Solving Op en Quasigroup Problems by Prop ositional Reasoning y Jieh Hsiang Hantao Zhang Department of Computer Science and Department of Computer Science Information Engeneering The University of Iowa National Taiwan University, Taipei, Taiwan Iowa City, IA 52242 [email protected] [email protected] Abstract. There are many op en problems quasigroups satisfying certain constraints is such in the study of quasigroups. Recently, au- an example. tomated techniques have b een employed to A quasigroup is simply a cancellative nite attack these op en problems. In this pap er, group oid, hS; i, where S is a set, a binary op- we show how a prop ositional satis ability prover is used to solve many op en problems eration on S . The cardinality of S , jS j, is called in quasigroups. Our success relies on a p ow- the order of the quasigroup. The \multiplication erful prop ositional prover called SATO and a table" for the op eration forms a Latin square, useful technique called the cyclic group con- of which each row and each column is a permu- struction. We provide detailed solutions to tation of S . Interest attaches to many classes of op en problems solved bySATO. quasigroups, partly b ecause they are very natural ob jects in their own right and partly b ecause of their relationships to design theory. 1 Intro duction In this pap er, we are interested in the prob- lems in quasigroups given by Fujita, Slaney and In the recent years, there has been considerable Bennett in their award-winning IJCAI pap er [3]. renewed interest in the prop ositional satis abil- The constraints in Table 1 are taken from [3]: ity problem (SAT). Because the SAT problem is Among the Latins squares satisfying these con- the rst known NP-complete problem, it is rel- straints, we are also interested in those squares atively easy to transform any NP-complete prob- with a hole, i.e., a subsquare (which itself is also lem in mathematics, computer science and electri- a Latin square) of the square is missing. cal engineering into the SAT problem. The SAT Quasigroups raise many combinatorial prob- problem is known to be dicult to solve in the- lems, some of which are often approached com- ory. However, contrary to the common p ercep- putationally. The usefulness of advanced auto- tion that transforming a problem into the SAT mated reasoning techniques to attack these prob- problem will not make the problem easier to solve, lems have been successfully demonstrated in [11 , many problems can in fact be solved more e ec- 3 , 7, 10, 6]. In 1990, Zhang rep orted a case of op en tively by aSAT problem solver than by a sp ecial problems in quasigroups solved using a constraint program. Needless to say, transforming a problem solving technique [11 ]. Subsequently, Fujita used into the SAT problem is much easier than writing MGTP, a mo del-generation based rst-order the- a sp ecial program. The existence problem of nite orem prover, and Slaney used FINDER, a pro- gram based on constraint solving, to solve several Partially supp orted by the National Science Founda- tion under Grants CCR-9202838 and CCR-9357851. op en problems in quasigroups [3 ]. Later, Stickel y Partially supp orted by the National Science Council and Zhang, indep endently, used their prop osi- under grant NSC-83-0408-E-002-012T. This pap er is organized as follows. In the next Table 1: Some constrains of quasigroups. section, we intro duce some problems in quasi- groups and show how to represent them in the Name Constraint prop ositional logic. In section 3, we present one QG1 x y = u; z w = u; v y = x; of our to ols to solve problems in quasigroups, i.e., v w = z ) x = z , y = w the cyclic group construction. In section 4, we QG2 x y = u; z w = u; y v = x; present solutions to some op en quasigroup prob- w v = z ) x = z , y = w lems. Section 5 concludes the pap er. QG3 (x y ) (y x)=x QG4 (x (y x)) y = x QG5 ((x y ) x) x = y 2 Quasigroup Problems (x y ) y = x (x y ) QG6 QG7 ((x y ) x) y = x Recall that a quasigroup is a pair hS; i where S is a nite set, a binary op eration on S and the \multiplication table" of forms a Latin tional provers, based on the Davis-Putnam algo- square. Without loss of generality, we assume rithm [2 ], to attack these problems and rep orted S = f0; 1; :::; v 1g,wherev is the order of hS; i. very