Delayed-logic and finite-state machines

FASER (Formal and Automatic Security Enforcement by Rewriting): An algebraic approach

“…In fact, it has been proved in [4] that the number of partial derivatives of P is smaller than ||P || + 1. Also, since any equation is the intersection of an element from PD(P ) and another from PD(Q), then the number of equations N is less than (||P || + 1) × (||Q|| + 1) and their resolutions can be done by using less than N times the Arden's Lemma [5].…”

“…• The system can be solved iteratively by using the Arden's Lemma [5]. Also, an extended version of the Arden Lemma shows that the solution of this system is X = A * B, where A * is computed as following:…”

Left omega algebras and regular equations

“…Arden's rule is a fundamental tool of language theory [1]. To determine, for instance, the language accepted by the automaton…”

“…Arden's rule is the classical tool for solving systems of recursive language equations [1]. A side condition is the negated empty word property which holds if a language does not contain the empty word.…”

On the origins of bisimulation and coinduction

“…Knaster-Tarski Theorem 2.9, about the existence of least and greatest fixed point for a monotone function on a complete lattice, or variations of this such as Theorem 2.11, are the starting point for all the works we mention. The earliest uses of fixed points in Computer Science, in the form of least fixed points, can be found in: recursive function theory, see for instance Rogers's book [1967] and references therein; formal language theory, as in the work of Arden [1960] and Ginsburg and Rice [1962]. However, distinguishing Computer Science from recursive function theory, the importance of fixed points in Computer Science really comes up only at the end of the 1960s, with four independent papers, roughly at the same time, by Dana Scott and Jaco de Bakker [1969], Hans Bekič [1969], David Park [1969], and Antoni Muzurkiewicz [1971] (however [Mazurkiewicz 1971] does not make explicit reference to fixed-point theory).…”