DOCSLIB.ORG

Cardinality of Set Partitions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cardinality of Set Partitions Cardinality of Set Partitions Lukas Bulwahn February 23, 2021 Abstract The theory’s main theorem states that the cardinality of set parti- tions of size k on a carrier set of size n is expressed by Stirling numbers of the second kind. In Isabelle, Stirling numbers of the second kind are defined in the AFP entry ‘Discrete Summation’ [1] through their well-known recurrence relation. The main theorem relates them to the alternative definition as cardinality of set partitions. The proof follows the simple and short explanation in Richard P. Stanley’s ‘Enumerative Combinatorics: Volume 1’ [2] and Wikipedia [3], and unravels the full details and implicit reasoning steps of these explanations. Contents 1 Set Partitions 1 1.1 Useful Additions to Main Theories ............... 2 1.2 Introduction and Elimination Rules ............... 2 1.3 Basic Facts on Set Partitions .................. 2 1.4 The Unique Part Containing an Element in a Set Partition . 4 1.5 Cardinality of Parts in a Set Partition ............. 5 1.6 Operations on Set Partitions .................. 5 2 Combinatorial Basics 6 2.1 Preliminaries ........................... 6 2.1.1 Injectivity and Disjoint Families ............ 6 2.1.2 Cardinality Theorems for Set.bind ........... 6 2.2 Third Version of Injectivity Solver ............... 7 3 Cardinality of Set Partitions 8 1 Set Partitions theory Set-Partition imports HOL−Library:Disjoint-Sets 1 HOL−Library:FuncSet begin 1.1 Useful Additions to Main Theories lemma set-eqI 0: assumes Vx: x 2 A =) x 2 B assumes Vx: x 2 B =) x 2 A shows A = B hproof i lemma comp-image: ((‘) f ◦ (‘) g) = (‘)(f o g) hproof i 1.2 Introduction and Elimination Rules The definition of partition-on is in HOL−Library:Disjoint-Sets. lemma partition-onI : assumes Vp: p 2 P =) p 6= fg assumes S P = A assumes Vp p 0: p 2 P =) p 0 2 P =) p 6= p 0 =) p \ p 0 = fg shows partition-on A P hproof i lemma partition-onE: assumes partition-on A P obtains Vp: p 2 P =) p 6= fg S P = A Vp p 0: p 2 P =) p 0 2 P =) p 6= p 0 =) p \ p 0 = fg hproof i 1.3 Basic Facts on Set Partitions lemma partition-onD4: partition-on A P =) p 2 P =) q 2 P =) x 2 p =) x 2 q =) p = q hproof i lemma partition-subset-imp-notin: assumes partition-on A P X 2 P assumes X 0 ⊂ X shows X 0 2= P hproof i lemma partition-on-Diff : assumes P: partition-on A P shows Q ⊆ P =) partition-on (A − S Q)(P − Q) hproof i 2 lemma partition-on-UN: assumes A: partition-on A B and B: Vb: b 2 B =) partition-on b (P b) shows partition-on A (S b2B: P b) hproof i lemma partition-on-notemptyI : assumes partition-on A P assumes A 6= fg shows P 6= fg hproof i lemma partition-on-disjoint: assumes partition-on A P assumes partition-on B Q assumes A \ B = fg shows P \ Q = fg hproof i lemma partition-on-eq-implies-eq-carrier: assumes partition-on A Q assumes partition-on B Q shows A = B hproof i lemma partition-on-insert: assumes partition-on A P assumes disjnt A X X 6= fg assumes A [ X = A 0 shows partition-on A 0 (insert X P) hproof i An alternative formulation of [[partition-on ?A ?P; disjnt ?A ?X; ?X 6= fg; ?A [ ?X = ?A 0]] =) partition-on ?A 0 (insert ?X ?P) lemma partition-on-insert 0: assumes partition-on (A − X) P assumes X ⊆ AX 6= fg shows partition-on A (insert X P) hproof i lemma partition-on-insert-singleton: assumes partition-on A P a 2= A insert a A = A 0 shows partition-on A 0 (insert fag P) hproof i lemma partition-on-remove-singleton: assumes partition-on A P X 2 PA − X = A 0 shows partition-on A 0 (P − fXg) hproof i 3 1.4 The Unique Part Containing an Element in a Set Parti- tion lemma partition-on-partition-on-unique: assumes partition-on A P assumes x 2 A shows 9 !X: x 2 X ^ X 2 P hproof i lemma partition-on-the-part-mem: assumes partition-on A P assumes x 2 A shows (THE X: x 2 X ^ X 2 P) 2 P hproof i lemma partition-on-in-the-unique-part: assumes partition-on A P assumes x 2 A shows x 2 (THE X: x 2 X ^ X 2 P) hproof i lemma partition-on-the-part-eq: assumes partition-on A P assumes x 2 XX 2 P shows (THE X: x 2 X ^ X 2 P) = X hproof i lemma the-unique-part-alternative-def : assumes partition-on A P assumes x 2 A shows (THE