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Decidable Logics Combining Heap Structures and Data

This talk presents a new logic for reasoning with programs manipulating both heap structures and data, focusing on deductive verification and SMT solvers. It addresses challenges like dynamically allocated memory and maintaining data structure invariants that require reasoning with unbounded heaps and data. The goal is to propose a logic that is decidable, expresses useful properties, and is efficient for real-world programs. The discussion includes related works and the STRAND logic for recursive data structures. Examples of data structures expressible in this logic are nested lists, cyclic linked lists, and threaded trees. The syntax and semantics of STRAND logic, formulation of verification conditions, and deciding STRAND fragments are explored. The talk also delves into strategies for extending or generalizing logic techniques. The techniques focus on maintaining balance between expressiveness, decidability, and efficiency in practical scenarios.

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Decidable Logics Combining Heap Structures and Data

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  1. Gennaro Parlato (LIAFA, Paris, France) Joint work with P. Madhusudan XiaokangQie Universityof Illinois at Urbana-Champaign DecidableLogicsCombiningHeapStructures and Data

  2. Whatisthis talk about… A newlogictoreasonwithprogramsthatmanipulateheap + data using • deductiveverification • SMT solvers • Program • … • new(x); • … • x->next=y->next->next; • … • y=nil; • … • x->data = y->data +1; • … • free(x); • … 75 157 14 5 15 23 1000 7 1 90

  3. It’s a difficult problem! • Analysisofprogramsthat • Dynamicallyallocatedmemory • Mantaindata-structureinvariants (ex. BinarySearchTree) requirestoreasonwithunboundedheapswithunbounded data • Thisroles out classicalcombinationoftheories • likeNelson-Oppenscheme

  4. Whatwouldbedesirable ? A logicthatis: • Decidable • Expressiveenoughto state usefulproperties • Efficientin practice on real-worldprograms Wetrytogive a solutionthathas a goodbalanceoverall the goals!

  5. Reletedworks • HAVOC (Lahiri, Qadeer: POPL’08) • Decidable:YES • Expressive:NO (doubly-linkedlistscannotbeexpressed) • Efficient: YES on simpleprograms • CSL (Bouajjani,Dragoi, Enea, Sighireanu: CONCUR’09) • Decidable:YES • Expressive:NO • extends HAVAC tohandleconstraints on sizesofstructures • stillweak (propertiesofbinarytreesofunboundeddepthcannotbeexpressed) • Efficient: ?(no implementationisprovided) • The technique to decide the logics is encoded in the syntax • Hard to extend or generalize

  6. STRANDLogic

  7. Recursive data structures Defines a set ofgraphsthatwhich are interpretedover regular trees R = ( ψTr, validnode, {nodea}a∈ Lv, {edgeb}b∈Le ) • Family ofbinary • predicates • Leis a finite set of • edgelabels • Family ofunary • predicates. • Lvis a finite set of • nodelabels labedgeb(u,v) =true <=> u,v∈ N ∧ edgeb(u,v)=true labnodea(v) =true <=> v∈ N ∧ nodea(v)=true { v ∈ V | validnode(v) } MSO formula on k-ary Σ-lab trees {(u,v) | b∈Le & edgeb(u,v) } Unary predicate Rdefines a classofgraphs For every tree T=(V,->) accepted by ψTr, graph(T) = ( N, E, labnodea, labedgeb)

  8. Examplesof data stuctures En (s,t)= leaf(s) ∧ leaf(t) ∧∃ z1, z2, z3. ( El(z3,z1) ∧ RightMostPath( z1, s) ∧ Er(z3,z2) ∧ LeftMostPath( z2, t) ) z3 z1 z2 t s • Some data structuresthat can beexpressed: • Nestedlists • Cyclic and doubly-linkedlists • Threadedtrees

