Boolean Polynomial Research Articles

On Complexity of Computation of Partial Derivatives of Boolean Functions Realized by Zhegalkin Polynomials

The combinational complexity of a system of partial derivatives in the basis of linear functions is established for a Boolean function of i>n variables that is realized by a Zhegalkin polynomial. An algorithm whose complexity equals 3^n - 2^n modulo 2 additions is proposed for computation of all partial derivatives of such a function from the coefficients of its Zhegalkin polynomial.

Parity OBDDs cannot be handled efficiently enough

Parity Ordered Binary Decision Diagrams (POBDDs) seem to be a promising alternative to OBDDs because they admit a more compact representation of Boolean functions and polynomial time algorithms for all important operations on Boolean functions. However, for the reduction of POBDDs only an algorithm based on Gaussian elimination is known. Since the reduction is used quite frequently, a linear time algorithm would be desirable. In this note it is proved that it is unlikely that such an algorithm exists because a linear time algorithm for POBDD reduction would imply a linear time algorithm for matrix rank computation.

On the complexity of completeness recognition of systems of Boolean functions realized in the form of Zhegalkin polynomials

On the complexity of completeness recognition of systems of Boolean functions realized in the form of Zhegalkin polynomials

Boolean differential equations

In this paper, we define two differentials D( f) and Δ( f) for a boolean polynomial f, with n variables. D( f), for instance, is chosen in such a way that its roots are exactly those of f( x) = m( f), where m( f) is the minimum of f on B n . Then D( f) ⩽ f. We first state different properties for the differentiation, which is shown to be a boolean algebraic operator, then solve three boolean differential equations, among which are: D( f) = θf( θ ϵ B), and the quadrature equation, D( f) = g.

Stone algebras, conditional events, and three valued logic

There are several equivalent ways to represent the set of conditional events, and in some the operations proposed by Goodman and Nguyen (1991) become much simpler, making the development of the theory much easier and much more concise. Such a development is carried out here using a representation whose relation to three-valued logic is analogous to that of Boolean algebras to two-valued logic, and in which the operations are simple and intuitive. There are many ways to extend the operations on events to operations on conditional events, but it is shown that there are only nine ways to extend intersection and nine ways to extend union so that the operations are Boolean polynomials of their arguments and are idempotent and commutative. Further, there is only one way to make such extensions so that the resulting structure is a bounded lattice extension of the space of events. These particular extensions turn out to be the operations proposed by Goodman and Nguyen.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A theory of conditional information with applications

The development of conditional propositions, deduction between conditionals, and boolean-like operations on conditionals, and their associated probabilities are here unified in terms of boolean relations of the form "b=0" defined on an initially relation-free algebra of boolean polynomials that transcends an initial domain of discourse. The conditional proposition (a|b) is assigned the conditional probability P(a|b), which is different from the probability that (a|b) is a tautology. The resulting algebraic techniques are demonstrated in several examples such as by simplifying a circular rule-based expert system and removing its circularity and by deriving logical and probability formulas for keeping communication lines open.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Applying Coding Theory to Sparse Interpolation

Fast interpolation algorithms are presented for sparse sums of characters on the product monoid $U^n $ with values in a field K in two special cases. In the first case, U is a finite cyclic group of order e, and K is a field that contains a root of unity of order e. Here linear codes over ${\mathbb{Z} / {e\mathbb{Z}}}$ can be used to construct sets of evaluation points that allow efficient interpolation by decoding Reed–Solomon codes. In the second case, K is the binary field $GF(2)$, and U is the multiplicative monoid of $GF(2)$. Here sparse sums of characters coincide with sparse Boolean polynomials and can be interpolated using the smallest set of evaluation points by decoding Reed–Muller codes.

Boolean polynomials and set functions

In this note, we consider the problem of maximizing an arbitrary Boolean polynomial of degree n. We “linearize” the problem and give a complete linear description of the resulting polytope. By examining the dual of the corresponding linear program, we find that the problem of maximizing an arbitrary set function defined on the set \\s{1,..., n\\s} is in one-to-one correspondence with our problem. Moreover, a known result concerning the polynomial solvability of quadratic Boolean functions is generalized to arbitrary Boolean polynomials.

A MIMD implementation of the Buchberger Algorithm for Boolean polynomials

In this note we present two methods to compute Gröbner basis in parallel, both based on Buchberger's sequential algorithm. A distributed memory MIMD computer (the FPS T40) gives experimental results obtained with the Boolean polynomials. The algorithms were implemented on the FPS T40 connected as a ring and as a hypercube of processors. The first implementation shows the interest of the parallelization. The second one, based on a divide and conquer strategy, has a behaviour very close to the sequential algorithm.

On the relation between resolution based and completion based theorem proving

Completion theorem proving, as proposed by J. Hsiang (1982), is based on the observation that proving a first order formula is equivalent to solving an equational system over a boolean polynomial ring. The latter can be accomplished, by completing the set of rewrite rules obtained from the equational system. This method's basic deduction rule is the generation of a new rule from a divergent critical pair obtained by superposition of two rules. This paper relates superposition to the resolution rule, which is the basis of resolution theorem proving, extending Dietrich's result on the correspondence between completion and resolution from Horn clauses to arbitrary clauses. It is shown that each completion deduction that follows Hsiang's N-strategy can be partitioned into sequences of superpositions that correspond to hyperresolution steps. However, such a correspondence does not exist for arbitrary completion deductions: There is a class of clause sets that preclude resolution refutations using each clause only once — a result which was given by R. Shostak — but admit completion refutations with this property, that is, such a completion refutation is shorter than any resolution refutation can be. Furthermore, we show by means of Shostak's theorem that the language of rings and ideals is well suited for short and elegant proofs of theorems about resolution deductions.

