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A Note on the Polynomial Form of Boolean Functions and Related Topics
August 1999 (vol. 48 no. 8)
pp. 860-864

Abstract—This note relates to a recently published partly tutorial article that presents some discussion of the polynomial form of Boolean functions and its applications based on the literature published in English and German. We show that a lot of the research in this area has also been done in Eastern Europe, and this note aims to present these unknown developments. The most recent work in this area is also described.

[1] W.G. Schneeweiss, “On the Polynomial Form of Boolean Functions: Derivations and Applications,” IEEE Trans. Computers, vol. 47, no. 2, pp. 217-221, Feb. 1998.
[2] P. Hammer and S. Rudeanu, Boolean Methods in Operations Research. Berlin: Springer, 1968.
[3] Z. Tosic, “Analytical Representations of$m$-Valued Logical Functions over the Ring of Integers Modulo m,” PhD thesis, Beograd, Yugoslavia: Univ. of Beograd Press, 1972.
[4] I.I. Zhegalkin, “On the Technique of Calculation the Sentences in Symbolic Logic,” Matem. Sbornik, vol. 34, pp. 9-28, 1927 (in Russian).
[5] R. Lechner, “Harmonic Analysis of Switching Functions,” Recent Developments in Switching Theory, A. Mukhopadhyay, ed. New York: Academic, 1971.
[6] I.S. Reed, “A Class of Multiple-Error-Correcting Codes and Their Decoding Scheme,” IRE Trans. Information Theory, vol. 4, pp. 38-42, 1954.
[7] D.E. Muller, “Applications of Boolean Algebra to Switching Circuit Design and to Error Detection,” IRE Trans. Electronic Computers, vol. 3, pp. 6-12, 1954.
[8] S. Puwar, “Polynomial Representation of Spectral Coefficients,” IEE Electronics Letters, vol. 28, no. 15, pp. 1,264-1,265, 1992.
[9] B.J. Falkowski, “On Polynomial Representation of Spectral Coefficients,” IEE Electronics Letters, vol. 29, no. 1, pp. 35-37, Jan. 1993.
[10] M.G. Karpovsky, Finite Orthogonal Series in Design of Digital Devices. New York: Wiley, 1976.
[11] P. Calingaert, “Switching Function Canonical Forms Based on Commutative and Associative Binary Operations,” Trans. AIEE, vol. 52, pp. 804-814, 1961.
[12] P. Davio, J.P. Deschamps, and A. Thayse, Discrete and Switching Functions. New York: McGraw-Hill, 1978.
[13] R.S. Stankovic, “A Note of the Relation between Reed-Muller Expansions and Walsh Transforms,” IEEE Trans. Electromagnetic Compatibility, vol. 24, no. 1, pp. 68-70, Feb. 1982.
[14] S. Agaian, J. Astola, and K. Egiazarian, Binary Polynomial Transforms and Nonlinear Digital Filters. New York: Marcel Dekker, 1995.
[15] R.S. Stankovic, M.R. Stojic, and M.S. Stankovic, Recent Developments in Abstract Harmonic Analysis with Applications in Signal Processing. Belgrade, Yugoslavia: Science Publisher, 1996.
[16] S.K. Kumar and M.A. Breuer, “Probabilistic Aspects of Boolean Switching Functions via a New Transform,” J. ACM, vol. 28, pp. 502-520, 1981.
[17] K.P. Parker and E.J. McCluskey, “Probabilistic Treatment of General Combinatorial Networks,” IEEE Trans. Computers, vol. 24, pp. 668-670, 1975.
[18] R.S. Stankovic, “Some Remarks about Spectral Transform Interpretation of MTBDDs and EVBDDs,” Proc. First IEEE Asia and South Pacific Design Automation Conf., pp. 385-390, Makuhari, Japan, Aug. 1995.
[19] N.N. Aizenberg and O.T. Trofimlyuk, “Conjunctive Transformation of Discrete Signals and Their Applications to Testing and Detection of Monotonic Boolean Functions,” Kibernetika, no. 1, pp. 138-139, 1981 (in Russian).
[20] S. Meldal,S. Sankar,, and J. Vera,“Exploiting locality in maintaining potential causality,” Proc. Tenth Annual ACM Symp. on Principles of Distributed Computing, pp. 231-239,New York, Aug. 1991. Washington, D.C.: ACM Press. Also Stanford Univ. Computer Systems Laboratory Technical Report No.CSL-TR-91-466.
