Boolean Reasoning Boolean Reasoning
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Redacted for Privacy Professor Donald Guthrie, Jr
AN ABSTRACT OF THE THESIS OF John Anthony Battilega for the DOCTOR OF PHILOSOPHY (Name) (Degree) in Mathematics presented on May 4, 1973 (Major) (Date) Title: COMPUTATIONAL IMPROVEMENTS TO BENDERS DECOMPOSITION FOR GENERALIZED FIXED CHARGE PROBLEMS Abstract approved Redacted for privacy Professor Donald Guthrie, Jr. A computationally efficient algorithm has been developed for determining exact or approximate solutions for large scale gener- alized fixed charge problems. This algorithm is based on a relaxa- tion of the Benders decomposition procedure, combined with a linear mixed integer programming (MIP) algorithm specifically designed to solve the problem associated with Benders decomposition and a com- putationally improved generalized upper bounding (GUB) algorithm which solves a convex separable programming problem by generalized linear programming. A dynamic partitioning technique is defined and used to improve computational efficiency.All component algor- ithms have been theoretically and computationally integrated with the relaxed Benders algorithm for maximum efficiency for the gener- alized fixed charge problem. The research was directed toward the approximate solution of a particular class of large scale generalized fixed charge problems, and extensive computational results for problemsof this type are given.As the size of the problem diminishes, therelaxations can be enforced, resulting in a classical Bendersdecomposition, but with special purpose sub-algorithmsand improved convergence pro- perties. Many of the results obtained apply to the sub-algorithmsinde- pendently of the context in which theywere developed. The proce- dure for solving the associated MIP is applicableto any linear 0/1 problem of Benders form, and the techniquesdeveloped for the linear program are applicable to any large scale generalized GUB implemen- tation. -
Laced Boolean Functions and Subset Sum Problems in Finite Fields
Laced Boolean functions and subset sum problems in finite fields David Canright1, Sugata Gangopadhyay2 Subhamoy Maitra3, Pantelimon Stanic˘ a˘1 1 Department of Applied Mathematics, Naval Postgraduate School Monterey, CA 93943{5216, USA; fdcanright,[email protected] 2 Department of Mathematics, Indian Institute of Technology Roorkee 247667 INDIA; [email protected] 3 Applied Statistics Unit, Indian Statistical Institute 203 B. T. Road, Calcutta 700 108, INDIA; [email protected] March 13, 2011 Abstract In this paper, we investigate some algebraic and combinatorial properties of a special Boolean function on n variables, defined us- ing weighted sums in the residue ring modulo the least prime p ≥ n. We also give further evidence to a question raised by Shparlinski re- garding this function, by computing accurately the Boolean sensitivity, thus settling the question for prime number values p = n. Finally, we propose a generalization of these functions, which we call laced func- tions, and compute the weight of one such, for every value of n. Mathematics Subject Classification: 06E30,11B65,11D45,11D72 Key Words: Boolean functions; Hamming weight; Subset sum problems; residues modulo primes. 1 1 Introduction Being interested in read-once branching programs, Savicky and Zak [7] were led to the definition and investigation, from a complexity point of view, of a special Boolean function based on weighted sums in the residue ring modulo a prime p. Later on, a modification of the same function was used by Sauerhoff [6] to show that quantum read-once branching programs are exponentially more powerful than classical read-once branching programs. Shparlinski [8] used exponential sums methods to find bounds on the Fourier coefficients, and he posed several open questions, which are the motivation of this work. -
