Today we are experiencing a major reconsideration of the computing
paradigm, as witnessed by the abundance and increasing frequency
of use of terms such as \em ambient intelligence, \em ubiquitous computing, \em disappearing computer, \em grid
computer, \em global computing and \em mobile ad-hoc
networks. Systems that can be described with such terms are of a
dynamic, with no clear physical boundary, nature and it seems that
it is impossible (or, at least, difficult) to define sharply a
number of important properties holding with certainty as well as
holding throughout the whole lifetime of the system.
One such system property, which is important for the viability of
a system, is \em trust. Our departure point is the assumption
that it seems very difficult to define static system properties
related to trust and expect that they hold eternally in the
rapidly changing systems falling under the new computing paradigm.
One should, rather, attempt to define trust in terms of properties
that hold with some limiting probability as the the system grows
and try to establish conditions that ensure that ``good''
properties hold \em almost certainly. Based on this viewpoint,
in this paper we provide a new framework for defining trust
through formally definable properties that hold, almost certainly,
in the limit in randomly growing combinatorial structures that
model ``shapeless'' computing systems (e.g. ad-hoc networks),
drawing on results that establish the threshold behavior of
predicates written in the first and second order logic.