Tautology testing with a generalized matrix reduction method

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.

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.

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.

Space Bounds for Resolution

On the hydra number of disconnected graphs

AI systems often need to deal with inconsistent information. For this reason since the early 2000s some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on three general information spaces: a relational database, a graph database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation.

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

Measures of quantity of information have been studied extensively for more than fifty years. The seminal work on information theory is by Shannon [67]. This work, based on probability theory, can be used in a logical setting when the worlds are the possible events. This work is also the basis of Lozinskii's work [48] for defining the quantity of information of a formula (or knowledgebase) in propositional logic. But this definition is not suitable when the knowledgebase is inconsistent. In this case, it has no classical model, so we have no “event” to count. This is a shortcoming since in practical applications (e.g. databases) it often happens that the knowledgebase is not consistent. And it is definitely not true that all inconsistent knowledgebases contain the same (null) amount of information, as given by the “classical information theory”. As explored for several years in the paraconsistent logic community, two inconsistent knowledgebases can lead to very different conclusions, showing that they do not convey the same information. There has been some recent interest in this issue, with some interesting proposals. Though a general approach for information theory in (possibly inconsistent) logical knowledgebases is missing. Another related measure is the measure of contradiction. It is usual in classical logic to use a binary measure of contradiction: a knowledgebase is either consistent or inconsistent. This dichotomy is obvious when the only deductive tool is classical inference, since inconsistent knowledgebases are of no use. But there are now a number of logics developed to draw non-trivial conclusions from an inconsistent knowledgebase. So this dichotomy is not sufficient to describe the amount of contradiction of a knowledgebase, one needs more fine-grained measures. Some interesting proposals have been made for this. The main aim of this paper is to review the measures of information and contradiction, and to study some potential practical applications. This has significant potential in developing intelligent systems that can be tolerant to inconsistencies when reasoning with real-world knowledge.

The paper presents an efficient algorithm for checking the satisfiability of a propositional linear time temporal logic formula, which may have past as well as future operators. This algorithm can be used to check validity of such formulas over all models as well as over computations of a finite-state program (model checking). Unlike previous theoretical presentations of a decision method for checking satisfiability or validity, whose first step is to construct the full set of all possible atoms of a tableau (satisfaction graph) and immediately pay the worst case exponential complexity price, the algorithm presented here builds the tableau incrementally. This means that the algorithm constructs only those atoms that are reachable from a possible initial atom, satisfying the formula to be checked.While incremental tableau construction for the future fragment of linear time temporal logic can be done in a single pass, the presence of past operators requires multiple passes that successively construct augmented versions of existing atoms, while still maintaining consistency and reachability.The proof of correctness of the algorithm is based on showing that any model of the considered formula is embedded as a path in the tableau at all the construction stages, and can be delineated when the construction terminates.The paper also describes an implementation of the algorithm with further attention to efficiency. This implementation is available as a support system for the book [8] under the name “temporal prover”. It has been used to verify all the propositional temporal formulas and to model-check all the finite-state programs appearing in the book.Keywordstemporal logicsatisfiability checkingvalidity checkingmodel checkingpast and future operatorsincremental tableauautomatic verification

The Complexity of Local Proof Search in Linear Logic: (Extended Abstract)

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.

A catalog of Boolean concepts

In the last decade, advances in satisfiability-modulo-theories (SMT) solvers have powered a new generation of software tools for verification and testing. These tools transform various program analysis problems into the problem of satisfiability of formulas in propositional or first-order logic, where they are discharged by SMT solvers, such as Z3 from Microsoft Research. This paper briefly summarizes four initiatives from Microsoft Research that build upon Z3 and move beyond first-order satisfaction: Fixed points—μZ is a scalable, efficient engine for discharging fixed point queries over recursive predicates with logical constraints, integrated in Z3; Interpolants—Interpolating Z3 uses Z3’s proof generation capability to generate Craig interpolants in the first-order theory of uninterpreted functions, arrays and linear arithmetic; Automata—The symbolic automata toolkit lifts classical automata analyses to work modulo symbolic constraints on alphabets; Polynomials—a new decision procedure for the existential theory of the reals allows efficient solving of systems of non-linear arithmetic constraints.KeywordsPoint QueryAbstract DomainSymbolic Model CheckLinear ArithmeticUninterpreted FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.