Abstract: Carboxy functional addition-polymerizable monoethylenically unsaturated monomers and their volatile amine salts are disclosed, along with addition copolymers useful in coatings prepared therefrom. Both monomers and polymers decompose on heating. The monomers are bisamides comprised of a monoethylenically unsaturated amide and malonic acid half amide joined through their amide functionalities by a divalent saturated aliphatic group having 1-4 carbon atoms.

Abstract: We prove that for almost all boolean functions f , the conmunication complexity of f on a linear array with p+1 processors is approximately p times its commuication complexity on a system with two processors. We use this result to develop a technique for establishing lower bounds on communication complexity on general networks by simulating them on linear arrays. Using this technique, we derive optimal lower bounds for ranking, distinctness, uniqueness and triangle-detection problems on the ring. The application of this technique to meshes and trees yields nontrivial near optimal lower bounds on the communicaton complexity of ranking and distinctness problems on these networks.

Abstract: It is shown how the well-known expansion theorem of Boolean (switching) functions theory can be used as the kernel of a powerful and extremely simple algorithm for producing a short disjoint products form of a Boolean function. Its efficiency may challenge other algorithms. Its ease of full documentation is a further positive feature, at least for teachers.

Abstract: In this paper we survey fundamental methods and results about the decision problem for classes of first order logical formulae. We begin with a historical account which tells the main steps in the development of the field from Hilbert's formulation of the Entscheidungs-problem to today. We then discuss in more detail a muthod due to Aanderaa and myself which builds upon and extends ideas of Turing and Buchi and is particularly well suited for logical descriptions of computational problems; we explain how by this method (and variants there of) structural properties of computation formalisms and of their describing formulae are intimately correlated in such a way that many recursion and complexity theoretical properties by this reduction are easily carried over from the combinatorial decision problems to the corresponding logical decision problems. As example we produce by slight and natural variations of that method uniform (and easy)proofs for: NP-resp. resp. n-resp. - completeness of the decision problem for propositional (Cook) resp. frist order logic (Church, Turing) resp. of the emptiness (Trachtenbrot, Buchi) resp. the infinity problem for first order iptctna, the characterization of the latter (Scholz's problem) as the NEXPTIME-acceptable sets (Bennett, Rodding, Schwichtenberg, Jones, Selman) resp. of the generalized spectra as the NP-sets (Fagin), simple axioms for essentially undecidable and Incomplete. theories, resp. satisfiable. formulae. without recursive. models describing enumeration programs for -unseparable r.e. sets, lower complexity bounds, and indeed completeness results for many natural solvable cases of first order logical decision problems as subrecursive analogues to the undecidable reduction classes, and other complexity results for first order or propositional logic problems like a natural logical characterization of network or Turing machine complexity o& boolean functions, which is strongly related to the P = NP-problem. Our main concern is to reveal the deep structural and combinatorial simularities between computations and logical deductions, which bring out explicitely the fundamental and uniform reason for many undecidability and complexity results for combinatorial and for logical decision problems (see the above cited examples).

Abstract: An extended Boolean retrieval strategy has previously been introduced in which the individual Boolean operators can be treated more or less strictly, depending on the perceived strength of association of the query terms. The extended Boolean system is illustrated by examples and evaluation output is used to demonstrate the effectiveness of the operations.

Abstract: The paper describes CAMP, a Computer Aided Minimization Procedure for Boolean functions. The procedure is based on theorems of switching theory and fully exploits the power of degree of adjacency. The program does not generate any superfluous prime implicant and all the essential and selective prime implicants are chosen with no or minimum iteration. For shallow functions consisting mainly of essential prime implicants (EPIs) and a few selective prime implicants (SPIs), CAMP produces the exact minimal sum of product form. For dense functions consisting of a large number of inter-connected cyclic SPI chains, the solution may not be exactly minimal, but near minimal.

Abstract: Fault trees are a major model for the analysis of system reliability. In particular, Boolean difference methods applied to fault trees provide a widely used measure of subsystem criticality. This paper generalizes the fault-tree model to time-varying systems and uses timedependent Boolean differences to analyze such systems. In particular, suitable partial Boolean differences provide maximal and minimal solution sets for sensitization conditions. A method of common-cause failure analysis based on partial time-dependent Boolean differences allows the study of failures due to repeated occurrences, at different times, of the same phenomenon. Such methods generalize to systems with repair, and under certain assumptions of independence, steady-state distributions can be used for the analysis of system faults. These methods are generally useful in reliability and sensitivity analysis.