X: x 2 X ^ X 2 P) = fy: 9 X2P: x 2 X ^ y 2 Xg hproof i lemma partition-on-all-in-part-eq-part: assumes partition-on A P assumes X 0 2 P shows fx 2 A: (THE X: x 2 X ^ X 2 P) = X 0g = X 0 hproof i lemma partition-on-part-characteristic: assumes partition-on A P assumes X 2 P x 2 X shows X = fy: 9 X2P: x 2 X ^ y 2 Xg hproof i lemma partition-on-no-partition-outside-carrier: assumes partition-on A P assumes x 2= A shows fy: 9 X2P: x 2 X ^ y 2 Xg = fg 4 hproof i 1.5 Cardinality of Parts in a Set Partition lemma partition-on-le-set-elements: assumes finite A assumes partition-on A P shows card P ≤ card A hproof i 1.6 Operations on Set Partitions lemma partition-on-union: assumes A \ B = fg assumes partition-on A P assumes partition-on B Q shows partition-on (A [ B)(P [ Q) hproof i lemma partition-on-split1: assumes partition-on A (P [ Q) shows partition-on (S P) P hproof i lemma partition-on-split2: assumes partition-on A (P [ Q) shows partition-on (S Q) Q hproof i lemma partition-on-intersect-on-elements: assumes partition-on (A [ C) P assumes 8 X 2 P: 9 x: x 2 X \ C shows partition-on C ((λX: X \ C) ‘P) hproof i lemma partition-on-insert-elements: assumes A \ B = fg assumes partition-on B P assumes f 2 A !E P shows partition-on (A [ B) ((λX: X [ fx 2 A: f x = Xg) ‘P)(is partition-on - ?P) hproof i lemma partition-on-map: assumes inj-on f A assumes partition-on A P shows partition-on (f ‘ A) ((‘) f ‘ P) hproof i lemma set-of-partition-on-map: 5 assumes inj-on f A shows (‘) ((‘) f ) ‘ fP: partition-on A Pg = fP: partition-on (f ‘ A) Pg hproof i end 2 Combinatorial Basics theory Injectivity-Solver imports HOL−Library:Disjoint-Sets HOL−Library:Monad-Syntax HOL−Eisbach:Eisbach begin 2.1 Preliminaries These lemmas shall be added to the Disjoint Set theory. 2.1.1 Injectivity and Disjoint Families lemma inj-on-impl-disjoint-family-on-singleton: assumes inj-on f A shows disjoint-family-on (λx: ff xg) A hproof i 2.1.2 Cardinality Theorems for Set.bind lemma card-bind: assumes finite S assumes 8 X 2 S: finite (f X) assumes disjoint-family-on f S shows card (S >>= f ) = (P x2S: card (f x)) hproof i lemma card-bind-constant: assumes finite S assumes 8 X 2 S: finite (f X) assumes disjoint-family-on f S assumes Vx: x 2 S =) card (f x) = k shows card (S >>= f ) = card S ∗ k hproof i lemma card-bind-singleton: assumes finite S assumes inj-on f S shows card (S >>= (λx: ff xg)) = card S hproof i 6 2.2 Third Version of Injectivity Solver Here, we provide a third version of the injectivity solver. The original first version was provided in the AFP entry ‘Spivey’s Generalized Recurrence for Bell Numbers‘. From that method, I derived a second version in the AFP entry ‘Cardinality of Equivalence Relations‘. At roughly the same time, Makarius improved the injectivity solver in the development version of the first AFP entry. This third version now includes the improvements of the second version and Makarius improvements to the first, and it further extends the method to handle the new cases in the cardinality proof of this AFP entry. As the implementation of the injectivity solver only evolves in the de- velopment branch of the AFP, the submissions of the three AFP entries that employ the injectivity solver, have to create clones of the injectivity solver for the identified and needed method adjustments. Ultimately, these three clones should only remain in the stable branches of the AFP from Is- abelle2016 to Isabelle2017 to work with their corresponding release versions. In the development version, I have now consolidated the three versions here. In the next step, I will move this version of the injectivity solver in the HOL−Library:Disjoint-Sets and it will hopefully only evolve further there. lemma disjoint-family-onI : assumes Vi j: i 2 I ^ j 2 I =) i 6= j =) (A i) \ (A j) = fg shows disjoint-family-on A I hproof i lemma disjoint-bind: VS T f g: (Vs t: S s ^ T t =) f s \ g t = fg) =) (fs: S sg >>= f ) \ (ft: T tg >>= g) = fg hproof i lemma disjoint-bind 0: VS T f g: (Vs t: s 2 S ^ t 2 T =) f s \ g t = fg) =) (S >>= f ) \ (T >>= g) = fg hproof i lemma injectivity-solver-CollectE: assumes a 2 fx: P xg ^ a 0 2 fx: P 0 xg assumes (P a ^ P 0 a 0) =) W shows W hproof i lemma