[21] G. De Micheli, Synthesis and Optimization of Digital Circuits. McGraw-Hill, 1994.
[22] B.J. Falkowski and C.H. Chang, “Generation of Multi-Polarity Arithmetic Transform from Reduced Representation of Boolean Functions,” Proc. IEEE Int'l Symp. Circuits and Systems, pp. 2,168-2,171, Seattle, Wash., May 1995.
[23] B.J. Falkowski and C.H. Chang, “Efficient Algorithms for the Calculation of Arithmetic Spectrum from OBDD and Synthesis of OBDD from Arithmetic Spectrum for Incompletely Specified Boolean Functions,” Proc. IEEE Int'l Symp. Circuits and Systems, vol. 1, pp. 197-200, London, May 1994.
[24] B.J. Falkowski and C.H. Chang, “Calculation of Arithmetic Spectra from Free Binary Decision Diagrams,” Proc. IEEE Int'l Symp. Circuits and Systems, pp. 1,764-1,767, Hong Kong, June 1997.
[25] B.J. Falkowski and C.H. Chang, “Mutual Conversions between Generalized Arithmetic Expansions and Free Binary Decision Diagrams,” IEE Proc. Circuits, Devices, and Systems, vol. 145, no. 4, pp. 219-228, Aug. 1998.
[26] B.J. Falkowski and C.H. Chang, “An Efficient Algorithm for the Calculation of Generalized Arithmetic and Adding Transforms from Disjoint Cubes of Boolean Functions,” VLSI Design, An Int'l J. Custom-Chip Design, Simulation and Testing, vol. 9, no. 2, pp. 135-146, Apr. 1999.
[27] K.D. Heidtmann,"Arithmetic spectrum applied to fault detection for combinational circuits," IEEE Trans. Computers, vol. 40, pp. 320-324, Mar. 1991.
[28] Y.T. Lai, M. Pedram, and S.B.K. Vrudhula, “EVBDD-Based Algorithms for Integer Linear Programming, Spectral Transformation, and Functional Decomposition,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 13, no. 8, pp. 959-975, Aug. 1994.
[29] R.S. Stankovic, Spectral Transform Decision Diagrams in Simple Questions and Simple Answers. Belgrade, Yugoslavia: Science Publisher, 1998.
[30] Z. Tosic, “Arithmetic Representations of Logic Functions,” Discrete Automata and Comm. Networks, (Diskretnyie avtomata i seti sviazi), Moscow: USSR Academy of Sciences, pp. 131-136, 1970 (in Russian).
[31] B.J. Falkowski, V.P. Shmerko, and S.N. Yanushkevich, “Arithmetical Logic—Its Status and Achievements,” Proc. Int'l Conf. Applications of Computer Systems, pp. 208-223, Szczecin, Poland, Nov. 1997.
[32] S. Yanushkevich, Logic Differential Calculus in Multi-Valued Logic Design. Szczecin, Poland: Technical Univ. of Szczecin Press, Scientific Works of Technical Univ. of Szczecin No. 537, Inst. of Computer Science and Information Systems, 1998.
[33] V.P. Shmerko and S.V. Mikhailov, “Review of Publications in the Former Soviet Union on Spectral Methods of Logic Data Processing and Logic Differential Calculus,” Proc. Fifth Int'l Workshop Spectral Techniques, pp. 48-54, Beijing, China, Mar. 1994.
[34] G. Boole, The Laws of Thought. London: Macmillan, 1854, reprint by New York: Dover, 1958.
[35] P.S. Poretski, “On the Method for Solving Logical Equations and on the Inverse Methods for Mathematical Logic,” Sobranie protokolov zasedanii fiz. mat., Kazan Univ., Kazan, Russia, vol. 2, pp. 161-330, 1884 (in Russian).
[36] V.D. Malyugin, “Switching Circuits Reliability,” Avtomatika i Telemekhanika, no. 9, pp. 1,375-1,383, 1964 (in Russian).
[37] V.N. Zadorozhny, “Realization of Logic Functions by Arithmetical Expressions,” Automatization for Computer Structure Analysis and Synthesis, Novosibirsk: Engineering Building Inst., Russia, 1978 (in Russian).
[38] V.D. Malyugin, “Representation of Boolean Functions by Arithmetic Polynomials,” Automation and Remote Control, vol. 43, no. 4,part. 1, pp. 496-504, Apr. 1982 (a translation of Avtomatika i Telemekhanika, no. 4, pp. 84-93, 1982).
