Patent Publication Number: US-10311125-B2

Title: Simplifying clauses for MAX-SAT

Description:
BACKGROUND 
     Technical Field 
     The present invention relates to simplifying clauses associated with logical variables for the purpose of solving a maximum satisfiability problem (MAX-SAT). 
     Description of the Related Art 
     An instance of MAX-SAT is a set of clauses associated with logical variables, where each of the clauses consists of a weight and a disjunction of one or more literals of the logical variables. A solution to the MAX-SAT instance is an assignment of values (TRUE or FALSE) to the variables that maximizes the sum of the weights of clauses satisfied by the assignment. When solving a MAX-SAT instance, inference rules are often used to simplify the set of clauses. Many such inference rules decide the value of a variable in a solution. See, for example, Larrosa, Javier et al. “A logical approach to efficient Max-SAT solving.”  Artif. Intell.  172 (2008): 204-233; Li, Chu Min et al. “New Inference Rules for Max-SAT.” Journal of Artificial Intelligence Research, Volume 30 Issue 1, September 2007, 321-359, 2007-09-01, ISSN: 1076-9757; and Heras, Federico et al. “New Inference Rules for Efficient Max-SAT Solving.” AAAI&#39;06 Proceedings of the 21st national conference on Artificial intelligence—Volume 1, 68-73, 2006-07-16, ISBN: 978-1-57735-281-5. 
     Practical applications of MAX-SAT are becoming increasingly numerous and diverse. For example, in addition to having applications in software package upgrading, software engineering, formal verification, software product lines, and bioinformatics, MAX-SAT is becoming a core component in cognitive applications. In view of such applicability, use of MAX-SAT is becoming increasingly widespread. However, actually solving a MAX-SAT instance requires significant time and resources (e.g. memory, processor load, etc.), even when existing inference rules are applied. 
     Meanwhile, MAX-SAT solvers exist, such as the open source SAT4J solver obtainable on the World Wide Web at sat4j.org, which can be used to solve MAX-SAT instances in many different fields. However, MAX-SAT solvers consume significant time and resources to find a solution to a MAX-SAT instance. 
     SUMMARY 
     Therefore, it is an object of an aspect of the innovations herein to provide a method capable of overcoming the above drawbacks accompanying the related art. The above and other objects can be achieved by the combinations recited in the claims. A first aspect of the innovations herein may include a method including obtaining a plurality of clauses associated with a plurality of logical variables, each of the clauses consisting of a weight and a disjunction of one or more literals of the logical variables, detecting (i) whether any clauses in the plurality of clauses other than a first clause (a∨b, w 11 ), a second clause (ā∨ b , w 12 ), a third clause (a∨c, w 21 ), a fourth clause (ā∨ c , w 22 ), and a fifth clause (a, w 0 ), where a is a first logical variable, b is a second logical variable, c is a third logical variable, and w 11 , w 12 , w 21 , and w 22  are weights, contain a literal of the first logical variable a and a non-zero weight and (ii) whether min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ), and simplifying the plurality of clauses on the basis of the detecting. The simplifying may include modifying the plurality of clauses according to the assumption that a≠b if the detecting indicates (i) that no clauses in the plurality of clauses other than the first, second, third, fourth, and fifth clauses contain a literal of the first logical variable a and a non-zero weight and (ii) that min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ). 
     A second aspect of the innovations herein may include a computer readable storage medium having instructions embodied therewith, the instructions executable by a processor to cause the processor to perform operations corresponding to the steps of the method of the first aspect. 
     A third aspect of the innovations herein may include an apparatus including the above computer readable storage medium of the second aspect and a processor operable to execute the instructions. 
     A fourth aspect of the innovations herein may include a method including obtaining a plurality of clauses associated with a plurality of logical variables, each of the clauses consisting of a weight and a disjunction of one or more literals of the variables, detecting (i) whether the plurality of clauses includes a first clause (a∨b, H) and a second clause (a∨c, H), where a is a first logical variable, b is a second logical variable, c is a third logical variable, and a weight of H signifies a hard clause that must be satisfied by any assignment of values to the plurality of logical variables, (ii) whether any hard clauses in the plurality of clause other than the first and second clauses contain the literal a, and (iii) whether w unit (ā)+w unit (b)≥X b  and w unit (ā)+w unit (c)≥X c , where w unit (ā) is the weight w 1  of a clause (ā, w 1 ) included in the plurality of clauses, w unit (b) is the weight w 2  of a clause (b, w 2 ) included in the plurality of clauses, w unit (c) is the weight w 3  of a clause (c, w 3 ) included in the plurality of clauses, X b  is the sum of the weights of all clauses in the plurality of clauses, other than hard clauses containing the literal a, that contain at least one of the literals a and  b , and X c  is the sum of the weights of all clauses in the plurality of clauses, other than hard clauses containing the literal a, that contain at least one of the literals a and  c , and simplifying the plurality of clauses on the basis of the detecting. The simplifying may include modifying the plurality of clauses according to the assumption that b=c if the detecting indicates (i) that the plurality of clauses includes the first and second clauses, (ii) that no hard clauses in the plurality of clauses other than the first and second clauses contain the literal a, and (iii) that w unit (ā)+w unit (b)≥X b  and w unit (ā)+w unit (c)≥X c . 
     A fifth aspect of the innovations herein may include a computer readable storage medium having instructions embodied therewith, the instructions executable by a processor to cause the processor to perform operations corresponding to the steps of the method of the fourth aspect. 
     A sixth aspect of the innovations herein may include an apparatus including the above computer readable storage medium of the fifth aspect and a processor operable to execute the instructions. 
     The summary clause does not necessarily describe all of the features of the embodiments of the present invention. The present invention may also be a combination or sub-combination of the features described above, including a combination of features from two or more of the aspects described above. The above and other features and advantages of the present invention will become more apparent from the following description of the embodiments, taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows an apparatus  100  according to an embodiment of the present invention. 
         FIG. 2  shows an example operational flow of the apparatus  100  according to an embodiment of the present invention. 
         FIG. 3  shows an example operational flow of step S 210  in  FIG. 2 . 
         FIG. 4  shows an alternative example operational flow of step S 210  in  FIG. 2 . 
         FIG. 5  shows an example operational flow of step S 420  in  FIG. 4 . 
         FIG. 6  shows an example graph, along with a corresponding example representation of a data structure. 
         FIG. 7  shows an alternative example operational flow of step S 420  in  FIG. 4 . 
         FIG. 8  shows an example graph, along with a corresponding example representation of a data structure. 
         FIG. 9  shows an example operational flow of step S 220  in  FIG. 2 . 
         FIG. 10  shows an example operational flow of step S 910  in  FIG. 9 . 
         FIG. 11  shows an example operational flow of step S 1010  in  FIG. 10 . 
         FIG. 12  shows an alternative example operational flow of step S 910  in  FIG. 9 . 
         FIG. 13  shows an example operational flow of step S 1210  in  FIG. 12 . 
         FIG. 14  shows an example operational flow of step S 930  in  FIG. 9 . 
         FIG. 15  shows an alternative example operational flow of step S 930  in  FIG. 9 . 
         FIG. 16  shows an example of a computer  1600  in which the apparatus  100 , the operational flow of  FIG. 2 , and/or other embodiments of the claimed invention may be wholly or partly embodied. 
     
