## Adding the axioms to Axiom: Towards a system of automated reasoning in Aldor (1998)

Venue: | Computing Laboratory, University of Kent |

Citations: | 6 - 1 self |

### BibTeX

@TECHREPORT{Poll98addingthe,

author = {Erik Poll and Simon Thompson},

title = {Adding the axioms to Axiom: Towards a system of automated reasoning in Aldor},

institution = {Computing Laboratory, University of Kent},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

A number of combinations of theorem proving and computer algebra systems have been proposed; in this paper we describe another, namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor -- the Axiom Library Compiler -- and show that with some modifications we can use the dependent types of the system to model a logic, under the Curry-Howard isomorphism. We give a number of example applications of the logic we construct. 1 Introduction Symbolic mathematical -- or computer algebra -- systems, such as Axiom [JS92], Maple and Mathematica, are in everyday use by scientists, engineers and indeed mathematicians, because they provide a user with techniques of, say, integration which far exceed those of the person themselves, and make routine many calculations which would have been impossible some years ago. These systems are, moreover, taught as standard tools within many university undergraduate programmes and are used in support of both ac...

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Citation Context ...uage is functionally based and provides higher order-functions, generators (which bear a strong relationship to list comprehensions) and other features of modern functional languages like Standard ML =-=[MTH90]-=- and Haskell [PH97], as well as being strongly typed. 2 2.1 The type system of Aldor Unusually among languages for computer algebra, but in keeping with the functional school, Aldor is strongly typed,... |

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Citation Context ...mple of the problems caused by the failure of Aldor to evaluate type expressions. 3.1 Types as values Another example is provided by trying to formalise the notion from (mathematical) category theory =-=[Mac72]-=-, namely that of a functor. Specifically we are thinking of the types of the language as forming a category, whose objects are the types themselves, and an arrow from A to B is a function of type A-?B... |

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Citation Context ...s in the sense of the programming language ML as the right tool for structuring mathematical theories in Mathematica, and [San95] notes similarities between the type system of Axiom, existential type =-=[MP88]-=-, and Haskell classes [WB89]. More closely related to our proposal here, it is interesting to note that constructive type theorists have added inductive types [PM93], giving their systems a more funct... |

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Citation Context ...r totally undefined proofs are not permitted. Consistency also depends on the strength of the type system itself; a sufficiently powerful type system will be inconsistent as shown by Girard's paradox =-=[Gir72]-=-. 5 Applications of an integrated logic If we identify a logic within Aldor, how can it be used? There are various applications possible; we outline some here and for others one can refer to the numbe... |

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Citation Context ...he Curry-Howard correspondence, logical propositions can be seen as types, and proofs can be seen as members of these types. Accounts of constructive type theories can be found in notes by Martin-Lof =-=[ML84]-=- amongst others [NPS90, Tho91]. Central to this correspondence are dependent types, which allow the representation of predicates and quantification. We can summarise the correspondence in a table Prog... |

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Citation Context ... a logic, under the Curry-Howard isomorphism. We give a number of example applications of the logic we construct. 1 Introduction Symbolic mathematical -- or computer algebra -- systems, such as Axiom =-=[JS92]-=-, Maple and Mathematica, are in everyday use by scientists, engineers and indeed mathematicians, because they provide a user with techniques of, say, integration which far exceed those of the person t... |

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Citation Context ...have added inductive types [PM93], giving their systems a more functional flavour, while functional programmers are showing an interest in dependent types [Aug97] and languages without nontermination =-=[Tur95]-=-. We see our work as part of that convergence, bringing type-theoretic ideas together with computer algebra systems, and thus providing a bridge between symbolic mathematics and theorem proving. Our f... |

26 | Symbolic Computation: Computer Algebra and Logic
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Citation Context ...hile asserting the validity of others. 6 Conclusion We have proposed a new way to combine -- or rather, to integrate -- computer algebra and theorem proving. Our proposal is similar to [BCZ96] 11 and =-=[Buc96]-=- in that theorem proving capabilities are incorporated in a computer algebra system. (In the classification of possible combinations of computer algebra and theorem proving of [CH96], all these are in... |

