Fusion in relational structures and the verification of monadic second-order properties

2002 ◽  
Vol 12 (2) ◽  
pp. 203-235 ◽  
Author(s):  
B. COURCELLE ◽  
J. A. MAKOWSKY
Keyword(s):  
Second Order ◽  
Algebraic Expression ◽  
Order Property ◽  
Unary Predicate ◽  
Free Graph ◽  
Relational Structures ◽  
Clear Cut ◽  
Algebraic Expressions ◽  
Common Framework ◽  
Fusion Operation

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

Metaontology

2018 ◽  
Author(s):  
Uriah Kriegel
Keyword(s):  
Second Order ◽  
Order Property ◽  
First Order ◽  
First Approximation

Brentano’s theory of judgment serves as a springboard for his conception of reality, indeed for his ontology. It does so, indirectly, by inspiring a very specific metaontology. To a first approximation, ontology is concerned with what exists, metaontology with what it means to say that something exists. So understood, metaontology has been dominated by three views: (i) existence as a substantive first-order property that some things have and some do not, (ii) existence as a formal first-order property that everything has, and (iii) existence as a second-order property of existents’ distinctive properties. Brentano offers a fourth and completely different approach to existence talk, however, one which falls naturally out of his theory of judgment. The purpose of this chapter is to present and motivate Brentano’s approach.

scholarly journals Examination of Semantic Structures Used by Teacher Candidates to Transform Algebraic Expressions into Verbal Problems

2021 ◽  
Vol 4 (3) ◽  
Author(s):  
Ayten Pinar Bal ◽  
Keyword(s):  
Teacher Candidates ◽  
Quantitative Research ◽  
Algebraic Expression ◽  
Study Group ◽  
State University ◽  
Algebraic Expressions ◽  
Purposeful Sampling ◽  
Division Problems ◽  
Addition And Subtraction ◽  
Research Types

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019-2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; It has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

On spectra of sentences of monadic second order logic with counting

2004 ◽  
Vol 69 (3) ◽  
pp. 617-640 ◽  
Author(s):  
E. Fischer ◽  
J. A. Makowsky
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Graph Grammars ◽  
Free Graph ◽  
Unary Relation ◽  
Tree Width ◽  
Binary Relation Symbol ◽  
Context Free ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Abstract.We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle.Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

scholarly journals Monadic Second-Order Logic, Graphs and Unfoldings of Transition Systems

1995 ◽  
Vol 2 (44) ◽  
Author(s):  
Bruno Courcelle ◽  
Igor Walukiewicz
Keyword(s):  
Transition System ◽  
Second Order ◽  
Order Logic ◽  
Order Property ◽  
Transition Systems ◽  
Graph Coverings ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We prove that every monadic second-order property of the unfolding<br />of a transition system is a monadic second-order property of the<br />system itself. We prove a similar result for certain graph coverings.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

Helping Students Make Sense of Algebraic Expressions: The Candy Shop

2002 ◽  
Vol 7 (9) ◽  
pp. 492-497
Author(s):  
Diana Underwood Gregg ◽  
Erna Yackel
Keyword(s):  
Algebraic Thinking ◽  
Instructional Sequence ◽  
Algebraic Expression ◽  
Instructional Approaches ◽  
Algebra For All ◽  
Algebraic Expressions ◽  
Experience Difficulty

Increasing emphasis on “Algebra for all” (NCTM 1997a, 1997b) compels educators to identify and address fundamental ideas that build the foundations for algebraic thinking and reasoning. Identifying these foundational concepts and developing appropriate instructional approaches are the focuses of our work. One area in which students often experience difficulty is adding and subtracting algebraic expressions. Although students may be able to memorize a procedure, such as “distribute the negative” when subtracting algebraic expressions, they are often unable to make sense of this procedure. Our work suggests that part of students' difficulty in this area is that they do not conceptualize an algebraic expression as a composite unit. In the paragraphs below, we explain what is meant by composite units and how this construct helped frame our development of an instructional sequence to help students make sense of, and find meaning in, algebraic expressions and operations on algebraic expressions.

scholarly journals Giant optical nonlinearity interferences in quantum structures

2019 ◽  
Vol 5 (10) ◽  
pp. eaaw7554
Author(s):  
S. Houver ◽  
A. Lebreton ◽  
T. A. S. Pereira ◽  
G. Xu ◽  
R. Colombelli ◽  
...  
Keyword(s):  
Quantum Cascade Lasers ◽  
Optical Nonlinearity ◽  
Second Order ◽  
Nonlinear Susceptibility ◽  
Complex Materials ◽  
Quantum Cascade ◽  
Clear Cut ◽  
Sensitive Tool ◽  
Cascade Lasers ◽  
Structure Properties

Second-order optical nonlinearities can be greatly enhanced by orders of magnitude in resonantly excited nanostructures. These resonant nonlinearities continually attract attention, particularly in newly discovered materials. However, they are frequently not as heightened as currently predicted, limiting their exploitation in nanostructured nonlinear optics. Here, we present a clear-cut theoretical and experimental demonstration that the second-order nonlinear susceptibility can vary by orders of magnitude as a result of giant destructive, as well as constructive, interference effects in complex systems. Using terahertz quantum cascade lasers as a model source to investigate interband and intersubband nonlinearities, we show that these giant interferences are a result of an unexpected interplay of the second-order nonlinear contributions of multiple light and heavy hole states. As well as of importance to understand and engineer the resonant optical properties of nanostructures, this advanced framework can be used as a novel, sensitive tool to elucidate the band structure properties of complex materials.

scholarly journals Asymptotic density in quasi-logarithmic additive number systems

2008 ◽  
Vol 144 (2) ◽  
pp. 267-287
Author(s):  
BRUNO NIETLISPACH
Keyword(s):  
Counting Function ◽  
Second Order ◽  
Asymptotic Density ◽  
Additive Number ◽  
Relational Structures ◽  
Number Systems ◽  
Density Theorems ◽  
Irreducible Elements

AbstractWe show that in quasi-logarithmic additive number systems$\mycal{A}$all partition sets have asymptotic density, and we obtain a corresponding monadic second-order limit law for adequate classes of relational structures. Our conditions on the local counting functionp(n) of the set of irreducible elements of$\mycal{A}$allow situations which are not covered by the density theorems of Compton [6] and Woods [15]. We also give conditions onp(n) which are sufficient to show the assumptions of Compton's result are satisfied, but which are not necessarily implied by those of Bell and Burris [2], Granovsky and Stark [8] or Stark [14].

Concurrent Transformations of Relational Structures

1986 ◽  
Vol 9 (1) ◽  
pp. 13-49
Author(s):  
Hartmut Ehrig ◽  
Annegret Habel ◽  
Barry K. Rosen
Keyword(s):  
Data Structures ◽  
Relational Data ◽  
Graph Grammars ◽  
Data Types ◽  
Relational Structures ◽  
Formal Framework ◽  
Common Framework ◽  
Abstract Data ◽  
Decomposition Of Transformations ◽  
Fundamental Theorems

This paper provides a common framework to study transformations of structures ranging from all kinds of graphs to relational data structures. Transformations of structures can be used as derivations of graphs in the sense of graph grammars, as update of relations in the sense of relational data bases, or even as operations on data structures in the sense of abstract data types. The main aim of the paper is to construct parallel and concurrent transformations from given sequential ones and to study sequentializability properties of complex transformations. The main results are three fundamental theorems concerning parallelism, concurrency and decomposition of transformations of structures. On one hand these results can be considered as a contribution to the study of con currency in graph grammars and on the other hand as a formal framework for consistent concurrent update of relational structures.

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