    
    
     DETAILED DESCRIPTION 
     Hereinafter, example embodiments of the present invention will be described. The embodiments should not be construed as limiting the scope of the invention, which is defined by the claims. The combinations of features described in the embodiments are not necessarily essential to the invention. 
       FIG. 1  shows an apparatus  100  according to an embodiment of the present invention. The apparatus  100  obtains a set of clauses representing a MAX-SAT instance and applies one or more inference rules. In accordance with the inference rules, the apparatus  100  detects various conditions and simplifies the clauses on the basis of the detection results. Using the simplified clauses, the apparatus  100  may then solve the MAX-SAT instance by determining an optimal assignment of values to the variables and sum the weights of clauses satisfied or unsatisfied by the optimal assignment. The apparatus  100  includes an input section  110 , a clause storage  120 , a detecting section  130 , a clause simplifying section  140 , an assignment determining section  150 , a weight summing section  160 , and an output section  170 . 
     The input section  110 , obtains a plurality of clauses associated with a plurality of logical variables, each of the clauses consisting of a weight and a disjunction of one or more literals of the logical variables. Each of the clauses may take the form (Z, w), where Z is the disjunction of one or more literals and w is the weight of the clause. Examples of clauses associated with the logical variables x 1 , x 2 , and x 3  are (x 1 ∨x 2 , w 1 ), ( x 1   ∨ x 2   ∨x 3 , w 2 ), ( x 3   , w 3 ), (T, w 4 ), etc., where “∨” is the symbol for disjunction (i.e. “OR”), the “literals” of a logical variable x refer to x and its negation  x , and “T” signifies a tautologically satisfied disjunction that is satisfied by any assignment of values to the plurality of logical variables. The weight w may be any non-negative number or “H”, where a weight of H signifies a hard clause that must be satisfied by any assignment of values to the logical variables. In this specification, a clause with weight w=0 is considered equivalent to the non-existence of the clause. Thus, generally, whether a clause of unspecified weight “exists” in a set of clauses is not meaningful, whereas the existence of a clause with specified non-zero weight is meaningful. 
     In some applications of MAX-SAT, the plurality of clauses may be associated with a graph having a plurality of vertices and a plurality of weighted edges connecting the vertices, such that each vertex of the plurality of vertices is represented by one of the logical variables. The graph may represent an arrangement of elements in a system. Non-limiting examples of systems include a computer system in which the elements may be processing elements, a communication network in which the elements may be communicating devices, a query execution system, a real or virtual structure or model, a process flow in which the elements are computation, decision, or other process nodes, or generally any spatial, temporal, or conceptual arrangement of components or items of information having a real-world application in a technical, business, or other practical setting. 
     In the example of the apparatus  100  shown in  FIG. 1 , the input section  110  includes a clause generating section  111 . The input section  110  may receive the graph from outside the apparatus  100 . For example, the graph can be received from an external storage or received from a computer or server through a network such as the Internet, WAN, and/or LAN. The clause generating section  111  may then generate the plurality of clauses based on the graph received from outside the apparatus. In this way, the input section  110  may obtain a plurality of clauses by generating the plurality of clauses based on an input graph. Alternatively or additionally, the input section  110  may obtain a plurality of clauses by receiving the plurality of clauses from outside the apparatus  100  in the same ways that the graph may be received. For example, the plurality of clauses can be received from an external storage or received from a computer or server through a network such as the Internet, WAN, and/or LAN. In a case where the input section  110  only receives clauses from outside the apparatus  100 , the clause generating section  111  can be omitted. 
     The input section  110  may receive data including a graph and/or a plurality of clauses through any combination of input device(s). For example, the input section  110  may be configured to receive mouse input, keyboard input, touchscreen input, eye tracking input, voice commands, and/or gestures. The input section  110  may receive the data from a remote user terminal or a remote user device. 
     The clause storage  120  stores the plurality of clauses obtained by the input section  110 . The clause storage  120  may store the plurality of clauses after one or more modifications have been applied to the plurality of clauses by the apparatus  100 . The clause storage  120  may store modified versions of the plurality of clauses in place of previous versions or in addition to previous versions. 
     The detecting section  130  detects various conditions of the plurality of clauses stored in the clause storage  120  in accordance with inference rules. 
     The clause simplifying section  140  simplifies the plurality of clauses stored in the clause storage  120  on the basis of the detecting by the detecting section  130 . The simplifying may include modifying the plurality of clauses according to assumptions that can be made on the basis of inference rules. For example, the clause simplifying section  140  may receive a detection result from the detecting section  130 , obtain the plurality of clauses stored in the clause storage  120 , and simplify the plurality of clauses to produce a simplified plurality of clauses based on the detection result. The clause simplifying section  140  may then store the simplified plurality of clauses in the clause storage  120  and/or provide the simplified plurality of clauses to the output section  170 . A simplified plurality of clauses output by the output section  170  in this way can be used downstream of the apparatus  100 , for example, by a separate MAX-SAT solver. 
     Depending on the detection results, the assumptions allowed by inference rules may or may not apply. In this specification, the meaning of simplifying the plurality of clauses on the basis of the detecting includes the meaning of simplifying or not depending on the detection result. In other words, if the clause simplifying section  140  receives a detection result and, in response to conditions being unmet, leaves the plurality of clauses unchanged, it can still be said that the clause simplifying section  140  simplified the plurality of clauses on the basis of the detecting. Likewise, the simplified plurality of clauses produced by the clause simplifying section  140  may be identical to the original plurality of clauses. That is, a plurality of clauses can be said to be a “simplified” plurality of clauses by virtue of the fact that detection results were checked and acted on by simplifying the clauses if appropriate. 
     The assignment determining section  150  determines, based on the simplified plurality of clauses, an optimal assignment of values to the plurality of logical variables, the optimal assignment of values determined so as to maximize the sum of the weights of clauses satisfied by the assignment. The assignment determining section  150  may determine one or more optimal assignments of values using known computational methods of MAX-SAT solvers, for example, the computational method of the open source SAT4J solver obtainable on the World Wide Web at sat4j.org. The assignment determining section  150  may provide the one or more optimal assignments of values to the output section  170 . 
     The weight summing section  160  may sum the weights of clauses satisfied by the optimal assignment of values determined by the assignment determining section  150 , the resulting sum representing the benefit of the optimal assignment of values. Alternatively, or additionally, the weight summing section  160  may sum the weights of clauses not satisfied by the optimal assignment of values determined by the assignment determining section  150 , the resulting sum representing the cost of the optimal assignment of values. In order to sum the weights, the weight summing section  160  may receive the plurality of clauses along with optimal assignment of values from the assignment determining section  150  as shown in  FIG. 1 . Alternatively, the weight summing section  160  may receive only the optical assignment of values from the assignment determining section  150  and may obtain the plurality of clauses directly from the clause storage  120 . The weight summing section  160  may provide the sum(s) to the output section  170 . 
     The output section  170  outputs one or more of the various outputs of the apparatus  100  for use by a downstream device or user. For example, the outputs may be stored, uploaded to a server, printed, displayed on a screen, or otherwise made available for viewing or analysis. The various outputs of the apparatus  100  output by the output section  170  may include, for example, the simplified plurality of clauses produced by the clause simplifying section  140 , one or more optimal assignments of values determined by the assignment determining section  150 , and/or one or more sums produced by the weight summing section  160 . 
     The output section  170  may output any of the various outputs to an external storage or to a computer or server through a network such as the Internet, WAN, and/or LAN. The outputting may include storing, uploading to a server, printing, displaying on a screen, or otherwise making the various outputs available for viewing or analysis. The output section  170  may output any of the various outputs through any output device or combination of output devices. For example, the output section  170  may be configured to provide still or moving visual output, audio output, or vibration or other touch-based output via a screen, speaker, printer, or other output device. The output section  170  may provide the various outputs to a remote user terminal or a remote user device. 
     The apparatus  100  shown in  FIG. 1  can obtain a set of clauses representing a MAX-SAT instance and apply one or more inference rules. In addition to known inference rules, the apparatus  100  can further simplify a MAX-SAT instance by the application of the following Inference Rules 1 and 2, shown informally in Table 1 below. Formal definitions and proofs of Inference Rules 1 and 2 can be found under the headings “Lemma 8 (Degree-2 Not-Equal Soft Clause Rule)” and “Lemma 10 (Degree-2 Hard Binary Clause Rule),” respectively, described in detail below. 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                   
                 Resulting  
               
