Showing papers on "Boolean function published in 1990"

Efficient implementation of a BDD package

Abstract: Efficient manipulation of Boolean functions is an important component of many computer-aided design tasks This paper describes a package for manipulating Boolean functions based on the reduced, ordered, binary decision diagram (ROBDD) representation The package is based on an efficient implementation of the if-then-else (ITE) operator A hash table is used to maintain a strong canonical form in the ROBDD, and memory use is improved by merging the hash table and the ROBDD into a hybrid data structure A memory function for the recursive ITE algorithm is implemented using a hash-based cache to decrease memory use Memory function efficiency is improved by using rules that detect when equivalent functions are computed The usefulness of the package is enhanced by an automatic and low-cost scheme for recycling memory Experimental results are given to demonstrate why various implementation trade-offs were made These results indicate that the package described here is significantly faster and more memory-efficient than other ROBDD implementations described in the literature

Fair Computation of General Functions in Presence of Immoral Majority

Abstract: This paper describes a method for n players, a majority of which may be faulty, to compute correctly, privately, and fairly any computable function f(x1,... ,xn) where xi, is the input of the i-th. player. The method uses as a building block an oblivious transfer primitive.Previous methods achieved these properties, only for boolean functions, which, in particular, precluded composition of such protocols.We also propose a simpler definition of security for multi-player protocols which still implies previous definitions of privacy and correctness.

Nonlinearity criteria for cryptographic functions

Abstract: Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design. The classification is set up in terms of the largest transformation group leaving a criterion invariant. In this respect two criteria turn out to be of special interest, the distance to linear structures and the distance to affine functions, which are shown to be invariant under all affine transformations. With regard to these criteria an optimum class of functions is considered. These functions simultaneously have maximum distance to affine functions and maximum distance to linear structures, as well as minimum correlation to affine functions. The functions with these properties are proved to coincide with certain functions known in combinatorial theory, where they are called bent functions. They are shown to have practical applications for block ciphers as well as stream ciphers. In particular they give rise to a new solution of the correlation problem.

Shared binary decision diagram with attributed edges for efficient Boolean function manipulation

Abstract: The efficiency of Boolean function manipulation depends on the form of representation of Boolean functions. Binary decision diagrams (BDDs) are graph representations proposed by S.B. Akers (1978) and R.E. Bryant (1985). BDDs have some properties which can be used to enable efficient Boolean function manipulation. The authors describe a technique of more efficient Boolean function manipulation that uses shared binary decision diagrams (SBDDs) with attributed edges. The implements include an ordering algorithm of input variables and a method of handling 'don't care'. A Boolean function manipulator using the above methods is developed and it is shown that the manipulator is very efficient in terms of speed and storage. >

NOVA: state assignment of finite state machines for optimal two-level logic implementation

Abstract: The problem of encoding the states of a synchronous finite state machine (FSM) so that the area of a two-level implementation of the combinational logic is minimized is addressed. As in previous approaches, the problem is reduced to the solution of the combinatorial optimization problems defined by the translation of the cover obtained by a multiple-valued logic minimization or by a symbolic minimization into a compatible Boolean representation. The authors present algorithms for this solution, based on a novel theoretical framework that offers advantages over previous approaches to develop effective heuristics. The algorithms are part of NOVA, a program for optimal encoding of control logic. Final areas averaging 20% less than other state assignment programs and 30% less than the best random solution have been obtained. Literal counts averaging 30% less than the best random solutions have been obtained. >

Hiding instances in multioracle queries

Abstract: Abadi, Feigenbaum, and Kilian have considered instance-hiding schemes [1]. Let f be a function for which no randomized polynomial-time algorithm is known; randomized polynomial-time machine A wants to query an oracle B for f to obtain f(x), without telling B exactly what x is. It is shown in [1] that, if f is an NP-hard function, A cannot query a single oracle B while hiding all but the size of the instance, assuming that the polynomial hierarchy does not collapse. This negative result holds for all oracles B, including those that are non-r.e.

