Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors

It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the B\"ohm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of B\"ohm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a class of algebraic $\lambda$-terms, encompassing both normalizable algebraic $\lambda$-terms and arbitrary ordinary $\lambda$-terms: the expansion of these is always normalizable, which guides the definition of a generalization of B\"ohm trees to this setting

A Proof of Strong Normalization for the Theory of Constructions Using a Kripke-Like Interpretation

We give a proof that all terms that type-check in the theory of contructions are strongly normalizing (under ß-reduction). The main novelty of this proof is that it uses a Kripke-like interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under ß-reduction), using the reducibility method

Strong Normalization of MLF via a Calculus of Coercions

MLF is a type system extending ML with first-class polymorphism as in system F. The main goal of the present paper is to show that MLF enjoys strong normalization, i.e. it has no infinite reduction paths. The proof of this result is achieved in several steps. We first focus on xMLF, the Church-style version of MLF, and show that it can be translated into a calculus of coercions: terms are mapped into terms and instantiations into coercions. This coercion calculus can be seen as a decorated version of system F, so that the simulation results entails strong normalization of xMLF through the same property of system F. We then transfer the result to all other versions of MLF using the fact that they can be compiled into xMLF and showing there is a bisimulation between the two. We conclude by discussing what results and issues are encountered when using the candidates of reducibility approach to the same problem

G-Matrix Equation in the Resonating-Group Method

The G-matrix equation is most straightforwardly formulated in the resonating-group method if the quark-exchange kernel is directly used as the driving term for the infinite sum of all the ladder diagrams. The inherent energy-dependence involved in the exchange term of the normalization kernel plays the essential role to define the off-shell T-matrix uniquely when the complete Pauli-forbidden state exists. We analyze this using a simple solvable model with no quark-quark interaction, and calculating the most general T-matrix in the formulation developed by Noyes and Kowalski. This formulation gives a certain condition for the existence of the solution in the Lippmann-Schwinger resonating-group method. A new procedure to deal with the corrections for the reduced masses and the internal-energy terms in the Lambda N - Sigma N coupled-channel resonating-group equation is proposed.Comment: 21 pages 0 figures, submitted to Prog. Theor. Phy

How one can obtain unambiguous predictions for the S-matrix in non-renormalizable theories

The usual Bogolyubov R-operation works in non-renormalizable theories in the same way as in renormalizable ones. However, in the non-renormalizable case, the counter-terms eliminating ultraviolet divergences do not repeat the structure of the original Lagrangian but contain new terms with a higher degree of fields and derivatives increasing from order to order of PT. If one does not aim to obtain finite off-shell Green functions but limits oneself only to the finiteness of the S-matrix, then one can use the equations of motion and drastically reduce the number of independent counter-terms. For example, it is possible to reduce all counter-terms to a form containing only operators with four fields and an arbitrary number of derivatives. And although there will still be infinitely many such counter-terms, in order to fix the arbitrariness of the subtraction procedure, one can normalize the on-shell 4-point amplitude, which must be known for arbitrary kinematics, plus the 6-point amplitude at one point. All other multiparticle amplitudes will be calculated unambiguously. Within the framework of perturbation theory, the number of independent counter-terms in a given order is limited, so does the number of normalization conditions. The constructed counter-terms are not absorbed into the normalization of a single coupling constant, the Lagrangian contains an infinite number of terms, but after fixing the arbitrariness, it allows one to obtain unambiguous predictions for observables.Comment: PDFLatex 13 pages, 4 figure

On bar recursion of types 0 and 1

For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by Tait [9]. The main tools of the proof are (i) the normalization theorem for (notations for) infinite terms and (ii) valuation functionals. Both are elaborated in [6]; for brevity some familiarity with this paper is assumed here. Using (i) and (ii) we reduce BRo.1 to ';-recursion with'; < co. From this the result follows by work of Tait [10], who gave a reduction of 2E-recursion to ';-recursion at a higher type. At the end of the paper we discuss a perhaps more natural variant of bar recursion introduced by Kreisel in [4]. Related results are due to rKeisel (in his appendix to [8]), who obtains results which imply, using the reduction given by Howard [2] of the constant of bar recursion of type '0 to the rule of bar recursion of type (0 ~ '0) ~ '0, that T is not closed under the rule of bar recursion of a type of level ~ 2, to Diller [1], who gave a reduction of BRo.1 to ';-recursion with'; bounded by the least (V-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type O. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions. Recall that a functional F of type 0 ~ (0 ~ '0) ~ (J is said to be defined by (the rule of) bar recursion of type '0 from Yand functionals G, H of the proper types i

The Quantum Transverse Field Ising Model on an Infinite Tree from Matrix Product States

We give a generalization to an infinite tree geometry of Vidal's infinite time-evolving block decimation (iTEBD) algorithm for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the Matrix Product State ansatz. We observe a second order phase transition, with certain key differences from the transverse field Ising model on an infinite spin chain. We also investigate a transverse field Ising model with a specific longitudinal field. When the transverse field is turned off, this model has a highly degenerate ground state as opposed to the pure Ising model whose ground state is only doubly degenerate.Comment: 28 pages, 23 figures, PDFlate