               
                 Rule 
                 Given Conditions 
                 Inference 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 1 
                 (i)  
                 no clauses in the MAX-SAT instance other than a 
                 a ≠ b 
               
               
                   
                   
                 first clause (a     b, w 11 ), a second clause (ā     b , 
                   
               
               
                   
                   
                 w 12 ), a third clause (a     c, w 21 ), a fourth clause (ā 
                   
               
               
                   
                   
                      c , w 22 ), and a fifth clause (a, w 0 ) contain a literal 
                   
               
               
                   
                   
                 of a and a non-zero weight; and 
                   
               
               
                   
                 (ii)  
                 min(w 11 ,w 12 ) ≥ w 0  + max(w 21 , w 22 ) 
                   
               
               
                 2 
                 (i) 
                 the MAX-SATinstance includes a first clause (a      
                 b = c 
               
               
                   
                   
                 b, H) and a second clause (a     c, H); 
                   
               
               
                   
                 (ii) 
                 no hard clauses in the plurality of clauses other 
                   
               
               
                   
                   
                 than the first and second clauses contain the literal 
                   
               
               
                   
                   
                 a; and 
                   
               
               
                   
                 (iii)  
                 w unit (ā) + w unit (b) ≥ X b  and w unit (ā) + w unit (c) ≥ X c , 
                   
               
               
                   
                   
                 where w unit (ā) is the weight w 1  of a clause (ā, w 1 ) 
                   
               
               
                   
                   
                 included in the plurality of clauses, w unit (b) is the 
                   
               
               
                   
                   
                 weight w 2  of a clause (b, w 2 ) included in the 
                   
               
               
                   
                   
                 plurality of clauses, w unit (c) is the weight w 3  of a 
                   
               
               
                   
                   
                 clause (c, w 3 ) included in the plurality of clauses, 
                   
               
               
