On connected Boolean functions

Cataloged from PDF version of article.A Boolean function is called (co-)connected if the subgraph of the Boolean hypercube induced by its (false) true points is connected; it is called strongly connected if it is both connected and co-connected. The concept of (co-)geodetic Boolean functions is de ned in a similar way by requiring that at least one of the shortest paths connecting two (false) true points should consist only of (false) true points. This concept is further strengthened to that of convexity where every shortest path connecting two points of the same kind should consist of points of the same kind. This paper studies the relationships between these properties and the DNF representations of the associated Boolean functions. ? 1999 Elsevier Science B.V. All rights reserved

Decision lists and related Boolean functions

AbstractWe consider Boolean functions represented by decision lists, and study their relationships to other classes of Boolean functions. It turns out that the elementary class of 1-decision lists has interesting relationships to independently defined classes such as disguised Horn functions, read-once functions, nested differences of concepts, threshold functions, and 2-monotonic functions. In particular, 1-decision lists coincide with fragments of the mentioned classes. We further investigate the recognition problem for this class, as well as the extension problem in the context of partially defined Boolean functions (pdBfs). We show that finding an extension of a given pdBf in the class of 1-decision lists is possible in linear time. This improves on previous results. Moreover, we present an algorithm for enumerating all such extensions with polynomial delay

An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are the amounts of time and memory used. In proof complexity, these resources correspond to the length and space of resolution proofs. There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution. Our collection of trade-offs cover almost the whole range of values for the space complexity of formulas, and most of the trade-offs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these trade-offs in fact extend to the exponentially stronger k-DNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question whether the k-DNF resolution systems form a strict hierarchy with respect to space in the affirmative. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formula F can be transformed by simple variable substitution into a new formula F' such that if F has the right properties, F' can be proven in essentially the same length as F, whereas on the other hand the minimal number of lines one needs to keep in memory simultaneously in any proof of F' is lower-bounded by the minimal number of variables needed simultaneously in any proof of F. Applying this theorem to so-called pebbling formulas defined in terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical reports TR09-034 and TR09-047, and it hence subsumes these two report

An Algorithmic Investigation of Conviction Narrative Theory: Applications in Business, Finance and Economics

The thesis aims to make conviction narrative theory (CNT) operational and test its validity via a combination of text analysis, network analysis and machine learning techniques. CNT is a theory of decision-making asserting that, when faced with uncertainty, agents are able to act by constructing narratives that yield conviction. The developed methodology is directed by CNT and therefore limits problems related to spurious correlations frequently encountered in studies using large datasets. The thesis provides empirical support of the theory and how it can be used to understand the economy and financial markets. The thesis develops a relative sentiment shift (RSS) methodology that captures emotional variables within text archives and also tests the extent to which these can be accurately measured, to establish causal economic and financial relationships hypothesised by the theory to exist on a macro level. Better-than-economic-consensus forecasts of the Michigan Consumer Confidence index, statistically significant explanatory power of real US GDP growth, evidence of causality from relative sentiment to the most widely used measure of financial market volatility, the VIX, are obtained in the process. On the micro level, the RSS methodology is applied to particular narratives to test theoretical expectations showing how it can be used to measure the emergence of phantastic object narratives, narratives for which anxiety substantially disappears despite the existence of conflicting evidence. To illustrate the importance of the overall ecology of narratives to understand shifts in macro sentiment and financial stability, as well as a means to qualitatively understand the relation between the macro and the micro approach, a form of dynamic content network analysis is applied. Using the narrative model, measures of the degree of formation of a dominant narrative are shown to correlate with RSS and Granger-cause indicators of financial stability, such as the VIX and the S&P 500 index

Computing on evolving social networks

Over the past decade, participation in social networking services has seen an exponential growth, so that nowadays most individuals are “virtually” connected to others anywhere in the world. Consistently, analysis of human social behavior has gained momentum in the computer science research community. Several well-known phenomena in the social sciences have been revisited in a computer science perspective, with a new focus on phenomena of emerging behavior, information diffusion, opinion formation and collective intelligence. Furthermore, the recent past has witnessed a growing interest in the dynamics of these phenomena and that of the underlying social structures. This thesis investigates a number of aspects related to the study of evolving social networks and the collective phenomena they mediate. We have mainly pursued three research directions. The first line of research is in a sense functional to the other two and concerns the collection of data tracking the evolution of human interactions in the physical space and the extraction of (time) evolving networks describing these interactions. A number of available datasets describing different kinds of social networks are available on line, but few involve physical proximity of humans in real life scenarios. During our research activity, we have deployed several social experiments tracking face-to-face human interactions in the physical space. The collected datasets have been used to analyze network properties and to investigate social phenomena, as further described below. A second line of research investigates the impact of dynamics on the analytical tools used to extract knowledge from social networks. This is clearly a vast area in which research in many cases is in its early stages. We have focused on centrality, a fundamental notion in the analysis and characterization of social network structure and key to a number of Web applications and services. While many social networks of interest (resulting from “virtual” or “physical” activity) are highly dynamic, many Web information retrieval algorithms were originally designed with static networks in mind. In this thesis, we design and analyze decentralized algorithms for computing and maintaining centrality scores over time evolving networks. These algorithms refer to notions of centrality which are explicitly conceived for evolving settings and which are consistent with PageRank in important cases. A further line of research investigates the wisdom of crowds effect, an important, yet not completely understood phenomenon of collective intelligence, whereby a group typically exhibits higher predictive accuracy than its single members and often experts. Phenomena of collective intelligence involve exchange and processing of information among individuals sharing some common social structure. In many cases of interest, this structure is suitably described by an evolving social network. Studying the interplay between the evolution of the underlying social structure and the computational properties of the resulting process is an interesting and challenging task. We have focused on the quantitative analysis of this aspect, in particular the effect of the network on the accuracy of prediction. To provide a mathematical characterization, we have revisited and modified a number of models of opinion formation and diffusion originally proposed in the social sciences. Experimental analysis using data collected from some of the social experiments we conducted allowed to test soundness of the proposed models. While many of these models seem to capture important aspects of the process of opinion formation in (physical) social networks, one variant we propose achieves higher predictive accuracy and is also robust to the presence of outliers