PELCR is an environment for lambda-terms reduction on parallel/distributed computing systems. The computation performed in this environment is a distributed graph rewriting and a major optimization to achieve efficient execution consists of a message aggregation technique exhibiting the potential for strong reduction of the communication overhead. In this paper we discuss the interaction between the effectiveness of aggregation and the schedule sequence of rewriting operations. Then we present a Priority Based (BP) scheduling algorithm well suited for the speci c aggregation technique. Results on a classical benchmark lambda-term demonstrate that PB allows PELCR to achieve up to 88% of the ideal speedup while executing on a shared memory parallel architecture.

We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following \cite{Girard95d}) which proofs of Elementary Linear Logic can be interpreted into. In order to extend geometry of interaction computation (the so called {\em execution formula}) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called {\em weak execution}). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs.

Let $q>1$. Initiated by P. Erd\H os et al. in \cite{ErdJooKom1}, several authors studied the numbers $l^m(q)=\inf \{y\ :\ y\in\Lambda_m,\ y\ne 0\}$, $m=1,2,\dots$, where $\Lambda_m$ denotes the set of all finite sums of the form $y=\eps_0 + \eps_1 q + \eps_2 q^2 + \dots + \eps_n q^n$ with integer coefficients $-m\le \eps_i \le m$. It is known (\cite{Bug}, \cite{ErdJooKom1}, \cite{ErdKom}) that $q$ is a Pisot number if and only if $l^m(q)>0$ for all $m$. The value of $l^1(q)$ was determined for many particular Pisot numbers, but the general case remains widely open. In this paper we determine the value of $l^m(q)$ in other cases.