                   
                   
                 X b  is the sum of the weights of all clauses in the 
                   
               
               
                   
                   
                 plurality of clauses, other than hard clauses 
                   
               
               
                   
                   
                 containing the literal a, that contain at least one of 
                   
               
               
                   
                   
                 the literals a and  b , and X c  is the sum of the 
                   
               
               
                   
                   
                 weights of all clauses in the plurality of clauses, 
                   
               
               
                   
                   
                 other than hard clauses containing the literal a, that 
                   
               
               
                   
                   
                 contain at least one of the literals a and  c   
               
               
                   
               
            
           
         
       
     
     Thus, according to Inference Rule 1, if conditions (i) and (ii) shown in the corresponding row of Table 1 are satisfied, the resulting inference that a≠b can be made. In other words, the clauses of the MAX-SAT instance can be simplified under the assumption that a≠b. Similarly, according to Inference Rule 2, if conditions (i), (ii), and (iii) shown in the corresponding row of Table 1 are satisfied, the resulting inference that b=c can be made. In other words, the clauses of the MAX-SAT instance can be simplified under the assumption that b=c. 
     By simplifying a MAX-SAT instance in accordance with one or both of Inference Rules 1 and 2, the apparatus  100  can decrease the time and resources necessary to find a solution. Thus, when applied in a practical setting in any number of technical fields that use MAX-SAT, the apparatus  100  can be used to produce practical results associated with such fields, e.g. test results, technical designs, actual products, etc., more quickly and efficiently. Moreover, considering the decreased time and resources necessary to find a solution to a MAX-SAT instance even in the abstract, the apparatus  100  represents an improvement to existing MAX-SAT solvers. That is, when implemented as a MAX-SAT solver (applicable to a wide range of technical fields), the apparatus  100  consumes less time and resources than existing MAX-SAT solvers. 
       FIG. 2  shows an example operational flow of the apparatus  100  according to an embodiment of the present invention. In the example shown in  FIG. 2 , the apparatus  100  performs the operations from S 210  to S 240 , but the apparatus  100  shown in  FIG. 1  is not limited to using this operational flow. Also, the operational flow in  FIG. 2  may be performed by a modified apparatus or a different apparatus that differs from the apparatus  100  shown in  FIG. 1 . 
     First, the apparatus  100  obtains a plurality of clauses associated with a plurality of logical variables, each of the clauses consisting of a weight and a disjunction of one or more literals of the logical variables (S 210 ). For example, the input section  110  of the apparatus  100  may obtain the plurality of clauses by receiving the plurality of clauses from outside the apparatus, or the clause generating section  111  of the input section  110  may generate the plurality of clauses based on a graph and the input section  110  may obtain the generated plurality of clauses. The input section  110  may store the obtained plurality of clauses in the clause storage  120 . 
     Next, the apparatus  100  applies one or more inference rules to the plurality of clauses (S 220 ). For example, the detecting section  130  of the apparatus  100  may detect whether conditions of one or more inference rules are met by the plurality of clauses stored in the clause storage  120 , and the clause simplifying section  140  of the apparatus  100  may simplify the plurality of clauses stored in the clause storage  120  on the basis of the detection. In this way, the apparatus  100  may apply Inference Rule 1 and/or Inference Rule 2. 
     Next, the apparatus  100  determines, based on the simplified plurality of clauses, an optimal assignment of values to the plurality of logical variables, the optimal assignment of values determined so as to maximize the sum of the weights of clauses satisfied by the assignment (S 230 ). For example, the assignment determining section  150  of the apparatus  100  may determine the optimal assignment by known methods. 
     Lastly, the apparatus  100  sums the weights of the clauses satisfied or not satisfied by the optimal assignment of values (S 240 ). For example, the weight summing section  160  of the apparatus  100  may sum the weights of the satisfied clauses so that the resulting sum represents the benefit of the assignment or may sum the weights of the unsatisfied clauses so that the resulting sum represents the cost of the assignment. 
     In the example operational flow shown in  FIG. 2 , step S 240  follows step S 230 . However, step S 240  may instead precede step S 230 , for example, if the sum of weights produced by the weight summing section  160  is used by the assignment determining section  150  to determine the optimal assignment of values. Or, steps S 230  and S 240  may occur simultaneously or may overlap, depending on the method used to determine the optimal assignment of values. 
       FIG. 3  shows an example operational flow of step S 210  in  FIG. 2 . First, the input section  110  of the apparatus  100  may obtain a plurality of clauses by receiving input clauses from outside the apparatus (S 310 ). Then, the input section  110  may store the obtained plurality of clauses in the clause storage  120  (S 320 ). For example, the input section  110  may produce or modify a data structure containing information of the plurality of clauses and store the data structure in the clause storage  120 . The data structure stored in the clause storage  120  may include, for each of the logical variables, a list of the clauses that contain a literal of the logical variable and a non-zero weight. 
       FIG. 4  shows an alternative example operational flow of step S 210  in  FIG. 2 . First, the input section  110  of the apparatus  100  may receive an input graph from outside the apparatus (S 410 ). Then, the clause generating section  111  may generate a plurality of clauses based on the graph (S 420 ). Then, the input section  110  may store the plurality of clauses thus obtained in the clause storage  120  (S 430 ). For example, the input section  110  may produce a data structure containing information of the plurality of clauses and store the data structure in the clause storage  120 . The data structure stored in the clause storage  120  may include, for each of the logical variables, a list of the clauses that contain a literal of the logical variable and a non-zero weight. 
