96 research outputs found
Easy lambda-terms are not always simple
A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ-terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ-calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N, it is possible to build (in a canonical way) a non-trivial filter model which equates the interpretation of M and N. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a non-empty co-r.e. (complement of a recursively enumerable) set of easy, but not simple easy, λ-terms
Ordered models of the lambda calculus
Answering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity (namely n-subtractivity) for partially ordered algebras. Finally we study the relation between n-subtractivity and relativized separation conditions in topological algebras, obtaining an incompleteness theorem for a general topological semantics of lambda calculus
Lattices of equational theories as Church algebras
Abstract. We introduce the class of Church algebras, which is general enough to compass all Boolean algebras, Heyting algebras and rings with unit. Using a new equational characterization of central elements, we prove that Church algebras satisfy a Stone representation theorem. We show that every lattice of equational theories is isomorphic to the congruence lattice of a suitable Church algebra, and we use this property to prove a meta-Stone representation theorem applicable to all varieties
Maximum-entropy theory of steady-state quantum transport
We develop a theoretical framework for describing steady-state quantum transport phenomena, based on the general maximum-entropy principle of nonequilibrium statistical mechanics. The general form of the many-body density matrix is derived, which contains the invariant part of the current operator that guarantees the nonequilibrium and steady-state character of the ensemble. Several examples of the theory are given, demonstrating the relationship of the present treatment to the widely used scattering-state occupation schemes at the level of the self-consistent single-particle approximation. The latter schemes are shown not to maximize the entropy, except in certain limits
The stack calculus
We introduce a functional calculus with simple syntax and operational
semantics in which the calculi introduced so far in the Curry-Howard
correspondence for Classical Logic can be faithfully encoded. Our calculus
enjoys confluence without any restriction. Its type system enforces strong
normalization of expressions and it is a sound and complete system for full
implicational Classical Logic. We give a very simple denotational semantics
which allows easy calculations of the interpretation of expressions.Comment: In Proceedings LSFA 2012, arXiv:1303.713
Boolean product representations of algebras via binary polynomials
We mimic the construction of guarded algebras and show how to extract a Church algebra out of the binary functions on an arbitrary algebra, containing a Church subalgebra of binary polynomial opera- tions. We put to good use the weak Boolean product representations of these Church algebras to obtain weak Boolean product representations of the original algebras. Although we cannot, in general, say much about the factors in these products, we identify a number of sufficient condi- tions for the stalks to be directly indecomposable. As an application, we prove that every skew Boolean algebra is a weak Boolean product of directly indecomposable skew Boolean algebras
On linear information systems
International audienc
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