Two-dimensional specification of universal quantification in agraphical database query language

A technique is proposed for specifying universal quantification and existential quantification (combined with negation) in a two-dimensional (graphical) database query language. Unlike other approaches that provide set operators to simulate universal quantification, this technique allows a direct representation of universal quantification. Syntactic constructs for specifying universal and existential quantifications, two-dimensional translation of universal quantification to existential quantification (with negation), and translation of existentially quantified two-dimensional queries to relational queries are presented. The resulting relational queries can be processed directly by many existing database systems. The authors claim that this technique renders universal quantifications easy to understand. To substantiate this claim, they provide a simple, easy-to-follow guideline for constructing universally quantified queries     

Axiomatising Tree-Interpretable Structures

We introduce the class of tree-interpretable structures which generalises the notion of a prefix-recognisable graph to arbitrary relational structures. We prove that every tree-interpretable structure is finitely axiomatisable in guarded second-order logic with cardinality quantifiers.     

Linguistic quantifiers modeled by Sugeno integrals

Since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them, linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning. This paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno's integral. This framework allows us to have some elegant logical properties of linguistic quantifiers. We compare carefully our new model of quantification and other approaches to linguistic quantifiers. A set of criteria for linguistic quantification was proposed in the previous literature. The relationship between these criteria and the results obtained in the present paper is clarified. Some simple applications of the Sugeno's integral semantics of quantifiers are presented.     

Quantifier-free logic for nondeterministic theories

We develop a quantifier-free logic for deriving consequences of multialgebraic theories. Multialgebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set-valued operations. Formulae are sequents over atoms allowing one to state set-inclusion or identity of 1-element sets (determinacy). We introduce a sound and weakly complete Rasiowa–Sikorski (R–S) logic for proving multialgebraic tautologies. We then extend this system for proving consequences of specifications based on translation of finite theories into logical formulae. Finally, we show how such a translation may be avoided—introduction of the specific cut rules leads to a sound and strongly complete Gentzen system for proving directly consequences of specifications. Besides giving examples of the general techniques of R–S and the specific cut rules, we improve the earlier logics for multialgebras by providing means to handle empty carriers (as well as empty result-sets) without the use of quantifiers, and to derive consequences of theories without translation into another format and without using general cut.     

Parallel Time and Quantifier Prefixes

We characterize the amount of alternation between blocks of Boolean quantifiers (having both existential and universal), blocks of real existential quantifiers, and blocks of real universal quantifiers that can be decided in parallel polynomial time over the reals. We do so under the assumption that blocks have a uniform bound on their size, both for the case of this bound to be polynomial and constant. On the way towards this characterization we prove a real version of Savitch’s Theorem.     

Relational analysis of (co)inductive predicates, (co)algebraic datatypes, and (co)recursive functions

We present techniques for applying a finite relational model finder to logical specifications that involve high-level definitional principles such as (co)inductive predicates, (co)algebraic datatypes, and (co)recursive functions. In contrast to previous work, which focused on algebraic datatypes and restricted occurrences of unbounded quantifiers in formulas, we can handle arbitrary formulas by means of a three-valued Kleene logic. The techniques form the basis of the counterexample generator Nitpick for Isabelle/HOL. As case studies, we consider formulas about an inductively defined context-free grammar, a functional implementation of AA trees, and a coalgebraic list datatype.     

Integrating Standard Dependency Schemes in QCSP Solvers

Quantified constraint satisfaction problems (QCSPs) are an extension to constraint satisfaction problems (CSPs) with both universal quantifiers and existential quantifiers.In this paper we apply variab...     

Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures

This paper presents results from two different areas. The first area is monadic second-order logic (MSO) over finite structures, in particular over the so-called grids. These are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors. For this logic, we study the expressive power of the alternation of existential and universal monadic second-order quantifiers, i.e., set quantifiers. In Matz et al. (Information and Computation, LICS’ 97, 1999, to appear) it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness. In this paper, we strengthen this result in several ways. Firstly, we show that there are MSO formulas that have a very restricted form of k+1 set quantifiers but are not equivalent to a formula with k quantifiers. Secondly, we show that if we fix the number of such alternations, the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one—this was previously known only for coloured grids. Thirdly, we investigate how an additional prefix of first-order (i.e., element) quantifiers influences the expressive power of MSO formulas. The second area that this paper is concerned with is two-dimensional formal language theory. We study how the alternation of (first- and monadic second-order) quantifications, on the one hand, relates to the dot-depth measure of two-dimensional (i.e., picture) languages, on the other hand. That measure is the two-dimensional version of the classical notion of dot-depth for (one-dimensional) starfree word languages. We show that the hierarchy induced by this dot-depth cuts through the hierarchy given by monadic second-order quantifications. In particular, beyond each level of the monadic second-order alternation hierarchy, there is a starfree picture language.     

The Expressive Power of Bijections over Weakly Arithmetized Structures

We investigate the expressive power of existential second-order formulas whose second-order quantifiers range over bijective unary functions. We show that as long as interpretations are taken over structures with a built-in linear order relation and an addition function, then quantifying over bijections is as expressive as quantifying over arbitrary unary functions. The originality of our result is that it remains true even if the first-order part of a formula contains exactly one variable (which is universally quantified). Our result immediately provides a new characterization of non-deterministic linear time on RAMs. It also permits us to derive a corollary on the Skolem normal form of first-order formulas over weakly arithmetized structures.     

