Introduction to Static Analysis

Sets, POSets, and Lattice

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SETS

Set

•Collection of objects (elements) that carry noparticular 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 elementswith a 0 remainder for division by 2

Set A includes integer elementswith 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: thefirst (resp., second) component comes fromthe 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 isreflexive, transitive, and anti-symmetric

•Examples?

Partial Ordering

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

•Examples?

–ancestor of

–phenomena causality

Note: Important to characterizesemantics of concurrent systems!

Note: Important to characterizesemantics of concurrent systems!

Total order

•Def.: Relation that is transitive, anti-symmetricand…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 upperbound u, lub ≤ u

lub, whenexist, is unique

Greatest lower bound (glb)

•Dual of lub…

Meet and Join

•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 andglbs. Also, Top = lub L and Bottom = glb L

Exercises

•Considering the following cases, answer thequestions:

–What is L?

–What is ≤?

–Is the (Hasse) diagram a lattice?

Example 1

Example 2

Example 3

Dataflow analysis encodes programinformation as an element in the lattice.Ordering denotes precision:

l0 ≤ l1 indicates that l1 carries moreinformation than l0.

Dataflow analysis encodes programinformation as an element in the lattice.Ordering denotes precision:

l0 ≤ l1 indicates that l1 carries moreinformation than l0.

Dataflow analysis encodes programinformation as an element in the lattice.Ordering denotes precision:

l0 ≤ l1 indicates that l1 carries moreinformation than l0.

Dataflow analysis encodes programinformation as an element in the lattice.Ordering denotes precision:

l0 ≤ l1 indicates that l1 carries moreinformation than l0.

Accumulation of information (i.e., progressin the analysis) conceptually corresponds toordered movements in the lattice. Butwhen to stop such iteractive process?

Accumulation of information (i.e., progressin the analysis) conceptually corresponds toordered movements in the lattice. Butwhen to stop such iteractive process?

Chain

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

Ascending (Descending) Chain

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

•Similar for descending chains

Ascending chain condition

•A poset satisfies ascending chain conditions iffall ascending chains stabilize

∞

…

satisfies ascendingchain condition, butnot descending

satisfies ascendingchain condition, butnot descending

Fix Points

•Consider a monotone function f : L => L

if L satisfies ascending chain conditionand f is monotonic increasing then existsn such that fn(Bottom)=fn+1(Bottom)

if L satisfies ascending chain conditionand f is monotonic increasing then existsn such that fn(Bottom)=fn+1(Bottom)

Lattices considered in program analysistypically have finite height. For example,due to finite number of program locationsand variable names.

Lattices considered in program analysistypically have finite height. For example,due to finite number of program locationsand variable names.

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•Recommended Bibliography

–Appendix A from “Principles of Program Analysis”,Nielson, Nielson and Hanking, Springer 2005

–First chapters from “Introduction to Lattices andOrder”, Davey and Priestley, Cambridge Press, 1990