Incremental Dead State Detection in Logarithmic Time

Identifying live and dead states in an abstract transition system is a recurring problem in formal verification; for example, it arises in our recent work on efficiently deciding regex constraints in SMT. However, state-of-the-art graph algorithms for maintaining reachability information incrementally (that is, as states are visited and before the entire state space is explored) assume that new edges can be added from any state at any time, whereas in many applications, outgoing edges are added from each state as it is explored. To formalize the latter situation, we propose guided incremental digraphs (GIDs), incremental graphs which support labeling closed states (states which will not receive further outgoing edges). Our main result is that dead state detection in GIDs is solvable in $O(\log m)$ amortized time per edge for $m$ edges, improving upon $O(\sqrt{m})$ per edge due to Bender, Fineman, Gilbert, and Tarjan (BFGT) for general incremental directed graphs. We introduce two algorithms for GIDs: one establishing the logarithmic time bound, and a second algorithm to explore a lazy heuristics-based approach. To enable an apples-to-apples experimental comparison, we implemented both algorithms, two simpler baselines, and the state-of-the-art BFGT baseline using a common directed graph interface in Rust. Our evaluation shows $110$-$530$x speedups over BFGT for the largest input graphs over a range of graph classes, random graphs, and graphs arising from regex benchmarks.Comment: 22 pages + reference

Non-perturbative corrections to mean-field behavior: spherical model on spider-web graph

We consider the spherical model on a spider-web graph. This graph is effectively infinite-dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of $\delta$-functions, and all the modes are localized. The fractional number of modes with frequency less than $\omega$ varies as $\exp (-C/\omega)$ for $\omega$ tending to zero, where $C$ is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time $t$ varies as $\exp(- C' t^{1/3})$, for large $t$, where $C'$ is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free-energy per site at temperature $T$, near and above the critical temperature $T_c$, also has an essential singularity of the type $\exp[ -K {(T - T_c)}^{-1/2}]$.Comment: substantially revised, a section adde

About adaptive state knowledge extraction for septic shock mortality prediction

The early prediction of mortality is one of the unresolved tasks in intensive care medicine. This contribution models medical symptoms as observations cased by transitions between hidden markov states. Learning the underlying state transition probabilities results in a prediction probability success of about 91%. The results are discussed and put in relation to the model used. Finally, the rationales for using the model are reflected: Are there states in the septic shock data

Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent tau in (2,3). In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices. We investigate the case when the speeds are equal and typical distances in a follow-up paper.Comment: 44 pages, 9 picture