Size Formula Research Articles

A note on the formula size of the “mod k” functions

We give an elegant construction of short formulas for some of the symmetric Boolean functions.

Decomposition of graphs and monotone formula size of homogeneous functions

We show that every graph on n nodes can be partitioned by a number of complete bipartite graphs with O(n 2/log n) nodes with no edge belonging to two of them. Since each partition corresponds directly to a monotone formula for the associated quadratic function we obtain an upper bound for the monotone formula size of quadratic functions. Our method extends to uniform hypergraphs with fixed range and thus to homogeneous functions with fixed length of prime implicants. Finally we give an example of a quadratic function where each monotone formula built from arbitrary partitions of the graph (double edges allowed) is not optimal. That means we disprove the single-level-conjecture for formulae.

Nearly optimal hierarchies for network and formula size

How large are the “gaps” in the complexity hierarchies for Boolean functions with respect to network size and formula size? A gap is a non-empty interval of integers none of which is the complexity of any Boolean function. It is shown for the most natural bases that there are no gaps at all over a broad range of values and that the largest gap anywhere is less than n.

A Lower Bound for the Formula Size of Rational Functions

We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Nečiporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Nečiporuk’s method to rational functions over finite fields. These results are then used to show that, over any field, the $n \times n$ determinant function has formula size at least $\Omega (n^3 )$. We thus have an algebraic analogue to the $\Omega (n^3 )$ lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 1537–1540].

New formula for gravitational radiation energy loss in general relativity.

Field contributions and the Gauss theorem for retarded functions lead to a new formula for gravitational radiation energy loss. It clarifies the applicability of the quadrupole formula for systems with ${\mathrm{T}}^{\mathrm{ik}}$-dominated motion. Applied to the axially symmetric two-body problem with equation of state \ensuremath{\epsilon}=\ensuremath{\epsilon}(P) for short free-fall times, the new formula yields the bulk-motion quadrupole-formula result. This contrasts our earlier larger flux with unconventional equation of state and that of critics claiming a quadrupole formula flux from internal motions. An order of magnitude estimate yields potential corrections of quadrupole formula size after \ensuremath{\sim}${10}^{2}$ yr for the binary pulsar.

Optimal decision trees and one-time-only branching programs for symmetric Boolean functions

Combinational complexity and depth are the most important complexity measures for Boolean functions. It has turned out to be very hard to prove good lower bounds on the combinational complexity or the depth of explicitly defined Boolean functions. Therefore one has restricted oneself to models where nontrivial lower bounds are easier to prove. Here decision trees, branching programs, and one-time-only branching programs are considered, where each variable may be tested on each path of computation only once. Efficient algorithms for the construction of optimal decision trees and optimal one-time-only branching programs for symmetric Boolean functions are presented. Furthermore, the following trade-off results are proved. An exponential lower bound on the decision tree complexity of some Boolean function is shown having linear formula size and linear one-time-only branching program complexity. Furthermore, a quadratic lower bound on the one-time-only branching program complexity of some Boolean function is shown having linear combinational complexity.

New lower bounds on the formula size of Boolean functions

In this paper we investigate the method of Nechiporuk [3] for deriving lower bounds on the formula size of Boolean functions. At first we prove non-linear lower bounds for functions which are related to the existence of a k-clique or a k-circle in a graph. Furthermore we determine the formula size of the "disjoint disjunction" of the outputs of the Boolean matrix product. Finally we show how useful the method may be in the case of monotone functions if the length of the prime implicants is bounded.

Size-depth tradeoff in non-monotone Boolean formulae

Formula size and depth are two important complexity measures of Boolean functions. We study the tradeoff between those two measures: We give an infinite set of Boolean functions and show for nearly each of them: There is no formula over "and", "or", "negation" computing it optimal with respect to both measures. That implies a logarithmic lower bound on circuit depth.

Size-depth tradeoff in monotone Boolean formulae

Formula size and depth are two important complexity measures of Boolean functions. We study the tradeoff between those two measures: We give an infinite set of Boolean functions and show for nearly each of them: There is no monotone formula computing it optimal with respect to both measures. We give a lower and upper bound on the product of size and depth of monotone formulae computing our functions. That implies, moreover, a logarithmic lower bound on circuit depth.

The circuit depth of symmetric boolean functions

Let S n be the set of symmetric Boolean functions of n arguments. The depth of every NAND circuit which evaluates a nonconstant function in S n is at least 2 log 2 n − α, where α = 2 log 23 − 1 ≅ 2 · 17. For the basis {NAND, →} a lower bound of 1 · 44log 2 n − β is obtained, where β = 3 − log φ 5 1 2 ≅ 1 · 328. [φ = (1 + 5 1 2 )/2 is the golden (Fibonacci) ratio.] The coefficient 1 · 44 is precisely given by log φ2. These results follow from general criteria relating circuit depth to the size of implicants. For certain symmetric functions, these lower bounds could not be derived from corresponding bounds on formula size. All but eight of the functions in S n require unate formulae of size Ω(n · log 2n) . Each of the eight exceptions has a formula of size at most 2 n.

A $2.5n$-Lower Bound on the Combinational Complexity of Boolean Functions

Consider the combinational complexity $L(f)$ of Boolean functions over the basis $\Omega = \{ f|f:\{ 0,1\} ^2 \to \{ 0,1\} \} $. A new method for proving linear lower bounds of size $2n$ is presented. Combining it with methods presented in Savage [13, (1974)] and Schnorr [18, (1974)], we establish for a special sequence of functions $f_n :\{ 0,1\} ^{n + 2\log (n) + 1} \to \{ 0,1\} :2.5n \leqq L(f) \leqq 6n$. Also a trade-off result between circuit complexity and formula size is derived.

Paper 12: A Comparison of Performance Predictions for Steadily Loaded Journal Bearings

Various predictions of the performance of steadily loaded circular journal bearings, running in the laminar regime, are compared. The values of load, friction, peak pressure, and oil flow given result from the iterative numerical solution of the Reynolds equation subject to a number of different boundary conditions, which attempt to allow for oil-film disruption. The boundary conditions considered are: (1) That only the positive pressure region of the 360-degree full-film solution contributes to load carrying. (2) That the full-film is bounded, both at breakdown and at build-up, by the conditions that pressure and pressure derivative are zero. (3) That the pressure is zero along the line of maximum film thickness, this being the assumed build-up boundary, and the breakdown occurs when the pressure and pressure derivative fall to zero. (4) That the build-up boundary arises when the flow balance, allowing for the oil feed system, indicates a full-film, breakdown occurring as in (2) and (3) above. The influence of mesh size, convergence limit, and finite difference formulae on the accuracy of the solutions is also discussed. Finally a comparison is made with the results from the analytical solutions due to Sommerfeld and Ocvirk, which show large errors in load and in peak pressure. This led the authors to devise a new and accurate procedure for the prediction of load and peak pressure for finite bearings using simple formulae.