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Well-Founded Linked Lists

Shuvendu K. Lahiri

Shaz Qadeer

Software Reliability Research

Microsoft Research

Motivation for analyzing linked lists

- Verify control, memory, and API safety of low-level systems code
- Integers
- Arrays
- Singly linked lists
- Doubly linked lists (acyclic and cyclic)

Motivation for analyzing linked lists

- Verify control, memory, and API safety of low-level systems code
- Integers
- Arrays
- Singly linked lists
- Doubly linked lists (acyclic and cyclic)
- Establish properties about linking structure and content
- Absence of null dereference, memory leaks
- All elements of a list have data value 0
- List1 and List2 are disjoint

//@ requires hd != null

//@ ensures v R(hd): v.data = 0

void acyclic_simple(Cell hd) {

Cell iter = hd;

while (iter != null) {

iter.data = 0;

iter = iter.next;

}

}

Example: Acyclic linked list iterationProblem

- Existing program analyses either lack scalability or precision for such programs/properties

Reasoning in first-order logic

- Can support many theories important for program verification
- Uninterpreted functions, linear arithmetic, arrays, quantifiers
- Reason about programs with a mix of scalar variables, arithmetic, arrays
- Powerful analysis engines
- Pioneering work by Nelson-Oppen[’79]
- Recent advances in SAT-based theorem provers

Program verification and first-order logic

- Automated software verification tools
- SLAM, BLAST, MAGIC,…
- ESC/JAVA, Boogie,..
- Perform symbolic reasoning for first-order logic
- Theorem provers to discharge verification conditions
- Operations for abstract interpretation (predicate abstraction, join, ..)
- Automatic abstraction-refinement

Linked lists and reach

x

- Class Cell {
- int data;
- Cell next;
- };

R(x)

R(u) = Set of cells reachable from u using next field

= {u, u.next, u.next.next,…}

Acyclic linked list iteration

//@ requires hd != null

//@ ensures v R(hd): v.data = 0

void acyclic_simple(Cell hd) {

Cell iter = hd;

while (iter != null) {

iter.data = 0;

iter = iter.next;

}

}

hd

iter

Visited = R(hd)\ R(iter)

ExampleLoop invariant

u Visited: u.data = 0

Reachability predicate

- Need to reason about reachability predicate
- e.g. u R(x): u.data = 0
- Need axioms to relate the field next and R
- However, reachability can’t be modeled in first-order logic
- Finite first-order axiomatization of reachability impossible

Motivation for this work

- Simple axioms may suffice for many examples
- Provide a first-order axiomatization of Reach
- Necessarily incomplete
- First investigated by Nelson [POPL’83]
- Enable list manipulating programs (also containing integers, arrays etc.) to be analyzed uniformly
- Can leverage first-order reasoning
- Predicate abstraction,…
- Abstraction refinement

Acyclic linked list iteration

//@ requires hd != null

//@ ensures v R(hd): v.data = 0

void acyclic_simple(Cell hd) {

Cell iter = hd;

while (iter != null) {

iter.data = 0;

iter = iter.next;

}

}

hd

iter

Visited = R(hd)\ R(iter)

ExampleLoop invariant

u Visited: u.data = 0

Axiom for reach:

u, v : v R(u)

(v = u (u.next null v R(u.next)))

Acyclic linked list iteration

//@ requires hd != null

//@ ensures v R(hd): v.data = 0

void acyclic_simple(Cell hd) {

Cell iter = hd;

while (iter != null) {

iter.data = 0;

iter = iter.next;

}

}

hd

iter

Visited = R(hd)\ R(iter)

ExampleLoop invariant

u Visited: u.data = 0

Axiom sufficient to prove the example

Axiom for reach:

u, v : v R(u)

(v = u (u.next null v R(u.next)))

Rest of the talk

- How to
- Handle cyclic lists
- Handle destructive updates
- Generate first-order axioms for Reach
- Well-founded linked lists
- How it makes the above tasks amenable
- Results
- Deciding ground fragment with Reach predicate

Cyclic linked list iteration

//@ requires hd points to a cyclic list

//@ ensures v R(hd): v.data = 0

void cyclic_simple(Cell hd) {

hd.data = 0;

iter = hd.next;

while (iter != hd) {

iter.data = 0;

iter = iter.next;

}

}

Part1: Cyclic List Traversalhd

iter

Visited = ?