[39] V.D. Malyugin, “Realization of Corteges of Boolean Functions by Linear Arithmetic Polynomials,” Automation and Remote Control, vol. 45, no. 2,part 1, pp. 239-245, 1984 (a translation of Avtomatika i Telemekhanika, no. 2, pp. 114-122, 1984).
[40] V.D. Malyugin, “Arithmetical Representations of Petri Nets,” Automation and Remote Control, vol. 85, no. 5, pp. 249-255, 1987 (a translation of Avtomatika i Telemekhanika, no. 5, pp. 156-164, 1987).
[41] V.L. Artyukhov, V.N. Kondratev, and A.A. Shalyto, “Generating Boolean Functions via Arithmetic Polynomials,” Automation and Remote Control, vol. 49, no.4,part 2, pp. 508-551, 1988 (a translation of Avtomatika i Telemekhanika, no. 4, pp. 138-147, 1988).
[42] V.P. Shmerko, “Synthesis of Arithmetic Forms of Boolean Functions Using the Fourier Transform,” Automation and Remote Control, vol. 50, no. 5,part 2, pp. 684-691, 1989 (a translation of Avtomatika i Telemekhanika, no. 5, pp. 134-142, 1989).
[43] B.J. Falkowski, “Forward and Inverse Transformations between Haar Wavelet and Arithmetic Functions,” IEE Electronics Letters, vol. 34, no. 11, pp. 1,084-1,085, May 1998.
[44] G.A. Kukharev, V.P. Shmerko, and S.N. Yanushkevich, Parallel Processing of Binary Data in VLSI. Minsk, Bielarus: Univ. Publisher, 1991 (in Russian).
[45] V.P. Shmerko and S.N. Yanushkevich, “Algorithms of Boolean Differential Calculus for Systolic Processors,” Automation and Remote Control, pp. 345-360, 1990 (a translation of Avtomatika i Telemekhanika, no. 3, pp. 31-40, 1990).
[46] A.A. Shalyto and V.N. Kondratev, Software Implementation of Logical Control Algorithms for Marine Microprocessor Systems. Leningrad: Inst. for Professional Development Publisher, 1990 (in Russian).
[47] V.D. Malyugin and V.V. Sokolov, “Intensive Logical Computation,” Automation and Remote Control, pp. 672-678, 1993 (a translation of Avtomatika i Telemekhanika, no. 4, pp. 160-167, 1993).
[48] V.D. Malyugin, Parallel Calculations by Means of Arithmetical Polynomials. Moscow: Physical and Mathematical Publishing Company, Russian Academy of Sciences, 1997 (in Russian).
[49] V.N. Kondratev and A.A. Shalito, “Realization of Systems of Boolean Functions by Linear Arithmetic Polynomials,” Automation and Remote Control, pp. 472-487, 1993 (a translation of Avtomatika i Telemekhanika, no. 3, pp. 135-151, 1993).
[50] B.J. Falkowski and C.H. Chang, “Properties and Methods of Calculating Generalized Arithmetic and Adding Transforms,” IEE Proc. Circuits, Devices, and Systems, vol. 144, no. 5, pp. 249-258, Oct. 1997.
[51] B.J. Falkowski and C.H. Chang, “Fast Generalized Arithmetic and Adding Transforms,” Proc. IFIP Int'l Conf. Very Large Scale Integration, pp. 723-728, Makuhari, Chiba, Japan, Aug. 1995.
[52] C.H. Chang and B.J. Falkowski, “Operations on Boolean Functions and Variables in Spectral Domain of Arithmetic Transform,” Proc. IEEE Int'l Symp. Circuits and Systems, vol. 4, pp. 400-403, Atlanta, Ga., May 1996.
[53] C.H. Chang and B.J. Falkowski, “Logical Manipulations and Design of Tributary Networks in the Arithmetic Spectral Domain,” IEE Proc. Computers and Digital Techniques, vol. 145, no. 5, pp. 347-356, Sept. 1998.
[54] S. Rahardja and B.J. Falkowski, “Family of Fast Transforms for Mixed Arithmetic Logic,” Proc. IEEE Int'l Symp. Circuits and Systems, vol. 4, pp. 396-399, Atlanta, Ga., May 1996.
[55] S. Rahardja and B.J. Falkowski, “Family of Fast Mixed Arithmetic Logic Transforms for Multiple-Valued Input Binary Functions,” Proc. IEEE Int'l Symp. Multiple-Valued Logic, pp. 24-29, Santiago de Compostella, Spain, May 1996.
