Position Automata for Kleene Algebra with Tests

Kleene algebra with tests (KAT) is an equational system that combines Kleene and Boolean algebras. One can model basic programming constructs and assertions in KAT, which allowed for its application in compiler optimization, program transformation and dataflow analysis. To provide semantics for KAT expressions, Kozen first introduced emph{automata on guarded strings}, showing that the regular sets of guarded strings plays the same role in KAT as regular languages play in Kleene algebra. Recently, Kozen described an elegant algorithm, based on ``derivatives'', to construct a deterministic automaton that accepts the guarded strings denoted by a KAT expression. This algorithm generalizes Brzozowski's algorithm for regular expressions and inherits its inefficiency arising from the explicit computation of derivatives. In the context of classical regular expressions, many efficient algorithms to compile expressions to automata have been proposed. One of those algorithms was devised by Berry and Sethi in the 80's (we shall refer to it as Berry-Sethi construction/algorithm, but in the literature it is also referred to as position or Glushkov automata algorithm). In this paper, we show how the Berry-Sethi algorithm can be used to compile a $KAT$ expression to an automaton on guarded strings. Moreover, we propose a new automata model for KAT expressions and adapt the construction of Berry and Sethi to this new model

Properties of a general quaternion-valued gradient operator and its applications to signal processing

The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternionvalued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm

A Computational Interpretation of Context-Free Expressions

We phrase parsing with context-free expressions as a type inhabitation problem where values are parse trees and types are context-free expressions. We first show how containment among context-free and regular expressions can be reduced to a reachability problem by using a canonical representation of states. The proofs-as-programs principle yields a computational interpretation of the reachability problem in terms of a coercion that transforms the parse tree for a context-free expression into a parse tree for a regular expression. It also yields a partial coercion from regular parse trees to context-free ones. The partial coercion from the trivial language of all words to a context-free expression corresponds to a predictive parser for the expression

Regularity of the Einstein Equations at Future Null Infinity

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.Comment: 43 pages, no figures, AMS-TeX. Minor revisions, updated to agree with published versio

Efficient Dynamic Access Analysis Using JavaScript Proxies

JSConTest introduced the notions of effect monitoring and dynamic effect inference for JavaScript. It enables the description of effects with path specifications resembling regular expressions. It is implemented by an offline source code transformation. To overcome the limitations of the JSConTest implementation, we redesigned and reimplemented effect monitoring by taking advantange of JavaScript proxies. Our new design avoids all drawbacks of the prior implementation. It guarantees full interposition; it is not restricted to a subset of JavaScript; it is self-maintaining; and its scalability to large programs is significantly better than with JSConTest. The improved scalability has two sources. First, the reimplementation is significantly faster than the original, transformation-based implementation. Second, the reimplementation relies on the fly-weight pattern and on trace reduction to conserve memory. Only the combination of these techniques enables monitoring and inference for large programs.Comment: Technical Repor

Testing the Equivalence of Regular Languages

The minimal deterministic finite automaton is generally used to determine regular languages equality. Antimirov and Mosses proposed a rewrite system for deciding regular expressions equivalence of which Almeida et al. presented an improved variant. Hopcroft and Karp proposed an almost linear algorithm for testing the equivalence of two deterministic finite automata that avoids minimisation. In this paper we improve the best-case running time, present an extension of this algorithm to non-deterministic finite automata, and establish a relationship between this algorithm and the one proposed in Almeida et al. We also present some experimental comparative results. All these algorithms are closely related with the recent coalgebraic approach to automata proposed by Rutten

All order I.R. finite expansion for short distance behavior of massless theories perturbed by a relevant operator

We consider here renormalizable theories without relevant couplings and present an I.R. consistent technique to study corrections to short distance behavior (Wilson O.P.E. coefficients) due to a relevant perturbation. Our method is the result of a complete reformulation of recent works on the field, and is characterized by a more orthodox treatment of U.V. divergences that allows for simpler formulae and consequently an explicit all order (regularization invariant) I.R. finitess proof. Underlying hypotheses are discussed in detail and found to be satisfied in conformal theories that constitute a natural field of application of this approach.Comment: 27 page

From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527