Finite State Evaluation of Logical Formulas : Jevons' Approach (1870) and Contemporary Description

In this paper, we describe a formal language for a class of logical expressions. We then present a Finite State Machine for recognition and evaluation of this language. The main interest of the language is its historical characteristic. This language invented by the British scholar W. Stanley JEVONS in 1865 is probably the earliest language in which expressions were evaluated by a Finite State Machine. The two outstanding contributions were the use of machinery to evaluate formulas and the evaluation of formulas with variables by several parallel evaluations with constants. The contribution of this paper is to present this ancient evaluation process in a contemporary framework, i.e. formal languages and finite state automata. The design of an evaluator is given in great detail. Introduction and Related Works The history of calculating machines is well known. Pascal and Babbage built machines that are considered as the mechanical ancestors of today’s computers. But computers do not only compute numbers, they can also perform symbolic evaluations. At a certain level of abstraction, we may consider mechanisms based on logical choices if . . . then . . . , or if . . . then . . . else . . . as the necessary complements of strictly arithmetic operations. The first mechanical machine making these choices to ” perform the logical inference ” was designed by William Stanley JEVONS in 1865 and published in 1870 ([14]). He had been a student of De Morgan. He was at that time becoming a professor of Logic (and of Political Economy) at the Owens College of Manchester. Figure 1 shows Jevons’ activity time w.r.t other well-known British logicians. His work has often been presented in the same terms as in the original paper: logical evaluation [1, 5, 6, 16]. Burris’ paper [5] gives interesting details about Jevons’ logic and about his machine. But none of these papers establishes relations ∗TIMA-CMP Lab. University of Grenoble, 46 Av. Felix Viallet, 38031 Grenoble CEDEX, France, E-mail: [email protected]