[56] S. Rahardja and B.J. Falkowski, “Application of Linearly Independent Arithmetic Transform in Testing of Digital Circuits,” IEE Electronics Letters, vol. 35, no. 5, pp. 363-364, Mar. 1999.
[57] D.A. Postelov and Z. Tosic, “Polynomial Representation in Mulivalued Logic,” Synthesis of Discrete Automata and Controllers, pp. 139-146, Moscow: Science Publisher, 1986 (in Russian).
[58] I. Strazdins, “The Polynomial Algebra of Multivalued Logic,” Algebra, Combinatorics and Logic in Computer Science, vol. 42, pp. 777-785, 1983.
[59] G.A. Kukharev, V.P. Shmerko, and E.N. Zaitseva, Algorithms and Systolic Processes for Multivalued Data Processing. Minsk, Bielarus: Science and Technology Publisher, 1990 (in Russian).
[60] E.N. Zaitseva, S.A. Mysovskikh, and V.M. Antonenko, ”Arithmetical Polynomial-Similar Forms of MVL-Functions,” Proc. Fifth Int'l Workshop Spectral Techniques, pp. 69-73, Beijing, China, Mar. 1994.
[61] E.N. Zaitseva, E.G. Kochergov, and A.A. Snarov, “Logic Polynomial-Similar Forms of MVL-Functions,” Proc. Fifth Int'l Workshop Spectral Techniques, pp. 53-68, Beijing, China, Mar. 1994.
[62] B.J. Falkowski and S. Rahardja, “Generalized Hybrid Arithmetic Canonical Expansions for Completely Specified Quaternary Functions,” IEE Proc. Circuits, Devices, and Systems, vol. 144, no. 4, pp. 201-208, Aug. 1997.
[63] M.U. Garaev and R.G. Faradzhev, “On an Analog of Fourier Expansion over Galois Fields and Its Applications to Problems of Generalized Sequential Machines,” Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Techn. I Math. Nauk., no. 6, pp. 1-68, 1965 (in Russian).
[64] R.G. Faradzhev, Linear Sequential Machines. Moscow: Soviet Radio Publisher, 1975 (in Russian).
[65] N.K. Aslanova and R.G. Faradzhev, “Arithmetic Representation of Functions in Multivalued Logic and a Parallel Algorithm for Finding Such Representations,” pp. 251-261, 1992 (a translation of Avtomatika i Telemekhanika, no. 2, pp. 120-131, 1992).
[66] V.P. Shmerko, V.M. Antonenko, and S.Y. Trushkin, “Method to Synthesize Linear Arithmetical Polynomials of MVL-Functions,” Proc. Fifth Int'l Workshop Spectral Techniques, pp. 55-58, Beijing, China, Mar. 1994.
[67] V.M. Antonenko, A.A. Ivanov, and V.P. Shmerko, “Linear Arithmetic Forms of k-Valued Logic Functions and Their Implementation on Systolic Arrays,” Automation and Remote Control, vol. 56, no. 3, pp. 419-432, 1995 (a translation of Avtomatika i Telemekhanika, no. 3, pp. 139-151, 1995).
[68] S.N. Yanushkevich, “Systolic Algorithms to Synthesize Arithmetical Polynomial Forms for k-Valued Logic Functions,” Automation and Remote Control, vol. 55, no. 5,part 2, pp. 715-729, 1994 (a translation of Avtomatika i Telemekhanika, no. 12, pp. 128-141, 1994).
[69] S.N. Yanushkevich, “Spectral and Differential Methods to Synthesize Polynomial Forms of MVL-Function on Systolic Arrays,” Proc. Fifth Int'l Workshop Spectral Techniques, pp. 78-83, Beijing, China, Mar. 1994.
[70] S.N. Yanushkevich, “Arithmetical Canonical Expansions of Boolean and MVL Functions as Generalized Reed-Muller Series,” Proc. IFIP WG 10.5 Workshop Applications of the Reed-Muller Expansions in Circuit Design, pp. 300-307, Makuhari, Chiba, Japan, Sept. 1995.

Index Terms:
Boolean functions, multiple-valued functions, polynomial forms, arithmetic transform, mixed arithmetic transform, arithmetic derivatives.
Citation:
Bogdan J. Falkowski, "A Note on the Polynomial Form of Boolean Functions and Related Topics," IEEE Transactions on Computers, vol. 48, no. 8, pp. 860-864, Aug. 1999, doi:10.1109/12.795128
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