Universal Algebra in Unimath
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EM-Training for Weighted Aligned Hypergraph Bimorphisms
EM-Training for Weighted Aligned Hypergraph Bimorphisms Frank Drewes Kilian Gebhardt and Heiko Vogler Department of Computing Science Department of Computer Science Umea˚ University Technische Universitat¨ Dresden S-901 87 Umea,˚ Sweden D-01062 Dresden, Germany [email protected] [email protected] [email protected] Abstract malized by the new concept of hybrid grammar. Much as in the mentioned synchronous grammars, We develop the concept of weighted a hybrid grammar synchronizes the derivations aligned hypergraph bimorphism where the of nonterminals of a string grammar, e.g., a lin- weights may, in particular, represent proba- ear context-free rewriting system (LCFRS) (Vijay- bilities. Such a bimorphism consists of an Shanker et al., 1987), and of nonterminals of a R 0-weighted regular tree grammar, two ≥ tree grammar, e.g., regular tree grammar (Brainerd, hypergraph algebras that interpret the gen- 1969) or simple definite-clause programs (sDCP) erated trees, and a family of alignments (Deransart and Małuszynski, 1985). Additionally it between the two interpretations. Seman- synchronizes terminal symbols, thereby establish- tically, this yields a set of bihypergraphs ing an explicit alignment between the positions of each consisting of two hypergraphs and the string and the nodes of the tree. We note that an explicit alignment between them; e.g., LCFRS/sDCP hybrid grammars can also generate discontinuous phrase structures and non- non-projective dependency structures. projective dependency structures are bihy- In this paper we focus on the task of training an pergraphs. We present an EM-training al- LCFRS/sDCP hybrid grammar, that is, assigning gorithm which takes a corpus of bihyper- probabilities to its rules given a corpus of discon- graphs and an aligned hypergraph bimor- tinuous phrase structures or non-projective depen- phism as input and generates a sequence dency structures. -
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. -
Linear Induction Algebra and a Normal Form for Linear Operators Laurent Poinsot
Linear induction algebra and a normal form for linear operators Laurent Poinsot To cite this version: Laurent Poinsot. Linear induction algebra and a normal form for linear operators. 2013. hal- 00821596 HAL Id: hal-00821596 https://hal.archives-ouvertes.fr/hal-00821596 Preprint submitted on 10 May 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Linear induction algebra and a normal form for linear operators Laurent Poinsot Universit´eParis 13, Sorbonne Paris Cit´e, LIPN, CNRS (UMR 7030), France, [email protected], http://lipn.univ-paris13.fr/~poinsot/ Abstract. The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e. in the category of modules over a commutative ring rather than in the category of sets, namely the free module generated by the integers. It also provides free structures of induction algebra and of monoid (in the category of modules). -
Domain Theory and the Logic of Observable Properties
Domain Theory and the Logic of Observable Properties Samson Abramsky Submitted for the degree of Doctor of Philosophy Queen Mary College University of London October 31st 1987 Abstract The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be com- putationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme: 1. A metalanguage is introduced, comprising • types = universes of discourse for various computational situa- tions. • terms = programs = syntactic intensions for models or points. 2. A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. 3. The metalanguage is also given a logical interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatizes the properties they satisfy. 2 4. The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. -
Algebraic Implementation of Abstract Data Types*
Theoretical Computer Science 20 (1982) 209-263 209 North-Holland Publishing Company ALGEBRAIC IMPLEMENTATION OF ABSTRACT DATA TYPES* H. EHRIG, H.-J. KREOWSKI, B. MAHR and P. PADAWITZ Fachbereich Informatik, TV Berlin, 1000 Berlin 10, Fed. Rep. Germany Communicated by M. Nivat Received October 1981 Abstract. Starting with a review of the theory of algebraic specifications in the sense of the ADJ-group a new theory for algebraic implementations of abstract data types is presented. While main concepts of this new theory were given already at several conferences this paper provides the full theory of algebraic implementations developed in Berlin except of complexity considerations which are given in a separate paper. The new concept of algebraic implementations includes implementations for algorithms in specific programming languages and on the other hand it meets also the requirements for stepwise refinement of structured programs and software systems as introduced by Dijkstra and Wirth. On the syntactical level an algebraic implementation corresponds to a system of recursive programs while the semantical level is defined by algebraic constructions, called SYNTHESIS, RESTRICTION and IDENTIFICATION. Moreover the concept allows composition of implementations and a rigorous study of correctness. The main results of the paper are different kinds of correctness criteria which are applied to a number of illustrating examples including the implementation of sets by hash-tables. Algebraic implementations of larger systems like a histogram or a parts system are given in separate case studies which, however, are not included in this paper. 1. Introduction The concept of abstract data types was developed since about ten years starting with the debacles of large software systems in the late 60's. -
