Efficiently Representing the Integer Factorization Problem Using Binary Decision Diagrams" (2017)
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An Introduction to Operad Theory
AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification. -
Convert Truth Table to Boolean Expression
Convert Truth Table To Boolean Expression Gustave often dozings hydrostatically when neap Caspar libelled readably and Frenchify her mare's-tails. Weidar enlarges consumedly while oven-ready Byron allaying penetratingly or matriculating contagiously. Micah hark her tart calumniously, she thresh it commensurately. Input boolean expressions can have iframes disabled in use boolean truth table to convert truth tables into one convenient source from. This page an example, are converted into single output line argument to convert a boolean function as the truth table lists all possible. This is converted to follow up for example to a bar, we occasionally miss an arrow at boolean. Converting Truth Tables into Boolean Expressions Boolean. So that we know that is either nand gates instead of expression to relate outputs are the faster than by. Truth table Rosetta Code. And a sale group- eliminating algorithm to headquarters the superintendent of enable input. The and converting from my two. Tutorial Performing Boolean Algebra inside an FPGA using Look-Up Tables LUTs. DeMorgan's Laws tell us how to negate a boolean expression and what it. Creating expressions. Gives a nested table that truth values of expr with the outermost level on possible combinations of the ai. Truth Table from All Boolean Functions of 2 Variables y 0. Algebra simplifications methods: converting truth table to these symbols, we have already. Boolean function CircuitVerse. S1 S2 Light he is living truth abuse for the logical connective. Boolean Functions and Truth Tables The meaning or braid of a logical expression tout a Boolean function from the set is possible assignments of truth values for. -
Multiplicative Complexity of Boolean Functions
Multiplicative Complexity of Boolean Functions Meltem S¨onmezTuran National Institute of Standards and Technology, Gaithersburg, MD WPI ECE Online Graduate Seminar Lecture March 17 2021 In this presentation, ... • Overview - Computer Security Division of NIST • Circuit Complexity Problem • Multiplicative Complexity • Three results • Multiplicative Complexity of Boolean functions with n ≤ 6 • Boolean functions with Multiplicative Complexity 1,2,3 and 4 • Multiplicative Complexity of Symmetric Boolean Functions • Research Directions 1 National Institute of Standards and Technology • Non-regulatory federal agency within U.S. Department of Commerce. • Founded in 1901, known as the National Bureau of Standards (NBS) prior to 1988. • Headquarters in Gaithersburg, Maryland, and laboratories in Boulder, Colorado. • Employs around 6,000 employees and associates. Computer Security Division (CSD) conducts research, development and outreach necessary to provide standards and guidelines, mechanisms, tools, metrics and practices to protect nation's information and information systems. 2 CSD Publications • Federal Information Processing Standards (FIPS): Specify approved crypto standards. • NIST Special Publications (SPs): Guidelines, technical specifications, recommendations and reference materials, including multiple sub-series. • NIST Internal or Interagency Reports (NISTIR): Reports of research findings, including background information for FIPS and SPs. • NIST Information Technology Laboratory (ITL) Bulletins: Monthly overviews of NIST's security and privacy publications, programs and projects. 3 Standard Development Process • International \competitions": Engage community through an open competition (e.g., AES, SHA-3, PQC, Lightweight Crypto). • Adoption of existing standards: Collaboration with accredited standards organizations (e.g., RSA, HMAC). • Open call for proposals: Ongoing open invitation (e.g., modes of operations). • Development of new algorithms: if no suitable standard exists (e.g., DRBGs). -
Binary Integer Programming and Its Use for Envelope Determination
