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STP: A Decision Procedure for Bit-vectors and Arrays

David L. Dill Stanford University. STP: A Decision Procedure for Bit-vectors and Arrays. Software analysis tools present unique challenges for decision procedures. Theories must match programming language semantics Operations are on bit-vectors, not integers

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STP: A Decision Procedure for Bit-vectors and Arrays

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  1. David L. Dill Stanford University STP: A Decision Procedure for Bit-vectors and Arrays

  2. Software analysis tools present unique challenges for decision procedures • Theories must match programming language semantics • Operations are on bit-vectors, not integers • Arrays (for modelling memories) • Must handle very large inputs with • Many array reads • Deeply nested array writes • Many linear equations • Many variables • Decision procedure is called many times.

  3. What went before • Series of decision procedures: SVC, CVC, CVCL • All of these had combinations of first-order theories • Equality • Uninterpreted functions and predicates • Boolean connectives • Linear arithmetic over real numbers (and integers, in the case of CVCL) • But not quantifiers. • CVCL was in use in EXE (or it’s predecessor – Dawson Engler research group)

  4. Combination of theories • The core strategy of SVC, CVC, CVCL was based on dynamically breaking down formulas into conjunctions of “atomic formulas” • Atomic formulas have no Boolean connectives (correspond to propositional variables). • Recursively assert/deny alpha (deny = assert negation of) • Simplify after assertion • When simplified formula is conjunction of literals, use special decision procedures.

  5. SAT vs. CVCL • CVC/CVCL used CHAFF-like SAT solver to choose splitting variables • … but puts lots of slow stuff in the inner loop!

  6. STP • CVCL was already used by Engler’s group, but was unfixably slow. • There existed many examples generated by Engler • Made correctness/performance testing easy. • Vijay Ganesh and I decide to try a different approach, inspired by UCLID (Seshia & Bryant). • Put SAT at the bottom, with unmodified inner loop. • Preprocess formula for higher-level reasoning (bit-vectors and arrays).

  7. STP • Decides satisfiability of formulas over • Bit-vector Terms • Constants • +, -, *, (signed) div, (signed) mod • Concatenation, Extraction • Left/Right Shift, Sign-extend, bitwise-Booleans • Array Terms • Read(Array, index) • Write(Array, index, val) • But no array equality • Predicates: =, signed & unsigned comparisons • If satisfiable, produces a model.

  8. Comparison with Saturn • STP is a separate “component” (can be stand-alone, or used through an API). • Programming language or other tool is separate • General input language (can define 23-bit bit-vector types if you want). • Signed/unsigned encoded in operators, not in data types. • No “points to”, heap, etc. • Implements signed/unsigned multiply, divide, remainder (but no floating point).

  9. Some projects using STP • STP has evolved (maintained by someone I don’t know in Australia) • Several projects have used it. • EXE : Bug Finder by Dawson Engler, Cristian Cadar and others at Stanford • Klee : Cadar, Dunbar, Engler • MINESWEEPER: Bug Finder by Dawn Song and her group at CMU • …

  10. Main Ideas of STP • Eager translation to CNF with word-level pre-processing • Theories not in “inner loop” of SAT solver, unlike Nelson-Oppen approaches (e.g. CVCL). • Abstraction-Refinement for arrays. • Laziness to counterbalance eagerness • Solve linear formulas mod 2n in P-time

  11. Bitvector theory • Data type: BV(n), where n is constant. • Almost all machine bitvector operations • Change length (sign extended and not). • Concatenate bitvectors, extract bits from BVs. • Signed and unsigned arithmetic: +, -, *, /, %, <,>, etc. • AND, OR, NOT, XOR, etc.

  12. Array theory • Array type: BV(n) -> B(m) • read(A, i) – value of A[i] • write(A, i, v) – copy of A updated at index i with value v. • No destructive modification – write returns a new array, which is updated old array. • Sometimes used to represent heap storage.

