Model Theory on Finite Structures∗
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Lecture Slides
Program Verification: Lecture 3 Jos´eMeseguer Computer Science Department University of Illinois at Urbana-Champaign 1 Algebras An (unsorted, many-sorted, or order-sorted) signature Σ is just syntax: provides the symbols for a language; but what is that language talking about? what is its semantics? It is obviously talking about algebras, which are the mathematical models in which we interpret the syntax of Σ, giving it concrete meaning. Unsorted algebras are the simplest example: children become familiar with them from the early awakenings of reason. They consist of a set of data elements, and various chosen constants among those elements, and operations on such data. 2 Algebras (II) For example, for Σ the unsorted signature of the module NAT-MIXFIX we can define many different algebras, such as the following: 1. IN, the algebra of natural numbers in whatever notation we wish (Peano, binary, base 10, etc.) with 0 interpreted as the zero element, s interpreted as successor, and + and * interpreted as natural number addition and multiplication. 2. INk, the algebra of residue classes modulo k, for k a nonzero natural number. This is a finite algebra whose set of elements can be represented as the set {0,...,k − 1}. We interpret 0 as 0, and for the other 3 operations we perform them in IN and then take the residue modulo k. For example, in IN7 we have 6+6=5. 3. Z, the algebra of the integers, with 0 interpreted as the zero element, s interpreted as successor, and + and * interpreted as integer addition and multiplication. 4. -
⊥N AS an ABSTRACT ELEMENTARY CLASS We Show
⊥N AS AN ABSTRACT ELEMENTARY CLASS JOHN T. BALDWIN, PAUL C. EKLOF, AND JAN TRLIFAJ We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺K such that (K, ≺K) satisfies ⊥ the properties described below. Here we deal with various classes ( N, ≺N ); the precise definition of the class of modules ⊥N is given below. A key innovation is ⊥ that A≺N B means A ⊆ B and B/A ∈ N. We define in the text the main notions used here; important background defini- tions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the ⊥ ⊥ class ( N, ≺N ) translate directly to algebraic properties of the class N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasingly strong conditions on the ring R, are: ⊥ Theorem 0.1. (1) Let R be any ring and N an R–module. If ( N, ≺N ) is an AEC then N is a cotorsion module. Conversely, if N is pure–injective, ⊥ then the class ( N, ≺N ) is an AEC. (2) Let R be a right noetherian ring and N be an R–module of injective di- ⊥ ⊥ mension ≤ 1. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
Ultraproducts and Elementary Classes *
MATHEMATICS ULTRAPRODUCTS AND ELEMENTARY CLASSES * BY H. JEROME KEISLER (Communicated by Prof. A. HEYTING at the meeting of May 27, 1961 INTRODUCTION In the theory of models of the first order predicate logic with identity we study the relationship between notions from logic and purely mathe matical objects, called relational systems, or (first order) structures, m, consisting of a set A and a sequence of relations on A corresponding to the predicate symbols of the logic. In such studies some of the most useful methods are those which permit us to form structures which have certain semantical properties. For example, the completeness theorem permits us to form a structure which satisfies a given consistent set of sentences. A more recent tool is the reduced product operation, which is particularly simple and direct from the mathematical point of view. Speaking very roughly, a reduced product Iliei ill:i/D of the structures mi, i El, is formed by first taking their direct product and then forming the quotient system determined in a certain way by the filter D on the index set 1. If D is an ultrafilter we have an ultraproduct (or prime reduced product), and if each m:i coincides with a fixed system ill: we have a reduced power, or ultrapower, ill: I j D. Ultraproducts have recently been applied to obtain new, comparatively direct mathematical proofs and at the same time stronger forms of a number of theorems in model theory. An outstanding example is the proof of the compactness theorem outlined in [33]. They have also been used to obtain several completely new results in the theory of models (for example, see [35]). -
Elementary Model Theory Notes for MATH 762
George F. McNulty Elementary Model Theory Notes for MATH 762 drawings by the author University of South Carolina Fall 2011 Model Theory MATH 762 Fall 2011 TTh 12:30p.m.–1:45 p.m. in LeConte 303B Office Hours: 2:00pm to 3:30 pm Monday through Thursday Instructor: George F. McNulty Recommended Text: Introduction to Model Theory By Philipp Rothmaler What We Will Cover After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development of first-order model theory. In the text this is the material covered beginning in Chapter 4. Our aim is cover most of the material in the text (although not all the examples) as well as some material that extends beyond the topics covered in the text (notably a proof of Morley’s Categoricity Theorem). The Work Once the introductory phase of the course is completed, there will be a series of problem sets to entertain and challenge each student. Mastering the problem sets should give each student a detailed familiarity with the main concepts and theorems of model theory and how these concepts and theorems might be applied. So working through the problems sets is really the heart of the course. Most of the problems require some reflection and can usually not be resolved in just one sitting. Grades The grades in this course will be based on each student’s work on the problem sets. Roughly speaking, an A will be assigned to students whose problems sets eventually reveal a mastery of the central concepts and theorems of model theory; a B will be assigned to students whose work reveals a grasp of the basic concepts and a reasonable competence, short of mastery, in putting this grasp into play to solve problems. -