Binary Integer Programming and its Use for Envelope Determination By Vladimir Y. Lunin1,2, Alexandre Urzhumtsev3,† & Alexander Bockmayr2 1 Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 140292 Russia 2 LORIA, UMR 7503, Faculté des Sciences, Université Henri Poincaré, Nancy I, 54506 Vandoeuvre-les-Nancy, France; [email protected] 3 LCM3B, UMR 7036 CNRS, Faculté des Sciences, Université Henri Poincaré, Nancy I, 54506 Vandoeuvre-les-Nancy, France; [email protected] † to whom correspondence must be sent Abstract The density values are linked to the observed magnitudes and unknown phases by a system of non-linear equations. When the object of search is a binary envelope rather than a continuous function of the electron density distribution, these equations can be replaced by a system of linear inequalities with respect to binary unknowns and powerful tools of integer linear programming may be applied to solve the phase problem. This novel approach was tested with calculated and experimental data for a known protein structure. 1. Introduction Binary Integer Programming (BIP in what follows) is an approach to solve a system of linear inequalities in binary unknowns (0 or 1 in what follows). Integer programming has been studied in mathematics, computer science, and operations research for more than 40 years (see for example Johnson et al., 2000 and Bockmayr & Kasper, 1998, for a review). It has been successfully applied to solve a huge number of large-scale combinatorial problems. The general form of an integer linear programming problem is max { cTx | Ax ≤ b, x ∈ Zn } (1.1) with a real matrix A of a dimension m by n, and vectors c ∈ Rn, b ∈ Rm, cTx being the scalar product of the vectors c and x. -
Making a Faster Curry with Extensional Types
Making a Faster Curry with Extensional Types Paul Downen Simon Peyton Jones Zachary Sullivan Microsoft Research Zena M. Ariola Cambridge, UK University of Oregon [email protected] Eugene, Oregon, USA [email protected] [email protected] [email protected] Abstract 1 Introduction Curried functions apparently take one argument at a time, Consider these two function definitions: which is slow. So optimizing compilers for higher-order lan- guages invariably have some mechanism for working around f1 = λx: let z = h x x in λy:e y z currying by passing several arguments at once, as many as f = λx:λy: let z = h x x in e y z the function can handle, which is known as its arity. But 2 such mechanisms are often ad-hoc, and do not work at all in higher-order functions. We show how extensional, call- It is highly desirable for an optimizing compiler to η ex- by-name functions have the correct behavior for directly pand f1 into f2. The function f1 takes only a single argu- expressing the arity of curried functions. And these exten- ment before returning a heap-allocated function closure; sional functions can stand side-by-side with functions native then that closure must subsequently be called by passing the to practical programming languages, which do not use call- second argument. In contrast, f2 can take both arguments by-name evaluation. Integrating call-by-name with other at once, without constructing an intermediate closure, and evaluation strategies in the same intermediate language ex- this can make a huge difference to run-time performance in presses the arity of a function in its type and gives a princi- practice [Marlow and Peyton Jones 2004]. -
CS228 Logic for Computer Science 2020 Lecture 7: Conjunctive
CS228 Logic for Computer Science 2020 Lecture 7: Conjunctive Normal Form Instructor: Ashutosh Gupta IITB, India Compile date: 2020-02-01 cbna CS228 Logic for Computer Science 2020 Instructor: Ashutosh Gupta IITB, India 1 Removing ⊕, ), and ,. We have seen equivalences that remove ⊕, ), and , from a formula. I (p ) q) ≡ (:p _ q) I (p ⊕ q) ≡ (p _ q) ^ (:p _:q) I (p , q) ≡ :(p ⊕ q) In the lecture, we will assume you can remove them at will. Commentary: Note that removal of ⊕ and , blows up the formula size. Their straight up removal is not desirable. cbna CS228 Logic for Computer Science 2020 Instructor: Ashutosh Gupta IITB, India 2 Topic 7.1 Negation normal form cbna CS228 Logic for Computer Science 2020 Instructor: Ashutosh Gupta IITB, India 3 Negation normal form(NNF) Definition 7.1 A formula is in NNF if : appears only in front of the propositional variables. Theorem 