Amplification and percolation (probabilistic Boolean functions)

The authors extend R.B. Boppana's results (1989) in two ways. They first show that his two lower bounds hold for general read-once formulae, not necessarily monotone, that may even include exclusive-or gates. They are then able to join his two lower bounds together and show that any read-once, not necessarily monotone, formula that amplifies (p-/sup 1///sub n/,p+/sup 1///sub n/) to (2/sup -n/,1-2/sup -n/) has size of at least Omega (n/sup alpha +2/). This result does not follow from Boppana's arguments and it shows that the amount of amplification achieved by L.G. Valiant (1984) is the maximal achievable using read-once formulae.<<ETX>>

[1]  A. N. Sokolov.  Partial decoding of monotonic Boolean functions , 1983 .

[2]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .

[3]  M.M. Halldorsson,et al.  Directed vs. undirected monotone contact networks for threshold functions , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[4]  Ronald L. Rivest.  The Necessity of Feedback in Minimal Monotone Combinational Circuits , 1977, IEEE Transactions on Computers.

[5]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[6]  Toshihide Ibaraki,et al.  A Lower Bound of the Number of Threshold Functions , 1965, IEEE Trans. Electron. Comput..

[7]  Koichiro Yamamoto.  Logarithmic order of free distributive lattice , 1954 .

[8]  Yu. A. Zuev,et al.  Estimating the efficiency of threshold representations of Boolean functions , 1988 .

[9]  I. Ninomiya.  A study of the structures of Boolean functions and its application to the synthesis of switching circutis. , 1962 .

[10]  N. Gallagher,et al.  An overview of median and stack filtering , 1992 .

[11]  Nicholas Pippenger,et al.  On Another Boolean Matrix , 1980, Theor. Comput. Sci..

[12]  Jaikumar Radhakrishnan,et al.  ΣΠΣ Threshold Formulas , 1994 .

[13]  Ingo Wegener,et al.  Switching Functions Whose Monotone Complexity is Nearly Quadratic , 1979, Theor. Comput. Sci..

[14]  Jürgen Tiekenheinrich.  A 4n-Lower Bound on the Monotone Network Complexity of a One-Output Boolean Function , 1984, Inf. Process. Lett..

[15]  M. Paterson,et al.  Optimal carry save networks , 1992 .

[16]  N. N. Katerinochkina.  Search for a maximum upper zero for a class of monotonic functions in k-valued logic☆ , 1981 .

[17]  Ingo Wegener.  A Counterexample to a Conjecture of Schnorr Referring to Monotone Networks , 1979, Theor. Comput. Sci..

[18]  Toshihide Ibaraki,et al.  Identifying 2-Monotonic Positive Boolean Functions in Polynominal Time , 1991, ISA.

[19]  Andrew Chi-Chih Yao,et al.  A lower bound for the monotone depth of connectivity , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[20]  N. N. Katerinochkina.  The problem of finding the maximum upper zero for a series of subclasses of monotonic Boolean functions , 1989 .

[21]  Randolph Church,et al.  Nunmerical analysis of certain free distributive structures , 1940 .

[22]  Paul E. Dunne,et al.  On Monotone Simulations of Nonmonotone Networks , 1989, Theor. Comput. Sci..

[23]  Uri Zwick,et al.  Faster circuits and shorter formulae for multiple addition, multiplication and symmetric Boolean functions , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[24]  E. A. Lamagna,et al.  Combinational Complexity of Some Monotone Functions , 1974, SWAT.

[25]  Ingo Wegener.  Boolean Functions Whose Monotone Complexity is of Size n2/log n , 1981, Theoretical Computer Science.

[26]  S. S. Marchenkov.  Existence of finite bases in closed classes of Boolean functions , 1984 .

[27]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[28]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[29]  Kurt Mehlhorn,et al.  Monotone Switching Circuits and Boolean Matrix Product , 1975, MFCS.

[30]  D. Lubell.  A Short Proof of Sperner’s Lemma , 1966 .

[31]  Giulia Galbiati.  M. J. Fischer: On the Complexity of 2-Output Boolean Networks , 1981, Theor. Comput. Sci..

[32]  P. Dunne.  Relationships between monotone and non-monotone network complexity , 1992 .

[33]  Robert O. Winder,et al.  Enumeration of Seven-Argument Threshold Functions , 1965, IEEE Trans. Electron. Comput..

