Showing papers on "Boolean function published in 1985"

Separating the polynomial-time hierarchy by oracles

Abstract: We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all the levels in the polynomial-time hierarchy are distinct, i.e., ΣkP,A is properly contained in Σk+1P,A for all k.

Boolean operations in solid modeling: Boundary evaluation and merging algorithms

Abstract: Solid modeling plays a key role in electromechanical CAD/CAM, three-dimensional computer graphics, computer vision, robotics, and other disciplines and activities that deal with spatial phenomena. Almost all contemporary solid modeling systems support Boolean operations akin to set intersection, union, and difference on solids. Boundary representations (face/edge/vertex structures) for solids defined through Boolean operations are generated in these modelers by using so-called boundary evaluation and boundary merging procedures. These are the most complex and delicate software modules in a solid modeler. This paper describes boundary evaluation algorithms used by the PADL solid modeling systems developed at the University of Rochester, and discusses other known approaches in terms of concepts that emerged from the PADL work, notably set membership classification and neighborhood manipulation.

On networks of noisy gates

Abstract: We show that many Boolean functions (including, in a certain sense, "almost all" Boolean functions) have the property that the number of noisy gates needed to compute them differs from the number of noiseless gates by at most a constant factor. This may be contrasted with results of von Neumann, Dobrushin and Ortyukov to the effect that (1) for every Boolean function, the number of noisy gates needed is larger by at most a logarithmic factor, and (2) for some Boolean functions, it is larger by at least a logarithmic factor.

Symbolic Manipulation of Boolean Functions Using a Graphical Representation

Abstract: In this paper we describe a data structure for representing Boolean functions and an associated set of manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to the representations of Lee and Akers, but with further restrictions on the ordering of decision variables in the graph. Although a function requires, in the worst case, a graph of size exponential in the number of arguments, many of the functions encountered in typical applications have a more reasonable representation. Our algorithms are quite efficient as long as the graphs being operated on do not grow too large. We present performance measurements obtained while applying these algorithms to problems in logic design verification.

A complexity theory based on Boolean algebra

Abstract: A projection of a Boolean function is a function obtained by substituting for each of its variables a variable, the negation of a variable, or a constant. Reducibilities among computational problems under this relation of projection are considered. It is shown that much of what is of everyday relevance in Turing-machine-based complexity theory can be replicated easily and naturally in this elementary framework. Finer distinctions about the computational relationships among natural problems can be made than in previous formulations and some negative results are proved.

Algorithmic aspects of the substitution decomposition in optimization over relations, set systems and Boolean functions

Abstract: In the last years, decomposition techniques have seen an increasing application to the solution of problems from operations research and combinatorial optimization, in particular in network theory and graph theory. This paper gives a broad treatment of a particular aspect of this approach, viz. the design of algorithms to compute the decomposition possibilities for a large class of discrete structures. The decomposition considered is thesubstitution decomposition (also known as modular decomposition, disjunctive decomposition, X-join or ordinal sum). Under rather general assumptions on the type of structure considered, these (possibly exponentially many) decomposition possibilities can be appropriately represented in acomposition tree of polynomial size. The task of determining this tree is shown to be polynomially equivalent to the seemingly weaker task of determining the closed hull of a given set w.r.t. a closure operation associated with the substitution decomposition. Based on this reduction, we show that for arbitrary relations the composition tree can be constructed in polynomial time. For clutters and monotonic Boolean functions, this task of constructing the closed hull is shown to be Turing-reducible to the problem of determining the circuits of the independence system associated with the clutter or the prime implicants of the Boolean function. This leads to polynomial algorithms for special clutters or monotonic Boolean functions. However, these results seem not to be extendable to the general case, as we derive exponential lower bounds for oracle decomposition algorithms for arbitrary set systems and Boolean functions.

Unbiased bits from sources of weak randomness and probabilistic communication complexity

Abstract: We introduce a general model for physical sources or weak randomness. Loosely speaking, we view physical sources as devices which output strings according to probability distributions in which no single string is too probable. The main question addressed is whether it is possible to extract alrnost unbiased random bits from such "probability bounded" sources. We show that most or the functions can be used to extract almost unbiased and independent bits from the output of any two independent "probability-bounded" sources. The number of extractable bits is within a constant factor of the information theoretic bound. We conclude this paper by establishing further connections between communication complexity and the problem discussed above. This allows us to show that most Boolean functions have linear communication complexity in a very strong sense.