7.1 For every formula F , there is a formula F 0 in NNF such that F ≡ F 0. Proof. Due to the equivalences, we can always push : under the connectives I Often we assume that the formulas are in NNF. I However, there are negations hidden inside ⊕, ), and ,. Sometimes, the symbols are also expected to be removed while producing NNF Exercise 7.1 Write an efficient algorithm to convert a propositional formula to NNF? Commentary: In our context, we will not ask one to remove e ⊕, ), and , during conversion to NNF. cbna CS228 Logic for Computer Science 2020 Instructor: Ashutosh Gupta IITB, India 4 Example :NNF Example 7.1 Consider :(q ) ((p _:s) ⊕ r)) ≡ q ^ :((p _:s) ⊕ r) ≡ q ^ (:(p _:s) ⊕ r) ≡ q ^ ((:p ^ ::s) ⊕ r) ≡ q ^ ((:p ^ s) ⊕ r) Exercise 7.2 Convert the following formulas into NNF I :(p ) q) I :(:((s ):(p , q))) ⊕ (:q _ r)) Exercise 7.3 Remove ), ,, and ⊕ before turning the above into NNF. -
Self-Similarity in the Foundations
Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . -
Solving Multiclass Learning Problems Via Error-Correcting Output Codes
Journal of Articial Intelligence Research Submitted published Solving Multiclass Learning Problems via ErrorCorrecting Output Co des Thomas G Dietterich tgdcsorstedu Department of Computer Science Dearborn Hal l Oregon State University Corval lis OR USA Ghulum Bakiri ebisaccuobbh Department of Computer Science University of Bahrain Isa Town Bahrain Abstract Multiclass learning problems involve nding a denition for an unknown function f x whose range is a discrete set containing k values ie k classes The denition is acquired by studying collections of training examples of the form hx f x i Existing ap i i proaches to multiclass learning problems include direct application of multiclass algorithms such as the decisiontree algorithms C and CART application of binary concept learning algorithms to learn individual binary functions for each of the k classes and application of binary concept learning algorithms with distributed output representations This pap er compares these three approaches to a new technique in which errorcorrecting co des are employed as a distributed output representation We show that these output representa tions improve the generalization p erformance of b oth C and backpropagation on a wide range of multiclass learning tasks We also demonstrate that this approach is robust with resp ect to changes in the size of the training sample the assignment of distributed represen tations to particular classes and the application of overtting avoidance techniques such as decisiontree pruning Finally we show thatlike -
Current Version, Ch01-03
Digital Logic and Computer Organization Neal Nelson c May 2013 Contents 1 Numbers and Gates 5 1.1 Numbers and Primitive Data Types . .5 1.2 Representing Numbers . .6 1.2.1 Decimal and Binary Systems . .6 1.2.2 Binary Counting . .7 1.2.3 Binary Conversions . .9 1.2.4 Hexadecimal . 11 1.3 Representing Negative Numbers . 13 1.3.1 Ten's Complement . 14 1.3.2 Two's Complement Binary Representation . 18 1.3.3 Negation in Two's Complement . 20 1.3.4 Two's Complement to Decimal Conversion . 21 1.3.5 Decimal to Two's Complement Conversion . 22 1.4 Gates and Circuits . 22 1.4.1 Logic Expressions . 23 1.4.2 Circuit Expressions . 24 1.5 Exercises . 26 2 Logic Functions 29 2.1 Functions . 29 2.2 Logic Functions . 30 2.2.1 Primitive Gate Functions . 31 2.2.2 Evaluating Logic Expressions and Circuits . 31 2.2.3 Logic Tables for Expressions and Circuits . 32 2.2.4 Expressions as Functions . 34 2.3 Equivalence of Boolean Expressions . 36 2.4 Logic Functional Completeness . 36 2.5 Boolean Algebra . 37 2.6 Tables, Expressions, and Circuits . 37 1 2.6.1 Disjunctive Normal Form . 38 2.6.2 Logic Expressions from Tables . 39 2.7 Exercises . 41 3 Dataflow Logic 43 3.1 Data Types . 44 3.1.1 Unsigned Int . 45 3.1.2 Signed Int . 45 3.1.3 Char and String . 45 3.2 Data Buses . 46 3.3 Bitwise Functions . 48 3.4 Integer Functions . -
Universal Functions Paul B
