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Understanding Ordered Sets and Lattices in Computer Science

This document covers the essential concepts of ordered sets and lattices, fundamental in the logical foundations of computer science. It explores order relations, including reflexive, antisymmetric, and transitive properties, along with definitions of total, partial, and strict orders. Key topics include majoring and minoring elements, lattice definitions, and properties such as completeness and complementarity. Additionally, it discusses applications of formal concept analysis (FCA) in knowledge processing and software engineering, demonstrating how these concepts aid in class hierarchy design and decision-making processes.

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Understanding Ordered Sets and Lattices in Computer Science

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  1. Ordered sets and Lattices CS 5303 – Logical Foundations of Computer Science

  2. Orders and pre-orders • These relations are very important in Computer Science as they give a mean to compare objects • Definition: an order relation is a reflexive, antisymmetric and transitive relation. A strict order is an irreflexive and transitive relation • Example: >, <, , , order in a tree

  3. Total order and partial order • Definition: if R is an order relation such that for a,a’ and aa’, then aRa’ or a’Ra, R is called a total order. • Example:  and  are total orders over ℕ or over the real numbers etc… Exercise: show that over the natural numbers, | (the division) is a partial order. Show that if E is a set, then  is a partial order over P(E) (show that both orders are not total).

  4. Pre-order • Definition: a pre-order relation is a transitive relation Exercise: Let E be a set and IdE the relation defined by (a,a’)IdE iff a=a’. Let R be q pre-order over E. Then show that IdE  (R  R-1) is an equivalence relation

  5. Ordered sets • An ordered set (E,) is a set E endowed with an order . • Example: (ℕ,) is an ordered set and so is (ℕ,|).

  6. Min and Max 1 • Definition: if E’ is a subset of and ordered set (E,), then an element of E is a majoring element of E’ if yx for all y in E’ • Minoring elements are defined similarly. • Maj(E’) and Min(E’) denotes the set of majoring and minoring elements. Exercise: show that Maj(E’)E’ and Min(E’)E’ have at most one element

  7. Min and Max 2 • Definition: if Maj(E’)E’ and Min(E’)E’ are non empty then their elements are called the maximum of E’ and minimum of E’ respectively. • Definition: an element x is the least greatest bound (sup) of E’ in E if for any y in E’ yx and if z in E is such that for any y in e’ yz then xz. The inf is defined similarly.

  8. Lattices • Definition: if (E,) is an ordered set, then it is a lattice if every pair of elements x,y admits an sup and an inf. • Notations: sup(x,y) is sometimes written x∨y and inf(x,y) is written x∧y Exercise: show that (P(E),) is a lattice.

  9. Properties • The inf and sup are 1. idempotent 2. commutative 3. associate 4. absorbing x∧(x∨y)=x = (y∧x)∨x • If inf and sup are distributive wrt one another, the lattice is called a distributive lattice

  10. Exercise: distributive lattice? ⊤ c a b ⊥

  11. Complemented lattice • Definition: a lattice is complemented iff • There is a minimum ⊥ and a maximum ⊤ that are distinct. • There exists an application :EE such that for all x in E, - x∧(x)= ⊥ - x∨(x)= ⊤ Exercise: Show that (P(E), ) is a complemented lattice

  12. Complete lattices • Definition: a lattice is called a complete lattice if every subset A of A admits a sup and an inf. Exercise: show that (P(E),) is a complete lattice.

  13. Application: formal concept analysis • FCA is a model to formalize the notions of concepts and conceptual hierarchies • Critical for conceptual data analysis and knowledge processing • Successfully used in decision engines, classification engines (e-mail), but also formal methods and patterns in software engineering, analysis of flight movements (Frankfurt), analysis of diabetic children (mcgill U.)

  14. Basics of FCA • Definition: a formal context K=(G,M,I) consists of a set G of objects, a set M of attributes and a binary relation I over GxM where gIm (or (g,m)I) reads object g has property m • I is called the incidence relation • Small contexts can be represented by cross tables

  15. Formal concepts • Definition: a formal concept of the context (G,M,I) is a pair (A,B) with AG, BM and A’=B, B’=A with • A’={m in M| gIm for all g in A} • B’={g in G | gIm for all m in B} • If (A1,B1) and (A2,B2) are concepts then A1 A2 defines an order on the concepts making B(G,M,I) the concept lattice

  16. Example (R. Godin – UQAM) Ref:http://www.info.uqam.ca/~godin/ Formal concept analysis based normal forms for class hierarchy design in object-oriented Software development

  17. Example 2 (Godin) • Designing and maintaining good class hierarchy large solution space, conflicting criteria, evolution • FCA  minimize redundancy, number of classes, multiple inheritance only if necessary, subclasses as specialization

  18. Example 3 (Godin) • G classes, objects • M instance variables, methods • I comes from analyst, code, patterns

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