Production systems are declarative, in that they do not explicitly specify the control flow. Yet, the concept of a production system does not include the definition of a given control strategy. The control between rules in a production rule program is, in practice, defined by each implementation of a production rule engine. Engines have traditionally been implemented using the Rete algorithm. Since the turn of the century, however, production systems have evolved into industrial products known

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... s Business Rules Management Systems (BRMS). BRMS have introduced new compilation and execution schemes, which are often called sequential in contrast with the incremental behavior of Rete. This change in execution scheme came with a change in semantics for rule programs. In this paper, we propose a formal description of the execution of production rule programs. Existing descriptions either ignore the control strategy, or assume a Rete semantics. Ours isolates the handling of rule eligibility in the control strategy, which allows us to describe the sequential execution semantics of rule programs, as well as the Rete semantics, and others.