injectivity-solver-prep-assms-Collect: assumes x 2 fx: P xg shows P x ^ P x hproof i lemma injectivity-solver-prep-assms: x 2 A =) x 2 A ^ x 2 A hproof i 7 lemma disjoint-terminal-singleton: Vs t X Y : s 6= t =) (X = Y =) s = t) =) fXg \ fY g = fg hproof i lemma disjoint-terminal-Collect: assumes s 6= t assumes Vx x 0: S x ^ T x 0 =) x = x 0 =) s = t shows fx: S xg \ fx: T xg = fg hproof i lemma disjoint-terminal: s 6= t =) (Vx x 0: x 2 S ^ x 0 2 T =) x = x 0 =) s = t) =) S \ T = fg hproof i lemma elim-singleton: assumes x 2 fsg ^ x 0 2 ftg obtains x = s ^ x 0 = t hproof i method injectivity-solver uses rule = insert method-facts; use nothing in h ((drule injectivity-solver-prep-assms-Collect j drule injectivity-solver-prep-assms)+)?; rule disjoint-family-onI ; ((rule disjoint-bind j rule disjoint-bind 0)+)?; (erule elim-singleton)?; (erule disjoint-terminal-singleton j erule disjoint-terminal-Collect j erule dis- joint-terminal); (elim injectivity-solver-CollectE)?; rule rule; assumption+ i end 3 Cardinality of Set Partitions theory Card-Partitions imports HOL−Library:Stirling Set-Partition Injectivity-Solver begin lemma set-partition-on-insert-with-fixed-card-eq: assumes finite A assumes a 2= A shows fP: partition-on (insert a A) P ^ card
Load more

Recommended publications
  • Fuzzy Logic : Introduction
    Fuzzy Logic : Introduction Debasis Samanta IIT Kharagpur [email protected] 07.01.2015 Debasis Samanta (IIT Kharagpur) Soft Computing Applications 07.01.2015 1 / 69 What is Fuzzy logic? Fuzzy logic is a mathematical language to express something. This means it has grammar, syntax, semantic like a language for communication. There are some other mathematical languages also known • Relational algebra (operations on sets) • Boolean algebra (operations on Boolean variables) • Predicate logic (operations on well formed formulae (wff), also called predicate propositions) Fuzzy logic deals with Fuzzy set. Debasis Samanta (IIT Kharagpur) Soft Computing Applications 07.01.2015 2 / 69 A brief history of Fuzzy Logic First time introduced by Lotfi Abdelli Zadeh (1965), University of California, Berkley, USA (1965). He is fondly nick-named as LAZ Debasis Samanta (IIT Kharagpur) Soft Computing Applications 07.01.2015 3 / 69 A brief history of Fuzzy logic 1 Dictionary meaning of fuzzy is not clear, noisy etc. Example: Is the picture on this slide is fuzzy? 2 Antonym of fuzzy is crisp Example: Are the chips crisp? Debasis Samanta (IIT Kharagpur) Soft Computing Applications 07.01.2015 4 / 69 Example : Fuzzy logic vs. Crisp logic Yes or No Crisp answer True or False Milk Yes Water A liquid Crisp Coca No Spite Is the liquid colorless? Debasis Samanta (IIT Kharagpur) Soft Computing Applications 07.01.2015 5 / 69 Example : Fuzzy logic vs. Crisp logic May be May not be Fuzzy answer Absolutely Partially etc Debasis Samanta (IIT Kharagpur) Soft Computing Applications
    [Show full text]
  • Mathematics Is Megethology
    17 Mathematics is megethology Mereology is the theory of the relation of part to whole, and kindred notions. Megethology is the result of adding plural quantification, as advocated by George Boolos in [1] and [2], to the language of mere­ ology. It is so-called because it turns out to have enough expressive power to let us express interesting hypotheses about the size of Reality. It also has the power, as John P. Burgess and A. P. Hazen have shown in [3], to simulate quantification over relations. It is generally accepted that mathematics reduces to set theory. In my book Parts of Classes, [6], I argued that set theory in turn reduces, with the aid of mereology, to the theory of singleton functions. I also argued (somewhat reluctantly) for a 'structuralist' approach to the theory of singleton functions. We need not think we have somehow achieved a primitive grasp of some one special singleton function. Rather, we can take the theory of singleton functions, and hence set theory, and hence mathematics, to consist of generalisations about all singleton functions. We need only assume, to avoid vacuity, that there exists at least one singleton function. But we need not assume even that. For it now turns out that if the size of Reality is right, there must exist a singleton function. All we need as a foundation for mathematics, apart from the framework of megethology, are some hypotheses about the size of Reality. First published in this form in Philosophia Mathematila 3 (1993), 3--23. This paper consists in part of near-verbatim excerpts from David Lewis, Parts of Classes (Oxford, Blackwell, 1991).
