Strong Convergence in Finite Model Theory

In [9] we introduced a new framework for asymptotic probabilities. in which a r i-add~t~ve measure is defined on the sample space of all sequences A =< dl.d?. . d, > of finite models. where the uniberse of sf,, is { I . 2. ...n ) . In this framework we Investigated the strong 0-1 law for sentences. whlch states that each sentence either holds in d,, eventually almost surely or fails In d,, eventually almost surely. In this paper we define the strong convergence la* for formulas. w h ~ c h carrles over the ideas of the strong 0-1 law to formulas w ~ t h free variables. and roughly states that for each formula @(x).the fraction of tuples a In sf". which satisfy the formula @(XI. as n tends to infinity almost surely has a l i m ~ t We show that the infinitary logic with finitely many var~ables has the strong convergence law for formulas for the uniform measure. and further character~ze the measures on random graphs for which the strong convergence law holds. $