Proving LTL Properties of Bitvector Programs and Decompiled Binaries

Abstract

There is increasing interest in applying verification tools to programs that have bitvector operations. SMT solvers, which serve as a foundation for these tools, have thus increased support for bitvector reasoning through bit-blasting and linear arithmetic approximations.

In this paper we show that similar linear arithmetic approximation of bitvector operations can be done at the source level through transformations. Specifically, we introduce new paths that over-approximate bitvector operations with linear conditions/constraints, increasing branching but allowing us to better exploit the well-developed integer reasoning and interpolation of verification tools. We show that, for reachability of bitvector programs, increased branching incurs negligible overhead yet, when combined with integer interpolation optimizations, enables more programs to be verified. We further show this exploitation of integer interpolation in the common case also enables competitive termination verification of bitvector programs and leads to the first effective technique for linear temporal logic (LTL) verification of bitvector programs. Finally, we provide an in-depth case study of decompiled (“lifted”) binary programs, which emulate X86 execution through frequent use of bitvector operations. We present a new tool DarkSea, the first tool capable of verifying reachability, termination and LTL of lifted binaries.

A Proof of Theorem 1

Proof

Induction on traces, showing equality on expression translation $T_E$ via induction on expressions/statements and then inclusion on statement translations $T_S$. First show that $T_E$ preserves traces equivalence. Structural induction on e, with base cases being constants, variables, etc. In the inductive case, for a bitvector operation $e_1 \otimes e_2$, assume $e_1,e_2$ has been (potentially) transformed to $e_1',e_2'$ (resp.) and that Lemma 1 holds for each $i\in \{1,2\}$: $\forall \sigma . [\![ e_i ]\!]\sigma =[\![ e_i' ]\!]\sigma$. Since $\otimes$ is deterministic, $[\![ e_1'\otimes e_2' ]\!]\sigma = [\![ e_1\otimes e_2 ]\!]\sigma$. Finally, applying the transformation to $\otimes$, we show that $[\![ T_E\{e_1'\otimes e_2'\} ]\!] = [\![ e_1'\otimes e_2' ]\!]$ again by Lemma 1. Next, for each statement s or relational condition c step, we prove $T_S$ preserves trace inclusion: that $[\![ s ]\!] \subseteq [\![ T_S\{s\} ]\!]$ or that $[\![ c ]\!] \subseteq [\![ T_S\{c\} ]\!]$. We do not recursively weaken conditional boolean expressions, which would require alternating strengthening/weakening. Thus, inclusion holds directly from Lemma 1.