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Pointer Analysis G. Ramalingam Microsoft Research, India A Constant Propagation Example x = 3; • x is always 3 here • can replace x by 3 y = 4; • and replace x+5 by 8 • and so on z = x + 5; A Constant Propagation Example With Pointers x = 3; • Is x always 3 here? *p = 4; z = x + 5; A Constant Propagation Example With Pointers p = &y; if (?) p = &x; x = 3; p = &x; x = 3; *p = 4; else *p = 4; z = x + 5; p = &y; z = x + 5; x = 3; x is always 3 *p = affect pointers4; x is always 4 z = x + analyses most program 5; x may be 3 or 4 (i.e., x is unknown in our lattice) A Constant Propagation Example With Pointers p = &y; if (?) p = &x; x = 3; p = &x; x = 3; *p = 4; else *p = 4; z = x + 5; p = &y; z = x + 5; x = 3; p always *p = 4; p always points-to y z = x + 5; points-to x p may point-to x or y Points-to Analysis • Determine the set of targets a pointer variable could point-to (at different points in the program) – “p points-to x” • “p stores the value &x” • “*p denotes the location x” – targets could be variables or locations in the heap (dynamic memory allocation) • p = &x; • p = new Foo(); or p = malloc (…); – must-point-to vs. may-point-to A Constant Propagation Example With Pointers *q = 3; Can *p denote the same location as *q? *p = 4; z = *q + 5; what values can this take? More Terminology • *p and *q are said to be aliases (in a given concrete state) if they represent the same location • Alias analysis – determine if a given pair of references could be aliases at a given program point – *p may-alias *q – *p must-alias *q Pointer Analysis • Points-To Analysis • Alias Analysis – may-point-to – may-alias – must-point-to – must-alias Points-To Analysis: A Simple Example p = &x; q = &y; if (?) { q = p; } x = &a; y = &b; z = *q; Points-To Analysis: A Simple Example p q x y z null null null null null p = &x; x null null null null q = &y; x y null null null if (?) { x y null null null q = p; x x null null null } x {x,y} null null null x = &a; x {x,y} a null null y = &b; x {x,y} a b null z = *q; x {x,y} a b {a,b} Points-To Analysis x = &a; y = &b; if (?) { How should we handle p = &x; this statement? } else { p = &y; Weak update Strong update } x: a y: b p: {x,y} a: null *x = &c; x: a y: b p: {x,y} a: c *p = &c; x: {a,c} y: {b,c} p: {x,y} a: c Questions • When is it correct to use a strong update? A weak update? • Is this points-to analysis precise? • What does it mean to say – p must-point-to x at pgm point u – p may-point-to x at pgm point u – p must-not-point-to x at u – p may-not-point-to x at u Points-To Analysis, Formally • We must formally define what we want to compute before we can answer many such questions Static Program Analysis • A static program analysis computes approximate information about the runtime behavior of a given program 1. The set of valid programs is defined by the programming language syntax 2. The runtime behavior of a given program is defined by the programming language semantics 3. The analysis problem defines what information is desired 4. The analysis algorithm determines what approximation to make Programming Language: Syntax • A program consists of – a set of variables Var – a directed graph (V,E,entry) with a distinguished entry vertex, with every edge labelled by a primitive statement • A primitive statement is of the form • x = null • x=y Omitted (for now) • Dynamic memory allocation • x = *y • Pointer arithmetic • x = &y; • Structures and fields • *x = y • Procedures • skip (where x and y are variables in Var) Example Program 1 x = &a; Vars = {x,y,p,a,b,c} x = &a y = &b; 2 if (?) { y = &b p = &x; 3 p = &x p = &y } else { 4 5 p = &y; skip skip } 6 *x = &c 7 *x = &c; *p = &c *p = &c; 8 Programming Language: Operational Semantics • Operational semantics == an interpreter (defined mathematically) • State – Data-State ::= Var -> (Var U {null}) – PC ::= V (the vertex set of the CFG) – Program-State ::= PC x Data-State • Initial-state: – (entry, \x. null) Example States Vars = {x,y,p,a,b,c} Initial data-state 1 x: N, y:N, p:N, a:N, b:N, c:N x = &a 2 Initial program-state y = &b <1, x: N, y:N, p:N, a:N, b:N, c:N > 3 p = &x p = &y 4 5 skip skip 6 *x = &c 7 *p = &c 8 Programming Language: Operational Semantics • Meaning of primitive statements – CS[stmt] : Data-State -> Data-State • CS[ x =y]s= • CS[ x = *y ] s = • CS[ *x = y ] s = • CS[ x = null ] s = • CS[ x = &y ] s = s[x y] Programming Language: Operational Semantics • Meaning of primitive statements – CS[stmt] : Data-State -> Data-State • CS[ x = y ] s = s[x s(y)] • CS[ x = *y ] s = s[x s(s(y))] • CS[ *x = y ] s = s[s(x) s(y)] must say what happens if null is • CS[ x = null ] s = s[x null] dereferenced • CS[ x = &y ] s = s[x y] Programming Language: Operational Semantics • Meaning of program – a transition relation on program-states – Program-State