Hypergraph Acyclicity and Propositional Model Counting

We show that the propositional model counting problem #SAT for CNF- formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore, we present a polynomial time algorithm that computes a disjoint branches decomposition of a given hypergraph if it exists and rejects otherwise. Finally, we show that some slight extensions of the class of hypergraphs with disjoint branches decompositions lead to intractable #SAT, leaving open how to generalize the counting result of this paper

Performance Evaluation and Optimization of Math-Similarity Search

Similarity search in math is to find mathematical expressions that are similar to a user's query. We conceptualized the similarity factors between mathematical expressions, and proposed an approach to math similarity search (MSS) by defining metrics based on those similarity factors [11]. Our preliminary implementation indicated the advantage of MSS compared to non-similarity based search. In order to more effectively and efficiently search similar math expressions, MSS is further optimized. This paper focuses on performance evaluation and optimization of MSS. Our results show that the proposed optimization process significantly improved the performance of MSS with respect to both relevance ranking and recall.Comment: 15 pages, 8 figure

Probabilistic Algorithmic Knowledge

The framework of algorithmic knowledge assumes that agents use deterministic knowledge algorithms to compute the facts they explicitly know. We extend the framework to allow for randomized knowledge algorithms. We then characterize the information provided by a randomized knowledge algorithm when its answers have some probability of being incorrect. We formalize this information in terms of evidence; a randomized knowledge algorithm returning ``Yes'' to a query about a fact \phi provides evidence for \phi being true. Finally, we discuss the extent to which this evidence can be used as a basis for decisions.Comment: 26 pages. A preliminary version appeared in Proc. 9th Conference on Theoretical Aspects of Rationality and Knowledge (TARK'03

Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets

In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in Multi-Agent Systems (CLIMA XI). To appea

Probabilistic Consensus of the Blockchain Protocol

We introduce a temporal epistemic logic with probabilities as an extension of temporal epistemic logic. This extension enables us to reason about properties that characterize the uncertain nature of knowledge, like “agent a will with high probability know after time s same fact”. To define semantics for the logic we enrich temporal epistemic Kripke models with probability functions defined on sets of possible worlds. We use this framework to model and reason about probabilistic properties of the blockchain protocol, which is in essence probabilistic since ledgers are immutable with high probabilities. We prove the probabilistic convergence for reaching the consensus of the protocol

How Many Topics? Stability Analysis for Topic Models

Topic modeling refers to the task of discovering the underlying thematic structure in a text corpus, where the output is commonly presented as a report of the top terms appearing in each topic. Despite the diversity of topic modeling algorithms that have been proposed, a common challenge in successfully applying these techniques is the selection of an appropriate number of topics for a given corpus. Choosing too few topics will produce results that are overly broad, while choosing too many will result in the "over-clustering" of a corpus into many small, highly-similar topics. In this paper, we propose a term-centric stability analysis strategy to address this issue, the idea being that a model with an appropriate number of topics will be more robust to perturbations in the data. Using a topic modeling approach based on matrix factorization, evaluations performed on a range of corpora show that this strategy can successfully guide the model selection process.Comment: Improve readability of plots. Add minor clarification

Randomisation and Derandomisation in Descriptive Complexity Theory

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures