Partial Order Theory in the Assessment of Environmental Chemicals: Formal Aspects of a Precautionary Pre-Selection Procedure
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Slides by Akihisa
Complete non-orders and fixed points Akihisa Yamada1, Jérémy Dubut1,2 1 National Institute of Informatics, Tokyo, Japan 2 Japanese-French Laboratory for Informatics, Tokyo, Japan Supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST Introduction • Interactive Theorem Proving is appreciated for reliability • But it's also engineering tool for mathematics (esp. Isabelle/jEdit) • refactoring proofs and claims • sledgehammer • quickcheck/nitpick(/nunchaku) • We develop an Isabelle library of order theory (as a case study) ⇒ we could generalize many known results, like: • completeness conditions: duality and relationships • Knaster-Tarski fixed-point theorem • Kleene's fixed-point theorem Order A binary relation ⊑ • reflexive ⟺ � ⊑ � • transitive ⟺ � ⊑ � and � ⊑ � implies � ⊑ � • antisymmetric ⟺ � ⊑ � and � ⊑ � implies � = � • partial order ⟺ reflexive + transitive + antisymmetric Order A binary relation ⊑ locale less_eq_syntax = fixes less_eq :: 'a ⇒ 'a ⇒ bool (infix "⊑" 50) • reflexive ⟺ � ⊑ � locale reflexive = ... assumes "x ⊑ x" • transitive ⟺ � ⊑ � and � ⊑ � implies � ⊑ � locale transitive = ... assumes "x ⊑ y ⟹ y ⊑ z ⟹ x ⊑ z" • antisymmetric ⟺ � ⊑ � and � ⊑ � implies � = � locale antisymmetric = ... assumes "x ⊑ y ⟹ y ⊑ x ⟹ x = y" • partial order ⟺ reflexive + transitive + antisymmetric locale partial_order = reflexive + transitive + antisymmetric Quasi-order A binary relation ⊑ locale less_eq_syntax = fixes less_eq :: 'a ⇒ 'a ⇒ bool (infix "⊑" 50) • reflexive ⟺ � ⊑ � locale reflexive = ... assumes -
Mathematical Tools MPRI 2–6: Abstract Interpretation, Application to Verification and Static Analysis
Mathematical Tools MPRI 2{6: Abstract Interpretation, application to verification and static analysis Antoine Min´e CNRS, Ecole´ normale sup´erieure course 1, 2012{2013 course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 1 / 44 Order theory course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 2 / 44 Partial orders Partial orders course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 3 / 44 Partial orders Partial orders Given a set X , a relation v2 X × X is a partial order if it is: 1 reflexive: 8x 2 X ; x v x 2 antisymmetric: 8x; y 2 X ; x v y ^ y v x =) x = y 3 transitive: 8x; y; z 2 X ; x v y ^ y v z =) x v z. (X ; v) is a poset (partially ordered set). If we drop antisymmetry, we have a preorder instead. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 4 / 44 Partial orders Examples of posets (Z; ≤) is a poset (in fact, completely ordered) (P(X ); ⊆) is a poset (not completely ordered) (S; =) is a poset for any S course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 5 / 44 Partial orders Examples of posets (cont.) Given by a Hasse diagram, e.g.: g e f g v g c d f v f ; g e v e; g b d v d; f ; g c v c; e; f ; g b v b; c; d; e; f ; g a a v a; b; c; d; e; f ; g course 1, 2012{2013 Mathematical Tools Antoine Min´e p. -
The Poset of Closure Systems on an Infinite Poset: Detachability and Semimodularity Christian Ronse
The poset of closure systems on an infinite poset: Detachability and semimodularity Christian Ronse To cite this version: Christian Ronse. The poset of closure systems on an infinite poset: Detachability and semimodularity. Portugaliae Mathematica, European Mathematical Society Publishing House, 2010, 67 (4), pp.437-452. 10.4171/PM/1872. hal-02882874 HAL Id: hal-02882874 https://hal.archives-ouvertes.fr/hal-02882874 Submitted on 31 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Portugal. Math. (N.S.) Portugaliae Mathematica Vol. xx, Fasc. , 200x, xxx–xxx c European Mathematical Society The poset of closure systems on an infinite poset: detachability and semimodularity Christian Ronse Abstract. Closure operators on a poset can be characterized by the corresponding clo- sure systems. It is known that in a directed complete partial order (DCPO), in particular in any finite poset, the collection of all closure systems is closed under arbitrary inter- section and has a “detachability” or “anti-matroid” property, which implies that the collection of all closure systems is a lower semimodular complete lattice (and dually, the closure operators form an upper semimodular complete lattice). After reviewing the history of the problem, we generalize these results to the case of an infinite poset where closure systems do not necessarily constitute a complete lattice; thus the notions of lower semimodularity and detachability are extended accordingly. -