Symmetric Approximations of Pseudo-Boolean Functions with Applications to Influence Indexes
SYMMETRIC APPROXIMATIONS OF PSEUDO-BOOLEAN FUNCTIONS WITH APPLICATIONS TO INFLUENCE INDEXES JEAN-LUC MARICHAL AND PIERRE MATHONET Abstract. We introduce an index for measuring the influence of the kth smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making. 1. Introduction Boolean and pseudo-Boolean functions play a central role in various areas of applied mathematics such as cooperative game theory, engineering reliability, and decision making (where fuzzy measures and fuzzy integrals are often used). In these areas indexes have been introduced to measure the importance of a variable or its influence on the function under consideration (see, e.g., [3, 7]). For instance, the concept of importance of a player in a cooperative game has been studied in various papers on values and power indexes starting from the pioneering works by Shapley [13] and Banzhaf [1]. These power indexes were rediscovered later in system reliability theory as Barlow-Proschan and Birnbaum measures of importance (see, e.g., [10]). In general there are many possible influence/importance indexes and they are rather simple and natural. For instance, a cooperative game on a finite set n = 1,...,n of players is a set function v∶ 2[n] → R with v ∅ = 0, which associates[ ] with{ any}coalition of players S ⊆ n its worth v S . -
Building the Signature of Set Theory Using the Mathsem Program
Building the Signature of Set Theory Using the MathSem Program Luxemburg Andrey UMCA Technologies, Moscow, Russia [email protected] Abstract. Knowledge representation is a popular research field in IT. As mathematical knowledge is most formalized, its representation is important and interesting. Mathematical knowledge consists of various mathematical theories. In this paper we consider a deductive system that derives mathematical notions, axioms and theorems. All these notions, axioms and theorems can be considered a small mathematical theory. This theory will be represented as a semantic net. We start with the signature <Set; > where Set is the support set, is the membership predicate. Using the MathSem program we build the signature <Set; where is set intersection, is set union, -is the Cartesian product of sets, and is the subset relation. Keywords: Semantic network · semantic net· mathematical logic · set theory · axiomatic systems · formal systems · semantic web · prover · ontology · knowledge representation · knowledge engineering · automated reasoning 1 Introduction The term "knowledge representation" usually means representations of knowledge aimed to enable automatic processing of the knowledge base on modern computers, in particular, representations that consist of explicit objects and assertions or statements about them. We are particularly interested in the following formalisms for knowledge representation: 1. First order predicate logic; 2. Deductive (production) systems. In such a system there is a set of initial objects, rules of inference to build new objects from initial ones or ones that are already build, and the whole of initial and constructed objects. 3. Semantic net. In the most general case a semantic net is an entity-relationship model, i.e., a graph, whose vertices correspond to objects (notions) of the theory and edges correspond to relations between them. -
Embedding and Learning with Signatures
Embedding and learning with signatures Adeline Fermaniana aSorbonne Universit´e,CNRS, Laboratoire de Probabilit´es,Statistique et Mod´elisation,4 place Jussieu, 75005 Paris, France, [email protected] Abstract Sequential and temporal data arise in many fields of research, such as quantitative fi- nance, medicine, or computer vision. A novel approach for sequential learning, called the signature method and rooted in rough path theory, is considered. Its basic prin- ciple is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. This approach relies critically on an embedding prin- ciple, which consists in representing discretely sampled data as paths, i.e., functions from [0, 1] to Rd. After a survey of machine learning methodologies for signatures, the influence of embeddings on prediction accuracy is investigated with an in-depth study of three recent and challenging datasets. It is shown that a specific embedding, called lead-lag, is systematically the strongest performer across all datasets and algorithms considered. Moreover, an empirical study reveals that computing signatures over the whole path domain does not lead to a loss of local information. It is concluded that, with a good embedding, combining signatures with other simple algorithms achieves results competitive with state-of-the-art, domain-specific approaches. Keywords: Sequential data, time series classification, functional data, signature. 1. Introduction Sequential or temporal data are arising in many fields of research, due to an in- crease in storage capacity and to the rise of machine learning techniques. An illustra- tion of this vitality is the recent relaunch of the Time Series Classification repository (Bagnall et al., 2018), with more than a hundred new datasets. -
Ultraproducts and Los's Theorem