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Citation Context ... as `crucial' while asserting the validity of others. 6 Conclusion We have proposed a new way to combine -- or rather, to integrate -- computer algebra and theorem proving. Our proposal is similar to =-=[BCZ96]-=- 11 and [Buc96] in that theorem proving capabilities are incorporated in a computer algebra system. (In the classification of possible combinations of computer algebra and theorem proving of [CH96], a... |

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Citation Context ...l, since in, for instance, an application of the form vectorSum(e) an arbitrary expression e:Nat will have to be evaluated. We therefore intend to use current work on terminating systems of recursion =-=[MA96]-=- as well as restrictions on the types of expression chosen for evaluation (as heuristics at least) to guide the form of evaluation that is supported. 4 Logic within Aldor In this section we discuss th... |

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Citation Context ...ction we can conjure up proofs of the logical requirements of (10); even in this case they appear as important documentation of requirements, and they are related to the lightweight formal methods of =-=[DK98]-=-. Alternatively we can build fully-fledged proofs as in the numerous implementations of constructive type theories mentioned above, or we can indeed adopt an intermediate position of proving propertie... |

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Citation Context ...uring mechanisms used in these different fields: [BJK + 97] argues for functors in the sense of the programming language ML as the right tool for structuring mathematical theories in Mathematica, and =-=[San95]-=- notes similarities between the type system of Axiom, existential type [MP88], and Haskell classes [WB89]. More closely related to our proposal here, it is interesting to note that constructive type t... |

10 | Classification of communication and cooperation mechanisms for logical and symbolic computation systems
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Citation Context ...o [BCZ96] 11 and [Buc96] in that theorem proving capabilities are incorporated in a computer algebra system. (In the classification of possible combinations of computer algebra and theorem proving of =-=[CH96], all thes-=-e are instance of the "subpackage" approach.) But the way in which we propose to do this is completely different: we propose to exploit the expressiveness of the type system of Axiom, using ... |

8 |
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Citation Context ... and thus to use the constructions of Axiom to handle the logic and represent proofs and propositions, in the same way as is done in theorem provers based on type theory such as Nuprl [C + 86] or Coq =-=[Cor95]-=-. This paper particularly explores the recent Axiom Library Compiler, Aldor [W + 95], which is unusual among computer algebra systems in being strongly typed, and moreover in having a very powerful ty... |

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Citation Context ...ral numbers, and such a method makes much more manageable a proof of the transitivity of `!' over Nat, say. Inductive types have been used extensively in the type theory community, see, for instance, =-=[PM93]-=-. 9 4.2 A logic within Aldor We need to examine whether the outline given in Section 4.1 amounts to a proper embedding of a logic within Aldor. We shall see that it places certain requirements on the ... |

6 | Computers, reasoning and mathematical practice
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Citation Context ...aive user of the system or indeed, if it occurs within a sufficiently complicated context, any user. Symbolic mathematics systems are also limited by their reliance on algebraic techniques. As Martin =-=[Mar98]-=- remarks, in performing operations of analysis it might be a precondition that a function be continuous; such a property cannot be guaranteed by a computer algebra system alone. 1 All this makes the c... |

3 |
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Citation Context ... used in support of both academic and commercial research. There are, however, drawbacks to the widespread use of automated support of complex mathematical tasks, which has been widely noted: Fateman =-=[Fat96]-=- gives the graphic example of systems which will assume that a 6= 0 on the basis that a = 0 has not been established. This can have potentially disastrous consequences for the naive user of the system... |

2 |
De Aldor `a Zermelo
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Citation Context ...Our future work will involve investigating the type system of Aldor, and integrating a logical approach to Aldor types; this is complementary to recent work on formalising the Aldor system within Coq =-=[Ale98]-=-. We also expect to apply our work in a case study with mathematical colleagues. Acknowledgements We are grateful to Stephen Watt of the University of Western Ontario and to Ursula Martin and her rese... |

1 |
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Citation Context ...ing to note that constructive type theorists have added inductive types [PM93], giving their systems a more functional flavour, while functional programmers are showing an interest in dependent types =-=[Aug97]-=- and languages without nontermination [Tur95]. We see our work as part of that convergence, bringing type-theoretic ideas together with computer algebra systems, and thus providing a bridge between sy... |