Boolean Reasoning: The Logic of Boolean Equations

Abstract: 1 Fundamental Concepts.- 1.1 Formulas.- 1.2 Propositions and Predicates.- 1.3 Sets.- 1.4 Operations on Sets.- 1.5 Partitions.- 1.6 Relations.- 1.7 Functions.- 1.8 Operations and Algebraic Systems.- 2 Boolean Algebras.- 2.1 Postulates for a Boolean Algebra.- 2.2 Examples of Boolean Algebras.- 2.2.1 The Algebra of Classes (Subsets of a Set).- 2.2.2 The Algebra of Propositional Functions.- 2.2.3 Arithmetic Boolean Algebras.- 2.2.4 The Two-Element Boolean Algebra.- 2.2.5 Summary of Examples.- 2.3 The Stone Representation Theorem.- 2.4 The Inclusion-Relation.- 2.4.1 Intervals.- 2.5 Some Useful Properties.- 2.6 n-Variable Boolean Formulas.- 2.7 n-Variable Boolean Functions.- 2.8 Boole's Expansion Theorem.- 2.9 The Minterm Canonical Form.- 2.9.1 Truth-tables.- 2.9.2 Maps.- 2.10 The Lowenheim-Muller Verification Theorem.- 2.11 Switching Functions.- 2.12 Incompletely-Specified Boolean Functions.- 2.13 Boolean Algebras of Boolean Functions.- 2.13.1 Free Boolean Algebras.- 2.14 Orthonormal Expansions.- 2.14.1 Lowenheim's Expansions.- 2.15 Boolean Quotient.- 2.16 The Boolean Derivative.- 2.17 Recursive Definition of Boolean Functions.- 2.18 What Good are "Big" Boolean Algebras?.- 3 The Blake Canonical Form.- 3.1 Definitions and Terminology.- 3.2 Syllogistic & Blake Canonical Formulas.- 3.3 Generation of BCF(f).- 3.4 Exhaustion of Implicants.- 3.5 Iterated Consensus.- 3.5.1 Quine's method.- 3.5.2 Successive extraction.- 3.6 Multiplication.- 3.6.1 Recursive multiplication.- 3.6.2 Combining multiplication and iterated consensus.- 3.6.3 Unwanted syllogistic formulas.- 4 Boolean Analysis.- 4.1 Review of Elementary Properties.- 4.2 Boolean Systems.- 4.2.1 Antecedent, Consequent, and Equivalent Systems.- 4.2.2 Solutions.- 4.3 Reduction.- 4.4 The Extended Verification Theorem.- 4.5 Poretsky's Law of Forms.- 4.6 Boolean Constraints.- 4.7 Elimination.- 4.8 Eliminants.- 4.9 Rudundant Variables.- 4.10 Substitution.- 4.11 The Tautology Problem.- 4.11.1 Testing for Tautology.- 4.11.2 The Sum-to-One Theorem.- 4.11.3 Nearly-Minimal SOP Formulas.- 5 Syllogistic Reasoning.- 5.1 The Principle of Assertion.- 5.2 Deduction by Consensus.- 5.3 Syllogistic Formulas.- 5.4 Clausal Form.- 5.5 Producing and Verifying Consequents.- 5.5.1 Producing Consequents.- 5.5.2 Verifying Consequents.- 5.5.3 Comparison of Clauses.- 5.6 Class-Logic.- 5.7 Selective Deduction.- 5.8 Functional Relations.- 5.9 Dependent Sets of Functions.- 5.10 Sum-to-One Subsets.- 5.11 Irredundant Formulas.- 6 Solution of Boolean Equations.- 6.1 Particular Solutions and Consistency.- 6.2 General Solutions.- 6.3 Subsumptive General Solutions.- 6.3.1 Successive Elimination.- 6.3.2 Deriving Eliminants from Maps.- 6.3.3 Recurrent Covers and Subsumptive Solutions.- 6.3.4 Simplified Subsumptive Solutions.- 6.3.5 Simplification via Marquand Diagrams.- 6.4 Parametric General Solutions.- 6.4.1 Successive Elimination.- 6.4.2 Parametric Solutions based on Recurrent Covers.- 6.4.3 Lowenheim's Formula.- 7 Functional Deduction.- 7.1 Functionally Deducible Arguments.- 7.2 Eliminable and Determining Subsets.- 7.2.1 u-Eliminable Subsets.- 7.2.2 u-Determining Subsets.- 7.2.3 Calculation of Minimal u-Determining Subsets.- 8 Boolean Identification.- 8.1 Parametric and Diagnostic Models.- 8.1.1 Parametric Models.- 8.1.2 The Diagnostic Axiom.- 8.1.3 Diagnostic Equations and Functions.- 8.1.4 Augmentation.- 8.2 Adaptive Identification.- 8.2.1 Initial and Terminal Specifications.- 8.2.2 Updating the Model.- 8.2.3 Effective Inputs.- 8.2.4 Test-Procedure.- 9 Recursive Realizations of Combinational Circuits.- 9.1 The Design-Process.- 9.2 Specifications.- 9.2.1 Specification-Formats.- 9.2.2 Consistent Specifications.- 9.3 Tabular Specifications.- 9.4 Strongly Combinational Solutions.- 9.5 Least-Cost Recursive Solutions.- 9.6 Constructing Recursive Solutions.- 9.6.1 The Procedure.- 9.6.2 An Implementation using BORIS.- A Syllogistic Formulas.- A.1 Absorptive Formulas.- A.2 Syllogistic Formulas.- A.3 Prime Implicants.- A.4 The Blake Canonical Form.