Universal Functions Paul B. Larson1 Arnold W. Miller Juris Stepr¯ans William A.R. Weiss Contents 1. Introduction 2 1.1. Cardinal characteristics 4 2. Borel universal functions 5 3. Universal functions and Martin's Axiom 9 4. Universal functions of special kinds 12 5. Abstract universal functions 18 5.1. Property R 28 6. Higher dimensional universal functions 30 7. Model-theoretic universality 43 8. Appendix 50 References 52 Abstract2 A function of two variables F (x; y) is universal if for every function G(x; y) there exists functions h(x) and k(y) such that G(x; y) = F (h(x); k(y)) for all x; y. Sierpi´nskishowed that assuming the Continuum Hypothesis there exists a Borel function F (x; y) which is universal. Assuming Mar- tin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. Here we show that it is consistent that for each α with 2 ≤ α < !1 there is a universal function of class α but none of class β < α. We show that it is consistent with ZFC that there is no universal function (Borel or not) on the reals, and we show that it is consistent that there is a universal function but no Borel universal function. We also prove some results concerning higher-arity 1Research supported in part by NSF Grants DMS-0801009 and DMS-1201494. Much of the writing of the paper was done while all four authors attended the thematic program on Forcing and its Applications at the Fields Institute in Fall 2012. -
Multiclass Classification Using Support Vector Machines
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2018 Multiclass Classification Using Support ectorV Machines Duleep Prasanna W. Rathgamage Don Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Artificial Intelligence and Robotics Commons, Other Applied Mathematics Commons, and the Other Statistics and Probability Commons Recommended Citation Rathgamage Don, Duleep Prasanna W., "Multiclass Classification Using Support ectorV Machines" (2018). Electronic Theses and Dissertations. 1845. https://digitalcommons.georgiasouthern.edu/etd/1845 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. MULTICLASS CLASSIFICATION USING SUPPORT VECTOR MACHINES by DULEEP RATHGAMAGE DON (Under the Direction of Ionut Iacob) ABSTRACT In this thesis we discuss different SVM methods for multiclass classification and introduce the Divide and Conquer Support Vector Machine (DCSVM) algorithm which relies on data sparsity in high dimensional space and performs a smart partitioning of the whole train- ing data set into disjoint subsets that are easily separable. A single prediction performed between two partitions eliminates one or more classes in a single partition, leaving only a reduced number of candidate classes for subsequent steps. The algorithm continues recur- sively, reducing the number of classes at each step, until a final binary decision is made between the last two classes left in the process. -
A Dynamical Approach Towards Collatz Conjecture
A DYNAMICAL APPROACH TOWARDS COLLATZ CONJECTURE PABLO CASTAÑEDA Abstract. The present work focuses on the study of the renowned Collatz conjecture, also known as the 3x + 1 problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of iterated numbers is given. Within the associated iterative map, characteristic periods for periodic orbits are identified. 1. Introduction The Collatz conjecture is a long standing open conjecture in number theory. Among many other names it is often known as the conjecture for the 3x + 1 problem, which concerns to an arithmetic procedure over integers. This conjecture is based on the Collatz function given by ( 3x + 1; if x ≡ 1 (mod 2) (1) C(x) = : x=2; if x ≡ 0 (mod 2) Notice that the Collatz problem concerns to the dynamical behavior of the above map for any positive integer x. Conjecture 1 (Collatz conjecture). Starting from any positive integer x, iterations of the function C(x) will eventually reach the number 1. Thus, iterations enter in a cycle, taking successive values f1; 4; 2g. The problem has been addressed from several viewpoints along nearly a Century (see e.g. [9] for an overview), which come from approaches of number theory, dynamical systems, ergodic theory, mathematical logic, and theory of computation as well as stochastic strategies. This paper consists of a numerical and dynamical hybrid analysis. The 3x + 1 problem have been considered as a discrete map C : Z ! Z. The map here proposed maps an interval in the real numbers into itself, yet the given map is discontinuous almost everywhere.