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Minimal polynomials for the conjunction of functions on disjoint variables can be very simple

Bernd Voigt, Ingo Wegener
1989 Information and Computation  
ACKNOWLEDGMENTS We thank Hans Ulrich Simon who asked the second author (in different terminology) whether the DCSH( A ) holds for minimal polynomials.  ...  The motivation for this question came from problems in the theory of learning within the field of artificial intelligence. March 14, 1988; ACCEPTED September 20, 1988 RECEIVED  ...  Remark. No explicit example is known of a Boolean function f~ B, with C(f x f) < 2. C(f). Remark.  ... 
doi:10.1016/0890-5401(89)90047-3 fatcat:erlme6eldfcvni3rwmro3zbgs4

On the Gap between the Complexity of SAT and Minimization for Certain Classes of Boolean Formulas

Ondrej Cepek, Stefan Gurský
2014 International Symposium on Artificial Intelligence and Mathematics  
It is also known that the decision version of Boolean minimization for CNF inputs is Σ2 complete. On the other hand there are several subclasses of CNFs (e.g.  ...  Thus, for both the general case and the above mentioned subclasses the gap between the complexity of SAT and minimization is exactly one level in the polynomial hierarchy.  ...  The second author gratefully acknowledges the support of the Charles University Grant Agency (grant No. 1390213).  ... 
dblp:conf/isaim/CepekG14 fatcat:kvypiio3orab7k6jzqpd2vhgka

Symbolic OBDD-Based Reachability Analysis Needs Exponential Space [chapter]

Beate Bollig
2010 Lecture Notes in Computer Science  
, y nn x j = (x j1 , . . . , x jn ) Boolean encoding of index i j The number of x π(i) -nodes in the minimal π-OBDD for a Boolean function f is the number of different subfunctions f |x π(1) =b 1 ,...  ...  Idea: A π-OBDD for a subfunction of a Boolean function f cannot be larger than a π-OBDD for f .  ...  Polynomial upper bound on the OBDD size of χ E (v i 1 ,...,in , v ℓ 1 ,...  ... 
doi:10.1007/978-3-642-11266-9_19 fatcat:dibjikvj6rdhdmmwlysur2ibpu

Minimization of pseudo-boolean functions by binary development

I.G. Rosenberg
1974 Discrete Mathematics  
An lteratlve method for zero-one minimization of integer polynomials, hnear m each variable, is outhned.  ...  It Js based on Camlon's method of binary developments with computations using multlphcatlon and sum mod 2.  ...  Rosenberg, Mmimlzatton of pseudo-Boolean functions Remark 3.4. The Boolean functionsg I are expressed as Boolean polynomials.  ... 
doi:10.1016/s0012-365x(74)80026-9 fatcat:pcfbg2ckpbc2baofomoe33auy4

On Quadratization of Pseudo-Boolean Functions [article]

Endre Boros, Aritanan Gruber
2014 arXiv   pre-print
We survey current term-wise techniques for quadratizing high-degree pseudo-Boolean functions and introduce a new one, which allows multiple splits of terms.  ...  We also introduce the first aggregative approach, which splits a collection of terms based on their common parts.  ...  One of the most frequently used technique is based on roof-duality [11] , and aims at finding in polynomial time a simpler form of the given quadratic minimization problem, by fixing some of the variables  ... 
arXiv:1404.6538v1 fatcat:id55upbdhrdrdcrrjpwzabymya

On quadratization of pseudo-Boolean functions

Endre Boros, Aritanan Gruber
2012 International Symposium on Artificial Intelligence and Mathematics  
In many applications of pseudo-Boolean optimization the objective function (1) is a higher degree multilinear polynomial.  ...  One of the most frequently used technique is based on roof-duality (Hammer, Hansen, and Simeone 1984) , and aims at finding in polynomial time a simpler form of the given quadratic minimization problem  ... 
dblp:conf/isaim/BorosG12 fatcat:gml432m77jh55n6kun7vqljdjy

Horn functions and submodular boolean functions

Oya Ekin, Peter L. Hammer, Uri N. Peled
1997 Theoretical Computer Science  
After providing a simple characterization of Horn functions (i.e., those Boolean functions that have a Horn DNF), we study in detail the special class of submodular functions.  ...  There is a one-to-one correspondence between the roots of a submodular function and the ideals of the associated partial preorder.  ...  Acknowledgements The authors gratefully acknowledge the partial support by the Office of Naval Research (grants N0001492F1375 and N0001492F4083) and by the Air Force Office of Scientific Research (grant  ... 
doi:10.1016/s0304-3975(96)00202-2 fatcat:q5getvzelrabvd3afcenpvwyyu