    [Show full text]
  • Definiteness and Determinacy
    Linguistics and Philosophy manuscript No. (will be inserted by the editor) Definiteness and Determinacy Elizabeth Coppock · David Beaver the date of receipt and acceptance should be inserted later Abstract This paper distinguishes between definiteness and determinacy. Defi- niteness is seen as a morphological category which, in English, marks a (weak) uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate (and thus denote an individual) or indeterminate (functioning as an existential quantifier). The latter option is observed in examples like `Anna didn't give the only invited talk at the conference', which, on its indeterminate reading, implies that there is nothing in the extension of `only invited talk at the conference'. The paper also offers a resolution of the issue of whether posses- sives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. Keywords definiteness · descriptions · possessives · predicates · type-shifting We thank Dag Haug, Reinhard Muskens, Luca Crniˇc,Cleo Condoravdi, Lucas
    [Show full text]
  • Solutions Are Available
    Math 436 Introduction to Topology Spring 2018 Examination 1 1. Define (a) the concept of the closure of a set, and (b) the concept of a basis for a given topology. Solution. The closure of a set A can be defined as the smallest closed set containing A, or as the intersection of all closed sets containing A, or as the union of A with the set of limit points of A. A basis for a given topology is a subset of such that every element of can be obtained as a union of elements of . In alternative language, a basis for a given topology is a collection of open sets such that every open set can be obtained by taking a union of some open sets belonging to the basis. 2. Give an example of a separable Hausdorff topological space .X; / with the property that X has no limit points. Solution. A point x is a limit point of X when every neighborhood of x intersects X ⧵ ^x`. This property evidently holds unless the singleton set ^x` is an open set. Saying that X has no limit points is equivalent to saying that every singleton set is an open set. In other words, the topology on X is the discrete topology. The discrete topology is automatically Hausdorff, for if x and y are distinct points, then the sets ^x` and ^y` are disjoint open sets that separate x from y. A minimal basis for the discrete topology consists of all singletons, so the discrete topology is separable if and only if the space X is countable.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:Cs/0410053V1 [Cs.DB] 20 Oct 2004 Oe.Teesrcue R Xeso of Extension Are Structures These Model
    An Extended Generalized Disjunctive Paraconsistent Data Model for Disjunctive Information Haibin Wang, Hao Tian and Rajshekhar Sunderraman Department of Computer Science Georgia State University Atlanta, GA 30302 email: {hwang17,htian1}@student.gsu.edu, [email protected] Abstract. This paper presents an extension of generalized disjunctive paraconsistent relational data model in which pure disjunctive positive and negative information as well as mixed disjunctive positive and nega- tive information can be represented explicitly and manipulated. We con- sider explicit mixed disjunctive information in the context of disjunctive databases as there is an interesting interplay between these two types of information. Extended generalized disjunctive paraconsistent relation is introduced as the main structure in this model. The relational algebra is appropriately generalized to work on extended generalized disjunctive paraconsistent relations and their correctness is established. 