Formalization of Incremental Simplex Algorithm by Stepwise Refinement

Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

“…He also formalized the abstract CDCL calculus by Nieuwenhuis et al and, together with Janičić [62], the more implementation-oriented calculus by Krstić and Goel [50]. As a milestone towards verified SMT solvers, Spasić and Marić [96] formalized the simplex algorithm in Isabelle. Thiemann extended this work to support incremental solving and provide unsatisfiable cores [63].…”

Verifying an Incremental Theory Solver for Linear Arithmetic in Isabelle/HOL

“…One way to prove Farkas' Lemma is by using the Fundamental Theorem of Linear Inequalities; this theorem can in turn be proved in the same way as the fact that the simplex algorithm terminates (see [15,Chapter 7]). Although Spasić and Marić have formalized a proof of termination for their simplex implementation [16], this is not sufficient to immediately prove Farkas' Lemma. Instead, our formalization of the result begins at the point where the simplex algorithm detects unsatisfiability in Phase 3, because this is the only point in the execution of the algorithm where Farkas coefficients can be computed directly from the available data.…”

“…The ultimate aim of this work is the optimization of the existing verified SMT solver, as it is quite basic: The current solver takes as input a quantifier free formula in the theory of linear rational arithmetic, translates it into disjunctive normal form (DNF), and then tries to prove unsatisfiability for each conjunction of literals with the verified simplex implementation of Spasić and Marić [16]. This basic solver has at least two limitations: It only works on small formulas, since the conversion to DNF often leads to an exponential blowup in the formula size; and the procedure is restricted to linear rational arithmetic, i.e., the existing formalization only contain results on satisfiability over Q, but not over R.…”

“…Therefore, in this paper we will extend the formalization of the simplex algorithm due to Spasić and Marić [16]. This will be an important milestone on the way to obtain a fully verified DPLL(T)-based SMT solver.…”

“…Chaieb and Nipkow verified quantifier elimination procedures (QEP) for dense linear orders and integer arithmetic [9], which are more widely applicable than the simplex algorithm. Spasić and Marić compared the QEPs with their implementation on a set of random quantifier-free formulas [16]. In these tests, their (and therefore our) simplex implementation outperforms the QEPs significantly.…”