Decidable Models of Recursive Asynchronous Concurrency

Asynchronously communicating pushdown systems (ACPS) that satisfy the empty-stack constraint (a pushdown process may receive only when its stack is empty) are a popular decidable model for recursive programs with asynchronous atomic procedure calls. We study a relaxation of the empty-stack constraint for ACPS that permits concurrency and communication actions at any stack height, called the shaped stack constraint, thus enabling a larger class of concurrent programs to be modelled. We establish a close connection between ACPS with shaped stacks and a novel extension of Petri nets: Nets with Nested Coloured Tokens (NNCTs). Tokens in NNCTs are of two types: simple and complex. Complex tokens carry an arbitrary number of coloured tokens. The rules of NNCT can synchronise complex and simple tokens, inject coloured tokens into a complex token, and eject all tokens of a specified set of colours to predefined places. We show that the coverability problem for NNCTs is Tower-complete. To our knowledge, NNCT is the first extension of Petri nets, in the class of nets with an infinite set of token types, that has primitive recursive coverability. This result implies Tower-completeness of coverability for ACPS with shaped stacks

Boundedness character of a max-type system of difference equations of second order

The boundedness character of positive solutions of the next max-type system of difference equations $$x_{n+1}=\max\left\{A,\frac{y_n^p}{x_{n-1}^q}\right\},\quad y_{n+1}=\max\left\{A,\frac{x_n^p}{y_{n-1}^q}\right\},\quad n\in\mathbb{N}_0,$$ with $\min\{A, p, q\}>0$, is characterized

Inequality Constraints in Recursive Economies

Dynamic models with inequality constraints pose a challenging problem for two major reasons: Dynamic Programming techniques often necessitate a non established differentiability of the value function, while Euler equation based techniques have problematic or unknown convergence properties. This paper aims to resolve these two concerns: An "envelope theorem" is presented that establishes the differentiability of any element in the convergent sequence of approximate value functions when inequality constraints may bind. As a corollary, convergence of an iterative procedure on the Euler equation, usually referred to as time iteration, is ascertained. This procedure turns out to be very convenient from a computational perspective; dynamic economic problems with inequality constraints can be solved reliably and extremely efficiently by exploiting the theoretical insights provided by the paper.Inequality constraints; Envelope theorem; Recursive methods; Time iteration

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the lambda-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to beta-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331