Parametrised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of V\"a\"an\"anen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture.Comment: Update includes refined result

Computing with and without arbitrary large numbers

In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some problems. However, this has only been shown so far for specific problems. We provide a characterization of the power of such extra inputs for general problems. To do so, we first correct a classical result by Simon and Szegedy (1992) as well as one by Simon (1981). In the former we show mistakes in the proof and correct these by an entirely new construction, with no great change to the results. In the latter, the original proof direction stands with only minor modifications, but the new results are far stronger than those of Simon (1981). In both cases, the new constructions provide the theoretical tools required to characterize the power of arbitrary large numbers.Comment: 12 pages (main text) + 30 pages (appendices), 1 figure. Extended abstract. The full paper was presented at TAMC 2013. (Reference given is for the paper version, as it appears in the proceedings.

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Making Classical Ground State Spin Computing Fault-Tolerant

We examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation. This model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions. In its most primitive form, systems constructed in this model cannot compute in an error free manner when working at non-zero temperature. However, by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free. We achieve this by using techniques in fault-tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature. We further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class Merlin-Arthur. This provides an interesting connection between the physical behavior of certain many-body spin systems and computational complexity.Comment: 24 pages, 1 figur

Large random simplicial complexes, I

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.Comment: 21 pages, 6 figure

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party to a smaller number $m$ of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures. A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.Comment: 15 pages, 3 embedded eps figure

Critical exponents for random knots

The size of a zero thickness (no excluded volume) polymer ring is shown to scale with chain length $N$ in the same way as the size of the excluded volume (self-avoiding) linear polymer, as $N^{\nu}$, where $\nu \approx 0.588$. The consequences of that fact are examined, including sizes of trivial and non-trivial knots.Comment: 4 pages, 0 figure

Clones with finitely many relative R-classes

For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A iff each one is a substitution instance of the other using operations from C. We study the clones for which there are only finitely many relative R-classes.Comment: 41 pages; proofs improved, examples adde

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

Tightness of slip-linked polymer chains

We study the interplay between entropy and topological constraints for a polymer chain in which sliding rings (slip-links) enforce pair contacts between monomers. These slip-links divide a closed ring polymer into a number of sub-loops which can exchange length between each other. In the ideal chain limit, we find the joint probability density function for the sizes of segments within such a slip-linked polymer chain (paraknot). A particular segment is tight (small in size) or loose (of the order of the overall size of the paraknot) depending on both the number of slip-links it incorporates and its competition with other segments. When self-avoiding interactions are included, scaling arguments can be used to predict the statistics of segment sizes for certain paraknot configurations.Comment: 10 pages, 6 figures, REVTeX