Superposition modulo theory
Evgeny Kruglov · Max Planck Digital Library · 2013
This thesis is about the Hierarchic Superposition calculus SUP(T) and its application to reasoning in hierarchic combinations FOL(T) of the free first-order logic FOL with a background theory T where the hierarchic calculus is refutationally complete or serves as a decision procedure. Particular hierarchic combinations covered in the thesis are the combinations of FOL and linear and non-linear arithmetic, LA and NLA resp. Recent progress in automated reasoning has greatly encouraged numerous applications in softand hardware verification and the analysis of complex systems. The applications typically require to determine the validity/unsatisfiability of quantified formulae over the combination of the free first-order logic with some background theories. The hierarchic superposition leverages both (i) the reasoning in FOL equational clauses with universally quantified variables, like the standard superposition does, and (ii) powerful reasoning techniques in such theories as, e.g., arithmetic, which are usually not (finitely) axiomatizable by FOL formulae, like modern SMT solvers do. The thesis significantly extends previous results on SUP(T), particularly: we introduce new substantially more effective sufficient completeness and hierarchic redundancy criteria turning SUP(T) to a complete or a decision procedure for various FOL(T) fragments; instantiate and refine SUP(T) to effectively support particular combinations of FOL with the LA and NLA theories enabling a fully automatic mechanism of reasoning about systems formalized in FOL(LA) or FOL(NLA).