  9. STRANDlogic (STRuctureANd Data) ∃xn. ∀ym. φ(xn,ym) φ is an Monadic Second Order (MSO) formula that combines • heapstructuresand • data, where the data-constraints are only allowed to refer to xnand ym Syntaxof STRAND • DATAExpression e::= data(x) | data(y) | c | g(e1,...,en) • φFormula φ::= γ(e1,...,en) | α(v1,...,vn) • | ¬φ | φ1 ∧φ2 | φ1 ∨φ2 | z ∈ S | ∃z. φ | ∀z. φ | ∃S. φ | ∀S. φ • ∀Formula ω::= φ | ∀y.ω • STRAND ψ::= ω|∃x.ψ MSO

  10. ExpressingBinarySearchTree (BST) left right right left right left right left right right right left left leftbranch(y1, y2) ≡ ∃z. (left(y1, z) ∧ z →∗ y2) rightbranch(y1, y2) ≡ ∃z. (right(y1, z) ∧ z →∗ y2) ψbst ≡ ∀y1∀y2. ( leftbranch(y1,y2) ⇒ data(y2) < data(y1) ) ∧ ( rightbranch(y1,y2) ⇒ data(y1) ≤ data(y2) )

  11. VerificationConditions in STRAND(bst-search procedure) bstSearch PRE: ψbst ∧ ∃x.( key(root)=k ) Node curr=root; LOOP-INV: ψbst ∧ ∃x.(reach(curr,x)∧key(curr)=k) while ( curr.key!=k && curr!=null ) { if (curr.key>k) curr=curr.left; else curr=curr.right; } POST: ψbst ∧ key(curr)=k

  12. Deciding STRANDfragments

  13. Overview Given • a linear block ofstatementss • a STRANDpre-conditionPand a post-conditionQexpressedas a Booleancombinationof STRANDformulasof the form • ∃xn.φ(xn), ∀ym. φ( ym) checking the validityof the Hoare triple {P} s {Q} reducesto thesatisfiabilityof STRAND • Satisfiabilityof STRANDisundecidable • Weidentify a decidablefragmentof STRANDsemanticallydefinedbyusing the notionofsatisfiability-presevingembeddings

  14. Romovingexistentialquantifications Let ψ=∃xn.∀ym. φ(xn, ym) be a STRAND formula over a recursively data-structure R • define a newrec. data-structureR’ thatencoperates the existentallyquantifiedvariablesofxnaspointers in R’ (unary predicate Valifor the varxi ) • define ψ’= ∀xn∀ym. ((∧i=1,…,n Vali(xi)) => φ(xn, ym)) Claim: ψ’ is satisfiable on R’ iffψ is satisfiable on R

  15. Decidability: finite modelproperty Wedefine a partialorderoverheapsthatcaptures the followingintuition S< T(Ssatisfiability-preservinglyembedsintoT) if - SissmallerthanT (fewernodes) - EachnodeofSismatchedwith a distinctnodeofT - no mutterhow data are associatedto the nodesofT if Tsatisfies ψthen also S satisfies ψby inhering the data-values T S

  16. STRANDdec • Checkingthesatisfiability • weignoreT and • checkonlyS • Satisfiabilityisdoneonly on the minimalmodels • STRANDdecis the classof the formulasthathas a finite set of minimal models • Satisfiabilityreducesto the satisfiabilityof the quantifier-freetheoryof the data-logic

  17. Definingsatisfiability-preservinglyembeddings: sub-models LetTbe a treedefining a graphof the givenRec. Data-structure A subset SofnodesofTisvalid: • Not-empty & least-ancestorclosed • The subtreedefinedbySalsodefinesa data-structure Submodels can bedefined in MSO

