Average-case complexity of detecting cliques

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 79-83).The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for k-CLIQUE, as this problem is known. Our results show that, in certain models of computation, solving k-CLIQUE in the average case requires Q(nk/4) resources (moreover, k/4 is tight). Here the models of computation are bounded-depth Boolean circuits and unbounded-depth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the well-studied Erdos-Renyi random graphs) are widely believed to be a source of computationally hard instances for clique problems (as Karp suggested in 1976). Our results are the first unconditional lower bounds supporting this hypothesis. For bounded-depth Boolean circuits, our average-case hardness result significantly improves the previous worst-case lower bounds of Q(nk/Poly(d)) for depth-d circuits. In particular, our lower bound of Q(nk/ 4 ) has no noticeable dependence on d for circuits of depth d ; k- log n/log log n, thus bypassing the previous "size-depth tradeoffs". As a consequence, we obtain a novel Size Hierarchy Theorem for uniform AC0 . A related application answers a longstanding open question in finite model theory (raised by Immerman in 1982): we show that the hierarchy of bounded-variable fragments of first-order logic is strict on finite ordered graphs. Additional results of this thesis characterize the average-case descriptive complexity of k-CLIQUE through the lens of first-order logic.by Benjamin Rossman.Ph.D

From Quantifier Depth to Quantifier Number: Separating Structures with k Variables

Given two $n$-element structures, $\mathcal{A}$ and $\mathcal{B}$, which can be distinguished by a sentence of $k$-variable first-order logic ($\mathcal{L}^k$), what is the minimum $f(n)$ such that there is guaranteed to be a sentence $\phi \in \mathcal{L}^k$ with at most $f(n)$ quantifiers, such that $\mathcal{A} \models \phi$ but $\mathcal{B} \not \models \phi$? We present various results related to this question obtained by using the recently introduced QVT games. In particular, we show that when we limit the number of variables, there can be an exponential gap between the quantifier depth and the quantifier number needed to separate two structures. Through the lens of this question, we will highlight some difficulties that arise in analysing the QVT game and some techniques which can help to overcome them. As a consequence, we show that $\mathcal{L}^{k+1}$ is exponentially more succinct than $\mathcal{L}^{k}$. We also show, in the setting of the existential-positive fragment, how to lift quantifier depth lower bounds to quantifier number lower bounds. This leads to almost tight bounds.Comment: 53 pages, 8 figures; added new result on the relative succinctness of finite variable logi

A Process Model of Non-Relativistic Quantum Mechanics

A process model of quantum mechanics utilizes a combinatorial game to generate a discrete and finite causal space upon which can be defined a self-consistent quantum mechanics. An emergent space-time and continuous wave function arise through a uniform interpolation process. Standard non-relativistic quantum mechanics (at least for integer spin particles) emerges under the limit of infinite information (the causal space grows to infinity) and infinitesimal scale (the separation between points goes to zero). This model is quasi-local, discontinuous, and quasi-non-contextual. The bridge between process and wave function is through the process covering map, which reveals that the standard wave function formalism lacks important dynamical information related to the generation of the causal space. Reformulating several classical conundrums such as wave particle duality, Schrodinger's cat, hidden variable results, the model offers potential resolutions to all, while retaining a high degree of locality and contextuality at the local level, yet nonlocality and contextuality at the emergent level. The model remains computationally powerful

Adding Threshold Concepts to the Description Logic EL

We introduce a family of logics extending the lightweight Description Logic EL, that allows us to define concepts in an approximate way. The main idea is to use a graded membership function m, which for each individual and concept yields a number in the interval [0,1] expressing the degree to which the individual belongs to the concept. Threshold concepts C~t for ~ in {,>=} then collect all the individuals that belong to C with degree ~t. We further study this framework in two particular directions. First, we define a specific graded membership function deg and investigate the complexity of reasoning in the resulting Description Logic tEL(deg) w.r.t. both the empty terminology and acyclic TBoxes. Second, we show how to turn concept similarity measures into membership degree functions. It turns out that under certain conditions such functions are well-defined, and therefore induce a wide range of threshold logics. Last, we present preliminary results on the computational complexity landscape of reasoning in such a big family of threshold logics