addition or subtraction, resulting in high-speed arithmetic operations. These number systems have a natural implementation using many-valued circuits. However, the practicality of these potential advantages heavily depends on the availability of circuit realizations, which must be compatible or competitive with present-day standard technologies. In addition to aiding in the design of electronic circuits, many-valued logic is used extensively to test circuits for faults and defects. Basically all known automatic test pattern generation (ATG) algorithms used for digital circuit testing require a simulator that can resolve 5-valued logic (0, 1, x, D, D'). The additional values—x, D, and D'—represent (1) unknown/uninitialized, (2) a 0 instead of a 1, and (3) a 1 instead of a 0. Research venues An IEEE International Symposium on Multiple-Valued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and verification. There is also a Journal of Multiple-Valued Logic and Soft Computing. See also Mathematical logic Degrees of truth Fuzzy logic Gödel logic Jaina seven-valued logic Kleene logic Kleene algebra (with involution) Łukasiewicz logic MV-algebra Post logic Principle of bivalence A. N. Prior Relevance logic Philosophical logic False dilemma Mu Digital logic MVCML, multiple-valued current-mode logic IEEE 1164 a nine-valued standard for VHDL IEEE 1364 a four-valued standard for Verilog Three-state logic Noise-based logic References Further reading General Augusto, Luis M. (2017). Many-valued logics: A mathematical and computational introduction. London: College Publications. 340 pages. . Webpage Béziau J.-Y. (1997), What is many-valued logic ? Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117–121. Malinowski, Gregorz, (2001), Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell. Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Many-valued Reasoning. Kluwer. S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001. Hájek P., (1998), Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as many-valued logic sui generis.) Specific Alexandre Zinoviev, Philosophical Problems of Many-Valued Logic, D. Reidel Publishing Company, 169p., 1963. Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325–373. Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press. Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht. Pavelka