Trivial Meet and Join Within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected]
Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 1-1-2014 Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected] Adam Hammett Bethel College - Mishawaka Published version. Electronic Journal of Combinatorics, Vol. 21, No. 3 (2014). Permalink. © 2014 The Authors. Used with permission. Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI, U.S.A. [email protected] Adam Hammett Department of Mathematical Sciences Bethel College Mishawaka, IN, U.S.A. [email protected] Submitted: Jan 22, 2014; Accepted: Jul 10, 2014; Published: Jul 21, 2014 Mathematics Subject Classifications: 05A05, 05A16, 06A20 Abstract The lattice of monotone triangles (Mn; 6) ordered by entry-wise comparisons is studied. Let τmin denote the unique minimal element in this lattice, and τmax the unique maximum. The number of r-tuples of monotone triangles (τ1; : : : ; τr) with minimal infimum τmin (maximal supremum τmax, resp.) is shown to asymptotically r−1 approach rjMnj as n ! 1. Thus, with high probability this event implies that one of the τi is τmin (τmax, resp.). Higher-order error terms are also discussed. Keywords: monotone triangles; permutations; square ice; alternating sign matri- ces; meet; join 1 Introduction and statement -
Programming Language Semantics a Rewriting Approach
Programming Language Semantics A Rewriting Approach Grigore Ros, u University of Illinois at Urbana-Champaign Chapter 2 Background and Preliminaries 15 2.6 Complete Partial Orders and the Fixed-Point Theorem This section introduces a fixed-point theorem for complete partial orders. Our main use of this theorem is to give denotational semantics to iterative (in Section 3.4) and to recursive (in Section 4.8) language constructs. Complete partial orders with bottom elements are at the core of both domain theory and denotational semantics, often identified with the notion of “domain” itself. 2.6.1 Posets, Upper Bounds and Least Upper Bounds Here we recall basic notions of partial and total orders, such as upper bounds and least upper bounds, and discuss several examples and counter-examples. Definition 14. A partial order, or a poset (from partial order set) (D, ) consists of a set D and a binary v relation on D, written as an infix operation, which is v reflexive, that is, x x for any x D; • v ∈ transitive, that is, for any x, y, z D, x y and y z imply x z; and • ∈ v v v anti-symmetric, that is, if x y and y x for some x, y D then x = y. • v v ∈ A partial order (D, ) is called a total order when x y or y x for any x, y D. v v v ∈ Here are some examples of partial or total orders: ( (S ), ) is a partial order, where (S ) is the powerset of a set S , that is, the set of subsets of S . -
Partial Orders — Basics
Partial Orders — Basics Edward A. Lee UC Berkeley — EECS EECS 219D — Concurrent Models of Computation Last updated: January 23, 2014 Outline Sets Join (Least Upper Bound) Relations and Functions Meet (Greatest Lower Bound) Notation Example of Join and Meet Directed Sets, Bottom Partial Order What is Order? Complete Partial Order Strict Partial Order Complete Partial Order Chains and Total Orders Alternative Definition Quiz Example Partial Orders — Basics Sets Frequently used sets: • B = {0, 1}, the set of binary digits. • T = {false, true}, the set of truth values. • N = {0, 1, 2, ···}, the set of natural numbers. • Z = {· · · , −1, 0, 1, 2, ···}, the set of integers. • R, the set of real numbers. • R+, the set of non-negative real numbers. Edward A. Lee | UC Berkeley — EECS3/32 Partial Orders — Basics Relations and Functions • A binary relation from A to B is a subset of A × B. • A partial function f from A to B is a relation where (a, b) ∈ f and (a, b0) ∈ f =⇒ b = b0. Such a partial function is written f : A*B. • A total function or just function f from A to B is a partial function where for all a ∈ A, there is a b ∈ B such that (a, b) ∈ f. Edward A. Lee | UC Berkeley — EECS4/32 Partial Orders — Basics Notation • A binary relation: R ⊆ A × B. • Infix notation: (a, b) ∈ R is written aRb. • A symbol for a relation: • ≤⊂ N × N • (a, b) ∈≤ is written a ≤ b. • A function is written f : A → B, and the A is called its domain and the B its codomain. -