No way to express Visited using R alone

- R for every cell in the cycle contains all the cells in the cycle

Cyclic linked list iteration

//@ requires hd points to a cyclic list

//@ ensures v R(hd): v.data = 0

void cyclic_simple(Cell hd) {

hd.data = 0;

iter = hd.next;

while (iter != hd) {

iter.data = 0;

iter = iter.next;

}

}

Cyclic List Traversalhd

iter

Visited = ?

Proving even null-dereference is non-trivial

Observation

- Usually, every cycle of “next” has at least one distinguished cell
- Usually, the “head” of the list
- This cell breaks the symmetry in the list
- For each linking field “f”, a subset of fields in the heap are heads
- Denoted by Hf
- Cells denoted by
- Always includes null

y

Rf(x)

x

Rf(z)

z

y

New Predicates Rf and Bf- Hf = Set of head cells for field f
- Rf(u)
- Set of cells u, u.f, u.f.f,…, until the first cell in H
- Bf(u)
- The first cell from the sequence u.f, u.f.f, …, that belongs to H
- The “block” for u

Bf(x) = null

Bf(x) = y

Bf(y) = x

Bf(z) = x

Well-founded heap

- Given Hf, a set of “head” cells for a field f
- Well-founded field f
- For any cell u, the sequence u.f, u.f.f, …, intersects with a cell in Hf
- Well-founded heap
- Every linking field f is well-founded wrt to Hf
- i.e., every f cycle has at least one Hf cell

Programming methodology

- Programmer must supply Hf
- Every mutation of the linking field f is required to preserve well-foundedness
- Restricted to programs maninpulating well founded heaps only
- Almost all list programs obey this restriction

Cyclic linked list iteration

//@ requires hd points to a cyclic list

//@ ensures v R(hd): v.data = 0

void cyclic_simple(Cell hd) {

hd.data = 0;

iter = hd.next;

while (iter != hd) {

iter.data = 0;

iter = iter.next;

}

}

Cyclic List Traversalhd

iter

Visited = ?

Cyclic linked list iteration

//@ requires hd H B(hd) = hd

//@ ensures v R(hd): v.data = 0

void cyclic_simple(Cell hd) {

hd.data = 0;

iter = hd.next;

while (iter != hd) {

iter.data = 0;

iter = iter.next;

}

}

Cyclic List Traversalhd

R(iter)

iter

Visited = (iter = hd)

? R(iter)

: R(hd) \ R(iter)

Loop invariant:

u Visited: u.data = 0

B(iter) = hd

Axioms Required

- Axiom for R
- v R (u) (v = u (u.next H v R(u.next))
- v R(u) (v = u (u.next null v R(u.next))
- Axiom for B
- B(u) = u.next H ? u.next :B(u.next)
- Able to prove the example (similar to acyclic case) with these axioms

Part 2: Destructive updates

- x.f := y
- Issues
- R, B need to be updated
- Since f is updated
- Destructive updates may make the heap ill-founded
- Flag such programs as bad

Updates to R, B (some cases)

- x.f := y

u

u

x

x

y

y

R(u) = R(u) \ R(x) {x} R(y)

R(u) = R(u) \ R(x) {x}

B(u) = y

B(u) = B(y)

Ensuring well-foundedness

- x.f := y

Orphan cycle: Cycle with no H cells

x

y

Add assert ( x R(y) y H ) before each x.f := y

Updating cells in Hf

- Hfis a program variable now
- Hf.Add(x)
- Adds the cell pointed to by x to Hf
- Useful when creating a cycle for the first time
- Hf.Remove(x)
- Removes the cell pointed to by x to Hf
- Remove one head when two cycles with a head each are fused
- Updates to Rf, Bf still remain quantifier-free

Summary: Instrumentation

- Quantifier-free updates to auxiliary variables R, B
- Similar to acyclic case [Dong & Su ‘95]
- Very difficult to update R for cyclic lists in general
- Instrumentation captures “well-foundedness” precisely
- The instrumented program goes wrong (violates an assertion)iff the source program
- goes wrong (violates some assertions), or
- heap of the source program becomes not well-founded

Part 3: Axioms Required

- Base axiom for R
- v R(u) (v = u (u.next H v R(u.next))
- Base axiom for B

B(u) = u.next H ? u.next :B(u.next)

- Fundamental axioms
- The axioms capture the intended meaning of R and Bin any finite and well-founded state of the program

Generating new axioms

- Not possible to express finiteness and well-foundedness in first-order logic
- Derive new axioms from the base axioms
- Using induction
- For well-founded heaps
- We provide an induction principle to generate derived axioms from base axioms

Induction principle

- Proposed axiom: u. P(u)
- To prove P(u) for any cell u in a finite well-founded heap
- Base Case
- u.f H P(u)
- Establish for all u at a distance 1 from H cells
- Induction Step
- u.f H (P(u.f) P(u))
- u.f has a shorter distance to H than u (well-founded induction)