comp etitive and new results [7 , 8, 10 ]. The The following clauses sp ecify a Latin square: For exp erimental results of this pap er are obtained by all elements x; y ; u; w 2 S , the program SATO (SAtis ability Testing Opti- mized) ([8 ], [10 ]) which is an ecient implemen- x u = y; x w = y ) u = w (1) tation of the Davis-Putnam algorithm. u x = y; w x = y ) u = w (2) Quasigroups raise very hard computational x y = u; x y = w ) u = w (3) problems. Many of these problems are simply (x y =0)__(x y = v 1) (4) intractable for to day's automated reasoning pro- grams. For instance, it remains op en if there exists It has b een shown in [7] that the following two a Latin square of order 10 satisfying QG2. While clauses are valid consequences of the ab ove clauses it is beyond the reach of the current reasoning and adding them into a prover can help to reduce techniques to do an exhaustive search, we may the searchspace. take advantage of techniques develop ed bymath- ematicians over decades for constructing quasi- (x 0=y ) __(x (v 1) = y ) (5) groups. One such technique we found very useful (0 x = y ) __((v 1) x = y ) (6) is a starter-adder-typ e construction. This tech- nique has b een used extensively byvarious math- In the following we denote by QGi(v ) a Latin ematicians (e.g., see [5 , 4, 1]). The main idea is square satisfying clauses (1){(6) plus the con- to generate a Latin square under a cyclic group straint QGi given in the intro duction for S = from the rst row and the rst column of the f0; :::; (v 1)g. In addition, the idemp otence law, square. This cyclic group technique o ers a signif- x x = x, and the constraint x (v 1) x 1, icant computation advantage: instead of searching which eliminates some isomorphic mo dels, are 2 for O (v )entries of a square of order v , only O (v ) used for each problem here. Further details on entries need to b e searched. If a prop ositional rea- these problems can b e found in [3 ] and [7 ]. 2 soning prover is used, only O (v ) prop ositional 3 To obtain prop ositional clauses, we simply in- variables instead of O (v ) variables are needed. stantiate variables in the clauses of QGi(v )bythe In general, to nd latin squares satisfying QG1{ 3 6 values of S and replace each equality a b = c QG7, only O (v ) clauses instead of O (v ) clauses by a prop ositional variable p . The number of are needed. Despite of the fact that this technique a;b;c the prop ositional clauses is thus decided by the is incomplete, we are able to nd many new Latin order of the quasigroup (i.e., v ) and the number squares using the SATO prover. Attach the condition that x and y cannot b e of distinct variables in a clause. However, for con- in X at the same time to clauses (1)-(3). straints QG3{QG7, we have to transform them into \ at" form rst. For example, the at form For the Davis-Putnam algorithm, it is relatively of QG5 is easier to work with incomplete groups of larger (x y = z ); (z x = w ) ) (w x = y ): holes. To search for a QGi(v; n), if we already have a QGi(m; n), wemay searchforaQGi(v; m)and It can be shown that the two \transp oses" of the then ll the QGi(m; n) into QGi(v; m) to obtain ab ove clause are also valid consequences of QG5: a QGi(v; n). This ll-in-hole technique is useful when lo oking for QGi(v; n). (w x = y ); (x y = z ) ) (z x = w ); There exist many op en problems regarding the (z x = w ); (w x = y ) ) (x y = z ): existence of QGi with holes given in Bennett and Exp erimental results show that when v > 10, Zhu's survey pap er [1 ]. We are able to nd new adding these \transp oses" in the input often im- mo dels for each typ e of QG1{QG7; these solved proves the p erformance of the Davis-Putnam al- quasigroups will b e given in section 4. gorithm. This is also true for QG3{QG7. For a class of Latin squares satisfying a given 3 Cyclic Group Construction constraint, we are often interested in those with holes. An incomplete Latin square is a Latin The prop ositional reasoning program we used to square with a single hole and is sp eci ed as attack quasigroup problems is called SATO (SAt- hS=X ; i, where X S and for any x ;x 2 X , 1 2 is abilityTesting Optimized) which is an ecient x x must be in X but is left undecided in the 1 2 implementation of the Davis-Putnam algorithm \multiplication table" of .