[34]  V. I. Terenkov.  The accuracy of algorithms for calculating estimates for tables generated by monotonic boolean functions , 1973 .

[35]  Johan Håstad,et al.  A Simple Lower Bound for Monotone Clique Using a Communication Game , 1992, Inf. Process. Lett..

[36]  William F. McColl,et al.  On the Planar Monotone Computation of Threshold Functions , 1985, STACS.

[37]  Ravi B. Boppana,et al.  Amplification of probabilistic boolean formulas , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[38]  Michael J. Fischer.  Hauptvortrag: The complexity of negation-limited networks - A brief survey , 1975, Automata Theory and Formal Languages.

[39]  Leslie M. Goldschlager,et al.  A Space Efficient Algorithm for the Monotone Planar Circuit Value Problem , 1980, Information Processing Letters.

[40]  Uri Zwick,et al.  Shrinkage of de Morgan formulae under restriction , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[41]  S. Mrowka,et al.  On The Cardinality of Closed Subsets , 1964 .

[42]  Beate Commentz-Walter.  Size-Depth Tradeoff in Boolean Formulas , 1978, ICALP.

[43]  Johan Håstad.  The shrinkage exponent is 2 , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[44]  J. A. Riley,et al.  Inessentiality in Minimal Networks and Formulas , 1962, IRE Trans. Electron. Comput..

[45]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[46]  Claus-Peter Schnorr,et al.  A Lower Bound on the Number of Additions in Monotone Computations , 1976, Theor. Comput. Sci..

[47]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[48]  Mauricio Karchmer,et al.  On proving lower bounds for circuit size , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[49]  Michael Ben-Or,et al.  A theorem on probabilistic constant depth Computations , 1984, STOC '84.

[50]  Wolfgang J. Paul.  Realizing Boolean Functions on Disjoint sets of Variables , 1976, Theor. Comput. Sci..

[51]  N. N. Katerinochkina.  Finding a maximal zero for some classes of monotone Boolean functions in Post's classification , 1990 .

[52]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[53]  Moshe Dubiner,et al.  Amplification and Percolation , 1992, FOCS 1992.

[54]  William F. McColl.  The Maximum Depth of Monotone Formulae , 1978, Inf. Process. Lett..

[55]  Ingo Wegener,et al.  On the Complexity of Slice Functions , 1984, Theor. Comput. Sci..

[56]  Andrew Chi-Chih Yao.  Circuits and local computation , 1989, STOC '89.

[57]  Claude E. Shannon,et al.  Reliable Circuits Using Less Reliable Relays , 1956 .

[58]  David R. Smith,et al.  Bounds on the Number of Threshold Functions , 1966, IEEE Trans. Electron. Comput..

[59]  Yu.A. Zuev,et al.  On the connection between linear inequalities and monotonic boolean functions , 1984 .

[60]  Leslie G. Valiant,et al.  Exponential lower bounds for restricted monotone circuits , 1983, STOC.

[61]  Douglas H. Wiedemann,et al.  A computation of the eighth Dedekind number , 1991 .

[62]  Sven Skyum.  A Measure in Which Boolean Negation is Exponentially Powerful , 1982 .

[63]  Paul. E. Dunne.  A 2.5 n lower bound on the monotone network complexity of T 3 n , 1985 .

[64]  Emil L. Post.  The two-valued iterative systems of mathematical logic , 1942 .

[65]  Nicholas Pippenger.  The realization of monotone Boolean functions (Preliminary Version) , 1976, STOC '76.

[66]  Claude E. Shannon,et al.  The Number of Two‐Terminal Series‐Parallel Networks , 1942 .

[67]  N. M. Riviere,et al.  Recursive formulas on free distributive lattices , 1968 .

[68]  Mike Paterson,et al.  Complexity of Monotone Networks for Boolean Matrix Product , 1974, Theor. Comput. Sci..

[69]  Robert O. Winder,et al.  Single stage threshold logic , 1961, SWCT.

[70]  Ingo Wegener,et al.  Relating Monotone Formula Size and Monotone Depth of Boolean Functions , 1983, Inf. Process. Lett..

[71]  Noam Nisan,et al.  The Effect of Random Restrictions on Formula Size , 1993, Random Struct. Algorithms.

[72]  Konrad Engel,et al.  Sperner theory in partially ordered sets , 1985 .

