Abstract
Cancellations are known to be helpful in efficient algebraic computation of polynomials over fields. We define a notion of cancellation in Boolean circuits and define Boolean circuits that do not use cancellation to be non-cancellative. Non-cancellative Boolean circuits are a natural generalization of monotone Boolean circuits. We show that in the absence of cancellation, Boolean circuits require super-polynomial size to compute the determinant interpreted over GF(2). This non-monotone Boolean function is known to be in P. In the spirit of monotone complexity classes, we define complexity classes based on non-cancellative Boolean circuits. We show that when the Boolean circuit model is restricted by withholding cancellation, P and popular classes within P are restricted as well, but classes NP and above remain unchanged.
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This work was supported by NSF grant CCR-9200878 and was done while the first author was at the College of Computing, Georgia Tech.
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References
N. Alon and R.B. Boppana, The monotone circuit complexity of Boolean functions, Combinatorica, 7 (1987), 1–22.
M. Ajtai and Y. Gurevich, Monotone versus positive, J. Assoc. Comput. Mach., 34:4 (1987), 1004–1015.
A. Borodin, S. Cook, P. Dymond, W. Ruzzo, M. Tompa, Two applications of inductive counting for complementation problems, SLAM J. Comput., 18 (1989), 559–578.
M. Grigni, Structure in monotone complexity, Ph.D. thesis, M.I.T., 1991.
M. Grigni and M. Sipser, Monotone separation of Logspace from NC1, Proc. 6th IEEE Structures (1991), 294–298.
M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1 (1981), 169–197.
J.E. Hopcroft and R.M. Karp, A n 5/2 algorithm for maximum matching in bipartite graphs, SIAM J. Comput. 2 (1973), 225–231.
N. Immerman, Nondeterministic space is closed under complement, SIAM J. Comput., 17 (1988), 935–938.
A. Markov, On the inversion complexity of systems of Boolean functions, Soviet Math. Doklady 4:3 (1963), 694–696.
N. Nisan, Lower bounds for non-commutative computation, Proc. 23nd annual ACM STOC (1991), 410–418.
A.A. Razborov, A lower bound on the monotone network complexity of the logical permanent, Mathematischi Zametki 37 (1985), 887–900.
R. Raz and A. Wigderson, Monotone circuits for matching require linear depth, Proc. 22nd annual ACM STOC (1990), 287–292.
M. Santha and C. Wilson Polynomial size constant depth circuits with a limited number of negations, Proc. 8th STACS (1991), 228–237.
J. E. Savage, The complexity of computing, R. E. Kreiger Publishing Co., Malabar, Florida, 1987.
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969) 354–356.
é. Tardos, The gap between monotone and non-monotone circuit complexity is exponential, Combinatorica 7 (1987), 141–142.
K. Tanaka and T. Nishino, On the complexity of negation-limited Boolean networks, Proc. 26th ACM Symposium on Theory of Computing (1994), 38–47.
L. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8 (1979), 189–201.
L. Valiant, Negation can be exponentially powerful, Theoretical Computer Science 12 (1980), 303–314.
H. Venkateswaran, Circuit definitions of nondeterministic complexity classes, SIAM J. Comput. 21 (1992) 655–670.
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© 1996 Springer-Verlag Berlin Heidelberg
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Sengupta, R., Venkateswaran, H. (1996). Non-cancellative Boolean circuits: A generalization of monotone Boolean circuits. In: Chandru, V., Vinay, V. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1996. Lecture Notes in Computer Science, vol 1180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62034-6_58
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DOI: https://doi.org/10.1007/3-540-62034-6_58
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