On the equational definition of the least prefixed point

✍ Scribed by Luigi Santocanale

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 265 KB
• Volume: 295
• Category: Article
• ISSN: 0304-3975
• DOI: 10.1016/s0304-3975(02)00412-7

✦ Synopsis

We propose a method to axiomatize by equations the least preÿxed point of an order preserving function. We discuss its domain of application and show that the Boolean modal -calculus has a complete equational axiomatization. The method relies on the existence of a "closed structure" and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed structure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for deÿning by equations the least preÿxed point. We stress the interplay between closed structures and ÿxed point operators by showing that the theory of Boolean modal -algebras is not a conservative extension of the theory of modal -algebras. The latter is shown to lack the ÿnite model property.