X Program-State – state1 state2 means that the execution of some edge in the program can transform state1 into state2 • Defining – (u,s) (v,s’) iff the program contains a control-flow edge u->v labelled with a statement stmt such that M[stmt]s = s’ Programming Language: Operational Semantics • A sequence of states s1s2 … sn is said to be an execution (of the program) iff – s1 is the Initial-State – si si+1 for 1 <= I < n • A state s is said to be a reachable state iff there exists some execution s1s2 … sn is such that sn = s. • Define RS(u) = { s | (u,s) is reachable } Programming Language: Operational Semantics All of this formalism for this one definition • Define RS(u) = { s | (u,s) is reachable } Ideal Points-To Analysis: Formal Definition • Let u denote a vertex in the CFG • Define IdealMustPT (u) to be { (p,x) | forall s in RS(u). s(p) == x } • Define IdealMayPT (u) to be { (p,x) | exists s in RS(u). s(p) == x } May-Point-To Analysis: Formal Requirement Specification May Point-To Analysis Compute R: V -> 2Vars’ such that R(u) IdealMayPT(u) (where Var’ = Var U {null}) For every vertex u in the CFG, compute a set R(u) such that R(u) { (p,x) | $sRS(u). s(p) == x } May-Point-To Analysis: Formal Requirement Specification Compute R: V -> 2Vars’ such that R(u) IdealMayPT(u) • An algorithm is said to be correct if the solution R it computes satisfies "uV. R(u) IdealMayPT(u) • An algorithm is said to be precise if the solution R it computes satisfies "uV. R(u) = IdealMayPT(u) • An algorithm that computes a solution R1 is said to be more precise than one that computes a solution R2 if "uV. R1(u) R2(u) Back To Our May-Point-To Algorithm p q x y z null null null null null p = &x; x null null null null q = &y; x y null null null if (?) { x y null null null q = p; x x null null null } x {x,y} null null null x = &a; x {x,y} a null null y = &b; x {x,y} a b null z = *q; x {x,y} a b {a,b} (May-Point-To Analysis) Algorithm A • Is this algorithm correct? • Is this algorithm precise? • Let’s first completely and formally define the algorithm. Algorithm A: A Formal Definition The “Data Flow Analysis” Recipe • Define semi-lattice of abstract-values – AbsDataState ::= Var -> 2Var’ – f1 f2 = \x. (f1 (x) f2 (x)) – bottom = \x.{} • Define initial abstract-value – InitialAbsState = \x. {null} • Define transformers for primitive statements • AS[stmt] : AbsDataState -> AbsDataState Algorithm A: A Formal Definition The “Data Flow Analysis” Recipe • Let st(v,u) denote stmt on edge v->u x(v) x(w) v w st(v,u) st(w,u) u x(u) • Compute the least-fixed-point of the following “dataflow equations” – x(entry) = InitialAbsState – x(u) = v->u AS(st(v,u)) x(v) Algorithm A: The Transformers • Abstract transformers for primitive statements – AS[stmt] : AbsDataState -> AbsDataState • AS[ x = y ] s = s[x s(y)] • AS[ x = null ] s = s[x {null}] • AS[ x = &y ] s = s[x {y}] • AS[ x = *y ] s = s[x s*(s(y))] where s*({v1,…,vn}) = s(v1) … s(vn) • AS[ *x = y ] s = ??? Correctness & Precision • We have a complete & formal definition of the problem. • We have a complete & formal definition of a proposed solution. • How do we reason about the correctness & precision of the proposed solution? Enter: The French Recipe (Abstract Interpretation) Concrete Domain • Concrete states: C • Semantics: For every statement st, CS[st] : C -> C a g 2Data-State 2Var x Var’ Points-To Analysis (Abstract Interpretation) MayPT(u) a RS(u) a IdealMayPT(u) 2Data-State 2Var x Var’ a(Y) = { (p,x) | exists s in Y. s(p) == x } IdealMayPT (u) = a ( RS(u) ) Approximating Transformers: Correctness Criterion c is said to be correctly approximated by a iff a(c) a c1 correctly a1 approximated by f f# c2 correctly a2 approximated by C A Approximating Transformers: Correctness Criterion concretization c1 a1 g f f# abstraction c2 a2 a requirement: C A f#(a1) ≥ a (f( g(a1)) Concrete Transformers • CS[stmt] : Data-State -> Data-State • CS[ x = y ] s = s[x s(y)] • CS[ x = *y ] s = s[x s(s(y))] • CS[ *x = y ] s = s[s(x) s(y)] • CS[ x = null ] s = s[x null] • CS*[stmt] : 2Data-State -> 2Data-State • CS*[st] X = { CS[st]s | s X } Abstract Transformers • AS[stmt] : AbsDataState -> AbsDataState • AS[ x = y ] s = s[x s(y)] • AS[ x = null ] s = s[x {null}] • AS[ x = *y ] s = s[x s*(s(y))] where s*({v1,…,vn}) = s(v1) … s(vn) • AS[ *x = y ] s = ??? Algorithm A: Tranformers Weak/Strong Update x: &y y: &x z: &a g x: {&y} y: {&x,&z} z: {&a} x: &y y: &z z: &a f *y = &b; f# *y = &b; x: &b y: &x z: &a a x: {&y,&b} y: {&x,&z} z: {&a,&b} x: &y y: &z z: &b Algorithm A: Tranformers Weak/Strong Update x: &y y: &x z: &a g x: {&y} y: {&x,&z} z: {&a} x: &y y: &z z: &a f *x = &b; f# *x = &b; x: &y y: &b z: &a a x: {&y} y: {&b} z: {&a} x: &y y: &b z: &a Algorithm A: Transformers Weak/Strong Update • Transformer for “*p = q”

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