Temporal relational data model

This paper incorporates a temporal dimension to nested relations. It combines research in temporal databases and nested relations for managing the temporal data in nontraditional database applications. A temporal data value is represented as a temporal atom; a temporal atom consists of two parts: a temporal set and a value. The temporal atom asserts that the value is valid over the time duration represented by its temporal set. The data model allows relations with arbitrary levels of nesting and can represent the histories of objects and their relationships. Temporal relational algebra and calculus languages are formulated and their equivalence is proved. Temporal relational algebra includes operations to manipulate temporal data and to restructure nested temporal relations. Additionally, we define operations to generate a power set of a relation, a set membership test, and a set inclusion test, which are all derived from the other operations of temporal relational algebra. To obtain a concise representation of temporal data (temporal reduction), collapsed versions of the set-theoretic operations are defined. Procedures to express collapsed operations by the regular operations of temporal relational algebra are included. The paper also develops procedures to completely flatten a nested temporal relation into an equivalent 1 NF relation and back to its original form, thus providing a basis for the semantics of the collapsed operations by the traditional operations on 1 NF relations     

A polynomial characterization of hypergraphs using the Ihara zeta function

The aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering. To this end, we develop a polynomial characterization for hypergraphs based on the Ihara zeta function. We investigate the flexibility of the polynomial coefficients for learning relational structures with different relational orders. Furthermore, we develop an efficient method for computing the coefficient set. Our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities, and thus can distinguish different relational orders, which is prone to ambiguity if the hypergraph Laplacian is used. In our experimental evaluation, we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph Laplacian.     

A Remark on Collective Quantification

We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize all collective quantification in natural language.     

Bottom-Up Induction of Feature Terms

The aim of relational learning is to develop methods for the induction of hypotheses in representation formalisms that are more expressive than attribute-value representation. Most work on relational learning has been focused on induction in subsets of first order logic like Horn clauses. In this paper we introduce the representation formalism based on feature terms and we introduce the corresponding notions of subsumption and anti-unification. Then we explain INDIE, a heuristic bottom-up learning method that induces class hypotheses, in the form of feature terms, from positive and negative examples. The biases used in INDIE while searching the hypothesis space are explained while describing INDIE's algorithms. The representational bias of INDIE can be summarised in that it makes an intensive use of sorts and sort hierarchy, and in that it does not use negation but focuses on detecting path equalities. We show the results of INDIE in some classical relational datasets showing that it's able to find hypotheses at a level comparable to the original ones. The differences between INDIE's hypotheses and those of the other systems are explained by the bias in searching the hypothesis space and on the representational bias of the hypothesis language of each system.     

Relaxing the universal quantifier of the division in fuzzy relational databases

In a previous paper, we presented an approach to calculate relational division in fuzzy databases, starting with the GEFRED model. This work centered on dealing with fuzzy attributes and fuzzy values and only the universal quantifier was taken into account since it is the inherent quantifier in classical relational division. In this paper, we present an extension of that division to relax the universal quantifier. With this new system we can use both absolute quantifiers and relative quantifiers irrespective of how the function of the fuzzy quantifier is defined. We also include a comparison with other fuzzy division approaches to relax the universal quantifier that have been published. Furthermore, in this paper we have extended the fuzzy SQL language to express any kind of fuzzy division. © 2001 John Wiley & Sons, Inc.     

The Hamiltonian Syllogistic

This paper undertakes a re-examination of Sir William Hamilton’s doctrine of the quantification of the predicate. Hamilton’s doctrine comprises two theses. First, the predicates of traditional syllogistic sentence-forms contain implicit existential quantifiers, so that, for example, All p is q is to be understood as All p is some q. Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, All p is all q. Hamilton attempted to provide a deductive system for his language, along the lines of the classical syllogisms. We show, using techniques unavailable to Hamilton, that such a system does exist, though with qualifications that distinguish it from its classical counterpart.     

Multiple Lyapunov functions and other analysis tools for switchedand hybrid systems

We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of “continuous switched” systems     

Monotonicity and Collective Quantification

This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and respects theirmonotonicity properties. However, some previously unnoticed factsindicate that monotonicity of plural determiners is not always preservedwhen they apply to collective predicates. We show that the proposedoperator describes this behavior correctly, and characterize themonotonicity of the collective determiners it derives. It is proved thatdeterminer fitting always preserves monotonicity properties ofdeterminers in their second argument, but monotonicity in the firstargument of a determiner is preserved if and only if it is monotonic inthe same direction in the second argument. We argue that this asymmetryfollows from the conservativity of generalized quantifiers innatural language.     

Efficient relational calculation for software analysis

Calculating with graphs and relations has many applications in the analysis of software systems, for example, the detection of design patterns or patterns of problematic design and the computation of design metrics. These applications require an expressive query language, in particular, for the detection of graph patterns, and an efficient evaluation of the queries even for large graphs. In this paper, we introduce RML, a simple language for querying and manipulating relations based on predicate calculus, and CrocoPat, an interpreter for RML programs. RML is general because it enables the manipulation not only of graphs (i.e., binary relations), but of relations of arbitrary arity. CrocoPat executes RML programs efficiently because it internally represents relations as binary decision diagrams, a data structure that is well-known as a compact representation of large relations in computer-aided verification. We evaluate RML by giving example programs for several software analyses and CrocoPat by comparing its performance with calculators for binary relations, a Prolog system, and a relational database management system.