The method for automatic theorem proving proposed in [Carnielli, W. A., Polynomial ring calculus for many-valued logics, Proceedings of the 35th International Symposium on Multiple-Valued Logic, IEEE Computer Society. Calgary, Canada (2005), 20-25], called Polynomial Ring Calculus, is an algebraic proof mechanism based on handling polynomials over finite fields. Although useful in general domains, as in first-order logic, certain non-truth-functional logics and even in modal logics (see [Agudelo, J. C., Carnielli, W. A., Polynomial Ring Calculus for Modal Logics: a new semantics and proof method for modalities, The Review of Symbolic Logic. 4 (2011), 150-170, URL: doi:10.1017/S1755020310000213]), the method is particularly apt for deterministic and non-deterministic many-valued logics, as shown here. The aim of the present paper is to show how the method can be extended to any finite-valued non-deterministic semantics, and also to explore the computational character of the method through the development of a software capable of translating provability in deterministic and non-deterministic finite-valued logical systems into operations on polynomial rings. © 2014 Elsevier B.V.305 1934Agudelo, J.C., Carnielli, W.A., Polynomial Ring Calculus for Modal Logics: A new semantics and proof method for modalities (2011) The Review of Symbolic Logic, 4, pp. 150-170. , 10.1017/S1755020310000213Avron, A., Non-deterministic Semantics for Logics with a Consistency Operator (2007) Journal of Approximate Reasoning., 45, pp. 271-287Bimbó, K., Dunn, J.M., Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi (2008) CSLI Lecture Notes, , CSLI PublicationsCarnielli, W.A., Possible-translations semantics for paraconsistent logics (1998) Frontiers in Paraconsistent Logic: Proceedings of the i World Congress on Paraconsistency, Ghent, p. 15972. , D. Batens, Kings College Publications 2000Carnielli, W.A., A polynomial proof system for Łukasiewicz logics (2001) Second Principia International Symposium, pp. 6-10Carnielli, W.A., Polynomial ring calculus for many-valued logics (2005) Proceedings of the 35th International Symposium on Multiple-Valued Logic, pp. 20-25. , IEEE Computer Society Calgary, CanadaCarnielli, W.A., Polynomial Ring Calculus for Logical Inference (2005) CLE E-Prints, 5, pp. 1-17. , ftp://ftp.cle.unicamp.br/pub/e-prints/vol.5,n.3,2005.pdfCarnielli, W.A., (2007) Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole, Model-Based Reasoning in Science, Technology, and Medicine, 64, pp. 349-364. , L. Magnani, P. Li (Eds.) Springer publisherCarnielli, W.A., Coniglio, M.E., Marcos, J., Logics of Formal Inconsistency (2007) Handbook of Philosophical Logic, 14, pp. 15-107. , D. Gabbay, F. Guenthner (Eds.)Carnielli, W.A., (2009) Formal Polynomials and the Laws of Form, the Multiple Dimensions of Logic, 54, pp. 2002-2012. , Y. Béziau, A. Costa-LeiteCarnielli, W.A., Formal polynomials, heuristics and proofs in logic, Logical Investigations (2010) Institute of Philosophy - Russian Academy of Sciences Publisher, 16, pp. 280-294. , A.S. Karpenko (Ed.)Carnielli, W.A., Proofs by handling polynomials: A tool for teaching logic and metalogic (2011) Proceedings of the Third International Congress on Tools for Teaching Logic, pp. 1-3. , Salamanca, SpainD'Agostino, M., (2013) Analytic Inference and the Informational Meaning of the Logical Operators, , Logique et Analyse, in printDunn, J.M., Intuitive semantics for first-degree entailments and coupled trees (1976) Philosophical Studies, 29, pp. 149-169Matulovic, M., (2013) Proofs in the Algibeira: Polynomials As A Universal Method of Proof, , Ph.D. Thesis State University of Campinas (UNICAMP) BrazilQuine, W.V.O., (1973) The Roots of Reference, , Open Court