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Statistical Properties of Simple Types
 Mathematical Structures in Computer Science
, 2000
"... We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n ! 1. ..."
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We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n ! 1.
Categorical Completeness Results for the SimplyTyped LambdaCalculus
 Proceedings of TLCA '95, Springer LNCS 902
, 1995
"... . We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the catego ..."
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. We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing betaeta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that betaeta conversion is complete for deriving all equalities between the (simplytyped) lambdadefinable...
Fixpoint Technique For Counting Terms In Typed lambda Calculus
"... : Typed calculus with denumerable set of ground types is considered. The aim of the paper is to show procedure for counting closed terms in long normal forms. It is shown that there is a surprising correspondence between the number of closed terms and the fixpoint solution of the polynomial equatio ..."
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: Typed calculus with denumerable set of ground types is considered. The aim of the paper is to show procedure for counting closed terms in long normal forms. It is shown that there is a surprising correspondence between the number of closed terms and the fixpoint solution of the polynomial equation in some complete lattice. It is proved that counting of terms in typed lambda calculus can be reduced to the problem of finding least fixpoint for the system of polynomial equations. The algorithm for finding the least fixpoint of the system of polynomials is considered. By the well known Curry Howard isomorphism the result can be interpreted as a method for counting proofs in the implicational fragment of the propositional intuitionistic logic. The problem of number of terms was studied but never published by Ben Yelles (see [3] ). Similarly in [4] it was proved that complexity of the question whether given type possess an infinite number of normal terms is polynomial space complete. 1. ...
Lambda Definability is Decidable for Second Order Types and for Regular Third Order Types
"... : It has been proved by Loader [1] that StatmanPlotkin conjecture (see [4] and [2]) fails. The Loader proof was done by encoding the word problem in the full type hierarchy based on the domain with 7 elements. The aim of this paper is to show that the lambda definability problem limited for second ..."
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: It has been proved by Loader [1] that StatmanPlotkin conjecture (see [4] and [2]) fails. The Loader proof was done by encoding the word problem in the full type hierarchy based on the domain with 7 elements. The aim of this paper is to show that the lambda definability problem limited for second order types and regular third order types is decidable in any finite domain. Obviously definability is decidable for 0 and 1 order types. As an additional effect of the result described we may observe that for certain types there is no finite grammar generating all closed terms. 1. Syntax of simple typed calculus We shall consider a simple typed lambda calculus with a single ground type O. The set TY PES is defined as follows: O is a type and if ø and ¯ are types then ø ! ¯ is a type. We will use the following notation: if ¯; ø1 ; ø2 ; :::; øn are types then by ø1 ! ø2 ! ::: ! øn ! ¯ we mean the type ø1 ! (ø2 ! ::: ! (øn ! ¯):::). Therefore, every type ø has the form ø1 ! ::: ! øn ! O....
SchwichtenbergStyle Lambda Definability is Undecidable.
"... We consider lambda definability problem over an arbitrary free algebra. There is a natural notion of primitive recursive function in such algebras and at the same time a natural notion of a  definable function. The paper answers the following question: for a given free algebra and a primitive recur ..."
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We consider lambda definability problem over an arbitrary free algebra. There is a natural notion of primitive recursive function in such algebras and at the same time a natural notion of a  definable function. The paper answers the following question: for a given free algebra and a primitive recursive function within this algebra decide whether this function is lambda definable. The paper shows that the question is undecidable if the algebra is infinite. The main part of the paper is dedicated to the algebra of numbers in which lambda definability is described by the Schwichtenberg theorem. The result for an arbitrary infinite free algebra has been obtained by a simple interpretation of numerical functions as recursive functions in this algebra. This result is a counterpart of the distinguished result of Loader [7] in which lambda terms are evaluated in finite domains for which undecidability of definability is proved.