Formal methods 6 - elements of algebra
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This document discusses elements of algebra used in formal methods and software engineering. It defines monoids as sets with an associative binary operation and identity, and provides examples. Groups are defined as monoids where every element has an inverse. It also discusses groupoids as structures with multiple objects and isomorphisms between them, and categories as structures with objects and arrows representing functions between objects. It provides examples of monoids, groups, groupoids, and categories to illustrate these concepts.
![Mappings of Monoids
Given monoids (A,opa
,za
) and (B,opb
,zb
), define a mapping f that
preserves structure:
● f: A → B - that is, defined on elements of A, mapping them to B
● f(x opa
y) = f(x) opb
f(y)
● f(za
) = zb
E.g.
● exp: (ℝ,+,0) → (ℝ,*,1)
● sum: List[Int] → (Int,+,0)](https://image.slidesharecdn.com/formalmethods-6-elementsofalgebra-140401210015-phpapp02/85/Formal-methods-6-elements-of-algebra-5-320.jpg)
Formal methods 6 - elements of algebra
- 1. Formal Methods in Software Lecture 6. Elements of Algebra Vlad Patryshev SCU 2014
- 2. In This Lecture ● monoid ● group ● groupoid ● category
- 3. Monoid An object X (e.g. a set, a type in programming language), binary operation Op, nullary operation Zero; Laws: ● Op is associative ● Zero is neutral re: Op
- 4. Monoid: examples ● Real numbers ℝ and multiplication; 1. is neutral element ● Natural numbers ℕ, max, 0 ● Sets and union; ∅ is neutral ● Lists, concatenation, empty list ● Predicates, conjunction, TRUE ● Given an X, functions: X → X, their composition, idX as neutral
- 5. Mappings of Monoids Given monoids (A,opa ,za ) and (B,opb ,zb ), define a mapping f that preserves structure: ● f: A → B - that is, defined on elements of A, mapping them to B ● f(x opa y) = f(x) opb f(y) ● f(za ) = zb E.g. ● exp: (ℝ,+,0) → (ℝ,*,1) ● sum: List[Int] → (Int,+,0)
- 6. Monoid of Endomorphisms Definition. Endomorphism is a function f:X → X Endomorphisms form a monoid, ({f:X → X}, ∘, idx ) If X is a finite set, the size is |X||X| , so we denote this monoid XX . Example: id ā b̄ swap a a a b b b b a b a
- 7. Group Group is a monoid (A,op,0) where each element has an inverse: ∀x∈A ∃y∈A ((x op y) = 0) ∧ ((y op x) = 0) Notation: y = x-1 , or y = inv(x) E.g. ● (ℤ,+,0); inv(x) = -x
- 8. Group of Isomorphisms In monoid ({f:X → X}, ∘, idx ) take only such functions that have an inverse they are called isomorphisms. f is an isomorphism if ∃g:X→X (f∘g = idx ) ∧ (g∘f = idx ) It is also known a bijection in the case when X is a set. Examples: (_+7):ℝ→ℝ; (-3.4*_):ℝ→ℝ
- 9. Group of Permutations Take a set of n elements, {0,1,2,...,n}, and its isomorphic endomorphisms They are called permutations. The group of all permutations on n elements is called An . |An | = n! Sorting n elements amounts to finding the right one out of n! elements. With no extra knowledge, binary search gives an estimate O(log(n!))=O(n log(n)) 0 1 2 po p1 p2 (p0 ,p1 ,p2 )
- 10. What if there’s more than one X? Have a bunch of objects, (X1 ,X2 ,...,Xn ) take all isomorphisms between them: invertible functions of kind f:Xi →Xj It is not a group, and not a monoid: 1. composition is only allowed between f:Xi →Xj and g:Xj →Xk 2. no common neutral element, but idi :Xi →Xi Still, have associativity, neutrality, inverses. It is called Groupoid
- 11. Examples of Groupoids ● a group is a groupoid ● X and Y are objects. Iso(X,X) ∪ Iso(Y,Y) is a groupoid ● ● Set A, and an identity on each element - this makes a discrete groupoid
- 12. What If Not Only Isomorphisms Have a bunch of objects, (X1 ,X2 ,...,Xn ) take functions between them, so that: 1. idi :Xi →Xi included, for each i; 2. if f:Xi →Xj and g:Xj →Xk are present, so is g∘f:Xi →Xk Have associativity, have neutrality; inverses optional. It is called Category
- 13. Examples of Categories ● Every monoid is a category (with just one object) ● Every groupoid is a category (all functions invertible) ● Category of all sets and their functions ● Category of all monoids and their functions ● Tiny things, like ○ Category 1 = ○ Category 1+1 = ○ Category 2 = ○ Category 3 =
- 14. Java as a Category Objects (Xi ): all types and classes (Integer, String, java.util.Date, Map<X,Y>) Functions: all imaginable static functions, plus methods, plus identities E.g. toString: Integer → String It is not an isomorphism: some strings are not results of toString. But it is an injection; injection is called monomorphism in Category Theory. Integer.parseInt: String → Integer - not even a function. Either have to restrict to representations of integers, or redefine “function”.