Measuring Inconsistency in a General Information Space

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

Space Bounds for Resolution

On the hydra number of disconnected graphs

We describe an incremental algorithm for computing interpolants for a pair ϕ A , ϕ B of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕ A ∧ ϕ B , and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

Cut formulas in propositional logic

An important step in data analysis is the exploration of data. For traditional relational databases one of the most powerful tools for performing such analysis is the relational database and the aggregates and rankings that they can compute: for instance, simple statistics such as the average number of links between two types of entities (relations) are easily computed using a query on a relational database and may already provide valuable information. However, for the exploration of graph data, relational databases may not be most practical and scalable. For instance, a statistic such as the shortest path between two given nodes cannot be computed by a relational database. Surprisingly, however, tools for querying graph and network databases are much less well developed than for relational data, and only recently an increasing number of studies are devoted to graph or network databases. Our position is that the development of such graph databases is important both to make basic graph mining easier and to prepare data for more complex types of analysis. An important component of such databases is the language that is used to enable aggregating queries, such as shortest path queries.In this paper, we propose an extension to a previously proposed query language. This extension allows for querying and analyzing databases by using aggregates and ranking. A notable feature of our language is that it also supports probabilistic graph queries by conceiving of such queries as aggregating queries. We demonstrate its value on a simple data analysis task.

On the expressivity of inconsistency measures

This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae has attracted interest mainly in the decision tree model. In the communication complexity model many interesting results deal with specific read-once formulae, such as disjointness and tribes. In this dissertation we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f )/c^d(f ) , where n(f ) is the number of variables and d(f ) is the depth of the formula, and they are optimal up to the constant in the base of the denominator. In the second part of the dissertation, we explore the applicability of the information-theoretic method in the number-on-the-forehead model. The work of Bar-Yossef, Jayram, Kumar & Sivakumar [BYJKS04] revealed a beautiful connection between Hellinger distance and two-party randomized communication protocols. Inspired by their work and motivated by the open questions in the number-on-the-forehead model, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party protocols. We provide a small toolbox that allows one to manipulate several Hellinger volume terms and also to lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. Finally, we show how to apply the new tools to obtain a lower bound on the informational complexity of the ANDk function.

In developing NoSQL databases, a major motivation is to achieve better efficient query performance compared with relational databases. The graph database is a NoSQL paradigm where navigation is based on links instead of joining tables. Links can be implemented as pointers, and following a pointer is a constant time operation, whereas joining tables is more complicated and slower, even in the presence of foreign keys. Therefore, link-based navigation has been seen as a more efficient query approach than using join operations on tables. Existing studies strongly support this assumption. However, query complexity has received less attention. For example, in enterprise information systems, queries are usually complex so data need to be collected from several tables or by traversing paths of graph nodes of different types. In the present study, we compared the query performance of a graph-based database system (Neo4j) and relational database systems (MySQL and MariaDB). The effect of different efficiency issues (e.g., indexing and optimization) were included in the comparison in order to investigate the most efficient solutions for different query types. The outcome is that although Neo4j is more efficient for simple queries, MariaDB is essentially more efficient when the complexity of queries increases. The study also highlighted how dramatically the efficiency of relational database has grown during the last decade.

In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan’s laws and double negation. This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF). Both tools make use of a rewrite strategy, which specifies how an exercise can be solved stepwise. Different types of feedback can be calculated automatically from such a strategy specification. We describe a strategy for constructing expert-like equivalence proofs, and present two techniques for improving the proofs that are generated by the strategy.Keywordspropositional logicequivalencese-learningfeedback

We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae, has attracted interest mainly in the decision tree model. In the communication complexity model, many interesting results deal with specific read-once formulae, such as DISJOINTNESS and TRIBES. In this paper we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f)/c^{d(f)}, where n(f) is the number of variables and d(f) the depth of the formula, and they are optimal up to the constant c in the denominator.

This paper shows that idempotents in finite rings of integers can act as Generalized Boolean Assignments (GBA's) by providing a completeness theorem. We introduce the notion of a generic Generalized Boolean Assignment. The mere propagation of such an assignment reveals feasibility (existence of a solution) of a formula in propositional logic. Then, we demystify this general concept by formulating the process on the bit-level: It turns out that propagation of a GBA only simulates bitwise (non-communicating) parallel computing. We capitalize on this by modifying the state-of-the-art local search Sat solver UnitWalk accordingly. This modification involves a more complicated parallelism.

Tautology testing with a generalized matrix reduction method

A catalog of Boolean concepts

The article considers non-standard semantics for syllogistic systems. In these systems, the formulas of propositional logic are the meanings of general terms, and the values of categorical statements are determined by the logical relationship between these formulas. If formula A is the meaning of term S, and if formula B is the meaning of term P, then the formula “SaP” is valid if and only if A entails B. Otherwise, the formula of the form “SoP” is valid. The “SeP” formula is valid if and only if A entails ¬B, otherwise the “SiP” formula is valid. This semantics is based on the intensional approach to the construction of syllogistic theories that goes back to Leibniz’s ideas. Firstly, the entailment relation by which categorical statements are interpreted allows, as Voishvillo has shown, an intensional interpretation: A entails B if the logical content of B is a part of the logical content of A. Secondly, propositional formulas can be considered as abbreviations for quantifier-free first-degree formulas with a single free variable, and such formulas, according to Voishvillo, state the contents of concepts. Therefore, in this semantics, the value of a syllogistic formula is determined not by the relationship between the extensions of the subject and the predicate, but by the relationship between their intensions. This article shows how different types of entailment affect the class of universally valid syllogistic formulas. Theories that use the relations of classical entailment and FDE-entailment have already been constructed. The classical entailment can be defined on the set of Carnap-style (consistent and complete) state-descriptions, and the first-degree relevant entailment can be defined on the set of generalized state-descriptions (the latter can be either contradictory or incomplete). In the article, semantics using two other relations: the Hao Wang-style entailment logic (defined on the set of all consistent state-descriptions) and its dual logic entailment (defined on the set of all complete state-descriptions) were built. The deductive capabilities of four syllogistic systems that use different types of logical entailments were compared.