Recent work has been done by many authors on infinitary versions of lambda calculus. Berarducci defines in =-=[5]-=- a new model of fi-calculus which is similar to the model of Bohm trees, but it does not identify all the unsolvable lambda terms. His method, that is based on an infinitary version of the lambda calc... |

5 |
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Citation Context ...culus which is similar to the model of Bohm trees, but it does not identify all the unsolvable lambda terms. His method, that is based on an infinitary version of the lambda calculus, is also used in =-=[6]-=- to obtain Church-Rosser extensions of the finitary lambda calculus. Another infinitary version of lambda calculus has been independently introduced by Kenneway, Klop, Sleep and Van de Vries in [25]. ... |

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Citation Context ...of cylindric and polyadic algebras are two early contributions to the algebraization of quantifier logics and have greatly influenced our work. The main references for cylindric algebras are [22] and =-=[23]-=-; for polyadic algebras it is [21]; see in particular [20]. We also mention here Nemeti [30]. It contains an extensive survey of the various algebraic versions of quantifier logics. The importance of ... |

3 |
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Citation Context ... been several attempts to reformulate the lambda calculus as a purely algebraic theory within the context of category theory: Obtu/lowicz and Wieger [31] via the algebraic theories of Lawvere; Adachi =-=[2]-=- via monads; Curien [11] via categorical combinators. There have also been several works that present an algebraic theory of the lambda calculus very close to lambda calculus in spirit. Locally finite... |

3 |
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(Show Context)
Citation Context ... models do not have an explicit algebraic structure. An abstractly defined class of algebras, called lambda substitution algebras, that is even closer in spirit to LAA's has been introduced by Diskin =-=[13, 14]-=-. The theories of cylindric and polyadic algebras are two early contributions to the algebraization of quantifier logics and have greatly influenced our work. The main references for cylindric algebra... |

2 |
Lambda term systems, Frame Inform Systems
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(Show Context)
Citation Context ... models do not have an explicit algebraic structure. An abstractly defined class of algebras, called lambda substitution algebras, that is even closer in spirit to LAA's has been introduced by Diskin =-=[13, 14]-=-. The theories of cylindric and polyadic algebras are two early contributions to the algebraization of quantifier logics and have greatly influenced our work. The main references for cylindric algebra... |

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Citation Context ...AA I A is rich if, for all finite-dimensional elements a; b 2 A and all x 2 I , we have that (8c 2 Zd A : (x:a)c = (x:b)c) ) a = b: Rich LAA I 's correspond roughly to rich polyadic Boolean algebras (=-=[20]-=-). Let V be an arbitrary variety of algebras and A 2 V . We recall that A is generic in V if an identity holds in A iff it holds in V (see Gratzer [19, p. 383]). We recall that CL is the equational th... |

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