Automating relatively complete verification of higher-order functional programs

Abstract

We present an automated approach to relatively completely verifying safety (i.e., reachability) property of higher-order functional programs. Our contribution is twofold. First, we extend the refinement type system framework employed in the recent work on (incomplete) automated higher-order verification by drawing on the classical work on relatively complete "Hoare logic like" program logic for higher-order procedural languages. Then, by adopting the recently proposed techniques for solving constraints over quantified first-order logic formulas, we develop an automated type inference method for the type system, thereby realizing an automated relatively complete verification of higher-order programs.

Automating relatively complete verification of higher-order functional programs

“…Section 6 discusses related work, and Section 7 concludes. The extended report [28] contains the omitted materials.…”

“…Therefore, a program either runs forever safely (due to CPS, a program cannot return), stops safely in the state safe by reaching a false assume, or aborts with an assertion failure. The typing rules for the simple type system are standard and are deferred to the extended report [28]. τ, σ ::= ⋆ | {u | θ} | x:σ → τ Figure 6.…”

“…(1) ∆ is a well-formed top-level type environment. 6 See the extended report [28] for the definition of a precise constant type. Also, non-precise but sound constant types, that over-approximate the actual semantics, are sufficient for soundness.…”

“…We defer the proof to the extended report [28]. The proof adopts the ideas from the work on relatively complete program logics for higher-order procedural languages [6][7][8] that instantiate the extra parameters by the base-type encoding of the function closures.…”

“…This generates constraints over the type templates, which, in turn, reduce to Horn-clause-like constraints over the predicate variables. (See the extended report [28] for the formal definition of the constraint generation rules.) Because SHP is recursion-free and linear, the generated Horn clauses are non-recursive, and we can obtain an equivalent FOL formula.…”