Reliability of Two-Stage Weighted-$bf k$-out-of-$n$Systems With Components in Common

Y. Chen, Q. Yang
2005 IEEE Transactions on Reliability  
This paper extends the existing one-stage weighted--out-of-model to two-stage weighted-k-out-of-models with components in common.  ...  Reliability bounds for systems with -dependent component failures are investigated based on the generated minimal cuts & minimal paths.  ...  1 The singular and plural of an acronym are always spelled the same. if it is functioning a term of a cut polynomial, which represents an -component minimal cut consisting of components . a term of a  ... 
doi:10.1109/tr.2005.853274 fatcat:bndh7t3ytbgplgupudd2neixi4

Efficient Determination of Gibbs Estimators with Submodular Energy Functions [article]

Boris Zalesky
2003 arXiv   pre-print
For that the decomposition of the functions by Boolean polynomials is used. The modified SFM algorithm for minimization of submodular Boolean polynomials is described.  ...  Combinatorial algorithms for minimization of functions of many variables, which take their values in finite totally ordered sets, are developed.  ...  After we represent the function V (j) by a Boolean polynomial and then minimize the polynomial.  ... 
arXiv:math/0304041v1 fatcat:reidej2mpbdwnlkhgmp4lged4a

Approximate evaluations of characteristic polynomials of Boolean functions

David Lee, Henryk Woźniakowski
2001 Theoretical Computer Science  
Let k( ; n) denote the minimal number of Boolean function evaluations needed to reduce the initial error by a factor of where n is the number of Boolean variables.  ...  Motivated by combinational circuit veriÿcation and testing, we study the approximate evaluation of characteristic polynomials of Boolean functions.  ...  Suppose that we use k Boolean function values for an approximate evaluation of a characteristic polynomial of a Boolean function of n variables.  ... 
doi:10.1016/s0304-3975(00)00151-1 fatcat:ak4cawhrczcv3eq2662ntlbqkq

Page 3364 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
A remark on minimal polynomials of Boolean functions. CSL ’88 (Duisburg, 1988), 372-383, Lecture Notes in Comput. Sci., 385, Springer, Berlin, 1989.  ...  that for the case of f and g being symmetric Boolean functions the minimal DNF of f(x) A g(y) is given by multiplying the minimal DNF representations of f(x) and g(y).  ... 

Representing Boolean Functions Using Polynomials: More Can Offer Less [chapter]

Yi Ming Zou
2011 Lecture Notes in Computer Science  
For this purpose, instead of an exact polynomial representation, usually the sign representation of a Boolean function is considered.  ...  In this paper, the basic methods of representing a Boolean function by polynomials are examined, and an alternative approach to this problem is proposed.  ...  is appropriate for the type of applications under consideration, or to use a combined approach to achieve the best result.  ... 
doi:10.1007/978-3-642-21111-9_32 fatcat:diytlldimbfcrbea2twre6funm

Representing Boolean Functions Using Polynomials: More Can Offer Less [article]

Yi Ming Zou
2013 arXiv   pre-print
For this purpose, instead of an exact polynomial representation, usually the sign representation of a Boolean function is considered.  ...  In this paper, the basic methods of representing a Boolean function by polynomials are examined, and an alternative approach to this problem is proposed.  ...  is appropriate for the type of applications under consideration, or to use a combined approach to achieve the best result.  ... 
arXiv:1307.1157v1 fatcat:t34h3t5yorbo5gecyf6piq5zu4

Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions

O. Ekin Karaşan
2009 Annals of Operations Research  
We consider the problem of dualizing a Boolean function f represented by a DNF.  ...  In its most general form, this problem is commonly believed not to be solvable by a quasi-polynomial total time algorithm.  ...  The author also acknowledges the support from the Turkish Academy of Science.  ... 
doi:10.1007/s10479-009-0637-x fatcat:2dke6pd2cbfx3g6233xjxbkegq

The Complexity of Minimizing FBDDs [chapter]

Detlef Sieling
1999 Lecture Notes in Computer Science  
Free Binary Decision Diagrams (FBDDs) are a data structure for the representation and manipulation of Boolean functions.  ...  In this paper it is shown that the existence of polynomial time approximation schemes for optimizing graph orderings or for minimizing FBDDs implies NP ZPP or NP P, respectively, and so such algorithms  ...  We remark that for OBDDs there are similar optimization problems, namely the computation of a minimal size OBDD for a function given by an OBDD and the computation of an optimal variable ordering for a  ... 
doi:10.1007/3-540-48340-3_23 fatcat:hmbdh6p5mvgtxa4bqd4ecr35we
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