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 . -
Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl
Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl To cite this version: Wolfram Kahl. Categories of Coalgebras with Monadic Homomorphisms. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. pp.151-167, 10.1007/978-3-662-44124-4_9. hal-01408758 HAL Id: hal-01408758 https://hal.inria.fr/hal-01408758 Submitted on 5 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl McMaster University, Hamilton, Ontario, Canada, [email protected] Abstract. Abstract graph transformation approaches traditionally con- sider graph structures as algebras over signatures where all function sym- bols are unary. Attributed graphs, with attributes taken from (term) algebras over ar- bitrary signatures do not fit directly into this kind of transformation ap- proach, since algebras containing function symbols taking two or more arguments do not allow component-wise construction of pushouts. We show how shifting from the algebraic view to a coalgebraic view of graph structures opens up additional flexibility, and enables treat- ing term algebras over arbitrary signatures in essentially the same way as unstructured label sets. -
A Guide to Self-Distributive Quasigroups, Or Latin Quandles
A GUIDE TO SELF-DISTRIBUTIVE QUASIGROUPS, OR LATIN QUANDLES DAVID STANOVSKY´ Abstract. We present an overview of the theory of self-distributive quasigroups, both in the two- sided and one-sided cases, and relate the older results to the modern theory of quandles, to which self-distributive quasigroups are a special case. Most attention is paid to the representation results (loop isotopy, linear representation, homogeneous representation), as the main tool to investigate self-distributive quasigroups. 1. Introduction 1.1. The origins of self-distributivity. Self-distributivity is such a natural concept: given a binary operation on a set A, fix one parameter, say the left one, and consider the mappings ∗ La(x) = a x, called left translations. If all such mappings are endomorphisms of the algebraic structure (A,∗ ), the operation is called left self-distributive (the prefix self- is usually omitted). Equationally,∗ the property says a (x y) = (a x) (a y) ∗ ∗ ∗ ∗ ∗ for every a, x, y A, and we see that distributes over itself. Self-distributivity∈ was pinpointed already∗ in the late 19th century works of logicians Peirce and Schr¨oder [69, 76], and ever since, it keeps appearing in a natural way throughout mathematics, perhaps most notably in low dimensional topology (knot and braid invariants) [12, 15, 63], in the theory of symmetric spaces [57] and in set theory (Laver’s groupoids of elementary embeddings) [15]. Recently, Moskovich expressed an interesting statement on his blog [60] that while associativity caters to the classical world of space and time, distributivity is, perhaps, the setting for the emerging world of information. -
Abstraction and Abstraction Refinement in the Verification Of
Abstraction and Abstraction Refinement in the Verification of Graph Transformation Systems Vom Fachbereich Ingenieurwissenschaften Abteilung Informatik und angewandte Kognitionswissenschaft der Unversit¨at Duisburg-Essen zur Erlangung des akademischen Grades eines Doktor der Naturwissenschaften (Dr.-rer. nat.) genehmigte Dissertation von Vitaly Kozyura aus Magadan, Russland Referent: Prof. Dr. Barbara K¨onig Korreferent: Prof. Dr. Arend Rensink Tag der m¨undlichen Pr¨ufung: 05.08.2009 Abstract Graph transformation systems (GTSs) form a natural and convenient specification language which is used for modelling concurrent and distributed systems with dynamic topologies. These can be, for example, network and Internet protocols, mobile processes with dynamic behavior and dynamic pointer structures in programming languages. All this, together with the possibility to visualize and explain system behavior using graphical methods, makes GTSs a well-suited formalism for the specification of complex dynamic distributed systems. Under these circumstances the problem of checking whether a certain property of GTSs holds – the verification problem – is considered to be a very important question. Unfortunately the verification of GTSs is in general undecidable because of the Turing- completeness of GTSs. In the last few years a technique for analysing GTSs based on approximation by Petri graphs has been developed. Petri graphs are Petri nets having additional graph structure. In this work we focus on the verification techniques based on counterexample-guided abstraction refinement (CEGAR approach). It starts with a coarse initial over-approxi- mation of a system and an obtained counterexample. If the counterexample is spurious then one starts a refinement procedure of the approximation, based on the structure of the counterexample. -
Semilattice Sums of Algebras and Mal'tsev Products of Varieties
Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S. -
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.