The Agda Universal Algebra Library Part 2: Structure Homomorphisms, Terms, Classes of Algebras, Subalgebras, and Homomorphic Images
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Kernel and Image
Math 217 Worksheet 1 February: x3.1 Professor Karen E Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. T Definitions: Given a linear transformation V ! W between vector spaces, we have 1. The source or domain of T is V ; 2. The target of T is W ; 3. The image of T is the subset of the target f~y 2 W j ~y = T (~x) for some x 2 Vg: 4. The kernel of T is the subset of the source f~v 2 V such that T (~v) = ~0g. Put differently, the kernel is the pre-image of ~0. Advice to the new mathematicians from an old one: In encountering new definitions and concepts, n m please keep in mind concrete examples you already know|in this case, think about V as R and W as R the first time through. How does the notion of a linear transformation become more concrete in this special case? Think about modeling your future understanding on this case, but be aware that there are other important examples and there are important differences (a linear map is not \a matrix" unless *source and target* are both \coordinate spaces" of column vectors). The goal is to become comfortable with the abstract idea of a vector space which embodies many n features of R but encompasses many other kinds of set-ups. A. For each linear transformation below, determine the source, target, image and kernel. 2 3 x1 3 (a) T : R ! R such that T (4x25) = x1 + x2 + x3. -
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. -
Math 120 Homework 3 Solutions
Math 120 Homework 3 Solutions Xiaoyu He, with edits by Prof. Church April 21, 2018 [Note from Prof. Church: solutions to starred problems may not include all details or all portions of the question.] 1.3.1* Let σ be the permutation 1 7! 3; 2 7! 4; 3 7! 5; 4 7! 2; 5 7! 1 and let τ be the permutation 1 7! 5; 2 7! 3; 3 7! 2; 4 7! 4; 5 7! 1. Find the cycle decompositions of each of the following permutations: σ; τ; σ2; στ; τσ; τ 2σ. The cycle decompositions are: σ = (135)(24) τ = (15)(23)(4) σ2 = (153)(2)(4) στ = (1)(2534) τσ = (1243)(5) τ 2σ = (135)(24): 1.3.7* Write out the cycle decomposition of each element of order 2 in S4. Elements of order 2 are also called involutions. There are six formed from a single transposition, (12); (13); (14); (23); (24); (34), and three from pairs of transpositions: (12)(34); (13)(24); (14)(23). 3.1.6* Define ' : R× ! {±1g by letting '(x) be x divided by the absolute value of x. Describe the fibers of ' and prove that ' is a homomorphism. The fibers of ' are '−1(1) = (0; 1) = fall positive realsg and '−1(−1) = (−∞; 0) = fall negative realsg. 3.1.7* Define π : R2 ! R by π((x; y)) = x + y. Prove that π is a surjective homomorphism and describe the kernel and fibers of π geometrically. The map π is surjective because e.g. π((x; 0)) = x. The kernel of π is the line y = −x through the origin. -
Discrete Topological Transformations for Image Processing Michel Couprie, Gilles Bertrand
Discrete Topological Transformations for Image Processing Michel Couprie, Gilles Bertrand To cite this version: Michel Couprie, Gilles Bertrand. Discrete Topological Transformations for Image Processing. Brimkov, Valentin E. and Barneva, Reneta P. Digital Geometry Algorithms, 2, Springer, pp.73-107, 2012, Lecture Notes in Computational Vision and Biomechanics, 978-94-007-4174-4. 10.1007/978-94- 007-4174-4_3. hal-00727377 HAL Id: hal-00727377 https://hal-upec-upem.archives-ouvertes.fr/hal-00727377 Submitted on 3 Sep 2012 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. Chapter 3 Discrete Topological Transformations for Image Processing Michel Couprie and Gilles Bertrand Abstract Topology-based image processing operators usually aim at trans- forming an image while preserving its topological characteristics. This chap- ter reviews some approaches which lead to efficient and exact algorithms for topological transformations in 2D, 3D and grayscale images. Some transfor- mations which modify topology in a controlled manner are also described. Finally, based on the framework of critical kernels, we show how to design a topologically sound parallel thinning algorithm guided by a priority function. 3.1 Introduction Topology-preserving operators, such as homotopic thinning and skeletoniza- tion, are used in many applications of image analysis to transform an object while leaving unchanged its topological characteristics. -
Kernel Methodsmethods Simple Idea of Data Fitting
