A Framework for the Verification of Certifying Computations

Abstract

Formal verification of complex algorithms is challenging. Verifying their implementations goes beyond the state of the art of current automatic verification tools and usually involves intricate mathematical theorems. Certifying algorithms compute in addition to each output a witness certifying that the output is correct.A checker for such a witness is usually much simpler than the original algorithmyet it is all the user has to trust. The verification of checkers is feasible with current tools and leads to computations that can be completely trusted. We describe a framework to seamlessly verify certifying computations. We use the automatic verifier VCC for establishing the correctness of the checker and the interactive theorem prover Isabelle/HOL for high-level mathematical properties of algorithms. We demonstrate the effectiveness of our approach by presenting the verification of typical examples of the industrial-level and widespread algorithmic library LEDA.

Efficient Certified Resolution Proof Checking

“…programs that check traces of unsatisfiability proofs and that have themselves been formally proven correct. However, all these approaches are limited in their scalability, essentially for one of two reasons: (1) information about deletion of learned clauses is not available nor used [40,14,15,2]; and (2) the formats used to provide proof traces by SAT solvers still require the checker to perform complex checking steps [20,41,21,43], which are very difficult to optimize.…”

“…def parse(line): ints = [int(s) for s in line.split() ] i0 = ints.index(0) return ints[0 ], set(ints[1:i0 ]), ints[i0+1:−1 ] def verify(file): cs = {} for id, c, ids in (parse(line) for line in file): if not id: # delete clauses for id in ids: del cs[id ] elif not ids: # add original clause cs[id ] = c else: # check & add learnt clause d = c.copy() for i in ids: e = cs[i ]−d if e: d. add(−e.pop()) # propagate assert not e # is unit? else: # empty clause reached cs[id ] = c if not c: return "VERIFIED" break return "NOT VERIFIED" import sys print(verify(open(sys.argv[1 ]))) Fully functional checker for the GRIT format written in Python.…”

A Graph Library for Isabelle

“…Based on that, we have formalized a characterization of Euler digraphs and a core part of a LEDA checker-algorithm for non-planarity of graphs. The library is already being used in other projects [1,8,20] and we expect it to become the standard for formalizations about directed graphs in Isabelle.…”

“…This work has been motivated by the verification effort on the LEDA graph library [13]. The LEDA library contains a number certifying graph algorithms and our library has been successfully used to verify the certificates generated by some of these algorithms [1,20].…”

“…The library is available in the Archive of Formal Proofs [18]. 1 It consists of around 7000 lines of Isabelle theories.…”