Incremental Closure for Systems of Two Variables Per Inequality

Subclasses of linear inequalities where each inequality has at most two variables are popular in abstract interpretation and model checking, because they strike a balance between what can be described and what can be efficiently computed. This paper focuses on the TVPI class of inequalities, for which each coefficient of each two variable inequality is unrestricted. An implied TVPI inequality can be generated from a pair of TVPI inequalities by eliminating a given common variable (echoing resolution on clauses). This operation, called result, can be applied to derive TVPI inequalities which are entailed (implied) by a given TVPI system. The key operation on TVPI is calculating closure: satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations. A closed system can be derived by repeatedly applying the result operator. The process of adding a single TVPI inequality to an already closed input TVPI system and then finding the closure of this augmented system is called incremental closure. This too can be calculated by the repeated application of the result operator. This paper studies the calculus defined by result, the structure of result derivations, and how derivations can be combined and controlled. A series of lemmata on derivations are presented that, collectively, provide a pathway for synthesising an algorithm for incremental closure. The complexity of the incremental closure algorithm is analysed and found to be O((n^2+m^2)lg(m)), where n is the number of variables and $m$ the number of inequalities of the input TVPI system.