On &Free Boolean Algebras*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 55 (1992) 265-284 265 North-Holland On &free Boolean algebras* SakaC Fuchino* * , Sabine Koppelberg* * Fachbereich Mathematik, II. Mathematisches Institut, Freie Uniuersitiit Berlin, Arnimallee 3, W-l&W Berlin 33, Germany Makoto Takahashi Department of Mathematics, College of Liberal Arts, Kobe University, 1-2-1 Tsurukabuto Nada, Kobe, 657 Japan Communicated by A. Prestel Received 18 April 1991 Abstract Fuchino, S., S. Koppelberg and M. Takahashi, On L,,- free Boolean algebras, Annals of Pure and Applied Logic 55 (1992) 265-284. We study L-,-freeness in the variety of Boolean algebras. It is shown that some of the theorems on L,,- free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L,,, -free Boolean algebras which have no counterpart in the varieties above. 0. Introduction For a cardinal K and a fixed variety 7f in a countable language, an algebra A in 7f is said to be L,,-free if A is L,,-equivalent to a free algebra in “Ir. L-,-free algebras in various varieties have been investigated by several authors (see e.g. [3,4,5, 161). In this paper we shall study the case of the variety of Boolean algebras. The most distinctive property of Boolean algebras in this connection is that every atomless (i.e., L,,-free) Boolean algebra of size w, has the property that the set of all countable free subalgebras is closed unbounded in the set of all countable subalgebras. -
Analysis of Boolean Functions and Its Applications to Topics Such As Property Testing, Voting, Pseudorandomness, Gaussian Geometry and the Hardness of Approximation
Analysis of Boolean Functions Notes from a series of lectures by Ryan O’Donnell Guest lecture by Per Austrin Barbados Workshop on Computational Complexity February 26th – March 4th, 2012 Organized by Denis Th´erien Scribe notes by Li-Yang Tan arXiv:1205.0314v1 [cs.CC] 2 May 2012 Contents 1 Linearity testing and Arrow’s theorem 3 1.1 TheFourierexpansion ............................. 3 1.2 Blum-Luby-Rubinfeld. .. .. .. .. .. .. .. .. 7 1.3 Votingandinfluence .............................. 9 1.4 Noise stability and Arrow’s theorem . ..... 12 2 Noise stability and small set expansion 15 2.1 Sheppard’s formula and Stabρ(MAJ)...................... 15 2.2 Thenoisyhypercubegraph. 16 2.3 Bonami’slemma................................. 18 3 KKL and quasirandomness 20 3.1 Smallsetexpansion ............................... 20 3.2 Kahn-Kalai-Linial ............................... 21 3.3 Dictator versus Quasirandom tests . ..... 22 4 CSPs and hardness of approximation 26 4.1 Constraint satisfaction problems . ...... 26 4.2 Berry-Ess´een................................... 27 5 Majority Is Stablest 30 5.1 Borell’s isoperimetric inequality . ....... 30 5.2 ProofoutlineofMIST ............................. 32 5.3 Theinvarianceprinciple . 33 6 Testing dictators and UGC-hardness 37 1 Linearity testing and Arrow’s theorem Monday, 27th February 2012 Rn Open Problem [Guy86, HK92]: Let a with a 2 = 1. Prove Prx 1,1 n [ a, x • ∈ k k ∈{− } |h i| ≤ 1] 1 . ≥ 2 Open Problem (S. Srinivasan): Suppose g : 1, 1 n 2 , 1 where g(x) 2 , 1 if • {− } →± 3 ∈ 3 n x n and g(x) 1, 2 if n x n . Prove deg( f)=Ω(n). i=1 i ≥ 2 ∈ − − 3 i=1 i ≤− 2 P P In this workshop we will study the analysis of boolean functions and its applications to topics such as property testing, voting, pseudorandomness, Gaussian geometry and the hardness of approximation. -
Boolean Like Semirings
Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 13, 619 - 635 Boolean Like Semirings K. Venkateswarlu Department of Mathematics, P.O. Box. 1176 Addis Ababa University, Addis Ababa, Ethiopia [email protected] B. V. N. Murthy Department of Mathematics Simhadri Educational Society group of institutions Sabbavaram Mandal, Visakhapatnam, AP, India [email protected] N. Amarnath Department of Mathematics Praveenya Institute of Marine Engineering and Maritime Studies Modavalasa, Vizianagaram, AP, India [email protected] Abstract In this paper we introduce the concept of Boolean like semiring and study some of its properties. Also we prove that the set of all nilpotent elements of a Boolean like semiring forms an ideal. Further we prove that Nil radical is equal to Jacobson radical in a Boolean like semi ring with right unity. Mathematics Subject Classification: 16Y30,16Y60 Keywords: Boolean like semi rings, Boolean near rings, prime ideal , Jacobson radical, Nil radical 1 Introduction Many ring theoretic generalizations of Boolean rings namely , regular rings of von- neumann[13], p- rings of N.H,Macoy and D.Montgomery [6 ] ,Assoicate rings of I.Sussman k [9], p -rings of Mc-coy and A.L.Foster [1 ] , p1, p2 – rings and Boolean semi rings of N.V.Subrahmanyam[7] have come into light. Among these, Boolean like rings of A.L.Foster [1] arise naturally from general ring dulity considerations and preserve many of the formal 620 K. Venkateswarlu et al properties of Boolean ring. A Boolean like ring is a commutative ring with unity and is of characteristic 2 with ab (1+a) (1+b) = 0. -