1 Introduction Relational data model was proposed by Ted Codd’s pioneering paper [1]. Re- lational data model is value-oriented and has rich set of simple, but powerful algebraic operators. Moreover, a strong theoretical foundation for the model is provided by the classical first-order logic [2]. This combination of a respectable theoretical platform, ease of implementation and the practicality of the model resulted in its immediate success, and the model has enjoyed being used by many database management systems. One limitation of the relational data model, however, is its lack of applicabil- arXiv:cs/0410053v1 [cs.DB] 20 Oct 2004 ity to nonclassical situations. These are situations involving incomplete or even inconsistent information. Several types of incomplete information have been extensively studied in the past such as null values [3,4,5], partial values [6], fuzzy and uncertain values [7,8], and disjunctive information [9,10,11].
    [Show full text]
  • Arxiv:1806.09506V1 [Cs.AI] 20 Jun 2018 Piiain Ed Efrpiain Pc Robotics
    OPTIMAL SEEDING AND SELF-REPRODUCTION FROM A MATHEMATICAL POINT OF VIEW. RITA GITIK Abstract. P. Kabamba developed generation theory as a tool for studying self-reproducing systems. We provide an alternative definition of a generation system and give a complete solution to the problem of finding optimal seeds for a finite self-replicating system. We also exhibit examples illustrating a connection between self-replication and fixed-point theory. Keywords Algorithm, directed labeled graph, fixed point, generation theory, optimization, seed, self-replication, space robotics. Introduction. Recent research for space exploration, such as [3], considers creating robot colonies on extraterrestrial planets for mining and manufacturing. However, sending such colonies through space is expensive [14], so it would be efficient to transport just one colony capable of self-replication. Once such a colony is designed, a further effort should be made to identify its optimal seeds, where a seed is a subset of the colony capable of manufacturing the whole colony. The optimization parameters might include the number of robots, the mass of robots, the cost of building robots on an extraterrestrial planet versus shipping them there, the time needed to man- ufacture robots at the destination, etc. Menezes and Kabamba gave a background and several algorithms for determining optimal seeds for some special cases of such colonies in [7], [8], and [9]. They also gave a lengthy discussion of the merits of the problem and an extensive reference list in [9]. In this paper, which originated in [5], we provide an alternative definition of a generation system, show how this definition simplifies and allows one to solve the problem of finding optimal seeds for a finite self-replicating system, and exhibit a connection between self-replication and fixed-point theory.
    [Show full text]
  • Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
    Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite.
    [Show full text]
  • From the Collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ
    From the collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ These documents can only be used for educational and research purposes (“Fair use”) as per U.S. Copyright law (text below). By accessing this file, all users agree that their use falls within fair use as defined by the copyright law. They further agree to request permission of the Princeton University Library (and pay any fees, if applicable) if they plan to publish, broadcast, or otherwise disseminate this material. This includes all forms of electronic distribution. Inquiries about this material can be directed to: Seeley G. Mudd Manuscript Library 65 Olden Street Princeton, NJ 08540 609-258-6345 609-258-3385 (fax) [email protected] U.S. Copyright law test The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or other reproduction is not to be “used for any purpose other than private study, scholarship or research.” If a user makes a request for, or later uses, a photocopy or other reproduction for purposes in excess of “fair use,” that user may be liable for copyright infringement. The Princeton Mathematics Community in the 1930s Transcript Number 27 (PMC27) © The Trustees of Princeton University, 1985 ROBERT SINGLETON (with ALBERT TUCKER) This is an interview of Robert Singleton at the Cromwell Inn on 6 June 1984. The interviewer is Albert Tucker.
