On Quasi-Interpretations, Blind Abstractions and Implicit Complexity

Quasi-interpretations are a technique to guarantee complexity bounds on first-order functional programs: with termination orderings they give in particular a sufficient condition for a program to be executable in polynomial time, called here the P-criterion. We study properties of the programs satisfying the P-criterion, in order to better understand its intensional expressive power. Given a program on binary lists, its blind abstraction is the nondeterministic program obtained by replacing lists by their lengths (natural numbers). A program is blindly polynomial if its blind abstraction terminates in polynomial time. We show that all programs satisfying a variant of the P-criterion are in fact blindly polynomial. Then we give two extensions of the P-criterion: one by relaxing the termination ordering condition, and the other one (the bounded value property) giving a necessary and sufficient condition for a program to be polynomial time executable, with memoisation.Comment: 18 page

Characterizations of Decomposable Dependency Models

Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to the well-known set characterizing dependency models that are isomorphic to undirected graphs. We also briefly discuss a potential application of our results to the problem of learning graphical models from data.Comment: See http://www.jair.org/ for any accompanying file

MaxSAT Resolution and Subcube Sums

We study the MaxRes rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which p-simulates MaxResW, can be viewed as a special case of the semialgebraic Sherali-Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums

Properties, Learning Algorithms, and Applications of Chain Graphs and Bayesian Hypergraphs

Probabilistic graphical models (PGMs) use graphs, either undirected, directed, or mixed, to represent possible dependencies among the variables of a multivariate probability distri- bution. PGMs, such as Bayesian networks and Markov networks, are now widely accepted as a powerful and mature framework for reasoning and decision making under uncertainty in knowledge-based systems. With the increase of their popularity, the range of graphical models being investigated and used has also expanded. Several types of graphs with dif- ferent conditional independence interpretations - also known as Markov properties - have been proposed and used in graphical models. The graphical structure of a Bayesian network has the form of a directed acyclic graph (DAG), which has the advantage of supporting an interpretation of the graph in terms of cause-effect relationships. However, a limitation is that only asymmetric relationships, such as cause and effect relationships, can be modeled between variables in a DAG. Chain graphs, which admit both directed and undirected edges, can be used to overcome this limitation. Today there exist three main different interpretations of chain graphs in the lit- erature. These are the Lauritzen-Wermuth-Frydenberg, the Andersson-Madigan-Perlman, and the multivariate regression interpretations. In this thesis, we study these interpreta- tions based on their separation criteria and the intuition behind their edges. Since structure learning is a critical component in constructing an intelligent system based on a chain graph model, we propose new feasible and efficient structure learning algorithms to learn chain graphs from data under the faithfulness assumption. The proliferation of different PGMs that allow factorizations of different kinds leads us to consider a more general graphical structure in this thesis, namely directed acyclic hypergraphs. Directed acyclic hypergraphs are the graphical structure of a new proba- bilistic graphical model that we call Bayesian hypergraphs. Since there are many more hypergraphs than DAGs, undirected graphs, chain graphs, and, indeed, other graph-based networks, Bayesian hypergraphs can model much finer factorizations and thus are more computationally efficient. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We also extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention. The framework of graphical models, which provides algorithms for discovering and analyzing structure in complex distributions to describe them succinctly and extract un- structured information, allows them to be constructed and utilized effectively. Two of the most important applications of graphical models are causal inference and information ex- traction. To address these abilities of graphical models, we conduct a causal analysis, comparing the performance behavior of highly-configurable systems across environmen- tal conditions (changing workload, hardware, and software versions), to explore when and how causal knowledge can be commonly exploited for performance analysis