A 2-phase method for the minimisation of boolean polynomials

The development of an algorithm for minimising Boolean polynomials is proposed. The algorithm operates in two phases. In the first phase a rapid search is conducted for ways to reduce the polynomial to a smaller form. In the second phase a deep search is conducted to further reduce the polynomial and eventually obtain the minimal form.

On the probability of a Boolean polynomial of events

On the probability of a Boolean polynomial of events

A compact method for the minimisation of boolean polynomials

An algorithm for minimising Boolean polynomials is developed. The algorithm can be programmed conveniently and although it makes use of existing knowledge of minimising methods, the algorithm is novel in that it involves no expansion of the number of terms of the polynomial other than by one term which is dropped before the next stage of the algorithm is reached

A Compilation Technique for Exact System Reliability

An algorithm is presented for evaluating exact system reliability without using a Boolean polynomial equation. A Boolean expression is used to define logical system configuration. The Boolean expression is converted to Polish postfix notation. System reliability is computed by converting the postfix notation into a sequence of arithmetic and stack operations. Errors are analyzed to identify a possible rounding or chopping error during floating point calculation for system reliability by digital computer.

Best Algorithms for Approximating the Maximum of a Submodular Set Function

A real-valued function z whose domain is all of the subsets of N = {1, …, n) is said to be submodular if z(S) + z(T) ≥ z(S ∪ T) + z(S ∩ T), ∀S, T ⊆ N, and nondecreasing if z(S) ≤ z(T), ∀S ⊂ T ⊆ N. We consider the problem maxS⊂N {z(S): |S| ≤ K, z submodular and nondecreasing, z(Ø) = 0}. Many combinatorial optimization problems can be posed in this framework. For example, a well-known location problem and the maximization of certain boolean polynomials are in this class. We present a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0, …, K − 1. For fixed K, the qth member of this family requires O(nq+1) computations and is guaranteed to achieve at least [Formula: see text] Our main result is that this is the best performance guarantee that can be obtained by any algorithm whose number of computations does not exceed O(nq+1).

Inverting and Minimalizing Path Sets and Cut Sets

This paper describes a technique for generating the minimal cuts from the minimal paths, or vice versa, for s-coherent systems. The process is a recursive 2-stage expansion based upon de Morgan's theorems; ie, it is the inversion of a Boolean polynomial having all common-valued (either all 0 or all 1) components, so that the inverse also has only common-valued components of the opposite sign. There are procedural short cuts and Quine-type absorptions; absorptions put the polynomial into its minimalized form. The number of stages of recursion is equal to the number of terms (minimal states) in the starting polynomial. The minimal states of the inverse form are the terms of the inverse polynomial after minimalization. Since the system is s-coherent and all components are common-valued in either the original or Inverse minimal forms, the lists of minimal states are unique.

Minimalization of Boolean polynomials, truth functions, and lattices.

Minimalization of Boolean polynomials, truth functions, and lattices.

System reliability analysis: A tutorial

This paper deals with the logical formulation of a system for purposes of reliability analyses and both exact and approximate methods of calculating the system reliability. The first part deals with the logical concepts, and the second part with probability calculations. The logical formulation in Part 1 starts from first principles. The universal set U of system states is a “Boolean algebra”. The “power set” P is the set of subsets of U . The subset of system success states or “paths” is a “lattice” within U , also an element of P , represented by a Boolean polynomial. The term of the polynomials are monomials which give the indicators for the “sublattices” or “complete subsets” of U within the lattice of paths. The optimal representation of this lattice polynomial is in the minimalized form; the terms of the minimalized lattice of paths are called the “minimal paths”. Similarly, the subset of system failure states or “cuts” is a lattice polynomial whose terms are sublattices within the lattice of cuts; the terms of the minimalized lattice polynomial are the “minimal cuts”. Logically consistent systems (also known as “coherent structures”) have certain properties with respect to the partial ordering of paths and cuts. The concepts of Boolean logic, minimalization, etc., apply to systems that do not have the “consistency” property, as well as to those that do have it. The second part of this paper deals with ways of using a minimalized lattice polynomial to derive a numerical-valued probability function from which to calculate the system reliability: also computer software, error bounds and approximations. The reliability is the probability that the actual state of the system is an element of the lattice of paths. The exact probability is derived from the minimalized lattice polynomial of paths by the method of inclusion-exclusion, also known as Poincaré's Theorem. Dually the probability of failure, or system unreliability, is derived from the minimalized lattice of cuts. Modularization and/or inversion can be helpful in keeping the probability function reasonably small in size. Computer software is available both to generate the probability polynomial and to calculate either the reliability of unreliability. Approximations can be used based upon simplifying assumptions which delete low probability terms from the function. Conservative error bounds are obtainable for some models. The approximation techniques include serializing methods (“single-point failures” and “parts count”), very large system approximations for fault-tree applications and the Esary—Proschan bounds.

Boolesche Minimalpolynome und �berdeckungsprobleme

The main difficulty in finding minimal Boolean polynomials for given switching functions comes from the evaluation of the table of prime implicants. We show the following results Finally we make some remarks about the complexity of algorithms, which--given the graph of a switching function--find a minimal polynomial of this function.

Generalized Completeness Theorem and Solvability of Systems of Boolean Polynomial Equations

Generalized Completeness Theorem and Solvability of Systems of Boolean Polynomial Equations