sets, posets, and lattice © marcelo d’amorim 2010

© Marcelo d’Amorim 2010

Sets, POSets, and Lattice

http://pan.cin.ufpe.br


SETS


Set

• Collection of objects (elements) that carry no particular order

• Two ways of expression– Intentional• A is the set of all odd integers

– Extensional• B = {2, 4, 5}

– Obs. Ø is the empty set


Set builder notation

A = {a | a: int, a % 2 = 0}

Set A includes integer elements with a 0 remainder for division by 2


Cardinality

• |A| denotes the number of elements in A


Set relations

• Membership a A iff A includes a∈• Subset A B iff a A implies a B⊂ ∈ ∈• Superset A B iff a B implies a A⊃ ∈ ∈


Set factories

• Intersection A ∩ B = {x | x A and x B}∈ ∈• Union A U B = {x | x ∈ A or x ∈ B}• Difference A - B = {x | x A and not (x B)}∈ ∈


Power Set

• Def.: The set of all subsets of A• Notations: P(A), 2A

• Example– A={1,2,3}, 2A={Ø,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}


Cartesian Product

• Def.: Set of ordered pairs from two sets: the first (resp., second) component comes from the first (resp., second) set.

• More formally: A x B = { (a,b) | a A and b B}∈ ∈• Notes– | A x B | = | A | x | B |– A2 is shorthand for A x A


Relation

• Def.: Particular subset of a cartesian product• Triple (A,B,G) defines relation R– A is the relation domain and B the codomain– G is the binary relationship

• Example: R = (Int,Int,<=) Int x Int = {(a,b)| a,b: Int, a <= b} ⊂• Notation aRb = (a,b) R∈


Relation properties

• Reflexive• Symmetric• Anti-symmetry• Transitive


Relation properties

• Reflexive: a. aRa∀

• Symmetric: a,b . aRb implies bRa∀

• Anti-symmetry: a,b . aRb and bRa implies a=b∀

• Transitive: a, b, c. aRb and bRc implies aRc∀


Function

• Def.: Relation R s.t. aRb and aRc implies b = c• Notation f : A => B• Mapping– Injective: f(a) = f(b) implies a = b– Surjective: y. _ . f(_) = y∀ ∃


PARTIAL ORDERING


Partial Ordering

• Def.: Relation ≤ : L x L => {true,false} that is reflexive, transitive, and anti-symmetric

• Examples?


Partial Ordering

• Def.: Relation ≤ : L x L => {true,false} that is reflexive, transitive, and anti-symmetric

• Examples?– ancestor of– phenomena causality

Note: Important to characterize semantics of concurrent systems!


Total order

• Def.: Relation that is transitive, anti-symmetric and…total

• Total: a, b. aRb or bRa∀

reflexivity implied


POSETS


POSET

• Set L with a partial ordering ≤ • Notation – (L, ≤) to describe POSET– l1 ≤ l2 to describe a particular pair

≤ is not necessarily the subset relation


Upper and Lower bounds

• For any subset Y of a poset L– u is an upper bound of Y if ∀ a . a ≤ u– l is a lower bound of Y if ∀ a . l ≤ a


Least upper bound (lub)

• lub is an upper bound s.t. for any other upper bound u, lub ≤ u

lub, when exist, is unique


Greatest lower bound (glb)

• Dual of lub…


Meet and Join

a b

d

f

g h

c

• d is the meet of {a,b,c}• f is the join of {g,h}


COMPLETE LATTICE


Complete lattice

• Poset (L,≤) s.t. all subsets of L have lubs and glbs. Also, Top = lub L and Bottom = glb L


Exercises

• Considering the following cases, answer the questions:– What is L?– What is ≤?– Is the (Hasse) diagram a lattice?


Example 1


Example 2


Example 3


Dataflow analysis encodes program information as an element in the lattice. Ordering denotes precision:l0 ≤ l1 indicates that l1 carries more information than l0.


Dataflow analysis encodes program information as an element in the lattice. Ordering denotes precision:l0 ≤ l1 indicates that l1 carries more information than l0.

Accumulation of information (i.e., progress in the analysis) conceptually corresponds to ordered movements in the lattice. But when to stop such iteractive process?


Chain

• A subset Y of L is a chain if the elements are totally ordered


Ascending (Descending) Chain

• A sequence [y0,y1,y2,…] denotes an ascending chain if y0 ≤ y1 ≤ y2 ≤ …

• Similar for descending chains


Ascending chain condition

• A poset satisfies ascending chain conditions iff all ascending chains stabilize

0

1

2

3

∞

…

satisfies ascending chain condition, but not descending


Fix Points

• Consider a monotone function f : L => L

if L satisfies ascending chain condition and f is monotonic increasing then exists n such that fn(Bottom)=fn+1(Bottom)


Lattices considered in program analysis typically have finite height. For example, due to finite number of program locations and variable names.


http://pan.cin.ufpe.br

• Recommended Bibliography– Appendix A from “Principles of Program Analysis”,

Nielson, Nielson and Hanking, Springer 2005– First chapters from “Introduction to Lattices and

Order”, Davey and Priestley, Cambridge Press, 1990

sets, posets, and lattice © marcelo d’amorim 2010

Documents

set marcelo damorim

posets marcelo damorim

bra marcelo damorim

lub u marcelo damorim

unique slide

b superset

b surjective

subset relation slide