       FIG. 5  shows an example operational flow of step S 420  in  FIG. 4 . The example of  FIG. 5  relates to an application of Inference Rule 1, namely solving a maximum cut problem of a graph. Given a graph having a plurality of vertices and a plurality of weighted edges connecting the vertices, the maximum cut problem is to find a subset of vertices among the plurality of vertices (maximum cut subset) such that a sum of the weights of the edges connecting vertices in the subset with vertices not in the subset is maximized. 
     After the input section  110  of the apparatus  100  has received an input graph in step S 410 , in step S 420  the clause generating section  111  may generate, based on the graph, a plurality of clauses associated with the graph such that each vertex of the plurality of vertices is represented by a logical variable and the plurality of clauses includes two clauses (u∨v, w uv ) and (ū∨ v , w uv ) for each edge of the plurality of weighted edges, where u and v are logical variables representing the vertices connected by the edge and w uv  is the weight associated with the edge. That is, the clause generating section  111  may generate (u∨v, w uv ) and (ū∨ v , w uv ) for each edge (S 510 , S 520 ). In the example operational flow of  FIG. 5 , step S 520  follows step S 510 . However, step S 520  may instead precede step S 510  or steps S 510  and S 520  may occur simultaneously or overlap. 
       FIG. 6  shows an example graph, along with a corresponding example representation of a data structure. In the upper portion of  FIG. 6 , the graph is shown, in which it can be seen that the graph includes five vertices a, b, c, d, and e (represented as points) and seven edges (represented as line segments) connecting the vertices. In the lower portion of  FIG. 6 , there is a representation of a data structure corresponding to the graph. The graph may be, for example, an input graph received by the input section  110  of the apparatus  100  in step S 410  of  FIG. 4 , and the data structure may be, for example, the data structure stored by the input section  110  in the clause storage  120  in step S 430  of  FIG. 4 . 
     The data structure shown in  FIG. 6  includes, for each of the logical variables, a list of the clauses that contain a literal of the logical variable and a non-zero weight. With the clause generating section  111  having generated (u∨v, w uv ) and (ū∨ v , w uv ) for each edge, the plurality of clauses associated with the graph shown in  FIG. 6  includes (a∨b, w ab ), (ā∨ b , w ab ), (a∨c, w ac ), (ā∨ c , w ac ), (a∨e, w ae ), (ā∨ē, w ae ), (b∨c, w bc ), ( b ∨ c , w bc ), (b∨e, w be ), ( b ∨ē, w be ), (c∨d, w cd ), ( c ∨ d ), (d∨e, w de ), and ( d ∨ē, w de ). Of these, (a∨b, w ab ), (ā∨ b , w ab ), (a∨c, w ac ), (ā∨ c , w ac ), (a∨e, w ae ), and (ā∨ē, w ae ) contain a literal of the logical variable a. Therefore, the data structure shown in  FIG. 6  includes the list (a∨b, w ab ), (ā∨ b , w ab ), (a∨c, w ac ),(ā∨ c ,w ac ), (a∨e, w ae ), and (ā∨ē, w ae ) for the logical variable a. As shown in  FIG. 6 , the list (a∨b, w ab ), (ā∨ b , w ab ), (a∨c, w ac ), (ā∨ c , w ac ), (a∨e, w ae ), and (ā∨ē, w ae ) is stored in association with the logical variable a. Other lists are similarly stored for each of the logical variables b, c, d, and e. As mentioned above, a clause with weight w=0 is considered equivalent to the non-existence of the clause, while the data structure stores only clauses having non-zero weight. Thus, although other clauses can be said to exist in the set of clauses, they do not appear in the data structure. Since each of the edges is represented in the data structure, it can be assumed that each of the weights w ab , w ac , w ae , w bc , w be , w cd , and w de  is non-zero. 
     The plurality of clauses generated for the graph as explained above can be used to solve the maximum cut problem. That is, if the value of each of the logical variables indicates whether the vertex represented by that logical variable is in the subset or not in the subset (e.g. TRUE=in the subset, FALSE=not in the subset), then a solution to the MAX-SAT instance consisting of the clauses generated for the graph yields a solution to the maximum cut problem. This is because each pair of clauses (u∨v, w uv ) and (ū∨ v , w uv ) associated with an edge contributes weight only when one of the two vertices of the edge is in the subset and the other is not, i.e. when the edge is “cut.” Therefore, when the assignment determining section  150  determines an optimal assignment of values to a plurality of logical variables associated with clauses that are associated with a graph, the determination of the optimal assignment may include determining a solution to the maximum cut problem of the graph, where the value of each of the logical variables in the determined optimal assignment indicates whether the vertex represented by the logical variable is in the maximum cut subset or not in the maximum cut subset. 
     By applying Inference Rule 1 to the MAX-SAT instance generated for the maximum cut problem as explained above with respect to  FIG. 6 , the MAX-SAT instance, and consequently the maximum cut problem, can be simplified. Therefore, the apparatus  100  can be used to more quickly and efficiently produce practical results in technical fields where the maximum cut problem is applied. 
       FIG. 7  shows an alternative example operational flow of step S 420  in  FIG. 4 . The example of  FIG. 7  relates to an application of Inference Rule 2, namely solving a minimum vertex cover problem of a graph. Given a graph having a plurality of vertices and a plurality of edges connecting the vertices, the minimum vertex cover problem is to find a set of vertices with minimum size among the plurality of vertices (minimum vertex cover set) such that, for each of the plurality of edges, the set of vertices includes at least one of the two vertices connected by the edge. 
     After the input section  110  of the apparatus  100  has received an input graph in step S 410 , in step S 420  the clause generating section  111  may generate, based on the graph, a plurality of clauses associated with the graph such that each vertex of the plurality of vertices is represented by a logical variable and the plurality of clauses includes a clause ( v , 1) for each vertex of the plurality of vertices and a clause (u∨v, H) for each edge of the plurality of edges, where u and v are logical variables representing the vertices connected by the edge. That is, the clause generating section  111  may generate ( v , 1) for each vertex (S 710 ) and (u∨v, H) for each edge (S 720 ). In the example operational flow of  FIG. 7 , step S 720  follows step S 710 . However, step S 720  may instead precede step S 710  or steps S 710  and S 720  may occur simultaneously or overlap. 