  13. Array theory Identities: read(write(A, i, v), j) = ite(i = j, v, read(A, j)) STP has a limited theory No comparison of whole arrays, e.g. write(A, i, v) = write(A, j, w) This makes things easier (see http://sprout.stanford.edu/PAPERS/LICS-SBDL-2001.pdf if you don’t like “easier”).

  14. Implementation • DAG representation of expressions • Same subexpression structure = same pointer. • Maintained by hashing. • No destructive operations on DAGs (modification requires new nodes). • Makes substitution, equality check very efficient. • Often log size of tree expression representation.

  15. DAGs • All recursive traversals must be “memoized” • Want traversal to be linear in size of DAG, not tree. • First thing to think about when functions don’t finish: “Maybe I messed up memoization.” • Updating nodes near the root less expensive than updating nodes near leaves.

  16. STP Architecture Input Formula Substitutions Simplifications Linear Solving Array Abstraction BitBlast CNF Coversion Refinement Loop SAT Solver Result

  17. Substitution is Important • Inputs often have many simple equations (in EXE, this is how constant arrays are defined): • x = 4 • A[4] = e • Early pass to substitute these • Allows constant evaluation • Enables other optimizations • Reduces non-constant indices in array reads.

  18. Word-level simplifications • Many simple local rewrites: • Bitwise Boolean identities (e.g. a AND !a = 00000, a XOR a = 11111, (a + b)[0:3] = a[0:3]+b[0:3], etc. • Generally, avoid distributive laws because they cause blow-up. • Be careful about “destroying sharing” in DAG. • Flatten trees of associative operators • Sort operands of commutative operation • Tweak ordering for easy simplification • Expressions numbered in order of creation (children < parents). Use this order to sort, but: • Put constants first (F AND a AND …), (1 + 3 + x +) • Arrange for x, !x to be adjacent (also x, -x)

  19. Array Reads v0 = t0 v1 = t1 . . . vn = tn (i1=i0) => v1=v0 (i2=i0) => v2=v0 (i2=i1) => v2=v1 … Replace array reads with fresh variables and axioms Read(A,i0) = t0 Read(A,i1) = t1 . . . Read(A,in) = tn • Problem : O(n2) axioms added, n is number of read indices • Lethal, if n is large: n = 10000, # of axioms: ~ 100 million • Blowup seems hard to avoid (e.g. UCLID). • This is “aliasing” from another perspective.

  20. Abstraction/Refinement in STP • Pose problem as a conjunction of formulas • E.g., instantiated array read axioms • Abstraction: solve for a proper subset of the formulas • E.g., omit array read axioms • Early exit if: • Unsatisfiable • Satisfiable and model actually satisfies unabstracted formula. • Otherwise, add some omitted formulas to the abstracted formula and solve again. • If at least one of these formulas is false in the model, that model will not be regenerated.

  21. Abstraction-Refinement Algorithm for Array Reads Input After Abstraction Read(A,0)=0 Read(A,i)=1 v0 = 0 vi = 1 SATSolver Counterexample i = 0 v0 = 0 vi = 1 Refinement Step: Add Axiom (i=0) => vi = 0 Rerun SATSolver i=1 vi=1 Check Input on Assignment False: Read(A,0)=0 Read(A,0)=1

  22. Experience with Read Abstraction-Refinement Works well, even in satisfiable cases • Satisfier often finds a model that minimizes aliasing • Few axioms need to be added during refinement • Typical number of refinement loops : < 3

  23. The Problemwith Array Writes • Standard transformation read(write(A, i, v), j) = ite(i=j, v, read(A, i)) <- “if-then-else” causes term blow-up • Many different read expressions share write sub-terms. • O(n*m) blow-up in expr DAG • n is write term nesting levels • m is number of read indices

  24. = R R j k W W i1 v1 A i0 The Problem with Array Writes If (i1=j) v1 elsif (i0=j) v0 else R(A,j) If (i1=k) v1 elsif(i0=k) v0 else R(A,j) R(W(W(A,i0,v0),i1,v1),j) = R(W(W(A,i0,v0),i1,v1),k) = = ite ite = = v1 v1 i1 i1 j k ite ite R = = v0 v0 A i0 i0 j k v0 j

  25. Write transformation Replace read(write(A, i, v), j) with a fresh variable (e.g., v0) and “axiom” v0 = ite(i=j, v, read(A, j)) Abstraction omits axiom.