KernelKernel MethodsMethods Simple Idea of Data Fitting Given ( xi,y i) i=1,…,n xi is of dimension d Find the best linear function w (hyperplane) that fits the data Two scenarios y: real, regression y: {-1,1}, classification Two cases n>d, regression, least square n<d, ridge regression New sample: x, < x,w> : best fit (regression), best decision (classification) 2 Primary and Dual There are two ways to formulate the problem: Primary Dual Both provide deep insight into the problem Primary is more traditional Dual leads to newer techniques in SVM and kernel methods 3 Regression 2 w = arg min ∑(yi − wo − ∑ xij w j ) W i j w = arg min (y − Xw )T (y − Xw ) W d(y − Xw )T (y − Xw ) = 0 dw ⇒ XT (y − Xw ) = 0 w = [w , w ,L, w ]T , ⇒ T T o 1 d X Xw = X y L T x = ,1[ x1, , xd ] , T −1 T ⇒ w = (X X) X y y = [y , y ,L, y ]T ) 1 2 n T −1 T xT y =< x (, X X) X y > 1 xT X = 2 M xT 4 n n×xd Graphical Interpretation ) y = Xw = Hy = X(XT X)−1 XT y = X(XT X)−1 XT y d X= n FICA Income X is a n (sample size) by d (dimension of data) matrix w combines the columns of X to best approximate y Combine features (FICA, income, etc.) to decisions (loan) H projects y onto the space spanned by columns of X Simplify the decisions to fit the features 5 Problem #1 n=d, exact solution n>d, least square, (most likely scenarios) When n < d, there are not enough constraints to determine coefficients w uniquely d X= n W 6 Problem #2 If different attributes are highly correlated (income and FICA) The columns become dependent Coefficients -
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. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
23. Kernel, Rank, Range
23. Kernel, Rank, Range We now study linear transformations in more detail. First, we establish some important vocabulary. The range of a linear transformation f : V ! W is the set of vectors the linear transformation maps to. This set is also often called the image of f, written ran(f) = Im(f) = L(V ) = fL(v)jv 2 V g ⊂ W: The domain of a linear transformation is often called the pre-image of f. We can also talk about the pre-image of any subset of vectors U 2 W : L−1(U) = fv 2 V jL(v) 2 Ug ⊂ V: A linear transformation f is one-to-one if for any x 6= y 2 V , f(x) 6= f(y). In other words, different vector in V always map to different vectors in W . One-to-one transformations are also known as injective transformations. Notice that injectivity is a condition on the pre-image of f. A linear transformation f is onto if for every w 2 W , there exists an x 2 V such that f(x) = w. In other words, every vector in W is the image of some vector in V . An onto transformation is also known as an surjective transformation. Notice that surjectivity is a condition on the image of f. 1 Suppose L : V ! W is not injective. Then we can find v1 6= v2 such that Lv1 = Lv2. Then v1 − v2 6= 0, but L(v1 − v2) = 0: Definition Let L : V ! W be a linear transformation. The set of all vectors v such that Lv = 0W is called the kernel of L: ker L = fv 2 V jLv = 0g: 1 The notions of one-to-one and onto can be generalized to arbitrary functions on sets. -
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. -
Positive Semigroups of Kernel Operators
Positivity 12 (2008), 25–44 c 2007 Birkh¨auser Verlag Basel/Switzerland ! 1385-1292/010025-20, published online October 29, 2007 DOI 10.1007/s11117-007-2137-z Positivity Positive Semigroups of Kernel Operators Wolfgang Arendt Dedicated to the memory of H.H. Schaefer Abstract. Extending results of Davies and of Keicher on !p we show that the peripheral point spectrum of the generator of a positive bounded C0-semigroup of kernel operators on Lp is reduced to 0. It is shown that this implies con- vergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators. Mathematics Subject Classification (2000). 47D06, 47B33, 35K05. Keywords. Positive semigroups, Kernel operators, asymptotic behaviour, trivial peripheral spectrum. 0. Introduction Irreducibility is a fundamental notion in Perron-Frobenius Theory. It had been introduced in a direct way by Perron and Frobenius for matrices, but it was H. H. Schaefer who gave the definition via closed ideals. This turned out to be most fruit- ful and led to a wealth of deep and important results. For applications Ouhabaz’ very simple criterion for irreduciblity of semigroups defined by forms (see [Ouh05, Sec. 4.2] or [Are06]) is most useful. It shows that for practically all boundary con- ditions, a second order differential operator in divergence form generates a positive 2 N irreducible C0-semigroup on L (Ω) where Ωis an open, connected subset of R . The main question in Perron-Frobenius Theory, is to determine the asymp- totic behaviour of the semigroup. If the semigroup is bounded (in fact Abel bounded suffices), and if the fixed space is non-zero, then irreducibility is equiv- alent to convergence of the Ces`aro means to a strictly positive rank-one opera- tor, i.e.