Peirce Algebras and Their Applications in Artificial Intelligence and Computational Linguistics

Renate Schmidt, Max-Planck-Institut für Informatik, Saarbrücken

Abstract. In [1] we present a two-sorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a set-forming operator on relations (the Peirce product of Boolean modules) and a relationforming operator on sets (a cylindrification operation). Peirce algebras provide useful formalisations for various fields in Computer Science. In this talk I focus on the application of Peirce algebras in artificial intelligence and computational intelligence. In particular, I show that the so-called terminological logics arising in knowledge representation (originating with a system called kl-one) have evolved a semantics best described as a calculus of relations interacting with sets [1,2,3]. In computational linguistics P. Suppes (1976,1979,1981) and M. Böttner (1992) use concrete Peirce algebras as a relational formalisation of the semantics of the English language. In [4] I link both these applications and show that Peirce algebra provides a useful bridge for utilising the linguistic investigations for the problem of finding adequate terminological representations for given information formulated in ordinary English.

References

[1] Brink, C., Britz, K. and Schmidt, R. A. (1994), Peirce Algebras. Formal Aspects of Computing 6, 1–20.

[2] Brink, C. and Schmidt, R. A. (1992), Subsumption Computed Algebraically. Computers and Mathematics with Applications 23(2–5), 329–342.

[3] Schmidt, R. A. (1991), Algebraic Terminological Representation. Master’s Thesis, University of Cape Town.

[4] Schmidt, R. A. (1993), Terminological Representation, Natural Language & Relation Algebra. In the Proceedings GWAI-92, Vol. 671 of LNAI, 357–371.