Abstract
We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.
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References
Avenhaus, J.: Correspondence by electronic mail, 2000.
Hillenbrand, T., Buch, A., Vogt, R. and Löchner, B.: Waldmeister, J. Automated Reasoning 18(2) (1997), 265–270.
Kunen, K.: Single axioms for groups, J. Automated Reasoning 9(3) (1992), 291–308.
McCune, W.: Otter, http://www.mcs.anl.gov/AR/otter/, 1994.
McCune, W.: Otter 3.0 reference manual and guide, Technical Report ANL-94/6, Argonne National Laboratory, Argonne, IL, 1994.
McCune, W.: Solution of the Robbins problem, J. Automated Reasoning 19(3) (1997), 263–276.
McCune, W.: Single axioms for Boolean algebra, Tech. Memo ANL/MCS-TM-243, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 2000.
McCune, W.: MACE 2.0 reference manual and guide, Tech. Memo ANL/MCS-TM-249, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 2001.
McCune, W. and Shumsky, O.: IVY: A preprocessor and proof checker for first-order logic, in M. Kaufmann, P. Manolios and J. Moore (eds), Computer-Aided Reasoning: ACL2 Case Studies, Kluwer Academic Publishers, 2000, Chapt. 16.
Meredith, C. A.: Equational postulates for the Sheffer stroke, Notre Dame J. Formal Logic 10(3) (1969), 266–270.
Meredith, C. A. and Prior, A. N.: Equational logic, Notre Dame J. Formal Logic 9 (1968), 212–226.
Padmanabhan, R. and McCune, W.: Single identities for ternary Boolean algebras, Comput. Math. Appl. 29(2) (1995), 13–16.
Padmanabhan, R. and Quackenbush, R. W.: Equational theories of algebras with distributive congruences, Proc. Amer. Math. Soc. 41(2) (1973), 373–377.
Sheffer, H.: A set of five independent postulates for Boolean algebras, with application to logical constants, Trans. Amer. Math. Soc. 14(4) (1913), 481–488.
Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: Case studies, J. Automated Reasoning 16(3) (1996), 223–239.
Veroff, R.: Short 2-bases for Boolean algebra in terms of the Sheffer stroke, Technical Report TR-CS-2000-25, Computer Science Department, University of New Mexico, Albuquerque,NM, 2000.
Wolfram, S.: Correspondence by electronic mail, 2000.
Wos, L.: Searching for circles of pure proofs, J. Automated Reasoning 15(3) (1995), 279–315.
Wos, L. and Pieper, G.: A Fascinating Country in the World of Computing: Your Guide to Automated Reasoning, World Scientific, Singapore, 1999.
Zhang, J. and Zhang, H.: SEM: A system for enumerating models, in Proc. IJCAI-95, Vol. 1, 1995, pp. 298–303.
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McCune, W., Veroff, R., Fitelson, B. et al. Short Single Axioms for Boolean Algebra. Journal of Automated Reasoning 29, 1–16 (2002). https://doi.org/10.1023/A:1020542009983
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DOI: https://doi.org/10.1023/A:1020542009983