Formal Programming Language Semantics Note 12 Complete Partial
Formal Programming Language Semantics note 12 CS4/MSc/TPG 18.11.02 Formal Programming Language Semantics note 12 Complete partial orders (CPOs) In this note we re-examine some of the ideas in the previous two notes from a slightly more abstract perspective. In particular, we are going to concentrate on the ideas behind the denotational semantics of while commands. By isolating the aspects of the situation which really made the semantics work, we will arrive at the basic concepts of a subject known as domain theory, which provides a general mathematical framework for much of denotational semantics. Complete partial orders. Let us write D for the set DCom = (S * S) from Note 9. Recall that we defined an element h ∈ D as the union of an increasing chain h0 ⊆ h1 ⊆ · · · . We now ask: what are the essential features of D that made this construction work? One reasonable answer (though not the only one!) is given by the following definitions: Definition 1 A partially ordered set, or poset, is a set D equipped with a binary relation v (called an order relation) such that the following hold for all x, y, z ∈ D: • x v x ( v is reflexive); • if x v y and y v z then x v z ( v is transitive); • if x v y and y v x then x = y ( v is antisymmetric). Definition 2 A complete partial order, or CPO, is a poset with the following prop- erty: every sequence of elements x0 v x1 v · · · in D has a limit or least upper F bound in D: that is, there is an element x ∈ D (written as i xi) such that • xi v x for all i (i.e. -
Domain Theory: an Introduction
Domain Theory: An Introduction Robert Cartwright Rebecca Parsons Rice University This monograph is an unauthorized revision of “Lectures On A Mathematical Theory of Computation” by Dana Scott [3]. Scott’s monograph uses a formulation of domains called neighborhood systems in which finite elements are selected subsets of a master set of objects called “tokens”. Since tokens have little intuitive significance, Scott has discarded neighborhood systems in favor of an equivalent formulation of domains called information systems [4]. Unfortunately, he has not rewritten his monograph to reflect this change. We have rewritten Scott’s monograph in terms of finitary bases (see Cartwright [2]) instead of information systems. A finitary basis is an information system that is closed under least upper bounds on finite consistent subsets. This convention ensures that every finite answer is represented by a single basis object instead of a set of objects. 1 The Rudiments of Domain Theory Motivation Programs perform computations by repeatedly applying primitive operations to data values. The set of primitive operations and data values depends on the particular programming language. Nearly all languages support a rich collection of data values including atomic objects, such as booleans, integers, characters, and floating point numbers, and composite objects, such as arrays, records, sequences, tuples, and infinite streams. More advanced languages also support functions and procedures as data values. To define the meaning of programs in a given language, we must first define the building blocks—the primitive data values and operations—from which computations in the language are constructed. Domain theory is a comprehensive mathematical framework for defining the data values and primitive operations of a programming language. -
On Comparability of Random Permutations
ON COMPARABILITY OF RANDOM PERMUTATIONS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Adam Hammett, B.S. ***** The Ohio State University 2007 Dissertation Committee: Approved by Dr. Boris Pittel, Advisor Dr. Gerald Edgar Advisor Dr. Akos Seress Graduate Program in Mathematics ABSTRACT Two permutations of [n] := {1, 2, . , n} are comparable in the Bruhat order if one can be obtained from the other by a sequence of transpositions decreasing the number of inversions. We show that the total number of pairs of permutations (π, σ) with π ≤ σ is of order (n!)2/n2 at most. Equivalently, if π, σ are chosen uniformly at random and independently of each other, then P (π ≤ σ) is of order n−2 at most. By a direct probabilistic argument we prove P (π ≤ σ) is of order (0.708)n at least, so that there is currently a wide qualitative gap between the upper and lower bounds. Next, emboldened by a connection with Ferrers diagrams and plane partitions implicit in Bressoud’s book [13], we return to the Bruhat order upper bound and show that for n-permutations π1, . , πr selected independently and uniformly at random, −r(r−1) P (π1 ≤ · · · ≤ πr) = O n , thus providing an extension of our result for pairs of permutations to chains of length r > 2. Turning to the related weak order “” – when only adjacent transpositions are ∗ admissible – we use a non-inversion set criterion to prove that Pn := P (π σ) is pn ∗ submultiplicative, thus showing existence of ρ = lim Pn . -
