On the freezing of variables in random constraint satisfaction problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.Comment: 32 pages, 7 figure

Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in<br> SAT-Based Planning

In Verification and in (optimal) AI Planning, a successful method is to formulate the application as boolean satisfiability (SAT), and solve it with state-of-the-art DPLL-based procedures. There is a lack of understanding of why this works so well. Focussing on the Planning context, we identify a form of problem structure concerned with the symmetrical or asymmetrical nature of the cost of achieving the individual planning goals. We quantify this sort of structure with a simple numeric parameter called AsymRatio, ranging between 0 and 1. We run experiments in 10 benchmark domains from the International Planning Competitions since 2000; we show that AsymRatio is a good indicator of SAT solver performance in 8 of these domains. We then examine carefully crafted synthetic planning domains that allow control of the amount of structure, and that are clean enough for a rigorous analysis of the combinatorial search space. The domains are parameterized by size, and by the amount of structure. The CNFs we examine are unsatisfiable, encoding one planning step less than the length of the optimal plan. We prove upper and lower bounds on the size of the best possible DPLL refutations, under different settings of the amount of structure, as a function of size. We also identify the best possible sets of branching variables (backdoors). With minimum AsymRatio, we prove exponential lower bounds, and identify minimal backdoors of size linear in the number of variables. With maximum AsymRatio, we identify logarithmic DPLL refutations (and backdoors), showing a doubly exponential gap between the two structural extreme cases. The reasons for this behavior -- the proof arguments -- illuminate the prototypical patterns of structure causing the empirical behavior observed in the competition benchmarks

Lost in translation: Toward a formal model of multilevel, multiscale medicine

For a broad spectrum of low level cognitive regulatory and other biological phenomena, isolation from signal crosstalk between them requires more metabolic free energy than permitting correlation. This allows an evolutionary exaptation leading to dynamic global broadcasts of interacting physiological processes at multiple scales. The argument is similar to the well-studied exaptation of noise to trigger stochastic resonance amplification in physiological subsystems. Not only is the living state characterized by cognition at every scale and level of organization, but by multiple, shifting, tunable, cooperative larger scale broadcasts that link selected subsets of functional modules to address problems. This multilevel dynamical viewpoint has implications for initiatives in translational medicine that have followed the implosive collapse of pharmaceutical industry &#x27;magic bullet&#x27; research. In short, failure to respond to the inherently multilevel, multiscale nature of human pathophysiology will doom translational medicine to a similar implosion