A promising approach to managing business operations is based on <i>business artifacts</i>, a.k.a. <i>business entities</i> (<i>with lifecycles</i>). These are key conceptual entities that are central to guiding the operations of a business, and whose content changes as they move through those operations. An artifact type includes both an <i>information model</i> that captures all of the business-relevant data about entities of that type, and a lifecycle model, that specifies the possible ways an entity of that type might progress through the business. Two recent papers have introduced and studied the <i>Guard-Stage-Milestone</i> (<i>GSM</i>) meta-model for artifact lifecycles. GSM lifecycles are substantially more declarative than the finite state machine variants studied in most previous work, and support hierarchy and parallelism within a single artifact instance. This paper presents the formal operational semantics of GSM, with an emphasis on how interaction between artifact instances is supported. Such interactions are supported both through testing of conditions against the artifact instances, and through events stemming from changes in artifact instances. Building on a previous result for the single artifact instance case, a key result here shows the equivalence of three different formulations of the GSM semantics for artifact instance interaction. One formulation is based on incremental application of ECA-like rules, one is based on two mathematical properties, and one is based on the use of first-order logic formulas.