Symbolic Verification of MOS Circuits

Abstract: The program MOSSYM simulates the behavior of a MOS circuit represented as a switch-level network symbolically. That is, during simulator operation the user can set an input to either 0, 1, or a Boolean variable. The simulator then computes the behavior of the circuit as a function of the past and present input variables. By using heuristically efficient Boolean function manipulation algorithms, the verification of a circuit by symbolic simulation can proceed much more quickly than by exhaustive logic simulation. In this paper we present our concept of symbolic simulation, derive an algorithm for switch-level symbolic simulation, and present experimental measurements from MOSSYM. 1This paper was presented at the 1985 Chapel Hill Conference on VLSI. 2This research was funded by the Defense Advanced Research Contracts Agency ARPA Order Number 3597.

A fast algorithm for the Boolean masking problem

Abstract: An algorithm is presented for the calculation of Boolean combinations between layers of a VLSI circuit layout. Each layer is assumed to contain only polygons, which are specified by their edges; the output is also polygonal. The algorithm runs in O((n + k)(r + log n)) time and O(nr) space, where n is the total number of edges on all layers, k is the number of edge intersections, and r is the number of layers. Also a number of restrictions on the general problem are discussed which lead to substantial improvements in the time bounds. It is proved that when the polygons are presented using a hierarchical description language the problem becomes NP-hard. Finally how this approach can be used to solve the i-contour problem of computational geometry and the hidden-line-elimination problem of computer graphics is discussed.

Amplification of probabilistic boolean formulas

Abstract: The amplification of probabilistic Boolean formulas refers to combining independent copies of such formulas to reduce the error probability. Les Valiant used the amplification method to produce monotone Boolean formulas of size O(n5.3) for the majority function of n variables. In this paper we show that the amount of amplification that Valiant obtained is optimal. In addition, using the amplification method we give an O(k4.3 n log n) upper bound for the size of monotone formulas computing the kth threshold function of n variables.

Computer Design and Architecture

Abstract: Introduction - computer system organization combinational logic synchronous sequential circuits memory and storage a simple computer - organization and programming a simple computer -hardware design input/output processor and system structures memory system enhancement control unit enhancement arithmetic/logic unit enhancement advanced architectures. Appendices: number systems and codes minimization of Boolean functions stack implementation.

Identification is easier than decoding

Abstract: Several questions related to the complexity of communication over channels with noise are addressed. We compare some of our results to wellknown results in information theory. In particular we compare the following two problems. Assuming that the communication channel between two processors P1 and P2 makes an error with probability e≫0, the identification problem is to determine whether P1 and P2 have the same n-bit integer. The decoding problem is for P2 to determine the n-bit integer of P1. For the latter problem we show that given any arbitrarily large constant λ≫0, there exists an e, 0≪e≪1/2, for which no scheme requiring less than λn bits of communication can guarantee (for large n) any bound q≪1 on the error probability. On the other hand, given any arbitrarily small constant γ≫0 and any e, 0≪e≪1/2, the identification problem can be solved with (1+γ)n bits of (one-way) communication with an error probability bounded by c2-αn, where c and α are positive constants. These techniques are extended to other problems, and a one-bit output Boolean function is shown to exhibit a similar behavior to that of the decoding problem regardless of how the input bits are partitioned among the two processors.

On one criterion of the optimality of an algorithm for evaluating monotonic Boolean functions

Abstract: A criterion of the optimality of an algorithm for evaluating monotonic Boolean functions, which is different from the Shannon criterion, is considered and its practical significance is proved. Upper and lower estimates are obtained for the efficiency of the algorithm for evaluating monotonic Boolean functions which is optimal with respect to the criterion which has been introduced. An algorithm is constructed for evaluating the class of monotonic Boolean functions which are generated by imcompatible systems of linear ineaqualities. This algorithm is optimal with respect to the criterion introduced in this paper, the Shannon criterion, and a number of other criteria subject to certain additional conditions.

Boolean Integral Calculus for Digital Systems

Abstract: The concept of Boolean integration is introduced and developed. When the changes in a desired function are specified in terms of changes in its arguments, then ways of "integrating" (i.e., realizing) the function, if it exists, are presented. Boolean integral calculus has applications in design of logic circuits.