       FIG. 8  shows an example graph, along with a corresponding example representation of a data structure. In the upper portion of  FIG. 8 , the graph is shown, in which it can be seen that the graph includes five vertices a, b, c, d, and e (represented as points) and seven edges (represented as line segments) connecting the vertices. In the lower portion of  FIG. 8  is a representation of a data structure corresponding to the graph. The graph may be, for example, an input graph received by the input section  110  of the apparatus  100  in step S 410  of  FIG. 4 , and the data structure may be, for example, the data structure stored by the input section  110  in the clause storage  120  in step S 430  of  FIG. 4 . 
     The data structure shown in  FIG. 8  includes, for each of the logical variables, a list of the clauses that contain a literal of the logical variable and a non-zero weight. With the clause generating section  111  having generated ( v , 1) for each vertex and (u∨v, H) for each edge, the plurality of clauses associated with the graph shown in  FIG. 8  includes (ā, 1), ( b , 1), ( c , 1), ( d , 1), (ē, 1), (a∨b, H), (a∨c, H), (a∨e, H), (b∨c, H), (b∨e, H), (c∨d, H), and (d∨e, H). Of these, (ā, 1), (a∨b, H), (a∨c, H), and (a∨e, H) contain a literal of the logical variable a. Therefore, the data structure shown in  FIG. 8  includes the list (ā, 1), (a∨b, H), (a∨c, H), and (a∨e, H) for the logical variable a. As shown in  FIG. 8 , the list (ā, 1), (a∨b, H), (a∨c, H), and (a∨e, H) is stored in association with the logical variable a. Other lists are similarly stored for each of the logical variables b, c, d, and e. As mentioned above, a clause with weight w=0 is considered equivalent to the non-existence of the clause, while the data structure stores only clauses having non-zero weight. Thus, although other clauses can be said to exist in the set of clauses, they do not appear in the data structure. 
     The plurality of clauses generated for the graph as explained above can be used to solve the minimum vertex cover problem. That is, if the value of each of the logical variables indicates whether the vertex represented by that logical variable is in the set or not in the set (e.g. TRUE=in the set, FALSE=not in the set), then a solution to the MAX-SAT instance consisting of the clauses generated for the graph yields a solution to the minimum vertex cover problem. This is because each clause ( v , 1) associated with a vertex contributes weight only when the vertex is not in the subset, such that maximizing the weight minimizes the number of vertices in the set, while the hard clauses (u∨v, H) guarantee that at least one of the two vertices connected by each edge is included in the set. Therefore, when the assignment determining section  150  determines an optimal assignment of values to a plurality of logical variables associated with clauses that are associated with a graph, the determination of the optimal assignment may include determining a solution to the minimum vertex problem of the graph, where the value of each of the logical variables in the determined optimal assignment indicates whether the vertex represented by the logical variable is in the minimum vertex cover set or not in the minimum vertex cover set. 
     By applying Inference Rule 2 to the MAX-SAT instance generated for the minimum vertex problem as explained above with respect to  FIG. 8 , the MAX-SAT instance, and consequently the minimum vertex cover problem, can be simplified. Therefore, the apparatus  100  can be used to more quickly and efficiently produce practical results in technical fields where the minimum vertex cover problem is applied. 
       FIG. 9  shows an example operational flow of step S 220  in  FIG. 2 . After the input section  110  of the apparatus  100  has obtained the plurality of clauses in step S 210 , e.g., by receiving the plurality of clauses from outside the apparatus in accordance with  FIG. 3  or by generating the plurality of clauses based on a graph in accordance with  FIG. 4 , the apparatus detects one or more conditions in accordance with one or more inference rules (S 910 ). For example, the detecting section  130  of the apparatus  100  may detect whether the plurality of clauses stored in the clause storage  120  meet condition(s) including those of Inference Rule 1 and/or Inference Rule 2. If the conditions of any rule (e.g. any rule set to be applied by the apparatus  100 ) are met (“Yes” at S 920 ), the apparatus  100  simplifies the clauses in accordance with the rule (S 930 ). For example, the clause simplifying section  140  of the apparatus  100  may simplify the plurality of clauses stored in the clause storage  120  in accordance with the rule and store the simplified plurality of clauses in the clause storage  120  in place of or in addition to the previous version. The operational flow then returns to step S 910 , where the detecting section  130  may again detect condition(s), this time on the basis of the simplified plurality of clauses stored in the clause storage  120 . Thus, the detecting by the detecting section  130  may be repeated after the simplification by the clause simplifying section  140  one or more times. If, at any time, the conditions are not met for any rule (“No” at S 920 ), the operational flow ends. 
       FIG. 10  shows an example operational flow of step S 910  in  FIG. 9 . In the example of  FIG. 10 , the conditions of Inference Rule 1 are detected with respect to the logical variable a (i.e. with a as the “first logical variable”). Namely, the detecting section  130  detects (i) whether any clauses in the plurality of clauses other than a first clause (a∨b, w 11 ), a second clause (ā∨ b , w 12 ), a third clause (a∨c, w 21 ), a fourth clause (ā∨ c , w 22 ), and a fifth clause (a, w 0 ), where a is a first logical variable, b is a second logical variable, c is a third logical variable, and w 11 , w 12 , w 21 , and w 22  are weights, contain a literal of the first logical variable a and a non-zero weight and (ii) whether min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ) (S 1010 ). The detecting section  130  may detect conditions (i) and (ii) of Inference Rule 1 simultaneously or sequentially in any order, may detect preliminary conditions that obviate the need to check one or both of conditions (i) and (ii), and/or may detect other, different conditions that are not identical to conditions (i) and (ii) but substantially satisfy conditions (i) and (ii). An example of such a condition that substantially satisfies condition (ii) may be a numerically similar inequality such as min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 )+1. Assuming that satisfaction of this inequality substantially satisfies condition (ii) for the set of clauses in question, detecting such similar inequality in place of condition (ii) is included in the meaning of detecting condition (ii). That is, as used herein, detecting a condition includes detecting a non-identical condition that substantially satisfies the condition. 
