3-SAT formulae are deened as nite sets of clauses, each clause being a disjunction of 3 literals over a set of boolean variables. Experiments on solving random 3-SAT formulae have provided strong evidence of a phase transition phenomenon. Explicitly, almost every random 3-SAT formula is satissable when its ratio: number of variables to number of clauses, denoted by c, is less than a value of about 4:25 and becomes unsatissable when this ratio exceeds 4:25. More formally, a critical value ! of the ratio c, the so-called satissability threshold, is thought to exist such that, for a suuciently large number n of variables and for any real ", the probability of satissability of a random 3-SAT formula, decreases from almost 1 to almost 0 as c increases from ! ? " to ! + ". Calculating the value of !, besides being a combinatorial challenge, has appeared as a way to understand better how the space of solutions of random formulae is structured. This is of practical importance for the design of eecient algorithms for solving SAT formulae. Up to now, estimates of ! from above and below have been produced from a `semantic' approach focusing on how a random 3-SAT formula can be satissed by binary assignments (possibly of a particular kind) to its variables. Thus, two lower bounds of ! were calculated : 1:63 and 3:003 2]. And starting from an easy upper bound, 5:19, three others were successively obtained : 4:76 3], 4:643 1] and 4:601 4]. However this semantic approach may be subject to diminishing returns in view of the complexity of the calculations for the latest bounds. In this paper, in order to estimate ! better, a new`syntactic' approach is proposed: This is suggested , e.g., simply by practical experience with a computer-based random generator and solver of formulae. The generator is completely ignorant of semantics, yet for large n, it produces (within a realistic timeframe) either only satissable formulae, for values of c a little below 4:25, or only unsatissable formulae, for values of c a little above 4:25. Only a certain class of formulae is likely to come out of the generator, thètypical' formulae. If we could describe (to a suucient degree) their syntactic structure, the convergence to 0 or to +1 of the expectation of the number of solutions, computed only for these typical formulae, should give a very good indication as …

An additive decomposition of a set $I$ of nonnegative integers is an expression of $I$ as the arithmetic sum of two other such sets. If the smaller of these has $p$ elements, we have a $p$- decomposition . If $I$ is obtained by randomly removing $n^{\alpha}$ integers from $\{0,\dots,n-1\}$, decomposability translates into a balls-and-urns problem, which we start to investigate (for large $n$) by first showing that the number of $p$-decompositions exhibits a threshold phenomenon as $\alpha$ crosses a $p$-dependent critical value. We then study in detail the distribution of the number of 2-decompositions. For this last case we show that the threshold is sharp and we establish the threshold function.

A constructive scheme for determining pure states at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of a partial decimation procedure and of the statistical properties of its output, i.e., a reduced Hamiltonian acting on a subset of the initial spins. The number of ground states (GS) in each state, the number of states, and the distances between GS are calculated and correspond to RSB predictions.

We prove the existence of the 3-XORSAT threshold, establishing its value as a function of the root of a transcendental equation.

We show that for c >= 2.4682, a random graph on n vertices with c n (1+o(1)) edges almost surely has no 3-colouring. This improves on the current best upper bound of 2.4947.

We present a new structural (or syntatic) approach for estimating the satisfiability threshold of random 3-SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3-colourability of random graphs.