Two Counterexamples Concerning the Scott Topology on a Partial Order
TWO COUNTEREXAMPLES CONCERNING THE SCOTT TOPOLOGY ON A PARTIAL ORDER PETER HERTLING Fakult¨atf¨urInformatik, Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany e-mail address: [email protected] Abstract. We construct a complete lattice Z such that the binary supremum function sup : Z × Z ! Z is discontinuous with respect to the product topology on Z × Z of the Scott topologies on each copy of Z. In addition, we show that bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X; Z) of continuous functions from X to Z where X may be any topological space and where on Z the Scott topology is considered. 1. Introduction In this note two counterexamples in the theory of partial orders are constructed. The first counterexample concerns the question whether the binary supremum function on a sup semilattice is continuous with respect to the product topology of the Scott topology on each copy of the sup semilattice. The second counterexample concerns the question whether bounded completeness of a complete lattice is inherited by the dcpo of continuous functions from some topological space to the complete lattice. The Scott topology on a partially ordered set (short: poset)(X; ) has turned out to v be important in many areas of computer science. Given two posets (X1; 1) and (X2; 2), v v one can in a natural way define a partial order relation × on the product X1 X2. This v × gives us two natural topologies on the product Z = X1 X2: × The product of the Scott topology on (X1; 1) and of the Scott topology on (X2; 2), • v v the Scott topology on (Z; ×). -
A Selection of Maximal Elements Under Non-Transitive Indifferences
Munich Personal RePEc Archive A selection of maximal elements under non-transitive indifferences Alcantud, José Carlos R. and Bosi, Gianni and Zuanon, Magalì Universidad de Salamanca, University of Trieste, University of Brescia 4 August 2009 Online at https://mpra.ub.uni-muenchen.de/16601/ MPRA Paper No. 16601, posted 10 Aug 2009 08:03 UTC A selection of maximal elements under non-transitive indifferences Jos´eCarlos R. Alcantud ∗,1 Facultad de Econom´ıay Empresa, Universidad de Salamanca, E 37008 Salamanca, Spain Gianni Bosi Facolt`adi Economia, Universit`adegli Studi di Trieste, Piazzale Europa 1, 34127 Trieste, Italy Magal`ıZuanon Facolt`adi Economia, Universit`adegli Studi di Brescia, Contrada Santa Chiara 50, 25122 Brescia, Italy Abstract In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of “undominated maximals” (cf., Peris and Subiza [7]). Provided that an agent’s binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alterna- tives is finite. In the case of semiorders, proceeding in this way is the same as using Luce’s selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known “sugar example” by Luce. Key words: Maximal element, Selection of maximals, Acyclicity, Interval order, Semiorder JEL Classification: D11. Preprint submitted to Elsevier August 4, 2009 1 Introduction Even though there are arguments to ensure the existence of maximal elements for binary relations in very general settings, this concept does not always explain choice under non-transitive indifference well. -
Generalized Quotients in Coxeter Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 308, Number 1, July 1988 GENERALIZED QUOTIENTS IN COXETER GROUPS ANDERS BJORNER AND MICHELLE L. WACHS ABSTRACT. For (W, S) a Coxeter group, we study sets of the form W/V = {w e W | l{wv) = l(w) + l(v) for all v e V}, where V C W. Such sets W/V, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when VCS (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in W/V, for general V C W, are lexicographically shellable. The Mobius function on W/V under Bruhat order takes values in { —1,0, +1}. For finite groups W, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order. Descent classes Dj = {w e W \ l{ws) < l(w) o s 6 I, for all s G S}, I C S, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics.