Cardinality Functions in Allegories ∗ Yasuo Kawahara a , Michael Winter B, ,1
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Logic Programming in Tabular Allegories∗
Logic Programming in Tabular Allegories∗ Emilio Jesús Gallego Arias1 and James B. Lipton2 1 Universidad Politécnica de Madrid 2 Wesleyan University Abstract We develop a compilation scheme and categorical abstract machine for execution of logic pro- grams based on allegories, the categorical version of the calculus of relations. Operational and denotational semantics are developed using the same formalism, and query execution is performed using algebraic reasoning. Our work serves two purposes: achieving a formal model of a logic programming compiler and efficient runtime; building the base for incorporating features typi- cal of functional programming in a declarative way, while maintaining 100% compatibility with existing Prolog programs. 1998 ACM Subject Classification D.3.1 Formal Definitions and Theory, F.3.2 Semantics of Programming Languages, F.4.1 Mathematical Logic Keywords and phrases Category Theory,Logic Programming,Lawvere Categories,Programming Language Semantics,Declarative Programming Digital Object Identifier 10.4230/LIPIcs.ICLP.2012.334 1 Introduction Relational algebras have a broad spectrum of applications in both theoretical and practical computer science. In particular, the calculus of binary relations [37], whose main operations are intersection (∪), union (∩), relative complement \, inversion (_)o and relation composition (;) was shown by Tarski and Givant [40] to be a complete and adequate model for capturing all first-order logic and set theory. The intuition is that conjunction is modeled by ∩, disjunction by ∪ and existential quantification by composition. This correspondence is very useful for modeling logic programming. Logic programs are naturally interpreted by binary relations and relation algebra is a suitable framework for algebraic reasoning over them, including execution of queries. -
Relations in Categories
Relations in Categories Stefan Milius A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Arts Graduate Program in Mathematics and Statistics York University Toronto, Ontario June 15, 2000 Abstract This thesis investigates relations over a category C relative to an (E; M)-factori- zation system of C. In order to establish the 2-category Rel(C) of relations over C in the first part we discuss sufficient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as defined by Lawvere. We give a significantly simplified proof for the main result of Pavlovi´c,namely that C Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof' that the category Map(Rel(C))⊆ is finitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization⊆ systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an (E¯; M¯ )-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a⊆ weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E; M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated. -
Knowledge Representation in Bicategories of Relations
Knowledge Representation in Bicategories of Relations Evan Patterson Department of Statistics, Stanford University Abstract We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a recent categorical framework for knowledge representation. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. In this paper, we investigate relational ologs both for their own sake and to gain insight into the relationship between the algebraic and logical approaches to knowledge representation. On a practical level, we show by example that relational ologs have a friendly and intuitive—yet fully precise—graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. Although we make extensive use of categorical language, this paper is designed to be self-contained and has considerable expository content. The only prerequisites are knowledge of first-order logic and the rudiments of category theory. 1. Introduction arXiv:1706.00526v2 [cs.AI] 1 Nov 2017 The representation of human knowledge in computable form is among the oldest and most fundamental problems of artificial intelligence. Several recent trends are stimulating continued research in the field of knowledge representation (KR). -
The Cardinality of a Finite Set
1 Math 300 Introduction to Mathematical Reasoning Fall 2018 Handout 12: More About Finite Sets Please read this handout after Section 9.1 in the textbook. The Cardinality of a Finite Set Our textbook defines a set A to be finite if either A is empty or A ≈ Nk for some natural number k, where Nk = {1,...,k} (see page 455). It then goes on to say that A has cardinality k if A ≈ Nk, and the empty set has cardinality 0. These are standard definitions. But there is one important point that the book left out: Before we can say that the cardinality of a finite set is a well-defined number, we have to ensure that it is not possible for the same set A to be equivalent to Nn and Nm for two different natural numbers m and n. This may seem obvious, but it turns out to be a little trickier to prove than you might expect. The purpose of this handout is to prove that fact. The crux of the proof is the following lemma about subsets of the natural numbers. Lemma 1. Suppose m and n are natural numbers. If there exists an injective function from Nm to Nn, then m ≤ n. Proof. For each natural number n, let P (n) be the following statement: For every m ∈ N, if there is an injective function from Nm to Nn, then m ≤ n. We will prove by induction on n that P (n) is true for every natural number n. We begin with the base case, n = 1. -
