72 research outputs found

    Fixed point combinators as fixed points of higher-order fixed point generators

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    Corrado B\"ohm once observed that if YY is any fixed point combinator (fpc), then Y(λyx.x(yx))Y(\lambda yx.x(yx)) is again fpc. He thus discovered the first "fpc generating scheme" -- a generic way to build new fpcs from old. Continuing this idea, define an fpc generator\textit{fpc generator} to be any sequence of terms G1,,GnG_1,\dots,G_n such that YFPCYG1GnFPC Y \in FPC \Rightarrow Y G_1 \cdots G_n \in FPC In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several robust classes of such generators, by examining their elementary properties like injectivity and (weak) constancy. We provide sufficient conditions for existence of fixed points of a given generator (G1,,Gn)(G_1,\cdots,G_n): an fpc YY such that Y=YG1GnY = Y G_1 \cdots G_n. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This statement generalizes Statman's conjecture on non-existence of "double fpcs": fixed points of the generator (G)=(λyx.x(yx))(G) = (\lambda yx.x(yx)) discovered by B\"ohm. Finally, we define and make a few observations about the monoid of (weak) fpc generators. This enables us to formulate new a conjecture about their structure

    On sets of terms with a given intersection type

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    We are interested in how much of the structure of a strongly normalizable lambda term is captured by its intersection types and how much all the terms of a given type have in common. In this note we consider the theory BCD (Barendregt, Coppo, and Dezani) of intersection types without the top element. We show: for each strongly normalizable lambda term M, with beta-eta normal form N, there exists an intersection type A such that, in BCD, N is the unique beta-eta normal term of type A. A similar result holds for finite sets of strongly normalizable terms for each intersection type A if the set of all closed terms M such that, in BCD, M has type A, is infinite then, when closed under beta-eta conversion, this set forms an adaquate numeral system for untyped lambda calculus. A number of related results are also proved

    Infinitary Rewriting Coinductively

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    We provide a coinductive definition of strongly convergent reductions between infinite lambda terms. This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept. As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types. The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step

    Axiomatizing the Quote

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    We study reflection in the Lambda Calculus from an axiomatic point of view. Specifically, we consider various properties that the quote operator must satisfy as a function on lambda terms. The most important of these is the existence of a definable left inverse, a so-called evaluator for the quote operator. Usually the quote operator encodes the syntax of a given term, and the evaluator proceeds by analyzing the syntax and reifying all constructors by their actual meaning in the calculus. Working in Combinatory Logic, Raymond Smullyan (1994) investigated which elements of the syntax must be accessible via the quote in order for an evaluator to exist. He asked three specific questions, to which we provide negative answers. On the positive side, we give a characterization of quotes which possess all of the desired properties, equivalently defined as being equitranslatable with a standard quote. As an application, we show that Scott\u27s coding is not complete in this sense, but can be slightly modified to be such. This results in a minimal definition of a complete quoting for Combinatory Logic

    Discriminating Lambda-Terms Using Clocked Boehm Trees

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    As observed by Intrigila, there are hardly techniques available in the lambda-calculus to prove that two lambda-terms are not beta-convertible. Techniques employing the usual Boehm Trees are inadequate when we deal with terms having the same Boehm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but nevertheless the terms are beta-inconvertible. To prove such beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey information of the tempo in which the data in the BT are produced. Boehm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda-terms. The corresponding equality is strictly intermediate between beta-convertibility and Boehm Tree equality, the equality in the model P-omega. An analogous approach pertains to Levy-Longo and Berarducci Trees. Our refined Boehm Trees find in particular an application in beta-discriminating fixed point combinators (fpc's). It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are obtained by postfixing the term SI, also known as Smullyan's Owl. We prove that all these newly generated fpc's are indeed new, by considering their clocked BT's. Even so, not all pairs of new fpc's can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257

    Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

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    The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees

    Fixed point combinators as fixed points of higher-order fixed point generators

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    Corrado B\"ohm once observed that if YY is any fixed point combinator (fpc), then Y(λyx.x(yx))Y(\lambda yx.x(yx)) is again fpc. He thus discovered the first "fpc generating scheme" -- a generic way to build new fpcs from old. Continuing this idea, define an fpc generator\textit{fpc generator} to be any sequence of terms G1,,GnG_1,\dots,G_n such that YFPCYG1GnFPC Y \in FPC \Rightarrow Y G_1 \cdots G_n \in FPC In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several robust classes of such generators, by examining their elementary properties like injectivity and (weak) constancy. We provide sufficient conditions for existence of fixed points of a given generator (G1,,Gn)(G_1,\cdots,G_n): an fpc YY such that Y=YG1GnY = Y G_1 \cdots G_n. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This statement generalizes Statman's conjecture on non-existence of "double fpcs": fixed points of the generator (G)=(λyx.x(yx))(G) = (\lambda yx.x(yx)) discovered by B\"ohm. Finally, we define and make a few observations about the monoid of (weak) fpc generators. This enables us to formulate new a conjecture about their structure
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