Decidable properties and optimal normalization in persistent term rewriting systems

Projet PARAWe introduce Persistent Term Rewriting Systems (PTRSs) by restriting ways of redex-creation during reductions in Orthogonal Term Rewriting Systems (OTRSs). In particular, Recursive (Applicative) Program Schemes (RPSs), considered as TRSs are persistent. We establish criteria for weak and strong normalization of terms in PTRSs and prove that they are decidable. The decidability of weak normalization implies immediately that RPSs do not have full computational power. We find a decidable, necessary and sufficient condition for syntactical equivalence of PTRSs. We also prove that the reducibility problem is decidable in PTRSs. Finally, we design an optimal, sequential normalizing strategy for PTRSs. In particular, if is a normalizable term in an RPS, then the number of steps in the optimal reduction of t to normal coincides with the minimum number of redex families contracted in reductions of t to normal form

Leftmost Outermost Revisited

We present an elementary proof of the classical result that the leftmost outermost strategy is normalizing for left-normal orthogonal rewrite systems. Our proof is local and extends to hyper-normalization and weakly orthogonal systems. Based on the new proof, we study basic normalization, i.e., we study normalization if the set of considered starting terms is restricted to basic terms. This allows us to weaken the left-normality restriction. We show that the leftmost outermost strategy is hyper-normalizing for basically left-normal orthogonal rewrite systems. This shift of focus greatly extends the applicability of the classical result, as evidenced by the experimental data provided

Distinguishing Bipartitite Orthogonal States using LOCC: Best and Worst Cases

Two types of results are presented for distinguishing pure bipartite quantum states using Local Operations and Classical Communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C^3 tensor C^3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in C^n tensor C^m. In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.Comment: 22 pages, published version. Some proofs rewritten for clarit

Statistics of Rare Events in Disordered Conductors

Asymptotic behavior of distribution functions of local quantities in disordered conductors is studied in the weak disorder limit by means of an optimal fluctuation method. It is argued that this method is more appropriate for the study of seldom occurring events than the approaches based on nonlinear $\sigma$-models because it is capable of correctly handling fluctuations of the random potential with large amplitude as well as the short-scale structure of the corresponding solutions of the Schr\"{o}dinger equation. For two- and three-dimensional conductors new asymptotics of the distribution functions are obtained which in some cases differ significantly from previously established results.Comment: 17 pages, REVTeX 3.0 and 1 Postscript figur

Lambda Calculus with Explicit Recursion

AbstractThis paper is concerned with the study ofλ-calculus with explicit recursion, namely of cyclicλ-graphs. The starting point is to treat aλ-graph as a system of recursion equations involvingλ-terms and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for first-order term rewriting. Surprisingly, now the confluence property breaks down in an essential way. Confluence can be restored by introducing a restraining mechanism on the substitution operation. This leads to a family ofλ-graph calculi, which can be seen as an extension of the family ofλσ-calculi (λ-calculi with explicit substitution). While theλσ-calculi treat the let-construct as a first-class citizen, our calculi support the letrec, a feature that is essential to reason about time and space behavior of functional languages and also about compilation and optimizations of program

Data-Oblivious Stream Productivity

We are concerned with demonstrating productivity of specifications of infinite streams of data, based on orthogonal rewrite rules. In general, this property is undecidable, but for restricted formats computable sufficient conditions can be obtained. The usual analysis disregards the identity of data, thus leading to approaches that we call data-oblivious. We present a method that is provably optimal among all such data-oblivious approaches. This means that in order to improve on the algorithm in this paper one has to proceed in a data-aware fashion