    [Show full text]
  • On Singleton Arc Consistency for Csps Defined by Monotone Patterns
    On Singleton Arc Consistency for CSPs Defined by Monotone Patterns∗ Clément Carbonnel1, David A. Cohen2, Martin C. Cooper3, and Stanislav Živný1 1 Department of Computer Science, University of Oxford, UK {clement.carbonnel,standa.zivny}@cs.ox.ac.uk 2 Royal Holloway, University of London, UK [email protected] 3 IRIT, University of Toulouse III, France [email protected] Abstract Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity Keywords and phrases constraint satisfaction problems, forbidden patterns, singleton arc con- sistency Digital Object Identifier 10.4230/LIPIcs.STACS.2018.20 1 Introduction The constraint satisfaction problem (CSP) is a declarative paradigm for expressing computa- tional problems. An instance of the CSP consists of a number of variables to which we need to assign values from some domain. Some subsets of the variables are constrained in that they are not permitted to take all values in the product of their domains. The scope of a constraint is the set of variables whose values are limited by the constraint, and the constraint relation is the set of permitted assignments to the variables of the scope.
    [Show full text]
  • Fuzzy Sets ( Type-1 and Type-2) and Their Applications
    Fuzzy Sets ( Type-1 and Type-2) and their Applications Presented by Prof. U. S. Tiwary, IIIT Allahabad (for self use only) Why Fuzzy Sets • It enables one to work in uncertain and ambiguous situations and solve ill-posed problems or problems with incomplete information Example : Fuzzy Image Processing (Humanlike) . Human visual system is perfectly adapted to handle uncertain information in both data and knowledge . It will be hard to define quantitatively how an object , such as a car, has to look in terms of geometrical primitives with exact shapes, dimensions and colors. We use descriptive language to define features that eventually are subject to a wide range of variations. 3 Fuzzy Reasoning and Probability • They are related , but complimentary to each other. • Say, for example , if we have to define the probability of appearance of an edge in few frames of images, we have to define, what is an edge. Certain threshold for rate of variation has to be taken, which may not be true for other images or noisy images. • Fuzzy logic, unlike probability, handles imperfection in the informational content of the event. 4 Two frameworks for Fuzzy Systems 1) Development based on Crisp mathematical model and fuzzifying some quantities : Model 1 : Fuzzy Mathematical Model Example : Fuzzy – K means clustering 2) Development based on Fuzzy Inference rules: Model 2 : Fuzzy Logical Model Example : Fuzzy decision Support System 1. Definition of fuzzy set • 1.1 Concept for fuzzy set – Definition (Membership function of fuzzy set) In fuzzy sets, each elements is mapped to [0,1] by membership function. A : X [0, 1] Where [0,1] means real numbers between 0 and 1 (including 0 and 1).
    [Show full text]
  • THE II~-SINGLETON Conjecfure L, the Universe of Constructible Sets
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 4, October 1990 THE II~-SINGLETON CONJECfURE SY D. FRIEDMAN INTRODUCTION L, the universe of constructible sets, was introduced by Godel to establish the relative consistency of the Axiom of Choice and Continuum Hypothesis with the usual axioms for set theory. The class of constructible sets forms the smallest inner model, by which we mean a subuniverse of the universe of all sets that contains all ordinal numbers and in which the axioms for set theory are true. This minimality for L is a consequence of the fact that constructible sets are totally absolute, in that they each can be defined from ordinal numbers via a definition that is independent of the inner model in which this definition is interpreted. Cohen produced nonconstructible sets using the method of forcing. However it is in the nature of this method to produce "generic" sets that cannot be explic- itly defined. Let us say that a set x is absolute if it can be defined from ordinal numbers via a definition that defines x in every inner model containing x. We are particularly interested in the case where x is a set of nonnegative integers, what we call a real. Thus we may ask: is there an absolute, nonconstructible real? Silver and Solovay produced such a real using large cardinal theory. If there is a measurable cardinal then the uncountable cardinal numbers WI' w2 ' ••• form a nonconstructible sequence. By listing sentences ¢I' ¢2 ' ... in the language of set theory (augmented by names for WI' w2 ' ••• ) they obtain the absolute, nonconstructible real ott = {n I ¢" is true in L}.
    [Show full text]