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2 Fundamenta Informaticae

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An extended Relational Algebra on Abstract Objects for summarizing answers to queries

Demolombe R.

Fundamenta Informaticae

2003

Vol. 57, nr 1

1--15

Answers to queries in terms of abstract objects are defined in the logical framework of first order predicate calculus. A partial algebraic characterisation of the supremum and of the infimum of abstract answers is given in an extended Relational Algebra of the Cylindric Algebra kind. Then, the form of queries is restricted in order to be able to compute answers without the cylindrification operator. For these restricted queries we give a technique to compute an upper bound and a lower bound of abstract answers using only the operators of the standard Relational Algebra.

Lambda abstraction algebras : coordinatizing models of lambda calculus

Pigozzi D., Salibra A.

Fundamenta Informaticae

1998

Vol. 33, nr 2

149-200

Lambda abstraction algebras are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order logic; they are intended as an alternative to combinatory algebras in this regard. Like combinatory algebras they can be defined by true identities and thus from a variety in the sense if universal algebra. One feature of lambda abstraction algebras that sts them apart from combinatory algebras is the way variables in the lambda calculus are abtracted; this provides each lambda abstraction algebra with an implicit coordinate system. Another peculiar feature in the algebraic reformulation of (b)-conversion as the definition of abstract substitution. Functional lambda abstraction algebras arise as the 'coordinatizations' of environment models or lambda models, the natural combinatory models of the lambda calculus. As in the case of cylindric and polyadic algebras, questions of the functional representation of various subclasses of lambda abstraction algebras are an important part of the theory. The main result of the paper is a stronger version of the functional representation theorem for locally finite lambda abstraction algebras, the algebraic analogue of the completeness theorem of lambda calculus. This result is used to study the connection between the combinatory models of the lambda calculus and lambda abstraction algebras. Two significant results of this kind are the existence of a strong categorical equivalence between lambda algebras and locally finite lambda abstraction algebras, and between lambda models and rich, locally finite lambda abstraction algebras.