Abstract: In this paper, a general lower bound on the monotone network complexity of semidisjoint bilinear forms is proved. By this method an n3/2 lower bound for the Boolean convolution is obtained. Up to now the best known lower bound for the Boolean convolution was of size n4/3 ( Blum 1981 , IEEE Annual Sympos. Found. Comput. Sci. 22, 101–108).

Abstract: The k-valued Kleene functions over Kleene algebra (instead of over Post algebra) are defined. Multivalued logic and fuzzy logic are studied in the light of lattice theory. It is shown that all Kleene k-valued logic (k = 3 or more) have the same algebraic structure as fuzzy logic through lattice isomorphism. As a by-product, an asymptotic formula and upper bound for enumeratinlg fuzzy switching functions of n variables is obtained. Some conditions for minimizing Kleene functions are discussed, and an efficient generalized Karnaught map method is described. An application is made of the above Kleene multivalued logic to the fault diagnosis of combinational circuits. The fault norm of a combinational circuit is introduced and it is shown that this norm contains all the information about the fault situation of the circuit, and can hence be used to solve such problems as fault analysis, fault diagnosis, and detection of circuit redundance and fault masking.

Abstract: 1. IntroducUon This paper investigates reductions between numerical functions with respect to Boolean circuit size or depth as complexity measures. We assume that the input is a binary fixed point representation of the argument, if there are n input bits, the circuit is supposed to compute the n most significant bits of the function value. A function ] is called rsduc#.bls to a function g, (jf ~ g) , iff, whenever g can be evaluated for an n-bit input with complexity C(n), then f can be evaluated with complexity O(U(n)), as well. UsualLy we will require that reductions are unifozlrct in the sense that they can be computed in logarithmic space or at least polynomial time. In this paper Theorem 1 will be uniform even with respect to logarithmic space. Two functions f, g are called e~u~g/e~tt (f ~//) if f ~g andg ~y. Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is givcn that copying is by permission of the Association for Computing Machincry. To copy otherwise, or to republish, requires a fcc and/or specific pcrmission. We will investigate reducibilities with respect to Boolean Some reducibilities among arithmetic functions are easy to demonstrate. Let rout. div. and sq denote multiplication , division, and squaring, respectively. Then the reduction of sq to mu/ is trivial, the converse one is possible because sit = ~(=+y)s _=s _ ~) (1) Since addition and subtraction can be done in linear size and logarithmic depth simultaneously (cf. [0]), i.e. optimally with respect to both complexity measures, we can use them for 'free'. So with (1) we have that Trtul ~ sq Additionally, from the equation z2 = 1 Z I i (~) z z+l it follows that and, thus sq ~ d/v The motivation to investigate reducibility and equivalence of functions is, of course, that lower or upper bounds, or both, for one function transfer to the 466

Abstract: With any form of a pseudo-Boolean function f , two graphs are associated: the conflict graph C( f ) and the disjunctive-normal-form graph D(f ). Various connections between these graphs and between them and max( f ) and min( f ) are given, including one between min( f ) and the independence number α( D(f) ), which improves a previous result. Finally, a characterization of D(f ) graphs is given.

Abstract: By a result of Berkowitz the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases. This result strengthens the importance of the theory of monotone circuits. We show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits. Here disjunction and conjunction are replaced by set union and set intersection. All known methods for proving lower bounds on the monotone complexity of Boolean functions do not work in their present form for slice functions. Furthermore we show that the canonical slice functions of the Boolean convolution, the Nechiporuk Boolean sums and the clique function can be computed with linear many gates.

Abstract: We present a linear characterization for the solution sets of propositional calculus formulas in conjunctive normal form. We obtain recursive definitions for the linear characterization similar to the basic recurrence relation used to define binomial coefficients. As a consequence, we are able to use standard combinatorial and linear algebra techniques to describe properties of the linear characterization.

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. 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.