Some derived axioms

- Transitivity
- R(u,v) R(v,w) R(u,w)
- Antisymmetry
- R(u,v) R(v,u) u = v
- Block
- R(u,v) v H u = v
- Bounded distinctness
- All cells in the set {u, u.f,…,} until the first H cell are distinct from each other
- Instantiate this for bounded sizes (e.g. 1)
- u.f H u u.f

Derived Axioms

- Set of axioms are fairly fundamental properties and fairly intuitive
- Can be easily proved from the base axioms using the induction principle
- Suffice for a large set of examples
- Otherwise derive new axioms using the base axioms and induction

Benefits of well-founded lists

- Set of required axioms almost similar to acyclic case
- Allows us to update Rf, Bf relations using simple quantifier-free formulas
- Provides an induction principle to establish derived axioms easily

Experimental setup

Instrumentation

Add R, B

+ Updates

+ Assertions

VC Generator

(UCLID)

Annotated

Source

Program

Theorem Prover

(UCLID)

Proved/Failure

Axioms for R, B

UCLID

- Verification system for systems modeled in first-order logic
- Bryant, Lahiri, Seshia, CAV’02
- Checking verification conditions
- Uses quantifier instantiation
- Uses various decision procedures for uninterpreted functions, arrays, arithmetic
- Inferring loop invariants with indexed predicate abstraction

Examples

- Simple_cyclic
- List traversal
- Reverse
- In place reversal of an acyclic list
- Sorted_insert
- Inserts a cell in a sorted list
- Requires arithmetic reasoning
- Set_union
- Merges two equivalence classes implemented as cyclic lists
- Dlist_remove
- Removes a cell from a cyclic doubly linked list

Experiments

- Proving Verification Conditions (VCs)
- Most examples take < 1 s
- Loop Invariant synthesis using indexed predicate abstraction

Synthesizing invariants by indexed predicate abstraction

- Flanagan & Qadeer POPL’02
- Lahiri & Bryant VMCAI ‘04
- Principle
- Provide a set of predicates P over state variables + “index variables” X
- Intuitively X contains heap cells, or indices into arrays
- e.g. P = {Rnext(u,v), Bnext (u) = v, a[i] < a[j] + 1, …}

X = {u,v,i,j,…}

- Theorem
- Indexed predicate abstraction constructs the strongest loop invariant of the form X: (P)
- is a Boolean combination of predicates in P

//@requires null Hnext

//@requires Bnext (l) = null

//@ensures Bnext (res) = null

//@ensures Rnext(res) = R0next (l)

Cell reverse (Cell l){

Cell curr = l;

Cell res = null;

while (curr != null){

Cell tmp = curr.next;

curr.next = res;

res = curr;

curr = tmp;

}

return res;

}

Predicates

X = {u}

P = {

u = null, u = curr, u = res, u = l0,

curr = null, l = l0,

Rnext(curr,u),

Rnext(res,u),

Rnext(l,u),

Hnext(u),

R0next(l0,u),

Bnext(u) = null

}

Tool constructs loop invariant in 0.57 sec

Synthesizing invariants in UCLIDResults with Predicate Abstraction

- Predicates provided manually

- Used Barcelogic tool for theorem proving
- Note: Results significantly improved from paper

Decision procedure for ground fragment

- Deciding ground formulas over
- Rf(u,v), ~Rf(u,v), u = f(v), u ≠ v, u Hf, u Hf,u = Bf (v)
- Reduce dependency on derived axioms
- A complete framework when VCs are quantifier-free
- Solving quantifier-free queries after instantiating quantifiers
- Result
- Checking satisfiability of a conjunction NP-complete

Related work

- First-order axiomatization of reachability
- Nelson ’83, Lev-Ami et al. ’05
- First-order reasoning without reachability
- Necula & McPeak ’05
- Shape analysis with 3-valued logic
- Sagiv et al. ’99, …
- TVLA
- Predicate abstraction for lists
- Dams et al. ’03, Balaban et al. ’05, Manevich et al. ’05, Bingham ’06
- Separation logic
- O’Hearn et al. ’01, Reynolds ’02,

Conclusions

- Two new predicates R, B for well-founded heaps
- Instrumentation of source program with auxiliary variables for the predicates
- First-order axiomatization
- New induction principle
- Simpler derived axioms
- Implementation
- Leverage first-order theorem provers
- Indexed predicate abstraction provides a uniform scheme for synthesizing invariants

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