       FIG. 11  shows an example operational flow of step S 1010  in  FIG. 10 .  FIG. 11  is an example of detecting preliminary conditions that obviate the need to check one or both of conditions (i) and (ii). An inspection of the conditions of Inference Rule 1 reveals that condition (i) cannot be satisfied with respect to the logical variable a if there are too many clauses containing a literal of a and a non-zero weight. That is, there can be at most five, since any more would mean that there exists some clause other than the first through fifth clauses that contains a literal of a and a non-zero weight. Thus, the detecting in step S 1010  may include confirming that the number of clauses in the list of clauses for the first logical variable a is not more than a threshold before detecting one or more of (i) whether any clauses in the plurality of clauses other than the first, second, third, fourth, and fifth clauses contain a literal of the first logical variable a and a non-zero weight and (ii) whether min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ). As shown in  FIG. 11 , when performing step S 1010 , if the number of clauses in the data structure for a≤a threshold, e.g. 5 (“Yes” at S 1110 ), the detecting section  130  checks conditions (i) and (ii) with respect to a (S 1120 ). Otherwise (“No” at S 1110 ), the operational flow of  FIG. 11  ends and the “detecting” step S 1010  is considered performed without the need to check conditions (i) and (ii). 
     In the example of  FIG. 11 , the preliminary condition represented by step S 1110  precedes the checking of conditions (i) and (ii) at step S 1120 . However, this is only one of many possible variants. For example, the detecting section  130  may check condition (ii) irrespective of the outcome of step S 1110 , such that a “No” result at step S 1110  only obviates the need to check condition (i). 
       FIG. 12  shows an alternative example operational flow of step S 910  in  FIG. 9 . In the example of  FIG. 12 , the conditions of Inference Rule 2 are detected with respect to the logical variable a (i.e. with a as the “first logical variable”). That is, the detecting section  130  detects (i) whether the plurality of clauses includes a first clause (a∨b, H) and a second clause (a∨c, H), where a is a first logical variable, b is a second logical variable, and c is a third logical variable, (ii) whether any hard clauses in the plurality of clause other than the first and second clauses contain the literal a, and (iii) whether w unit (ā)+w unit (b)≥X b  and w unit (ā)+w unit (c)≥X c , where w unit (ā) is the weight w 1  of a clause (ā, w 1 ) included in the plurality of clauses, w unit (b) is the weight w 2  of a clause (b, w 2 ) included in the plurality of clauses, w unit (c) is the weight w 3  of a clause (c, w 3 ) included in the plurality of clauses, X b  is the sum of the weights of all clauses in the plurality of clauses, other than hard clauses containing the literal a, that contain at least one of the literals a and  b , and X c  is the sum of the weights of all clauses in the plurality of clauses, other than hard clauses containing the literal a, that contain at least one of the literals a and  c  (S 1210 ). Similarly to when detecting the conditions of Inference Rule 1, the detecting section  130  may detect conditions (i)-(iii) of Inference Rule 2 simultaneously or sequentially in any order, may detect preliminary conditions that obviate the need to check one or more of conditions (i)-(iii), and/or may detect other, different conditions that are not identical to conditions (i)-(iii) but substantially satisfy conditions (i)-(iii). 
       FIG. 13  shows an example operational flow of step S 1210  in  FIG. 12 .  FIG. 13  is an example of detecting preliminary conditions that obviate the need to check one or more of conditions (i)-(iii). An inspection of the conditions of Inference Rule 2 reveals that condition (ii) cannot be satisfied with respect to the logical variable a if there are too many hard clauses containing the literal a. That is, there can be at most two, since any more would mean that there exists some hard clause other than the first and second clauses that contains the literal a. Thus, the detecting in step S 1210  may include confirming that the number of hard clauses in the list of clauses for the first logical variable a is not more than a threshold before detecting one or more of (i) whether the plurality of clauses includes the first and second clauses, (ii) whether any hard clauses in the plurality of clauses other than the first and second clauses contain the literal a, and (iii) whether w unit (ā)+w unit (b)≥X b  and w unit (ā)+w unit (c)≥X c . As shown in  FIG. 13 , when performing step S 1210 , if the number of hard clauses in the data structure for a≤a threshold, e.g. 4 (“Yes” at S 1310 ), the detecting section  130  checks conditions (i)-(iii) with respect to a (S 1320 ). (A threshold of 4 would allow for two hard clauses containing the literal a and two hard clauses containing the literal ā.) Otherwise (“No” at S 1310 ), the operational flow of  FIG. 13  ends and the “detecting” step S 1210  is considered performed without the need to check conditions (i)-(iii). 
     In the example of  FIG. 13 , the preliminary condition represented by step S 1310  precedes the checking of conditions (i)-(iii) at step S 1320 . However, this is only one of many possible variants. For example, the detecting section  130  may check condition (i) irrespective of the outcome of step S 1310 , such that a “No” result at step S 1310  only obviates the need to check conditions (ii) and (iii). 