[73]  N. N. Katerinochkina.  THE SEARCH FOR THE MAXIMUM UPPER ZERO OF SOME SUBCLASSES OF MONOTONIC BOOLEAN FUNCTIONS , 1978 .

[74]  Valery A. Vardanian,et al.  On the length of single dynamic tests for monotone Boolean functions , 1985, International Symposium on Fundamentals of Computation Theory.

[75]  Michael J. Fischer,et al.  Omega(n log n) Lower Bounds on Length of Boolean Formulas , 1982, SIAM J. Comput..

[76]  R. E. Krichevskii,et al.  Complexity of Contact Circuits Realizing a Function of Logical Algebra , 1964 .

[77]  Russell Impagliazzo,et al.  Communication complexity towards lower bounds on circuit depth , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[78]  A. Razborov.  Lower bounds on monotone complexity of the logical permanent , 1985 .

[79]  Alexander A. Razborov,et al.  Applications of matrix methods to the theory of lower bounds in computational complexity , 1990, Comb..

[80]  Mike Paterson.  New bounds on formula size , 1977, Theoretical Computer Science.

[81]  Juraj Hromkovic.  On the number of monotonic functions from two-valued logic to k-valued logic , 1985, Kybernetika.

[82]  R. Dedekind,et al.  Über Zerlegungen von Zahlen Durch Ihre Grössten Gemeinsamen Theiler , 1897 .

[83]  Yu.A. Zuev,et al.  The lower bound of the number of inequalities which represent a monotonic boolean function of n variables , 1983 .

[84]  Michael J. Fischer,et al.  Boolean Matrix Multiplication and Transitive Closure , 1971, SWAT.

[85]  Ingo Wegener,et al.  Properties of Complexity Measures for PRAMs and WRAMs , 1986, MFCS.

[86]  Koichi Yamamoto.  Note on the Order of Free Distributive Lattices , 1953 .

[87]  Saburo Muroga,et al.  Enumeration of Threshold Functions of Eight Variables , 1970, IEEE Transactions on Computers.

[88]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[89]  Paul E. Dunne,et al.  The Complexity of Boolean Networks , 1988 .

[90]  A. I. Zelichenko.  The correspondence of monotonic boolean functions to systems of linear inequalities , 1979 .

[91]  Alexander A. Razborov,et al.  On Small Depth Threshold Circuits , 1992, SWAT.

[92]  Pavel Pudlák.  Bounds for Hodes-Specker theorem , 1983, Logic and Machines.

[93]  Ingo Wegener.  The critical complexity of all (monotone) Boolean functions and monotone graph properties , 1985, FCT.

[94]  Ravi B. Boppana,et al.  Threshold Functions and Bounded Depth Monotone Circuits , 1986, J. Comput. Syst. Sci..

[95]  M. den Nijs,et al.  A relation between the temperature exponents of the eight-vertex and q-state Potts model , 1979 .

[96]  Edmund A. Lamagna,et al.  The Complexity of Monotone Networks for Certain Bilinear Forms, Routing Problems, Sorting, and Merging , 1979, IEEE Transactions on Computers.

[97]  Vaughan R. Pratt,et al.  The power of negative thinking in multiplying Boolean matrices , 1974, STOC '74.

[98]  Harold N. Shapiro.  On the counting problem for monotone boolean functions , 1970 .

[99]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[100]  Ioannis Pitas,et al.  Nonlinear Digital Filters - Principles and Applications , 1990, The Springer International Series in Engineering and Computer Science.

[101]  Claus-Peter Schnorr.  Computation of the Boolean Matrix-Vector, AND/OR-Produkt in Average Time O(m + nlnn) , 1992, Informatik.

[102]  Ingo Wegener,et al.  Symmetric Functions in AC0A Can Be Computed in Constant Depth With Very Small Size , 1990, MFCS.

[103]  Paul Erdös,et al.  Extremal problems among subsets of a set , 1974, Discret. Math..

[104]  Saburo Muroga,et al.  Lower Bounds of the Number of Threshold Functions and a Maximum Weight , 1962, IEEE Trans. Electron. Comput..

[105]  A. Razborov.  On submodular complexity measures , 1992 .

[106]  A. A. Sapozhenko.  The search for a maximum upper zero of a monotone function on ranked sets , 1991 .

[107]  Éva Tardos,et al.  The gap between monotone and non-monotone circuit complexity is exponential , 1988, Comb..