Planar circuits have short specifications

Abstract: A counting argument is used to establish a lower bound of Ω(2n) on the planar circuit size of almost all n-argument Boolean functions. The counting argument exploits the fact that planar circuits can be more concisely specified than general circuits.

The critical complexity of all (monotone) Boolean functions and monotone graph properties

Abstract: CREW-PRAM's build a powerful model of parallel computers. Cook/Dwork/Reischuk proved that the CREW-PRAM complexity of Boolean functions is bounded below by logbc(f) where b ≈ 4.79 and c(f) is the critical complexity of f. This lower bound is often even tight. For a class of functions F the critical complexity c(F), the minimum of all c(f) where f ∈ F, is the best general lower bound on the critical complexity of all f ∈ F. We determine the critical complexity of the set of all nondegenerate Boolean functions and all monotone nondegenerate Boolean functions up to a small additive term. And we compute exactly the critical complexity of the class of all monotone graph properties proving partially a conjecture of Turan.

A Method for Improving Cascode-Switch Macro Wirability

Abstract: In this paper, a problem in macro design using cascode-switch tree logic is studied. It involves selecting specific tree instantiations of Boolean functions and input variable assignments to maximize the alignment of variables between adjacent trees. An algorithm to find optimal solutions based on the principle of optimality is proposed. Although in general it is not a polynomial time algorithm, it runs sufficiently fast for our practical application. Finally we prove the problem is NP-complete, thus the existence of polynomial time algorithms is indeed unlikely.

The McBoole Logic Minimizer

Abstract: A new logic minimization algorithm is presented. It finds a minimal cover for a multiple-output boolean function expressed as a list of cubes. A directed graph is used to speed up the selection of a minimal cover. Covering cycles are partitioned and branched independently to reduce greatly the branching depth. The resulting minimized list of cubes is guaranteed to be minimal in the sense that no cover with less cubes can exist. The dont care at output is handled properly. This algorithm was implemented in C under UNIX BSD4.2. An extensive comparison with ESPRESSO IIC shows that the new algorithm is particularly attractive for functions with less than 20 input and 20 output variables.

Binary-decision graphs for implementation of Boolean functions

Abstract: Binary decision graphs provide the most efficient means for computing Boolean functions by program, and also relate to hardware implementation In the present paper, formal properties of binary decision graphs are considered, and rules are provided for the anticipation of path lengths and vertex requirements, which indicate computation time and program length, respectively Rules for the construction of efficient (but not truly optimal) binary decision graphs are provided, these rules are then applied in design examples

Replaceability and computational equivalence for monotone Boolean functions

Abstract: Replacement rules have played an important role in the study of monotone boolean function complexity. In this paper, notions of replaceability and computational equivalence are formulated in an abstract algebraic setting, and examined in detail for finite distributive lattices — the appropriate algebraic context for monotone boolean functions. It is shown that when computing an element f of a finite distributive lattice D, the elements of D partition into classes of computationally equivalent elements, and define a quotient of D in which all intervals of the form [t ∧ f, t ∨ f] are boolean. This quotient is an abstract simplicial complex with respect to ordering by replaceability. Other results include generalisations and extensions of known theorems concerning replacement rules for monotone boolean networks. Possible applications of computational equivalence in developing upper and lower bounds on monotone boolean function complexity are indicated, and new directions of research both abstract mathematical and computational, are suggested.

PHIPLA--A New Algorithm for Logic Minimization

Abstract: PHIPLA, a new algorithm for logic minimization, is presented. The algorithm sets out to find optimal sum-of-products representations for a set of Boolean functions, thus contributing to area minimization of the Programmable Logic Array corresponding to the set of functions. The results of a comparative study of PHIPLA and two other algorithms, SPAM and PRESTOL-II, are presented. From these results it is concluded that PHIPLA generates representations which are competitive with those generated by SPAM and PRESTOL-II, whilst the algorithm is extremely fast for small problems (up to 12 variables).

Multithreshold logic circuits implemented with operational amplifiers

Abstract: The paper presents a procedure for the synthesis or multithreshold circuits with up to four thresholds. The circuit has three operational amplifiers as its kernel. Given the weight-threshold vector of the boolean function which has to be obtained, the values of the different elements of the circuit are easily obtained. Noise immunity has been computed and has an acceptable value. One example of synthesis is included.