Interprocedural Abstract Interpretation of Block Structured Languages with Nested Procedures, Aliasing and Recursivity

We are now going to abstract the Call and Return primitives. Let (g, r) be the current abstract stack. We use the following notations: g = ~ P , c , h , ~ , ~ / r = {(gi, ui)}iEl , , = r(g) E ~ ~ [ . ] The abstract store v = v[p] thus corresponds to the current generalized control point g, and p is the set of locations accessible to procedure P. In order to deal with the locations created during procedure calls, we will use the function A +t,_,u + : SVal [p-] --+ S--V-~[# +] which takes and abstract store defined o v e r / t and inserts the new locations in #+ / ~ with undefined values. The function At,_~-#,+ does the same but assigns the value of the location ~ to the new locations. On the opposite, the function A~+,u_ : SVal [it+] ~ SVal[p-] forgets every information about the locations in #+ # . A formal definition is given below. Def in i t i on 16 For any sets # C_ it +, let T(I~ +) be an upper approzimation of the lattice "P(Iz + --* SVal) , a ± and 7 ± being the abstraction and meaning functions. The functions A + A"+,u_, and A ~ " u + are defined by: ~x+ (P ) = ,~+({, , e (it+ ~ SVal) : % e : ( e ) } ) / t d.t+ A~+,,,_(p) = a-({%,:a E ,,/+(e)}) zX~-#,+ (P) = a+({oE (~+ --, SVal): %,e "t-(P) ^V2~+ E (it+ / t ) : a(,~+) = a(.~-)}) 7 . 1 P r o c e d u r e c a l l s Let us call {g', r ~} the abstract stack after the procedure call. The generalized control point g~ = (P ~, c~, h ~, g~, ~ ) is easily determined using the formal parameter binding function II, and the environment ~. We call # ' the set of locations accessible to P~. We have # ' = #sUpvUPAUPL,~ ~ ~ ' where P3 = #$ contains the locations shared between procedure P and procedure P~, namely the locations accessible to plo (for pl° < p by construction), and t i #v , PA and p~ are the locations that are local to Pt and respectively belong to Value, Alias and Local. The second element r ~ is defined by: r' = p,,(r[/g, ~/ / /g," / ] U {/g', v'/}) The first abstract store ~ is what remains of the abstract store v after the call to procedure P~. Remember that all the shared locations in #$ must be "erased" to be consistent with our abstraction ~. This is achieved through the use of A ~ , ~ , where PH = P -PS is the set of "hidden" locations of procedure P.