Built-in Treatment of an Axiomatic Floating-Point Theory for SMT Solvers

Abstract

The treatment of the axiomatic theory of floating-point numbers is out of reach of current SMT solvers, especially when it comes to automatic reasoning on approximation errors. In this paper, we describe a dedicated procedure for such a theory, which provides an interface akin to the instantiation mechanism of an SMT solver. This procedure is based on the approach of the Gappa tool: it performs saturation of consequences of the axioms, in order to refine bounds on expressions. In addition to the original approach, bounds are further refined by a constraint solver for linear arithmetic. Combined with the natural support for equalities provided by SMT solvers, our approach improves the treatment of goals coming from deductive verification of numerical programs. We have implemented it in the Alt-Ergo SMT solver.

Deciding floating-point logic with abstract conflict driven clause learning

“…While theorem proving approaches have the potential to be sound and complete, they require substantial manual work, although sophisticated (but incomplete) strategies exist to automate substeps of the proof, e.g., [2]. A preliminary attempt to integrate such techniques with SMT solvers has recently been proposed in [18].…”

Building Better Bit-Blasting for Floating-Point Problems

“…It instantiates a series of theorems about floating-point numbers until a sufficient error bound is determined. Although its saturation process is naïve, it is fast and effective, especially when directed by a more conventional SMT solver [20]. Why3 [9] uses an axiomatisation of floating-point numbers based on reals when producing verification conditions for provers that only support real arithmetic.…”

“…We would have liked to benchmark iSAT3 [51], Coral [53] and Gappa [23], but they do not provide SMT-LIB front-ends and there are no automatic translators we are aware of to their native input language. Binaries for FPCS [43] and Alt-Ergo/Gappa [20] were not available.…”