Finding the optimal variable ordering for binary decision diagrams

Abstract: The ordered binary decision diagram is a canonical representation for Boolean functions, presented by R.E. Bryant (1985) as a compact representation for a broad class of interesting functions derived from circuits. However, the size of the diagram is very sensitive to the choice of ordering on the variables; hence, for some applications, such as differential cascode voltage switch (DCVS) trees, it becomes extremely important to find the ordering leading to the most compact representation. An algorithm for this problem with time complexity O(n/sup 2/3/sup n/) is presented. This represents an improvement over the previous best algorithm. >

Harmonic Analysis of Polynomial Threshold Functions

Abstract: The analysis of linear threshold Boolean functions has recently attracted the attention of those interested in circuit complexity as well as ofthose interested in neural networks. Here a generalization oflinear threshold functions is defined, namely, polynomial threshold functions, and its relation to the class of linear threshold functions is investigated. A Boolean function is polynomial threshold if it can be represented as a sign function ofa polynomial that consists ofa polynomial (in the number ofvariables) number ofterms. The main result ofthis paper is showing that the class ofpolynomial threshold functions (which is called PT1 is strictly contained in the class ofBoolean functions that can be computed by a depth 2, unbounded fan-in polynomial size circuit of linear threshold gates (which is called LT2). Harmonic analysis ofBoolean functions is used to derive a necessary and sufficient condition for a function to be an S-threshold function for a given set S of monomials. This condition is used to show that the number of different S-threshold functions, for a given S, is at most 2 t'/ 1)lsl. Based on the necessary and sufficient condition, a lower bound is derived on the number of terms in a threshold function. The lower bound is expressed in terms of the spectral representation of a Boolean function. It is found that Boolean functions having an exponentially small spectrum are not polynomial threshold. A family of functions is exhibited that has an exponentially small spectrum; they are called "semibent" functions. A function is constructed that is both semibent and symmetric to prove thatPT is properly contained in LT2.

Chortle: a technology mapping program for lookup table-based field programmable gate arrays

Abstract: Field Programmable Gate Arrays are new devices that combine the versatility of a Gate Array with the user-programmability of a PAL. This paper describes an algorithm for technology mapping of combinational logic into Field Programmable Gate Arrays that use lookup table memories to realize combinational functions. It is difficult to map into lookup tables using previous techniques because a single lookup table can perform a large number of logic functions, and prior approaches require each function to be instantiated separately in a library. The new algorithm, implemented in a program called Chortle uses the fact that a K-input lookup table can implement any Boolean function of K-inputs, and so does not require a library-based approach. Chortle takes advantage of this complete functionality to evaluate all possible decompositions of the input Boolean network nodes. It can determine the optimal (in area) mapping for fanout-free trees of combinational logic. In comparisons with the MIS II technology mapper, on MCNC-89 Logic Synthesis benchmarks Chortle achieves superior results in significantly less time. 1

Applications of matrix methods to the theory of lower bounds in computational complexity

Abstract: We present some criteria for obtaining lower bounds for the formula size of Boolean functions. In the monotone case we get the boundn Ω(logn) for the function “MINIMUM COVER” using methods considerably simpler than all previously known. In the general case we are only able to prove that the criteria yield an exponential lower bound when applied to almost all functions. Some connections with graph complexity and communication complexity are also given.

Verifying Temporal Properties of Sequential Machines Without Building their State Diagrams

Abstract: This paper presents the algorithm we have developed for proving that a finite state machine holds some properties expressed in temporal logic. This algorithm does not require the building of the state-transition graph nor the transition relation of the machine, so it overcomes the limits of the methods that have been proposed in the past. The verification algorithm presented here is based on Boolean function manipulations, which are represented by typed decision graphs. Thanks to this canonical representation, all the operations used in the algorithm have a polynomial complexity, expect for one called the computation of the “critical term”. The paper proposes techniques that reduce the computational cost of this operation.