  18. Definingsatisfiability-preservinglyembeddings: abstractingdata Let ψ=∀yn. φ(yn) be a STRAND formula, and γ1, …,γrbe the atomic relational formulas of the data-logic. • Example: ψsorted : ∀y1,y2. ( (y1->*y2 ) => (data(y1) ≤ data(y2)) ) Define the pure MSO formula ψ’=∀yn. ∃b1, …, br.φ’ (yn,b1, …, br) where φ’ is obtainedfromφ by replacingγiwith the predicate bi • Example: ψsorted : ∀y1,y2. ( (y1->*y2 ) => data(y1) ≤ data(y2) ) ψ’sorted : ∀y1,y2.∃b. ( (y1->*y2 ) => b) • Claim: ifψ is satisfiable on a recursive data-structure R thenψ’ is also satisfiable on R. • The other direction doesnothold OVER-APPROXIMATION

  19. MinModels &Satisfiability-PreservingEmbeddings T • ψ=∀yn. φ(yn) • ψ’=∀yn. ∃b1, …, br.φ’ (yn,b1, …, br) • MinModelψ= \* pure MSO formula *\ • ψTr// the tree defines a valide heap T • ∧interpret(ψ’) // ψ’ holds true on T • ∧ // NO satisfiability-preserving embeddings in T • ¬∃S. ( NonEmpty(S) ∧ ValidSubModel(S) • ∧ (∀ym. ∀b1, …, br. • (∧i∈ [m] (yi∈ S) ∧ interpret(φ’ (yn,b1, …, br)) • => interpret( S, φ’ (yn,b1, …, br) ) • ) ) S

  20. Decision procedure • TransformMinModelstotree-automatonTA MinModels -> TA • Checkfinitenessfor TA • Extract all trees accepted by TA: TA -> T1, …, Tt • For eachTi build the finite graph Gi Ti -> Gi • Create a quantifier-free formula λi for GiGi -> λi • a data-variableforeachnode • ∀’s are “expanded” in conjunctions • ∃’s are “expanded” in disjunctions • data-constraints in STRAND are directlywritten in the data-logic • The quantifier-free formula∨i=1,…,tλiis a pure data-logic formula that is satisfiable iff the original formula ψ is satisfiable on R

  21. ProgramVerificationwith STRAND

  22. idea Hoare-triples: ( (R, Pre), P, Post ) • R • Pre ∈ STRAND∃,∀ • P: • Node t = newhead; • newhead = head; • head = head.next; • newhead.next = t; • Post ∈STRAND∃,∀ ⇒ Is ψ ∈ Strand Satisfiable over RP ? • Idea: capture the entirecomputationofPstartingfrom a particularrecursivedata-structureRusing a single data-structureRP

  23. The reductionto the satisfiability Error = ∨i ∈[m](PreRp ∧∧j∈[i −1] πj ∧ errori ) ViolatePost = PreRp∧ ∧j∈[m] πj∧¬PostRp Theorem The Hoare-triple (R, Pre, P, Post) does not hold iff the STRAND formula Error ∨ ViolatePost is satisfiable on the trail RP

  24. Evaluation

  25. Experiments

  26. bst-search procedure bstSearch (pre: ψbst ∧ ∃x.(key(root) = k)) Node curr=root; (loop-inv: ψbst ∧∃x.(reach(curr,x)∧key(curr)=k)) while ( curr.key!=k && curr!=null ){ if (curr.key>k) curr=curr.left; else curr=curr.right; } (post: ψbst ∧ key(curr) = k)

  27. MONA formula forbst-search

  28. Webpageof STRAND http://cs.uiuc.edu/~qiu2/strand/ • Code • Experiments • Tableof the experiments

  29. Conclusions & Future Work

  30. Conclusions WehavepresentedSTRAND a logicforreasoningaboutstructures and data whichhasgoodbalanceamong • Decidability • Expressiveness • Efficiency • Webelievethis work • breaksnewground in combiningstructures and data • maypave the way fordefiningdecidablefragmentsofotherlogics

  31. Future Work • Alternative to MONA? • Syntacticdecidablefragments • Back-and-forth connection between the structural part and the data part

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