Unification of Boolean Differential Rings Is Unitary

Grubmüller, Fabian Lukas (10. September 2024): Unification of Boolean Differential Rings Is Unitary. Bachelorarbeit, Fakultät für Mathematik, Informatik und Statistik, Ludwig-Maximilians-Universität München.

Abstract

The theory of Boolean differential rings is a natural extension of the theory of Boolean rings, that additionaly provides an abstract notion of differential. Boolean rings are important and extensively studied concepts arising naturally in many parts of mathematics, especially logic, and computer science. One important result is that the theory of Boolean rings has the unitary unification type. We show that the unification of Boolean differential rings can be reduced to the unification of Boolean rings and that the theory of Boolean differential rings also has the unitary unification type, and we provide an algorithm that calculates a most general unifier. We also show that terms of Boolean differential rings have a flat normal form similar to the polynomial form of terms of Boolean rings and that terms of Boolean differential rings correspond to terms of Boolean rings in a way that respects both equivalences.

• Dokumententyp: LMU München: Studienabschlussarbeit
• Keywords: unification theory; Boolean differential rings; Boolean differential algebras
• Fakultät: Mathematik, Informatik und Statistik > Informatik > Ausgewählte Abschlussarbeiten
• Institut oder Departement: Institut für Informatik
• Themengebiete: 000 Informatik, Informationswissenschaft, allgemeine Werke > 004 Informatik ; 500 Naturwissenschaften und Mathematik > 510 Mathematik
• Sprache: Englisch

Literaturliste

B. Steinbach and C. Posthoff, Logic Functions and Equations, 3rd ed. Springer, Cham, 2022. doi: 10.1007/978-3-030-88945-6.

B. Steinbach and C. Posthoff, Boolean Differential Equations. Morgan & Claypool Publishers, 2013. doi: 10.2200/S00511ED1V01Y201305DCS042.

F. Weitkämper, “Axiomatizing Boolean Differentiation,” 2021, Springer Interna­ tional Publishing. doi: 10.1007/978-3-030-68071-8_4.

U. Martin and T. Nipkow, “Boolean Unification – The Story So Far,” Journal of Symbolic Computation, vol. 7, pp. 275–293, 1989, doi: 10.1016/ S0747-7171(89)80013-6.

H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 3rd ed. Springer, Cham, 2021. doi: 10.1007/978-3-030-73839-6.

F. Baader and T. Nipkow, Term Rewriting and All That. Cambridge University Press, 1998. doi: 10.1017/CBO9781139172752.

U. Martin and T. Nipkow, “Unification in Boolean Rings,” Journal of Automated Reasoning, vol. 4, pp. 381–396, 1988, doi: 10.1007/3-540-16780-3_115.