Abstract: In cases where simple data validation techniques are inadequate and optimization policies relatively complex (e.g., in health and medical systems), a Boolean optimization algorithm can be used to report errors accurately and unambiguously. The algorithm is presented in the context of a data-validating software module that uses an LR(1)-parser. The algorithm's precision makes it of potential use for the retrieval of records that nearly satisfy a query.

Abstract: The most compact algebraic means of classification of boolean functions is the self-dualized (SD) classification, which subsumes the weaker NP and NPN algebraic procedures. However, an alternative non-algebraic means of function classification is the use of spectral coefficient data assembled in positive, canonic form. Here we show that invariance operations performed in the spectral domain are equivalent to the operations involved in the algebraic SD classification procedure.

Abstract: We characterize the optimal networks for a simultaneous computation of AND and NOR over the base of all 16 Boolean operators We show that the optimal networks for AND and NOR are precisely the networks that consist of a disjoint union of an optimal network for AND with an optimal network for NOR

Abstract: It is known that all unate lines of an internally unate realisation of a boolean function are syndrome-testable. For other lines in a circuit there have been a number of suggestions for hardware modifications to ensure that a circuit is syndrome-testable. The letter proposes, as an alternative to hardware modification, augmenting the syndrome testing with auto-correlation testing.

Abstract: Area and computation time are considered to be important measures with which VLSI circuits are evaluated. In this paper, the area-time complexity for nontrivial n-input m-output Boolean functions, such as a decoder and an encoder, is studied with a model similar to Brent-Kung's model. A lower bound on area-time-product (ATαaα.≥1) for these functions is shown: for example, ATα= ω(2n. nα-l) for an n-input 2V-output decoder, and ATα= ω( n . logα-1n) for an n-input ⌈log n⌉-output encoder. The results shown in this paper are complementary to those by Brent-Kung or Thompson, and are useful for a class of functions of rather simple structures, e.g., a priority encoder, a comparator, and symmetric functions.

Abstract: The random logic portion of a chip Implementing a set of Boolean functions and sequential circuits usually represents a major contribution to chip area Obviously, there are many circuits which realize the same Boolean function Unfortunately, at present there is no general theory that provides designers (and design automation programs) with lower bounds for total area, for gate oxide area, and for delay time of logic Implementations in Integrated systems Therefore, the main task for the computer optimization program appears In choice of the circuit with the most convenient layoutDIADES is a design automation system with register-transfer level description on its Input and CIF file on output [5] The digital circuit can be described in both behavioral and structural mode A set of successive compilations and hardware implementing and optimizing transformations create the description of the network on the level of logic gates and pass transistors As the output of hardware compilation from the higher level, this description is usually nonoptimal and thus is next optimized by recursive technology independent transformations based on Boolean algebra (like A*O = O, A*A = A, etc) Inverters are also inserted into long chains of AND or OR gates, being the results of iterative circuits' compilation [4]The next stages are: technology dependant optimization of the logic network, and network's layout It is assumed that the resultant network is multilevel, consists of complex negative gates, and is realized in semiregular Weinberger-style gate matrix layout The logic minimization method intended for layout minimization is described in this paperIt is assumed in our silicon compiler, that logic is constructed of n-channel, polysilicon gate MOSFET ratioless complex gates, performing any negative function By negative function we understand negation of positive function, while positive function is any combination of AND and OR functors [1] Functions for evaluation of circuit's performance parameters such as total area, area of gate oxide, and gate delay time are used These functions are defined in terms of basic technology and selected topology parameters The method can be adapted to other technologies

Abstract: A method is presented for obtaining an algebraic representation of boolean functions. The proposed method consists in considering the boolean function as a training set for a pattern recognition system in which the variables arc of the binary type. The orthogonal Rademacher-Walsh polynomials are used as a basis for the probability density functions in a Bayes classifier.

Abstract: In this report the theory of Boolean functions (of indicator variables) is developed in an extremely economical way to give sufficient insight into modern algorithms for the determination of disjunctive normal forms consisting of pairwise disjoint terms (DDNF's). After that it is shown how easily DDNF's lend to stochastic analysis especially to the determination of the probability of a Boolean function being 1 (which corresponds to system unavailability in fault tree analysis) and of the mean time a Boolean function is O viz. 1 (which corresponds to MTBF viz. MTTR in fault tree analysis.)