Flag algebras

2007 ◽  
Vol 72 (4) ◽  
pp. 1239-1282 ◽  
Author(s):  
Alexander A. Razborov
Keyword(s):  
Model Theory ◽  
Simple Proof ◽  
Edge Density ◽  
Universal Theory ◽  
Extremal Combinatorics ◽  
Finite Models ◽  
Flag Algebras ◽  
Formal Calculus ◽  
And Function

AbstractAsymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen sub-model of N with ∣M∣ elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1,N),…, p(Mh,N), where M1,…, M1, are fixed “template” models and ∣N∣ grows to infinity?In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.

scholarly journals A finite model-theoretical proof of a property of bounded query classes within PH

2004 ◽  
Vol 69 (4) ◽  
pp. 1105-1116 ◽  
Author(s):  
Leszek Aleksander Kołodziejczyk
Keyword(s):  
Normal Form ◽  
Model Theory ◽  
Finite Model ◽  
Sufficient Condition ◽  
Finite Model Theory ◽  
Finite Models ◽  
Constructible Function ◽  
Simple Sufficient Condition

Abstract.We use finite model theory (in particular, the method of FM-truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove:Let m ≥ 2. Then:(A) If there exists k such that NP⊆ Σm TIME(nk)∩ Πm TIME(nk), then for every r there exists kr such that :(B) If there exists a superpolynomial time-constructible function f such that NTIME(f), then additionally .This strengthens a result by Mocas [M96] that for any r, .In addition, we use FM-truth definitions to give a simple sufficient condition for the arity hierarchy to be strict over finite models.

Démonstration du théorème sur la suite spectrale des fibrés au sens de Kan

1968 ◽  
Vol 64 (2) ◽  
pp. 293-297
Author(s):  
Gheorghe Lusztig ◽  
Henri Moscovici
Keyword(s):  
Spectral Sequence ◽  
Model Theory ◽  
Simple Proof ◽  
Fibre Space

AbstractWe give a relatively simple proof of the theorem on the spectral sequence of a fibre-space in the sense of Kan. The method consists in the reduction to locally trivial fibre-spaces by minimal fibrings and in the use of a very simple acyclic model theory (without degeneracies).

scholarly journals RELIGION IN THE THEORIES OF CLASSICAL SOCIOLOGY

2020 ◽  
pp. 307-325
Author(s):  
Armin Jašarević
Keyword(s):  
Social Behavior ◽  
Theoretical Analysis ◽  
Universal Theory ◽  
Research Topics ◽  
Essential Condition ◽  
Theory Of Religion ◽  
Social Events ◽  
History Of ◽  
And Function ◽  
Classical Sociology

As the science about society, from its very beginning, sociology has dealt with religion and its importance and function in a society. Social events and changes that have taken place in the area of Europe have contributed to bringing religion in the focus of many scholars, which shows that in the overall history of humankind it has been one of the unavoidable research topics. This research aimed at showing how classical sociologists (Comte, Marx, Durkheim and Weber) approached the phenomenon of religion. The stances of the aforementioned scholars are presented by the means of a method of theoretical analysis. The findings indicate that all scholars approach the phenomenon of religion differently. Thus, for instance Comte, as a founder of sociology, embodies a positivist discourse through which he promotes the universal theory of religion. Unlike his contemporaries, Durkheim, claims that religion is an unavoidable society factor and that it presents an essential condition for social integration. Contrary to Durkheim, Marx argues that religion is the alienation and opium of the ruling masses who use it to establish balance. Weber, adopting a systematic sociology approach to religion, analyses comparatively religious and social behavior, and claims that religion is a radical response to specific life situations.

Strong 0-1 laws in finite model theory

2000 ◽  
Vol 65 (4) ◽  
pp. 1686-1704
Author(s):  
Wafik Boulos Lotfallah
Keyword(s):  
Model Theory ◽  
Sample Space ◽  
Finite Model ◽  
Additive Measure ◽  
Finite Model Theory ◽  
Finite Models ◽  
The Universe ◽  
New Framework

AbstractWe introduce a new framework for asymptotic probabilities of sentences, in which we have a σ-additive measure on the sample space of all sequences A = {} of finite models, where the universe of is {1,2, …, n}. and use this framework to strengthen 0-1 laws for logics.