From Categories to Allegories
Final dialgebras : from categories to allegories Citation for published version (APA): Backhouse, R. C., & Hoogendijk, P. F. (1999). Final dialgebras : from categories to allegories. (Computing science reports; Vol. 9903). Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1999 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Monomorphism - Wikipedia, the Free Encyclopedia
Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G. -
A New Characterization of Injective and Surjective Functions and Group Homomorphisms
A new characterization of injective and surjective functions and group homomorphisms ANDREI TIBERIU PANTEA* Abstract. A model of a function f between two non-empty sets is defined to be a factorization f ¼ i, where ¼ is a surjective function and i is an injective Æ ± function. In this note we shall prove that a function f is injective (respectively surjective) if and only if it has a final (respectively initial) model. A similar result, for groups, is also proven. Keywords: factorization of morphisms, injective/surjective maps, initial/final objects MSC: Primary 03E20; Secondary 20A99 1 Introduction and preliminary remarks In this paper we consider the sets taken into account to be non-empty and groups denoted differently to be disjoint. For basic concepts on sets and functions see [1]. It is a well known fact that any function f : A B can be written as the composition of a surjective function ! followed by an injective function. Indeed, f f 0 id , where f 0 : A Im(f ) is the restriction Æ ± B ! of f to Im(f ) is one such factorization. Moreover, it is an elementary exercise to prove that this factorization is unique up to an isomorphism: i.e. if f i ¼, i : A X , ¼: X B is 1 ¡ Æ ±¢ ! 1 ! another factorization for f , then we easily see that X i ¡ Im(f ) and that ¼ i ¡ f 0. The Æ Æ ± purpose of this note is to study the dual problem. Let f : A B be a function. We shall define ! a model of the function f to be a triple (X ,i,¼) made up of a set X , an injective function i : A X and a surjective function ¼: X B such that f ¼ i, that is, the following diagram ! ! Æ ± is commutative. -
Generic Programming with Adjunctions
Generic Programming with Adjunctions Ralf Hinze Department of Computer Science, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD, England [email protected] http://www.cs.ox.ac.uk/ralf.hinze/ Abstract. Adjunctions are among the most important constructions in mathematics. These lecture notes show they are also highly relevant to datatype-generic programming. First, every fundamental datatype| sums, products, function types, recursive types|arises out of an adjunc- tion. The defining properties of an adjunction give rise to well-known laws of the algebra of programming. Second, adjunctions are instrumental in unifying and generalising recursion schemes. We discuss a multitude of basic adjunctions and show that they are directly relevant to program- ming and to reasoning about programs. 1 Introduction Haskell programmers have embraced functors [1], natural transformations [2], monads [3], monoidal functors [4] and, perhaps to a lesser extent, initial alge- bras [5] and final coalgebras [6]. It is time for them to turn their attention to adjunctions. The notion of an adjunction was introduced by Daniel Kan in 1958 [7]. Very briefly, the functors L and R are adjoint if arrows of type L A → B are in one-to- one correspondence to arrows of type A → R B and if the bijection is furthermore natural in A and B. Adjunctions have proved to be one of the most important ideas in category theory, predominantly due to their ubiquity. Many mathemat- ical constructions turn out to be adjoint functors that form adjunctions, with Mac Lane [8, p.vii] famously saying, \Adjoint functors arise everywhere." The purpose of these lecture notes is to show that the notion of an adjunc- tion is also highly relevant to programming, in particular, to datatype-generic programming. -
Abstract Relation-Algebraic Interfaces for Finite Relations Between Infinite
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector The Journal of Logic and Algebraic Programming 76 (2008) 60–89 www.elsevier.com/locate/jlap Relational semigroupoids: Abstract relation-algebraic interfaces for finite relations between infinite types Wolfram Kahl ∗ Department of Computing and Software, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 Available online 13 February 2008 Abstract Finite maps or finite relations between infinite sets do not even form a category, since the necessary identities are not finite. We show relation-algebraic extensions of semigroupoids where the operations that would produce infinite results have been replaced with variants that preserve finiteness, but still satisfy useful algebraic laws. The resulting theories allow calculational reasoning in the relation-algebraic style with only minor sacrifices; our emphasis on generality even provides some concepts in theories where they had not been available before. The semigroupoid theories presented in this paper also can directly guide library interface design and thus be used for principled relation-algebraic programming; an example implementation in Haskell allows manipulating finite binary relations as data in a point-free relation-algebraic programming style that integrates naturally with the current Haskell collection types. This approach enables seamless integration of relation-algebraic formulations to provide elegant solutions of problems that, with different data organisation, are awkward to tackle. © 2007 Elsevier Inc. All rights reserved. AMS classification: 18B40; 18B10; 03G15; 04A05; 68R99 Keywords: Finite relations; Semigroupoids; Determinacy; Restricted residuals; Direct products; Functional programming; Haskell 1. Introduction The motivation for this paper arose from the desire to use well-defined algebraic interfaces for collection datatypes implementing sets, partial functions, and binary relations. -