The use of observability and external don't cares for the simplification of multi-level networks

Abstract: An algorithm is given for computing subsets of the observability don't cares at the nodes of a multilevel Boolean network. These subsets are based on an extension of the methods introduced by S. Muroga et al. (IEEE Trans. on Computers, Oct. 1989) for computing compatible sets of permissible functions (CSPFs) at the nodes of networks composed of NOR gates. The extensions presented are in four directions: an arbitrary logic function is allowed at any node, the don't cares are expressed in terms of both primary inputs and intermediate variables, a new ordering scheme is used. and maximal CSPFs are computed. These ideas are incorporated in an algorithm designed to take full advantage of the power of two-level minimization in multilevel logic synthesis systems. This has been implemented in MIS-II, and results are presented that demonstrate the effectiveness of these techniques. >

On the ability of the optimal perceptron to generalise

Abstract: A linearly separable Boolean function is derived from a set of examples by a perceptron with optimal stability. The probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method.

Technology mapping using Boolean matching and don't care sets

Abstract: The authors describe a new approach to technology mapping where matchings are recognized by means of Boolean operations. The matching algorithm uses tautology checking based on Shannon decompositions. They show how to use the symmetry and unateness properties to speed-up the Boolean matching algorithm. They examine how don't care information can be used during Boolean matching. The algorithms have been implemented in program Ceres and tested on the 1989 MCNC benchmark circuits. >

Improving a Network Generalization Ability by Selecting Examples

Abstract: We show that the generalization ability of simple Perceptron-like devices is strongly enhanced by allowing the network itself to select the training examples. Analytic and numerical results are obtained for the Hebb and for the optimal Perceptron learning rule, respectively.

An improved minimizing algorithm for sum of disjoint products (reliability theory)

Abstract: The Abraham-Locks-revised (ALR) sum-of-disjoint products (SDP) algorithm is an efficient method for obtaining a system reliability formula. The author describes a minor modification of the ALR algorithm called the Abraham-Locks-Wilson (ALW) method. The new feature is an alternative method of ordering paths and terms. ALW obtains a shorter disjoint system formula on a test example than any previous SDP method and allows small computational savings in processing large paths of complex networks. As there are different ways to obtain a reliability formula it is useful to use an approach which yields the smallest formula relative to computational effort expended. The extra effort in ordering the terms should be reasonably small and usually leads to improved efficiency in the later stages of the algorithm. ALW allows the analyst to operate in a more efficient way on many problems, particularly if the overlap ordering is used in the early stages of processing but is probably ignored for terms that contain a majority of the Boolean variables. >

A parallel algorithm for constructing binary decision diagrams

Abstract: A parallel algorithm for constructing binary decision diagrams is described. The algorithms treats binary decision graphs as minimal finite automata. The automation for a Boolean function with AND as its main operation (OR operation) is obtained by forming the intersection (union) of the regular sets associated with its operands. The union and intersection operations are implemented by a product construction on the minimal automata for the regular sets. After each product construction step the automaton must be reminimized. The parallel algorithm is designed so that it is possible to find the minimal representations for several Boolean operations in parallel. The level of each operation is determined. Operations at the same level can be performed in parallel without any communication between processors. If there are relatively few operations in one level, then the product generation step is divided into several suboperations and the results are merged. >

Algorithms for library-specific sizing of combinational logic

Abstract: Examined is a problem of choosing the proper sizes from a cell library for the logic elements of a Boolean network to meet timing constraints on the propagation delay along every path from the primary input to the primary output. It is shown that, if the Boolean network has a tree topology, there exists a pseudo-polynomial time algorithm for finding the optimal solution to this problem. A backtracking-based algorithm for finding feasible solutions for networks that are not trees is also suggested and evaluated. >

Enumerating boolean functions of cryptographic significance

Abstract: In this paper we describe applications of functions from GF(2) m onto GF(2) n in the design of encryption algorithms. If such a function is to be useful it must satisfy a set of criteria, the actual definition of which depends on the type of encryption technique involved. This in turn means that it is important to ensure that the selected criteria do not restrict the choice of function too severely, i.e., the set of functions must be enumerated. We discuss some of the possible sets of criteria and then give partial results on the corresponding enumeration problems. Many open problems remain, some of them corresponding to well-known hard enumeration questions.