Finite model theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Model Theory ◽  
Classical Model ◽  
Expressive Power ◽  
Order Logic ◽  
Finite Model ◽  
Descriptive Complexity ◽  
Finite Model Theory ◽  
First Order ◽  
Finite Models ◽  
Finite Structures

This final chapter unites ideas from both model theory and complexity theory. Finite model theory is the part of model theory that disregards infinite structures. Examples of finite structures naturally arise in computer science in the form of databases, models of computations, and graphs. Instead of satisfiability and validity, finite model theory considers the following finite versions of these properties. • A first-order sentence is finitely satisfiable if it has a finite model. • A first-order sentence is finitely valid if every finite structure is a model. Finite model theory developed separately from the “classical” model theory of previous chapters. Distinct methods and logics are used to analyze finite structures. In Section 10.1, we consider various finite-variable logics that serve as useful languages for finite model theory. We define variations of the pebble games introduced in Section 9.2 to analyze the expressive power of these logics. Pebble games are one of the few tools from classical model theory that is useful for investigating finite structures. In Section 10.2, it is shown that many of the theorems from Chapter 4 are no longer true when restricted to finite models. There is no analog for the Completeness and Compactness theorems in finite model theory. Moreover, we prove Trakhtenbrot’s theorem which states that the set of finitely valid first-order sentences is not recursively enumerable. Descriptive complexity is the subject of 10.3. This subject describes the complexity classes discussed in Chapter 7 in terms of the logics introduced in Chapter 9. We prove Fagin’s theorem relating the class NP to existentional second-order logic. We prove the Cook–Levin theorem as a consequence of Fagin’s Theorem. This theorem states that the Satisfiability Problem for Propositional Logic is NP-complete. We conclude this chapter (and this book) with a section describing the close connection between logic and the P = NP problem. In this section, we discuss appropriate logics for the study of finite models. First-order logic, since it describes each finite model up to isomorphism, is too strong. For this reason, we must weaken the logic. It may seem counter-intuitive that we should gain knowledge by weakening our language.

From “Metabelian ℚ-Vector Spaces” to New ω-Stable Groups

1996 ◽  
Vol 2 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Group Theory ◽  
Model Theory ◽  
Stability Theory ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Universal Theory ◽  
Direct Power ◽  
Metabelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

The aim of this paper is to describe (without proofs) an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and group theory (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for group theory). However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups.

scholarly journals Minimizing the number of 5-cycles in graphs with given edge-density

2019 ◽  
Vol 29 (1) ◽  
pp. 44-67
Author(s):  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Bernard Lidický ◽  
Oleg Pikhurko
Keyword(s):  
Upper Bound ◽  
Stability Result ◽  
Edge Density ◽  
Flag Algebras ◽  
Image Position

AbstractMotivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

scholarly journals Nonclassical mythology on the cognitive problems and capabilities of myth ontology

2021 ◽  
pp. 1-10
Author(s):  
Andrey Vladimirovich Stavitskiy
Keyword(s):  
Scientific Research ◽  
Universal Theory ◽  
Cognitive Research ◽  
Classical Mythology ◽  
The Creation ◽  
And Function ◽  
Cognitive Problems

This article is dedicated to the cognitive and epistemological peculiarities of the myth, which can be better understood relying on the principles and approaches of nonclassical science. The article discloses the key reasons for incomprehension of the myth by science, and explains the ways for its overcoming based on the broader sense of the myth. Within the framework of this paradigm, myth is viewed as a cultural universal, where mythmaking is a quality and function of consciousness. Such myth long ago has transcended the representations and formulas of classical mythology, and requires different attitude, considering the latest research (A. A. Gagaev, A. M. Lobok, V. M. Naydysh, and others). These studies indicate that modern (nonclassical) myth can be grasped only with consideration of latest discoveries in psychology, semiotics, and cognitive research, which prove that people have always been engaged in mythmaking, not just at the dawn of humanity. This means that on the agenda of scientific research of myth is the problem of existence of another nonclassical mythology and the creation of the universal theory of myth. All major elaborations of the leading theories of myth of the XX century are successfully synthesized within the universal theory of myth in accordance with the principle of mutual complementarity. This topic is increasingly relevant, and opens up new opportunities for science, fundamentally changing its perception of myth.

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