[108]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[109]  Claude E. Shannon,et al.  A symbolic analysis of relay and switching circuits , 1938, Transactions of the American Institute of Electrical Engineers.

[110]  Johan Håstad,et al.  On the power of small-depth threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[111]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[112]  Jaikumar Radhakrishnan.  Better bounds for threshold formulas , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[113]  Edward J. Coyle,et al.  Stack filters , 1986, IEEE Trans. Acoust. Speech Signal Process..

[114]  G. P. Tonoyan.  The successive splitting of vertices of an n-dimensional unit cube into chains and decoding problems of monotonic boolean functions , 1979 .

[115]  Ingo Wegener,et al.  Proving lower bounds of the monotone complexity of Boolean functions , 1983, Logic and Machines.

[116]  V. M. Khrapchenko.  The complexity of the realization of symmetrical functions by formulae , 1972 .

[117]  Jaikumar Radhakrishnan,et al.  Directed Monotone Contact Networks for Threshold Functions , 1994, Inf. Process. Lett..

[118]  Qian-Ping Gu,et al.  Amplification of Bounded Depth Monotone Read-Once Boolean Formulae , 1991, SIAM J. Comput..

[119]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[120]  E. Gilbert.  Lattice Theoretic Properties of Frontal Switching Functions , 1954 .

[121]  V. A. Vardanyan.  Complexity of single dynamic tests for monotone Boolean functions , 1987 .

[122]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[123]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[124]  M. Sipser,et al.  Monotone complexity , 1992 .

[125]  Eiichi Goto,et al.  Some Theorems Useful in Threshold Logic for Enumerating Boolean Functions , 1962, IFIP Congress.

[126]  Ingo Wegener.  Best Possible Asymptotic Bounds on the Depth of Monotone Functions in Multivalued Logic , 1982, Inf. Process. Lett..

[127]  J. Friedman.  Constructing O(n log n) Size Monotone Formulae for the k-th Elementary Symmetric Polynomial of n Boolean Variables , 1984, FOCS.

[128]  Yu. I. Zhuravlev,et al.  An approach to optimization in a class of parametric recognition algorithms , 1976 .

[129]  Alexander A. Razborov,et al.  On the method of approximations , 1989, STOC '89.

[130]  A. Razborov,et al.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: Lower estimates of the dimension of schemes of bounded depth in the basis \{\&,\vee,\oplus\} , 1986 .

[131]  L. Lovász,et al.  The Work of A. A. Razborov , 1990 .

[132]  Leslie G. Valiant.  Negation is Powerless for Boolean Slice Functions , 1986, SIAM J. Comput..

[133]  Michael E. Saks,et al.  Size-depth trade-offs for threshold circuits , 1993, STOC.

[134]  Joel Friedman.  Constructing O(n log n) Size Monotone Formulae for the k-th Threshold Function of n Boolean Variables , 1986, SIAM J. Comput..

[135]  Yu. A. Zuev.  Representations of Boolean functions by systems of linear inequalities , 1985 .

[136]  S. Muroga,et al.  Theory of majority decision elements , 1961 .

[137]  Harry Kesten,et al.  Percolation Theory and First-Passage Percolation , 1987 .

[138]  O. B. Lupanov.  On the Influence of the Depth of Formulas on Their Complexity , 1971 .

[139]  A. D. Korshunov.  The Number and the Structure of Typical Sperner and K-Non-Separable Families of Subsets of a Finite Set , 1987, FCT.

[140]  Vince Grolmusz,et al.  A weight-size trade-off for circuits with MOD m gates , 1994, STOC '94.

[141]  Jürgen Weiss,et al.  An n^3/2 Lower Bound on the Monotone Network Complexity of the Boolean Convolution , 1984, Inf. Control..

[142]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[143]  Toshihide Ibaraki,et al.  The Maximum Latency and Identification of Positive Boolean Functions , 1994, ISAAC.

[144]  L. H. Harper,et al.  On the complexity of the marriage problem , 1972 .

[145]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

[146]  Leonard M. Adleman,et al.  Two theorems on random polynomial time , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[147]  Ingo Wegener,et al.  A new lower bound on the monotone network complexity of Boolean sums , 1980, Acta Informatica.

[148]  R. Tarjan.  Complexity of monotone networks for computing conjunctions , 1976 .