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Lecture 10 Abstract Interpretation using Fixpoints. Notation. Partial order: Least upper bound: lub , sup, Greatest lower bound: glb , inf , Least fixpoint : lfp Complete lattice: always exist. Tarski’s fixpoint theorem. (A , ) complete lattice
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Notation Partial order: Least upper bound: lub, sup, Greatest lower bound: glb, inf, Least fixpoint: lfp Complete lattice: always exist
Tarski’sfixpoint theorem (A, ) complete lattice G : A A monotonic function Postfixp = { x | G(x) x } Then G has least fixpoint, lfp(G) and it’s Postfixp So, if G(x) x then x Postfixp, so lfp(G) x Let G be also -continuous. Then { Gn ( ) | n 0 } = lfp(G)
Fixpoints for Recursive Definitions Given monotonic G in complete lattice, If G(P) P then lfp(G) P Natural numbers: lfp under adding successors: G(P) = {0} {x+1|x P} lfp(G) = {0,1,2, … } (natural numbers) Take a property PP(x), let P = {x | PP(x)}What does G(P) P mean?{0} U {x+1|x P} P 0 P x P x+1 P If so, we get that all natural numbers are in P
Structural Recursion Trees = lfp(G) where G(P) = { Leaf } U { Node(t1,x,t2) | t1P, t2P} To prove all trees satisfy property, prove for the set P of trees satisfying property that G(P) P. { Leaf } U { Node(t1,x,t2) | t1 P /\ t2 P} P Structural induction! Every inductively defined set is given as lfp(G) for some G, and this gives us an induction principle.
Concrete post Init,Good S r SxS Goal: show sp(Init,r*) Good Approach: find I S such that sp(Init,r*) I Good Define post : 2S2S post(P) = Init U sp(P,r) • post is monotonic function on complete lattice 2S so it has a least fixpoint. This is sp(Init,r*) • Consequence: if post(P) P, then sp(Init,r*) P • Remark: post is also -continuous and join morphism: post(UC)=U{post(x)|xC}so this is also equal to U{ postn({}) | n ≥ 0} Strategy: find P such that post(P) P and then check that P Good
Example k=1; while(k < 100) { k=k+3 }; assert(k <= 255) k=1; loop {assume(k < 100); k=k+3}; assume(k>=100); assert(k <= 255) r = {(k,k’) | (k < 100 /\ k’ = k + 3) } Approximating sp({1},r*) post(P) = {1} U sp(P,r) = {1} U {k+3|k P, k < 100} postn({}): {}, {1}, {1,4}, …, {1,…,97},{1,…,97,100}, {1,…,97,100}
Approximate post Let apost be approximation of post, that is, post(P) apost(P) for all sets P (e.g. take interval enclosing a set) If we find P such that apost(P) P, then post(P) apost(P) P Summary: all we need is • apost that approximes post • a postfix point P of apost, ideally as small as possible, hoping to be able to prove P Good
Domain: Representation of Invariants apost will work only on certain invariants (e.g. only intervals) Represent these invariants as another set A. Define concretization : A 2S post# : A A(post#(a)) = apost((a)) Then post(P) apost(P) becomes post((a)) (post#(a)) This is the soundness condition.
Postfixpoint in Abstract Domain We have found a postfixpoint(a) if (post#(a)) (a) How to effectively check if this inclusion holds? Define an ordering on domain that is computable and satisfies: a1 a2 (a1) (a2) If post#(a) a , then (post#(a)) (a) is monotonic. We need postfixpoint of Define such that it is ordering, the nicer ordering the better (e.g. complete lattice).
Interval Approximation post(P) = {1} U {k+3|k P /\ k < 100} post#() = [1,1] post#([a,b]) = [min(1,a+3),min(b+3,102)] Soundness: post([a,b]) post#([a,b]) {1}U{k+3|a ≤ k ≤ b, k <100} [min(1,a+3),min(b+3,102)] (post#)n(): , [1,1], [1,4], …, [1,97], lfp(post#) = [1,102] We proved automatically sp({1},r*) [1,102]
How to define post# ? One way: Galois comes to rescue Suppose s.t. (, ) is Galois connection. Then post((a)) (post#(a)) is equivalent to ( on right = on left) (post((a)) post#(a) So we can define: post#(a) = (post((a)) Note: such post# is monotonic
Finding fixpoint of post# Compute post# iteration: (post#)n() If for some n (post#)n()= (post#)n+1()then for this n lfp(post#) = (post#)n() To help make this happen • make post# monotonic • design domain so post# iteration terminates in finitely many steps
Interval Domain A A = , intervals w/ finite and infinite bounds Partial order a1, a2 A
Galois Connection: Monotonicity : A 2S ([L,U]) = {x | L ≤ x ≤ U} : 2S A (c) = [min(c),max(c)] Prove , are monotonic
Galois Connection Condition : A 2S ([L,U]) = {x | L ≤ x ≤ U} : 2S A (c) = [min(c),max(c)] Galois connection condition c(a) (c) a Prove it for case where c is bounded, a is [L,U]
Multiple variables Wish to track interval for each variable We track not [L,U] but ([L1,U1],[L2,U2]) If program state is (x,y), define (([L1,U1],[L2,U2])) = {(x,y) | } (p) = ([L1,U1],[L2,U2]) L1 = U1 = L2 = U2 =
Product of Partial Orders (Ai, ≤i) partial orders for iJ (A, ≤) given by A = {f : J UiJAi , i. f(i)Ai} f,gA ordered by f ≤ g i. f(i)≤ig(i) Then (A, ≤) is a partial order. Moreover: If (Ai, ≤i) all have lub, then so does (A, ≤). If (Ai, ≤i) all have glb, then so does (A, ≤).
