Forays into Sequential Composition and Concatenation in Eagle

Abstract

The run-time verification logic Eagle is equipped with two forms of binary cut operator, sequential composition (;) and concatenation (·). Essentially, a concatenation formula F ₁ ·F ₂ holds on a trace if that trace can be cut into two non-overlapping traces such that F ₁ holds on the first and F ₂ on the second. Sequential composition differs from concatenation in that the two traces must overlap by one state. Both cut operators are non-deterministic in the sense that the cutting point is not uniquely defined. In this paper we establish that sequential composition and concatenation are equally expressive. We then extend Eagle with deterministic variants of sequential composition and concatenation. These variants impose a restriction on either the left or right operand so that the cut point defines either the shortest or longest possible satisfiable cut trace. Whilst it is possible to define such deterministic operators recursively within Eagle, such definitions based on the non-deterministic cut operators impose a complexity penalty. By augmenting Eagle’s evaluation calculus for the deterministic variants, we establish that the asymptotic time and space complexity of on-line monitoring for the variants with deterministic restrictions applied to the left operand is no worse than the asymptotic time and space complexity of the sub-formulæ.