The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary

Commentary on Jakab's Ineffability of Qualia

Zoltan Jakab has presented an interesting conceptual analysis of the ineffability of qualia in a functionalist and classical cognitivist framework. But he does not want to commit himself to a certain metaphysical thesis on the ontology of consciousness or qualia. We believe that his strategy has yielded a number of highly relevant and interesting insights, but still suffers from some minor inconsistencies and a certain lack of phenomenological and empirical plausibility. This may be due to some background assumptions relating to the theory of mental representation employed. Jakab's starting assumption is that there is no linguistic description of a given experience such that understanding the description would result in someone who has never had the experience being described undergoing an experience of that type. (In terms of the well-known Mary case: No description could reveal what colors are like to Mary.) This is what Jakab means by the ineffability of qualia. And this is Jakab's explanation: Understanding in the standard sense involves our linguistic- conceptual abilities; but our linguistic-conceptual abilities are notinvolved in undergoing simple sensory experiences; so they cannot deliver knowledge by acquaintance, which means linguistic descriptions of sensory experiences cannot result in someone who understands the description undergoing the experience being described. (We do not agree with the assumption that our linguistic-conceptual abilities are not at all involved in undergoing simple sensory experiences; such processes can be involved in undergoing simple sensory experiences, but they need not be the only thinginvolved in undergoing simple sensory experiences; in undergoing simple sensory experiences something else is involved which cannot be captured by descriptions. The crucial point is that descriptions do not give us knowledge by acquaintance.

RelBAC: Relation Based Access Control

TheWeb 2.0, GRID applications and, more recently, semantic desktop applications are bringing the Web to a situation where more and more data and metadata are shared and made available to large user groups. In this context, metadata may be tags or complex graph structures such as file system or web directories, or (lightweight) ontologies. In turn, users can themselves be tagged by certain properties, and can be organized in complex directory structures, very much in the same way as data. Things are further complicated by the highly unpredictable and autonomous dynamics of data, users, permissions and access control rules. In this paper we propose a new access control model and a logic, called RelBAC (for Relation Based Access Control) which allows us to deal with this novel scenario. The key idea, which differentiates RelBAC from the state of the art, e.g., Role Based Access Control (RBAC), is that permissions are modeled as relations between users and data, while access control rules are their instantiations on specific sets of users and objects. As such, access control rules are assigned an arity which allows a fine tuning of which users can access which data, and can evolve independently, according to the desires of the policy manager(s). Furthermore, the formalization of the RelBAC model as an Entity-Relationship (ER) model allows for its direct translation into Description Logics (DL). In turn, this allows us to reason, possibly at run time, about access control policies

Modality and expressibility

When embedding data are used to argue against semantic theory A and in favor of semantic theory B, it is important to ask whether A could make sense of those data. It is possible to ask that question on a case-by-case basis. But suppose we could show that A can make sense of all the embedding data which B can possibly make sense of. This would, on the one hand, undermine arguments in favor of B over A on the basis of embedding data. And, provided that the converse does not hold—that is, that A can make sense of strictly more embedding data than B can—it would also show that there is a precise sense in which B is more constrained than A, yielding a pro tanto simplicity-based consideration in favor of B. In this paper I develop tools which allow us to make comparisons of this kind, which I call comparisons of potential expressive power. I motivate the development of these tools by way of exploration of the recent debate about epistemic modals. Prominent theories which have been developed in response to embedding data turn out to be strictly less expressive than the standard relational theory, a fact which necessitates a reorientation in how to think about the choice between these theories

Approximating Holant problems by winding

We give an FPRAS for Holant problems with parity constraints and not-all-equal constraints, a generalisation of the problem of counting sink-free-orientations. The approach combines a sampler for near-assignments of "windable" functions -- using the cycle-unwinding canonical paths technique of Jerrum and Sinclair -- with a bound on the weight of near-assignments. The proof generalises to a larger class of Holant problems; we characterise this class and show that it cannot be extended by expressibility reductions. We then ask whether windability is equivalent to expressibility by matchings circuits (an analogue of matchgates), and give a positive answer for functions of arity three

Synthesis of Stochastic Flow Networks

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network, and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. Stochastic flow networks can be easily implemented by DNA-based chemical reactions, with promising applications in molecular computing and stochastic computing. In this paper, we address a fundamental synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability transformation dates back to von Neumann's 1951 work and was followed, among others, by Knuth and Yao in 1976. Most existing works have been focusing on the "simulation" of target distributions. In this paper, we design optimal-sized stochastic flow networks for "synthesizing" target distributions. It shows that when each splitter has two outgoing edges and is unbiased, an arbitrary rational probability \frac{a}{b} with a\leq b\leq 2^n can be realized by a stochastic flow network of size n that is optimal. Compared to the other stochastic systems, feedback (cycles in networks) strongly improves the expressibility of stochastic flow networks.Comment: 2 columns, 15 page