  26. Abstraction-Refinement Algorithmfor Array Writes R(W(A,i,v),j)= 0 R(W(A,i,v),k)=1 i = j /= k v /= 0 After Abstraction v1=0 v2=1 i = j /=k v/=0 SATSolver v1=0, v2=1 i = j =0, k=1, v = 1 False: R(W(A,0,1),0)=0 R(W(A,0,1),1)=1 0 = 0 /= 1 1 /= 0 Refinement Step Add Axiom to SAT v1=ite(i=j,v,R(A,j)) UNSAT Check model on original formula

  27. Experimental Results:Array Writes Examples courtesy Dawn Song (CMU) and David Molnar (Berkeley)

  28. Algorithm for SolvingLinear Bit-vector Equations • Inspired by Barrett et al., DAC 1998 • Basic Idea in STP • Solve for a variable and substitute it away • If cannot eliminate a whole variable, eliminate as many bits as possible. • Previous Work • Mostly variants of Gaussian Elimination • Solve-and-substitute is more convenient in a general decision procedure.

  29. Algorithm for Solving Linear Bit-vector Equations (3 bits) 3x + 4y + 2z = 0 2x + 2y + 2 = 0 4y + 2x + 2z = 0 Solve for x in first eqn: 3-1 mod 8 = 3, (3 bits) 2y + 4z + 2 = 0 4y + 6z = 0 Substitute x x = 4y + 2z

  30. Algorithm for SolvingLinear Bit-vector Equations All Coeffs Even No Inverse (3 bits) 2y + 4z + 2 = 0 4y + 6z = 0 (2 bits) y[1:0] + 2z[1:0] + 1 = 0 2y[1:0] + 3z[1:0] = 0 Divide by 2 Ignore high-order bits

  31. Algorithm for SolvingLinear Bit-vector Equations (2 bits) y[1:0] + 2z[1:0] + 1 = 0 2y[1:0] + 3z[1:0] = 0 Solve for y[1:0] (2 bits) y[1:0]=2z + 3 (2 bits) 3z[1:0] + 2 = 0 Substitute y[1:0]

  32. Algorithm for SolvingLinear Bit-vector Equations (2 bits) 3z[1:0] + 2 = 0 Solve for z[1:0] Solution (3 bits): z[1:0] = 2 y[1:0] = 2z[1:0] + 3 = 3 y = y’ @ 2 z = z’ @ 3 x = 4y + 2z (2 bits) z[1:0]=2

  33. Experimental Results: Solver for Linear Equations

  34. Equivalence checking of block cipher implementations • Problem: Prove correctness of block ciphers (e.g., AES). • Constant number of loop iterations • No interesting heap usage • Approach: • Given two implementations, AES1 and AES2 • Turn them into big expressions by unrolling loops, etc. • Prove that AES1(x) ≠ AES2(x) is unsatisfiable.

  35. How can this possibly work? • Round 1 • Round 1 Many block ciphers consist of a fixed sequence of “rounds”. Implementations of rounds in two algorithms may vary, but bits “between” rounds are equal. So, we only have to prove individual rounds are equivalent. • Round 2 • Round 2 • Round 3 • Round 3 • Round 4 • Round 4

  36. Equivalence checking Equal for many test inputs. Only try to prove Equivalence when nodes pass this test. •  •  inputs

  37. Equivalence checking Prove equal using STP •  • 

  38. Equivalence checking Replace b by a everywhere in DAG. This makes higher-level expressions more similar. •  •  inputs

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