142 research outputs found
A Finite Model Property for Intersection Types
We show that the relational theory of intersection types known as BCD has the
finite model property; that is, BCD is complete for its finite models. Our
proof uses rewriting techniques which have as an immediate by-product the
polynomial time decidability of the preorder <= (although this also follows
from the so called beta soundness of BCD).Comment: In Proceedings ITRS 2014, arXiv:1503.0437
An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus
This paper brings together two lines of research: implicit characterization
of complexity classes by Linear Logic (LL) on the one hand, and computation
over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a
fixed ring structure K we define an extension of Terui's light affine
lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We
show that this calculus captures the polynomial time function class FP(K):
every typed term can be evaluated in polynomial time and conversely every
polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR
Roma, N.57 (11/2004), november 200
Uniform Proofs of Normalisation and Approximation for Intersection Types
We present intersection type systems in the style of sequent calculus,
modifying the systems that Valentini introduced to prove normalisation
properties without using the reducibility method. Our systems are more natural
than Valentini's ones and equivalent to the usual natural deduction style
systems. We prove the characterisation theorems of strong and weak
normalisation through the proposed systems, and, moreover, the approximation
theorem by means of direct inductive arguments. This provides in a uniform way
proofs of the normalisation and approximation theorems via type systems in
sequent calculus style.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
The Rooster and the Syntactic Bracket
We propose an extension of pure type systems with an algebraic presentation
of inductive and co-inductive type families with proper indices. This type
theory supports coercions toward from smaller sorts to bigger sorts via
explicit type construction, as well as impredicative sorts. Type families in
impredicative sorts are constructed with a bracketing operation. The necessary
restrictions of pattern-matching from impredicative sorts to types are confined
to the bracketing construct. This type theory gives an alternative presentation
to the calculus of inductive constructions on which the Coq proof assistant is
an implementation.Comment: To appear in the proceedings of the 19th International Conference on
Types for Proofs and Program
Checking Zenon Modulo Proofs in Dedukti
Dedukti has been proposed as a universal proof checker. It is a logical
framework based on the lambda Pi calculus modulo that is used as a backend to
verify proofs coming from theorem provers, especially those implementing some
form of rewriting. We present a shallow embedding into Dedukti of proofs
produced by Zenon Modulo, an extension of the tableau-based first-order theorem
prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the
verification of programs in both academic and industrial projects. The purpose
of our embedding is to increase the confidence in automatically generated
proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
On sets of terms with a given intersection type
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
TichĂ˝ and Fictional Names
The paper examines two possible analyses of fictional names within Pavel Tichý’s Transparent Intensional Logic. The first of them is the analysis actually proposed by Tichý in his (1988) book The Foundations of Frege’s Logic. He analysed fictional names in terms of free variables. I will introduce, explain, and assess this analysis. Subsequently, I will explain Tichý’s notion of individual role (office, thing-to-be). On the basis of this notion, I will outline and defend the second analysis of fictional names. This analysis is close to the approach known in the literature as role realism (the most prominent advocates of this position are Nicholas Wolterstorff, Gregory Currie, and Peter Lamarque)
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