4,548 research outputs found
Lambda theories of effective lambda models
A longstanding open problem is whether there exists a non-syntactical model
of untyped lambda-calculus whose theory is exactly the least equational
lambda-theory (=Lb). In this paper we make use of the Visser topology for
investigating the more general question of whether the equational (resp. order)
theory of a non syntactical model M, say Eq(M) (resp. Ord(M)) can be
recursively enumerable (= r.e. below). We conjecture that no such model exists
and prove the conjecture for several large classes of models. In particular we
introduce a notion of effective lambda-model and show that for all effective
models M, Eq(M) is different from Lb, and Ord(M) is not r.e. If moreover M
belongs to the stable or strongly stable semantics, then Eq(M) is not r.e.
Concerning Scott's continuous semantics we explore the class of (all) graph
models, show that it satisfies Lowenheim Skolem theorem, that there exists a
minimum order/equational graph theory, and that both are the order/equ theories
of an effective graph model. We deduce that no graph model can have an r.e.
order theory, and also show that for some large subclasses, the same is true
for Eq(M).Comment: 15 pages, accepted CSL'0
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
The Broadest Necessity
In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given
On the completeness of quantum computation models
The notion of computability is stable (i.e. independent of the choice of an
indexing) over infinite-dimensional vector spaces provided they have a finite
"tensorial dimension". Such vector spaces with a finite tensorial dimension
permit to define an absolute notion of completeness for quantum computation
models and give a precise meaning to the Church-Turing thesis in the framework
of quantum theory. (Extra keywords: quantum programming languages, denotational
semantics, universality.)Comment: 15 pages, LaTe
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