Homomorphism Problems for First-Order Definable Structures
Homomorphism Problems for First-Order Definable Structures∗ Bartek Klin1, Sławomir Lasota1, Joanna Ochremiak2, and Szymon Toruńczyk1 1 University of Warsaw 2 Universitat Politécnica de Catalunya Abstract We investigate several variants of the homomorphism problem: given two relational structures, is there a homomorphism from one to the other? The input structures are possibly infinite, but definable by first-order interpretations in a fixed structure. Their signatures can be either finite or infinite but definable. The homomorphisms can be either arbitrary, or definable with parameters, or definable without parameters. For each of these variants, we determine its decidability status. 1998 ACM Subject Classification F.4.1 Mathematical Logic–Model theory, F.4.3 Formal Lan- guages–Decision problems Keywords and phrases Sets with atoms, first-order interpretations, homomorphism problem Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2016.? 1 Introduction First-order definable sets, although usually infinite, can be finitely described and are therefore amenable to algorithmic manipulation. Definable sets (we drop the qualifier first-order in what follows) are parametrized by a fixed underlying relational structure A whose elements are called atoms. We shall assume that the first-order theory of A is decidable. To simplify the presentation, unless stated otherwise, let A be a countable set {1, 2, 3,...} equipped with the equality relation only; we shall call this the pure set. I Example 1. Let V = {{a, b} | a, b ∈ A, a 6= b } , E = { ({a, b}, {c, d}) | a, b, c, d ∈ A, a 6= b ∧ a 6= c ∧ a 6= d ∧ b 6= c ∧ b 6= d ∧ c 6= d } . -
Almost Recursively Enumerable Sets
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 164, February 1972 ALMOST RECURSIVELY ENUMERABLE SETS BY JOHN W. BERRYO) Abstract. An injective function on N, the nonnegative integers, taking values in TV, is called almost recursive (abbreviated a.r.) if its inverse has a partial recursive exten- sion. The range of an a.r. function /is called an almost recursively enumerable set in general; an almost recursive set if in addition/is strictly increasing. These are natural generalizations of regressive and retraceable sets respectively. We show that an infinite set is almost recursively enumerable iff it is point decomposable in the sense of McLaughlin. This leads us to new characterizations of certain classes of immune sets. Finally, in contrast to the regressive case, we show that a.r. functions and sets are rather badly behaved with respect to recursive equivalence. 1. Introduction and notation. Almost recursive functions, almost recursive sets and almost recursively enumerable sets were introduced by Vuckovic in [7]. For basic facts about retraceable sets the reader is referred to [3]; for regressive sets to [2] and [1]. Throughout this paper, N denotes the set of nonnegative integers. The word "set" will usually mean "subset of TV," and the complement TV—Aof a set A is denoted by A. Iff is a partial function on TVwe denote its domain and range by S(/) and p{f) respectively. Tre denotes the eth partial recursive function in the Kleene enumeration, i.e. ■ne{x)^U{p.yTx(e, x, y)). we denotes 8(TTe).As a pairing function we use j(x, y) = ^[(x+y)2 + 3y + x], and k(x), l(x) are recursive functions which satisfy x =j{k(x), l(x)) for all x, and k(j(x, y)) = x, l(j(x, y))=y for all x and y. -
Minicourse on the Axioms of Zermelo and Fraenkel
Minicourse on The Axioms of Zermelo and Fraenkel Heike Mildenberger January 15, 2021 Abstract These are the notes for a minicourse of four hours in the DAAD-summerschool in mathematical logic 2020. To be held virtually in G¨ottingen,October 10-16, 2020. 1 The Zermelo Fraenkel Axioms Axiom: a plausible postulate, seen as evident truth that needs not be proved. Logical axioms, e.g. A∧B → A for any statements A, B. We assume the Hilbert axioms, see below. Now we focus on the non-logical axioms, the ones about the underlying set the- oretic structure. Most of nowadays mathematics can be justified as built within the set theoretic universe or one of its extensions by classes or even a hierarchy of classes. In my lecture I will report on the Zermelo Fraenkel Axioms. If no axioms are mentioned, usually the proof is based on these axioms. However, be careful, for example nowaday’s known and accepted proof of Fermat’s last theorem by Wiles uses “ZFC and there are infinitely many strongly inaccessible cardinals”. It is conjectured that ZFC or even the Peano Arithmetic will do. The language of the Zermelo Fraenkel axioms is the first order logic L (τ) with signature τ = {∈}. The symbol ∈ stands for a binary relation. However, we will work in English and thus expand the language by many defined notions. If you are worried about how this is compatible with the statement “the language of set theory is L({∈})” you can study the theory of language expansions by defined symbols. E.g., you may read page 85ff in [15].