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

Classical and Quantum Consistency of the DGP Model

We study the Dvali-Gabadadze-Porrati model by the method of the boundary effective action. The truncation of this action to the bending mode \pi consistently describes physics in a wide range of regimes both at the classical and at the quantum level. The Vainshtein effect, which restores agreement with precise tests of general relativity, follows straightforwardly. We give a simple and general proof of stability, i.e. absence of ghosts in the fluctuations, valid for most of the relevant cases, like for instance the spherical source in asymptotically flat space. However we confirm that around certain interesting self-accelerating cosmological solutions there is a ghost. We consider the issue of quantum corrections. Around flat space \pi becomes strongly coupled below a macroscopic length of 1000 km, thus impairing the predictivity of the model. Indeed the tower of higher dimensional operators which is expected by a generic UV completion of the model limits predictivity at even larger length scales. We outline a non-generic but consistent choice of counterterms for which this disaster does not happen and for which the model remains calculable and successful in all the astrophysical situations of interest. By this choice, the extrinsic curvature K_{\mu\nu} acts roughly like a dilaton field controlling the strength of the interaction and the cut-off scale at each space-time point. At the surface of Earth the cutoff is \sim 1 cm but it is unlikely that the associated quantum effects be observable in table top experiments.Comment: 26 pages, 1 eps figur