An entropy measure for the complexity of multi-output Boolean functions

Abstract: Entropy measures are examined in view of the current logic synthesis methodology. The complexity of a Boolean function can be expressed in terms of computational work. Experimental data are presented in support of the entropy definition of computational work based upon the input-output description of a Boolean function. These data show a linear relationship between the computational work and the average number of literals in a multilevel implementation. The investigation includes single-output and multioutput function with and without don't care states. The experiments conducted on a large number of randomly generated functions showed that the effect of don't cares is to reduce the computational work. For several finite state machine benchmarks, the computational work gave a good estimate of the size of the circuit. Circuit delay is shown to have a nonlinear relationship to the computational work. >

A Growth Algorithm for Neural Network Decision Trees

Abstract: This paper explores the application of neural network principles to the construction of decision trees from examples We consider the problem of constructing a tree of perceptrons able to execute a given but arbitrary Boolean function defined on Ni input bits We apply a sequential (from one tree level to the next) and parallel (for neurons in the same level) learning procedure to add hidden units until the task in hand is performed At each step, we use a perceptron-type algorithm over a suitable defined input space to minimize a classification error The internal representations obtained in this way are linearly separable Preliminary results of this algorithm are presented

Segmentation as the search for the best description of the image in terms of primitives

Abstract: A paradigm is presented for the segmentation of images into piecewise continuous patches. Data aggregation is performed via model recovery in terms of variable-order bivariate polynomials using iterative regression. All the recovered models are candidates for the final description of the data. Selection of the models is achieved through a maximization of the quadratic Boolean problem. The procedure can be adapted to prefer certain kinds of descriptions (one which describes more data points, or has smaller error, or has a lower order model). A fast optimization procedure for model selection is discussed. The approach combines model extraction and model selection in a dynamic way. Partial recovery of the models is followed by the optimization (selection) procedure where only the best models are allowed to develop further. The results are comparable with the results obtained when using the selection module only after all the models are fully recovered, while the computational complexity is significantly reduced. The procedure was tested on real range and intensity images. >

Estimation of power dissipation in CMOS combinational circuits

Abstract: It is shown that a simplified model of power dissipation relates maximizing dissipation to maximizing gate output activity, appropriately weighted to account for differing load capacitances. To find the input or input sequence that maximizes the weighted activity, algorithms are given for transforming the problem to a weighted max-satisfiability problem, and then exact and approximate algorithms for solving weighted max-satisfiability are given. That is, transformations are presented that convert a logic description into a multiple-output Boolean function of the input vector or vector sequence, where each output of the Boolean function is associated with a logic gate output transition. Algorithms for constructing the Boolean function for dynamic CMOS and static CMOS, which take into account dissipation due to glitching, are presented. Finally, efficient exact and approximate methods for solving the generated weighted max-satisfiability problem are presented. >

A spectral lower bound technique for the size of decision trees and two-level AND/OR circuits

Abstract: A universal lower-bound technique for the size and other implementation characteristics of an arbitrary Boolean function as a decision tree and as a two-level AND/OR circuit is derived. The technique is based on the power spectrum coefficients of the n dimensional Fourier transform of the function. The bounds vary from constant to exponential and are tight in many cases. Several examples are presented. >

The complexity of graph problems for succinctly represented graphs

Abstract: Highly regular graphs can be represented advantageously by some kind of description shorter than the full adjacency matrix; a natural succinct representation is by means of a boolean circuit computing the adjacency matrix as a boolean function The complexity of the decision problems for several graph-theoretic properties changes drastically when this succinct representation is used to present the input We close up substantially the gaps between the known lower and upper bounds for these succinct problems, in most cases matching optimally the lower and the upper bound

Learning boolean functions in an infinite attribute space

Abstract: This paper presents a theoretical model for learning Boolean functions in domains having a large, potentially infinite number of attributes. The model allows an algorithm to employ a rich vocabulary to describe the objects it encounters in the world without necessarily incurring time and space penalties so long as each individual object is relatively simple. We show that many of the basic Boolean functions learnable in standard theoretical models, such as conjunctions, disjunctions, K-CNF, and K-DNF, are still learnable in the new model, though by algorithms no longer quite so trivial as before. The new model forces algorithms for such classes to act in a manner that appears more natural for many learning scenarios.

Efficient algorithm for canonical Reed-Muller expansions of Boolean functions

Abstract: The paper presents a new method for computing all 2n canonical Reed-Muller forms (RMC forms) of a Boolean function. The method constructs the coefficients directly and no matrix-multiplication is needed. It is also usable for incompletely specified functions and for calculating a single RMC form. The method exhibits a high degree of parallelism.