Ultraproducts andLo´s’sTheorem: A Category-Theoretic Analysis by Mark Jonathan Chimes Dissertation presented for the degree of Master of Science in Mathematics in the Faculty of Science at Stellenbosch University Department of Mathematical Sciences, University of Stellenbosch Supervisor: Dr Gareth Boxall March 2017 Stellenbosch University https://scholar.sun.ac.za Declaration By submitting this dissertation electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly otherwise stated), that reproduction and publication thereof by Stellenbosch University will not infringe any third party rights and that I have not previously in its entirety or in part submitted it for obtaining any qualification. Signature: . Mark Jonathan Chimes Date: March 2017 Copyright c 2017 Stellenbosch University All rights reserved. i Stellenbosch University https://scholar.sun.ac.za Abstract Ultraproducts andLo´s’sTheorem: A Category-Theoretic Analysis Mark Jonathan Chimes Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Dissertation: MSc 2017 Ultraproducts are an important construction in model theory, especially as applied to algebra. Given some family of structures of a certain type, an ul- traproduct of this family is a single structure which, in some sense, captures the important aspects of the family, where “important” is defined relative to a set of sets called an ultrafilter, which encodes which subfamilies are considered “large”. This follows fromLo´s’sTheorem, namely, the Fundamental Theorem of Ultraproducts, which states that every first-order sentence is true of the ultraproduct if, and only if, there is some “large” subfamily of the family such that it is true of every structure in this subfamily. -
Elementary Equivalence Versus Isomorphism
Elementary equivalence versus Isomorphism by Florian Pop at Bonn Lecture 1: Introduction and motivation / The Question It is one of the basic questions of Algebra to \classify" algebraic objects, like for instance fields, up to isomorphism. There are several aspects of this question, the main one being a formalized definition of \classifying". Rather than riding on such themes, we will start by giving two typical examples: 1) The isomorphy type of a finite field K is given by its cardinality jKj, i.e., if K and L are such fields, then K =∼ L iff jKj = jLj. 1) The isomorphy type of an algebraically closed field K is determined by two invariants: (i) Absolute transcendence degree td(K), (ii) The characteristic p = char(K) ≥ 0. In other words, if K and L are algebraically closed fields, then K =∼ L iff td(K) = td(L) and char(K) = char(L). Nevertheless, if we want to classify fields K up to isomorphism even in an \only a little bit" more general context, then we run into very serious difficulties... A typical example here is the attempt to give the isomorphy types of real closed fields (it seems first tried by Artin and Schreier). A real closed field Kpis \as close as possible" to its algebraic closure K, as [K : K] = 2, and K = K[ −1]. These fields were introduced by Artin in his famous proof of Hilbert's 17th Problem. Roughly speaking, the real closed fields are the fields having exactly the same algebraic properties as the reals R. One knows quite a lot about real closed fields, see e.g. -
Elementary Equivalence, Partial Isomorphisms, and Scott-Karp Analysis
Elementary Equivalence, Partial Isomorphisms, and Scott-Karp analysis 1 These are self-study notes I prepared when I was trying to understand the subject. 1 Elementary equivalence and Finite back and forth We begin by analyzing the power of finitary analysis of countable structures. In doing so, we introduce finitary versions of all the tools we will see later. This helps to clarify the need for the more powerful infinitary tools. Definition 1.1 Let A and B be two s-structures. A partial embedding between two structures A and A is a map p whose domain is a finite subset of A and range is a finite subset of B that preserves constants and relations. Namely, p has the following properties: (i) It is injective, (ii) For every constant c 2 s, cA 2 dom(p) and p(cA ) = cB , and (iii) For every n-ary relation R 2 s, and all a1;:::;an 2 dom(p), A B R (a1;:::;an) , R (p(a1);:::; p(an)): The above definition is purely structural. We could define a partial embedding using language as follows: A partial embedding between two s-structures A and B is a finite relation A B f(a1;b1);··· ;(an;bn)g [ f(c ;c ): c a constant in sg ⊂ A × B such that the simple extensions (A ;a1;:::;an) and (B ;b1;:::;bn) satisfy the same atomic sen- tences. Equivalently, for all j atomic, A j= j(a1;:::;an) , B j= j(b1;:::;bn). This definition allows us to define an extension of an embedding p as an embedding q whose graph contains the graph of p. -
Second Order Logic
Second Order Logic Mahesh Viswanathan Fall 2018 Second order logic is an extension of first order logic that reasons about predicates. Recall that one of the main features of first order logic over propositional logic, was the ability to quantify over elements that are in the universe of the structure. Second order logic not only allows one quantify over elements of the universe, but in addition, also allows quantifying relations over the universe. 1 Syntax and Semantics Like first order logic, second order logic is defined over a vocabulary or signature. The signature in this context is the same. Thus, a signature is τ = (C; R), where C is a set of constant symbols, and R is a collection of relation symbols with a specified arity. Formulas in second order logic will be over the same collection of symbols as first order logic, except that we can, in addition, use relational variables. Thus, a formula in second order logic is a sequence of symbols where each symbol is one of the following. 1. The symbol ? called false and the symbol =; 2. An element of the infinite set V1 = fx1; x2; x3;:::g of variables; 1 1 k 3. An element of the infinite set V2 = fX1 ; x2;:::Xj ;:::g of relational variables, where the superscript indicates the arity of the variable; 4. Constant symbols and relation symbols in τ; 5. The symbol ! called implication; 6. The symbol 8 called the universal quantifier; 7. The symbols ( and ) called parenthesis. As always, not all such sequences are formulas; only well formed sequences are formulas in the logic.