Understanding Program Analysis with Set Constraints
Explore the concept of program analysis with set constraints, delving into techniques like set-variable-based analysis, constant propagation, and constraint graphs. Learn about term constraints, additional implicit constraints, and function calls in the context of set-constraint based analysis. Gain insights into how constraint resolution plays a key role in determining program variables and values using set constraints.
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Program Analysis with Set Constraints Ravi Chugh
Set-constraint based analysis Another technique for computing information about program variables Phase 1: constraint generation Create set variables corresponding to program Add inclusion constraints between these sets Usually a local, syntax-directed process (ASTs vs CFGs) Phase 2: constraint resolution Solve for values of all set variables Extends naturally to inter-procedural analysis
Constant propagation Want to determine whether x and y are constant values when they are used int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } We will build a flow- insensitive analysis
Set constraints Terms t := c (constant) (set variable) (constructed term) | X | C(t1,...,tn) Constraints t1 t2 Constructors C(v1,...,vn) is an n-arg ctor C with variances vi viis either + (covariant) or (contravariant) Covariance corresponds to forwards flow Contravariance corresponds to backwards flow (set inclusion)
Additional constraints Implicit constraints added by following rules: 1) Transitivity if t1 t2and t2 t3then t1 t3 2) Variance through constructed terms if C(...,ti,...) C(...,ui,...) then ti uifor covariant positions of C ui tifor contravariant positions of C
Constraint graphs 1 X 1 X X Y Y B C A Ctor Ctor(A,B,C) Ctor(D,E,F) where Ctor(+,-,+) E F D Ctor
Function calls Define ctor Fun(-,+) for one input/one output To encode a function def/call: int z = id(2); Fun(i,r) id Fun(2,z) By contravariance, the actual 2 flows to i By covariance, the return value of id flows to z r i Fun id z 2 Fun
int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... }
int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... }
Fun(i,r1) abs int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } i Fun r1 abs
Fun(i,r1) abs int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } i Fun r1 abs
Fun(i,r1) abs i r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } i Fun r1 abs
Fun(i,r1) abs i r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } i Fun r1 abs
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } T i Fun r1 abs
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } T i Fun r1 abs
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id T i j Fun Fun r1 r2 abs id
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id T i j Fun Fun r1 r2 abs id
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 T i j Fun Fun r1 r2 abs id
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 T i j Fun Fun r1 r2 abs id
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 1 a 2 b T i j Fun Fun r1 r2 abs id a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 1 a 2 b T i j Fun Fun r1 r2 abs id a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 1 a 2 b abs Fun(a,x) T i j Fun Fun r1 r2 abs id x Fun a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 1 a 2 b abs Fun(a,x) T i j Fun Fun r1 r2 abs id x Fun a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 1 a 2 b abs Fun(a,x) id Fun(b,y) T i j Fun Fun r1 r2 abs id x y Fun Fun a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 ?? x ?? y 1 a 2 b abs Fun(a,x) id Fun(b,y) T i j Fun Fun r1 r2 abs id x y Fun Fun a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 {1,T} x ?? y 1 a 2 b abs Fun(a,x) id Fun(b,y) T i j Fun Fun r1 r2 abs id x y Fun Fun a b 1 2
Fun(i,r1) abs i r1 T r1 int abs(int i) { if (...) { return i; } else { return i; } } int id(int j) { return j; } void main() { int a = 1, b = 2; int x = abs(a); int y = id(b); ... use x ... ... use y ... } Fun(j,r2) id j r2 {1,T} x {2} y 1 a 2 b abs Fun(a,x) id Fun(b,y) T i j Fun Fun r1 r2 abs id x y Fun Fun a b 1 2
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 int i = 1; int *p = &i; *p = 2; int j = *p;
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 Ref int i = 1; int *p = &i; *p = 2; int j = *p;
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 Ref int i = 1; int *p = &i; *p = 2; int j = *p; p
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 Ref int i = 1; int *p = &i; *p = 2; int j = *p; p 2 Ref
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 Ref int i = 1; int *p = &i; *p = 2; int j = *p; p j Ref 2 Ref
Pointers Handle pointers with a Ref(-,+) constructor Two args correspond to set and get operations i 1 Ref int i = 1; int *p = &i; *p = 2; int j = *p; p j Ref 2 Ref
More on functions This encoding supports higher-order functions Passing around Fun terms just like constants Function pointers also work int (*funcPtr)(int); int id(int j) { return j }; funcPtr = &id; int x = (*funcPtr)(0); j Fun id
More on functions This encoding supports higher-order functions Passing around Fun terms just like constants Function pointers also work int (*funcPtr)(int); int id(int j) { return j }; funcPtr = &id; int x = (*funcPtr)(0); j Fun id id Ref
More on functions This encoding supports higher-order functions Passing around Fun terms just like constants Function pointers also work int (*funcPtr)(int); int id(int j) { return j }; funcPtr = &id; int x = (*funcPtr)(0); j Fun id Ref funcPtr
More on functions This encoding supports higher-order functions Passing around Fun terms just like constants Function pointers also work int (*funcPtr)(int); int id(int j) { return j }; funcPtr = &id; int x = (*funcPtr)(0); j Fun id Ref funcPtr Ref x 0 Fun
Context (in)sensitivity Multiple call sites int x = id(1); int y = id(2); {1,2} x{1,2} y r id j Fun x y 1 2 Fun Fun
Context sensitivity Multiple call sites int x = id1(1); int y = id2(2); Option 1: Specialization Each call idi gets a new copy of id {1} x {2} y r2 Fun j2 r1 Fun j1 id2 id1 y 2 Fun x 1 Fun Eliminates smearing but increases graph size
Context sensitivity Option 2: Unique labeled edges for each call site Not using Fun constructor j r [1 ]1 [2 ]2 1 x 2 y There is flow only if there is a path that spells a substring of a well-bracketed string [a[b]b]a and [a]a[b are valid; [a[b]a]b is not For both options, if there are higher-order functions or function pointers, need a first pass to compute pointer targets
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); r j Fun id o.f o.g Fldf Fldg o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); r j Fun id o.f o.g Fldf Fldg o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); Fldg r j Fun id r3 p Fun o.f o.g Fldf Fldg readG o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); Fldg r j Fun id r3 p Fun o.f o.g Fldf Fldg w Fun readG Fldf o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); Fldg r j Fun id r3 p Fun o.f o.g Fldf Fldg w Fun readG Fldf o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); Fldg r j Fun id r3 p Fun o.f o.g Fldf Fldg w Fun readG z Fun Fldf o 3 Fldf 4 Fldg
Field sensitivity For each field f, define Fldf(-,+)constructor obj o = { f:3; g:4 }; int readG(obj p) { return p.g; } int w = id(o.f); int z = readG(o); Fldg r j Fun id r3 p Fun o.f o.g Fldf Fldg w Fun readG z Fun Fldf o 3 Fldf 4 Fldg
Scalability Constraint graph for entire program is in memory Even for flow-insensitive analyses, this can become a bottleneck Even worse for flow-sensitive analyses Techniques for analyzing parts of program in isolation and storing summaries of their observable effects
Summary Set constraints a natural way to express various program analyses Constant propagation, pointer analysis Closure analysis Receiver class analysis, prototype-based inheritance Information flow Rich literature on solving systems of constraints Non-trivial to extend to flow-sensitive or summary-based analyses Interference between functions and references
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