sets, posets, and lattice © marcelo d’amorim 2010
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© Marcelo d’Amorim 2010
Sets, POSets, and Lattice
http://pan.cin.ufpe.br
© Marcelo d’Amorim 2010
SETS
© Marcelo d’Amorim 2010
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
© Marcelo d’Amorim 2010
Set builder notation
A = {a | a: int, a % 2 = 0}
Set A includes integer elements with a 0 remainder for division by 2
© Marcelo d’Amorim 2010
Cardinality
• |A| denotes the number of elements in A
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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⊃ ∈ ∈
© Marcelo d’Amorim 2010
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)}∈ ∈
© Marcelo d’Amorim 2010
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}}
© Marcelo d’Amorim 2010
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
© Marcelo d’Amorim 2010
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∈
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Relation properties
• Reflexive• Symmetric• Anti-symmetry• Transitive
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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∀
© Marcelo d’Amorim 2010
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∀ ∃
© Marcelo d’Amorim 2010
PARTIAL ORDERING
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Partial Ordering
• Def.: Relation ≤ : L x L => {true,false} that is reflexive, transitive, and anti-symmetric
• Examples?
© Marcelo d’Amorim 2010
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!
© Marcelo d’Amorim 2010
Total order
• Def.: Relation that is transitive, anti-symmetric and…total
• Total: a, b. aRb or bRa∀
reflexivity implied
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POSETS
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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
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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
© Marcelo d’Amorim 2010
Least upper bound (lub)
• lub is an upper bound s.t. for any other upper bound u, lub ≤ u
lub, when exist, is unique
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Greatest lower bound (glb)
• Dual of lub…
© Marcelo d’Amorim 2010
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}
© Marcelo d’Amorim 2010
COMPLETE LATTICE
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Complete lattice
• Poset (L,≤) s.t. all subsets of L have lubs and glbs. Also, Top = lub L and Bottom = glb L
© Marcelo d’Amorim 2010
Exercises
• Considering the following cases, answer the questions:– What is L?– What is ≤?– Is the (Hasse) diagram a lattice?
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Example 1
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Example 2
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Example 3
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Dataflow analysis encodes program information as an element in the lattice. Ordering denotes precision:l0 ≤ l1 indicates that l1 carries more information than l0.
© Marcelo d’Amorim 2010
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?
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Chain
• A subset Y of L is a chain if the elements are totally ordered
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Ascending (Descending) Chain
• A sequence [y0,y1,y2,…] denotes an ascending chain if y0 ≤ y1 ≤ y2 ≤ …
• Similar for descending chains
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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
© Marcelo d’Amorim 2010
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)
© Marcelo d’Amorim 2010
Lattices considered in program analysis typically have finite height. For example, due to finite number of program locations and variable names.
© Marcelo d’Amorim 2010
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
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