Probabilistic Boolean Logic, Arithmetic and Architectures
PROBABILISTIC BOOLEAN LOGIC, ARITHMETIC AND ARCHITECTURES A Thesis Presented to The Academic Faculty by Lakshmi Narasimhan Barath Chakrapani In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Computer Science, College of Computing Georgia Institute of Technology December 2008 PROBABILISTIC BOOLEAN LOGIC, ARITHMETIC AND ARCHITECTURES Approved by: Professor Krishna V. Palem, Advisor Professor Trevor Mudge School of Computer Science, College Department of Electrical Engineering of Computing and Computer Science Georgia Institute of Technology University of Michigan, Ann Arbor Professor Sung Kyu Lim Professor Sudhakar Yalamanchili School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Gabriel H. Loh Date Approved: 24 March 2008 College of Computing Georgia Institute of Technology To my parents The source of my existence, inspiration and strength. iii ACKNOWLEDGEMENTS आचायातर् ्पादमादे पादं िशंयः ःवमेधया। पादं सॄचारयः पादं कालबमेणच॥ “One fourth (of knowledge) from the teacher, one fourth from self study, one fourth from fellow students and one fourth in due time” 1 Many people have played a profound role in the successful completion of this disser- tation and I first apologize to those whose help I might have failed to acknowledge. I express my sincere gratitude for everything you have done for me. I express my gratitude to Professor Krisha V. Palem, for his energy, support and guidance throughout the course of my graduate studies. Several key results per- taining to the semantic model and the properties of probabilistic Boolean logic were due to his brilliant insights. -
Dynamic Functions in Dyalog
Direct Functions in Dyalog APL John Scholes – Dyalog Ltd. [email protected] A Direct Function (dfn) is a new function definition style, which bridges the gap between named function expressions such as and APL’s traditional ‘header’ style definition. Simple Expressions The simplest form of dfn is: {expr} where expr is an APL expression containing s and s representing the left and right argument of the function respectively. For example: A dfn can be used in any function context ... ... and of course, assigned a name: Dfns are ambivalent. Their right (and if present, left) arguments are evaluated irrespective of whether these are subsequently referenced within the function body. Guards A guard is a boolean-single valued expression followed by . A simple expression can be preceded by a guard, and any number of guarded expressions can occur separated by s. Guards are evaluated in turn (left to right) until one of them yields a 1. Its corresponding expr is then evaluated as the result of the dfn. A guard is equivalent to an If-Then-Else or Switch-Case construct. A final simple expr can be thought of as a default case: The s can be replaced with newlines. For readability, extra null phrases can be included. The parity example above becomes: Named dfns can be reviewed using the system editor: or , and note how you can comment them in the normal way using . The following example interprets a dice throw: Local Definition The final piece of dfn syntax is the local definition. An expression whose principal function is a simple or vector assignment, introduces a name that is local to the dfn. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Predicate Logic
