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Parallel Computation of Boolean Gr Obner Bases 1 Introduction

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Parallel Computation of Boolean Gr Obner Bases 1 Introduction Parallel computation of Bo olean Grobner bases y z Y. Sato and A. Suzuki Department of Computer Science Ritsumeikan University y [email protected] z [email protected] Abstract Bo olean rings are isomorphic to direct pro ducts of a Galois eld GF 2. Using these structures, b o olean Grobner bases can b e constructed by a parallel computation algorithm. We implemented this metho d using a parallel logic programming language KLIC. In this pap er, we rep ort on this work. 1 Intro duction In p olynomial rings over commutative Von Neumann regular rings, we can construct Grobner bases using sp ecial kinds of monomial reductions[W 89]. This metho d is implemented as a Grobner bases computation algorithm in \SET CONSTRAINT SOLVER Version 1.0"[Sa 97, Sb 97]. Meanwhile any commutative Von Neumann regular ring is known to be iso- morphic to a sub-ring of a direct pro duct of elds [SW 75], and a Grobner basis of its p olynomial ring can be characterized in terms of a Grobner ba- sis of a p olynomial ring over each eld. See Theorem 4.1 for more detailed description. With this prop erty, we can also construct a Grobner basis of a p olynomial ring over a commutativeVon Neumann regular ring by computing a Grobner basis in a p olynomial ring over each eld indep endently. When the structures of direct pro ducts are simple, this metho d seems more ecient, esp ecially under the environment we can use parallel computations. Finite b o olean rings are among the simplest examples of commutative Von Neumann regular rings. They are isomorphic to nite direct pro ducts of a Galois eld GF 2. Polynomial rings we are handling in [Sa 97] are essentially p olynomial rings over nite b o olean rings. So, the ab ove metho d is exp ected to b e more ecient than the original one. We implemented this metho d as a parallel computation algorithm of Grobner bases in \SET CONSTRAINT SOLVER Version 2.0" [S 99]. In this pap er, we intro duce this parallel computation algorithm, and rep ort on exp erimental results we got through the actual parallel computations. In section 2, we give a short description of the theory of Grobner bases in p olynomial rings over commutative Von Neumann regular rings with mini- mum results for understanding our work. The reader is referred to [W 89] or [S 98] for more detailed comprehensive description. In section 3, we give a quick review for our set constraint solver based on Grobner bases. In section 4, we describ e our parallel computation algorithm. In section 5, we give several data we got through our exp eriments. 2 Grobner bases A commutative ring R with identity 1 is called a Von Neumann regular ring if it has the following prop erty: 2 8a 2 R 9b 2 Ra b = a: 1 2 For such b, a = ab and a = ab are uniquely determined and satisfy 1 2 aa = a, aa = a and a = a . Note that every direct pro duct of elds is Von Neumann regular ring. Con- versely, any Von Neumann regular ring is known to be isomorphic to a sub- ring of a direct pro duct of elds[SW 75]. We assume R is a Von Neumann regular ring in this section. Note that any sub-ring of R generated by a nite subset of R b ecomes a No etherean ring, although R is not generally a No etherean ring. Therefore there exists a Grobner basis for any nitely generated ideal in p olynomial rings over R . We can even have an algorithm to construct a Grobner basis of the ideal gener- ated by a given nite set of p olynomials. This algorithm is essentially same as Buchb erger's algorithm except that we have to de ne sp ecial monomial reductions in our p olynomial rings. In the following unless mentioned, Greek letters , , are used for terms, Roman letters a, b, c for elements of R and f , g , h for p olynomials over R . Throughout this section wework in a p olynomial ring over R and assume that some total admissible order on the set of terms is given. The largest term of f is denoted by ltf and its co ecient by lcf . The largest monomial of f i.e. lcf ltf is denoted by lmf , the rest of f i.e. f lmf is denoted by rmf . De nition 2.1 For a p olynomial f = a + g with lmf =a , a monomial reduction ! is f de ned as follows: 1 b + h ! b + h ba a + g f where ab 6= 0 and b need not b e the largest monomial of b + h. A monomial reduction ! by a set F of p olynomials is also de ned naturally. F Using this monomial reduction, we can construct a Grobner basis of the ideal generated by a given nite set of p olynomials. Though the algorithm is al- most same as the Buchb erger's algorithm for p olynomial rings over elds, we have to take account of the following observation. In a p olynomial ring over a eld, the equivalence relation $ induced by the F monomial reduction ! for a set F of p olynomials coincides with the equiv- F alence relation induced by the ideal F generated by F . In our p olynomial ring, however, these two equivalence relations are in general not equal. De nition 2.2 A p olynomial f is called boolean closed if lcf f = f . lcf f is called a boolean closure of f and denoted by bcf . Note that the b o olean closure of any p olynomial is b o olean closed. Theorem 2.1 Let F be a set of b o olean closed p olynomials. Then the equivalence relation $ coincides with the equivalence relation induced by the ideal F . F Since lmf =lmbcf , we can construct a set of b o olean closed p olynomials H from any given set of p olynomials F such that F =H. H is also called a boolean closure of F . Though such H is not determined uniquely,we abuse the notation bcF to denote one of such H . Using our monomial reductions, Grobner bases are de ned as follows. De nition 2.3 A nite set G of p olynomials is called a Grobner basis, if it satis es the fol- lowing two prop erties. f $ g i f g 2 G for each p olynomial f and g . G ! has aChurch Rosser prop erty, G i.e. for each p olynomial f and g , f $ g i there exists G a p olynomial h such that f ! h and g ! h. G G De nition 2.4 0 0 For each pair of p olynomials f = a + f and g = b + g , where lmf = a , lmg = b and GC D ; = 1, the p olynomial b f a g = 0 0 b f a g is called the S-p olynomial of f and g and denoted by SP f; g. We can also characterize Grobner bases in terms of S-p olynomials as in p oly- nomial rings over elds. Theorem 2.2 Let G be a nite set of b o olean closed p olynomials. Then G is a Grobner basis i SP f; g ! 0 for any pair f and g of p olynomials G in G. It is crucial that each p olynomial of G is b o olean closed. The theorem do es not hold otherwise. This theorem enables us to construct a Grobner basis G for a given nite set F of p olynomials such that G= F. Algorithm Let F = bcF . 0 while there exist f; g 2 F such that SP f; g 6=0 i F i F = bcF [fSP f; g g i+1 i F i otherwise G = F i Reduced Grobner bases are de ned naturally and can be constructed from any Grobner bases immediately. We have the following prop erty. Theorem 2.3 Let G be a reduced Grobner basis; then any element of G is b o olean closed. In p olynomial rings over elds, reduced Grobner bases serve us as canonical forms of Grobner bases, however we have to be careful in our p olynomial rings. De nition 2.5 A p olynomial f is called monic if it satis es lcf =lcf . De nition 2.6 A reduced Grobner basis G is called a strati ed Grobner basis, when it satis- es the following two prop erties. Ever element of G is monic. ltf 6= ltg for any distinct element f and g of G. Theorem 2.4 A strati ed Grobner basis is determined uniquely. That is two strati ed 0 0 Grobner bases G and G such that G= Gmust be identical. Let us conclude this section with showing that there exist another typ es of canonical forms of Grobner bases. De nition 2.7 A Grobner basis G each element of which is monic and b o olean closed is called an optimal Grobner basis if it has the following three prop erties. 1 G is a minimal Grobner basis, i.e. G ffg is not a Grobner basis of the ideal G for any element f of G. 2 For any element f of G, rmf is not reducible by ! . G 3 For each element f and g of G,ifltf jltg i.e.
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