Predicate Logic Laura Kovács Recap: Boolean Algebra and Propositional Logic 0, 1 True, False (other notation: t, f) boolean variables a ∈{0,1} atomic formulas (atoms) p∈{True,False} boolean operators ¬, ∧, ∨, fl, ñ logical connectives ¬, ∧, ∨, fl, ñ boolean functions: propositional formulas: • 0 and 1 are boolean functions; • True and False are propositional formulas; • boolean variables are boolean functions; • atomic formulas are propositional formulas; • if a is a boolean function, • if a is a propositional formula, then ¬a is a boolean function; then ¬a is a propositional formula; • if a and b are boolean functions, • if a and b are propositional formulas, then a∧b, a∨b, aflb, añb are boolean functions. then a∧b, a∨b, aflb, añb are propositional formulas. truth value of a boolean function truth value of a propositional formula (truth tables) (truth tables) Recap: Boolean Algebra and Propositional Logic 0, 1 True, False (other notation: t, f) boolean variables a ∈{0,1} atomic formulas (atoms) p∈{True,False} boolean operators ¬, ∧, ∨, fl, ñ logical connectives ¬, ∧, ∨, fl, ñ boolean functions: propositional formulas (propositions, Aussagen ): • 0 and 1 are boolean functions; • True and False are propositional formulas; • boolean variables are boolean functions; • atomic formulas are propositional formulas; • if a is a boolean function, • if a is a propositional formula, then ¬a is a boolean function; then ¬a is a propositional formula; • if a and b are boolean functions, • if a and b are propositional formulas, then a∧b, a∨b, aflb, añb are -
Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions
Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions Section 10. 1, Problems: 1, 2, 3, 4, 10, 11, 29, 36, 37 (fifth edition); Section 11.1, Problems: 1, 2, 5, 6, 12, 13, 31, 40, 41 (sixth edition) The notation ""forOR is bad and misleading. Just think that in the context of boolean functions, the author uses instead of ∨.The integers modulo 2, that is ℤ2 0,1, have an addition where 1 1 0 while 1 ∨ 1 1. AsetA is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S,it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers ℕ,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L. -
Positive Fragments of Coalgebraic Logics∗
Logical Methods in Computer Science Vol. 11(3:18)2015, pp. 1–51 Submitted Feb. 23, 2014 www.lmcs-online.org Published Sep. 22, 2015 POSITIVE FRAGMENTS OF COALGEBRAIC LOGICS ∗ ADRIANA BALAN a, ALEXANDER KURZ b, AND JIRˇ´I VELEBIL c a University Politehnica of Bucharest, Romania e-mail address: [email protected] b University of Leicester, United Kingdom e-mail address: [email protected] c Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic e-mail address: [email protected] Abstract. Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn’s result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories Set of sets and Pos of posets: Every functor Set → Pos has a Pos-enriched left Kan extension Pos → Pos. Functors arising in this way are said to have a presentation in discrete arities. In the case that Set → Pos is actually Set-valued, we call the corresponding left Kan extension Pos → Pos its posetification. A set functor preserves weak pullbacks if and only if its posetification preserves exact squares. -
1.1 Logic 1.1.1
Section 1.1 Logic 1.1.1 1.1 LOGIC Mathematics is used to predict empirical reality, and is therefore the foundation of engineering. Logic gives precise meaning to mathematical statements. PROPOSITIONS def: A proposition is a statement that is either true (T) or false (F), but not both. Example 1.1.1: 1+1=2.(T) 2+2=5.(F) Example 1.1.2: A fact-based declaration is a proposition, even if no one knows whether it is true. 11213 is prime. 1isprime. There exists an odd perfect number. Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed. Chapter 1 FOUNDATIONS 1.1.2 Example 1.1.3: Logical analysis of rhetoric begins with the modeling of fact-based natural language declarations by propositions. Portland is the capital of Oregon. Columbia University was founded in 1754 by Romulus and Remus. If 2+2 = 5, then you are the pope. (a conditional fact-based declaration). Example 1.1.4: A statement cannot be true or false unless it is declarative. This excludes commands and questions. Go directly to jail. What time is it? Example 1.1.5: Declarations about semantic tokens of non-constant value are NOT proposi- tions. x+2=5. Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed. Section 1.1 Logic 1.1.3 TRUTH TABLES def: The boolean domain is the set {T,F}. Either of its elements is called a boolean value. An n-tuple (p1,...,pn)ofboolean values is called a boolean n-tuple.