       FIGS. 10-13  are only simple examples of sub-flows step S 910  in  FIG. 9 , each illustrating the detection of conditions for only a single instance of an inference rule (e.g. Inference Rule 1 for logical variable “a”; note that an instance of an inference rule is also referred to as a rule herein, such that a plurality of rules may refer to multiple instances of the same rule). As noted above, step S 910  may include the detection of conditions for a plurality of rules, including rules other than Inference Rule 1 and Inference Rule 2. In such cases, step  910  may include both the sub-flow of  FIG. 10  and the sub-flow of  FIG. 12  as well as other sub-flows for other rules. Rules may have overlapping conditions, and such combined sub-flows need not be performed sequentially. Similarly, the sub-flows of  FIGS. 11 and 13  (and corresponding sub-flows for other rules) may be combined with many variations, including the detection of preliminary conditions that simultaneously obviate the need to check conditions of multiple rules. 
       FIG. 14  shows an example operational flow of step S 930  in  FIG. 9 . In the example of  FIG. 14 , it is assumed for simplicity that the apparatus  100  has only performed detection of the conditions of Inference Rule 1 with respect to the logical variable a in step S 910 . If the conditions of Inference Rule 1 are satisfied, the apparatus  100  simplifies the clauses in accordance with the rule. That is, if the detecting by the detecting section  130  indicates (i) that no clauses in the plurality of clauses other than the first clause (a∨b, w 11 ), the second clause (ā∨ b , w 12 ), the third clause (a∨c, w 21 ), the fourth clause (ā∨ c , w 22 ), and the fifth clause (a, w 0 ) contain a literal of the first logical variable a and a non-zero weight and (ii) that min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ) (“Yes” at S 1410 ), the clause simplifying section  140  modifies the plurality of clauses according to the assumption that a≠b. Under the assumption that a≠b, it can be inferred, for example, that a=b, such that the first clause (a∨b, w 11 ) and the second clause (ā∨ b , w 12 ) are automatically satisfied. Thus, the clause simplifying section  140  may replace any clauses having non-zero weight from among the first, second, third, fourth, and fifth clauses with a replacement first clause (T, w r11 ), a replacement second clause (T, w r12 ), a replacement third clause ( b ∨c, w r21 ) or (a∨c, w r21 ), a replacement fourth clause (b∨ c , w r22 ) or (ā∨ c , w r22 ), and a replacement fifth clause ( b , w r0 ) or (a, w r0 ), respectively, where w r11 , w r12 , w r21 , w r22 , and w r0  are substantially equal to w 11 , w 12 , w 21 , w 22 , and w 0 , respectively. Replacing the third, fourth, and fifth clauses with the replacement third clause (a∨c, w r21 ), the replacement fourth clause (ā∨ c , w r22 ), and the replacement fifth clause (a, w r0 ) may include leaving the third, fourth, and fifth clauses unchanged (S 1420 ). Based on the assumption of a≠b (a= b ), in step S 1420  the clause simplifying section  140  may further replace, in clauses having non-zero weight, every a with  b  or every b with ā so that one of a and b can be eliminated from the clauses having non-zero weight in the plurality of clauses. 
     The clause simplifying section  140  may store the simplified plurality of clauses in the clause storage  120  in place of or in addition to the previous version. If the simplified plurality of clauses is simplified on the basis of Inference Rule 1 as explained above, the plurality of clauses may include the first replacement clause (T, w r11 ) and the second replacement clause (T, w r12 ), which have tautologically satisfied disjunctions T and weights substantially equal or equal to w 11  and w 12 , respectively. Therefore, when the weight summing section  160  refers to the simplified clauses to sum the weights of clauses satisfied or unsatisfied by the optimal assignment of values in step S 240  of  FIG. 2 , the weight summing section  160  may include or not include w 11  and w 12  in the resulting sum accordingly. Alternatively, the clause simplifying section  140  may not store clauses having tautologically satisfied disjunctions T and may instead separately provide the weight information of such clauses (e.g. w 11  and w 12 ) to the weight summing section  160 . By any of these or other methods, when the weight summing section  160  sums the weights of clauses satisfied by the optimal assignment of values, the resulting sum may include w 11  and w 12  if the detecting indicates (i) that no clauses in the plurality of clauses other than the first, second, third, fourth, and fifth clauses contain a literal of the first logical variable a and a non-zero weight and (ii) that min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ) in accordance with Inference Rule 1. Similarly, when the weight summing section  160  sums the weights of clauses not satisfied by the optimal assignment of values, the resulting sum may not include w 11  or w 12  if the detecting indicates (i) that no clauses in the plurality of clauses other than the first, second, third, fourth, and fifth clauses contain a literal of the first logical variable a and a non-zero weight and (ii) that min(w 11 ,w 12 )≥w 0 +max(w 21 , w 22 ) in accordance with Inference Rule 1. 
       FIG. 15  shows an alternative example operational flow of step S 930  in  FIG. 9 . In the example of  FIG. 15 , it is assumed for simplicity that the apparatus  100  has only performed detection of the conditions of Inference Rule 2 with respect to the logical variable a in step S 910 . If the conditions of Inference Rule 2 are satisfied, the apparatus  100  simplifies the clauses in accordance with the rule. That is, if the detecting by the detecting section  130  indicates (i) that the plurality of clauses includes the first clause (a∨b, H) and the second clause (a∨c, H), (ii) that no hard clauses in the plurality of clauses other than the first and second clauses contain the literal a, and (iii) that w unit (ā)+w unit (b)≥X b  and w unit (ā)+w unit (c)≥X c  (“Yes” at S 1510 ), the clause simplifying section  140  modifies the plurality of clauses according to the assumption that b=c. Under the assumption that b=c, the first clause (a∨b, H) and the second clause (a∨c, H) are equivalent. Thus, the clause simplifying section  140  may remove one of the first and second clauses. 
       FIGS. 14 and 15  are only simple examples of sub-flows of step S 930  in  FIG. 9 , each illustrating the simplifying of a plurality of clauses in response to the detection of conditions for only a single inference rule. As noted above, step S 910  may include the detection of conditions for a plurality of rules, including rules other than Inference Rule 1 and Inference Rule 2. In such cases, step  930  may include both the sub-flow of  FIG. 14  and the sub-flow of  FIG. 15  as well as other sub-flows for other rules. Rules may yield overlapping or redundant inferences, and such combined sub-flows need not be performed sequentially. 
       FIG. 16  shows an example of a computer  1600  in which the apparatus  100 , the operational flow of  FIG. 2 , and/or other embodiments of the claimed invention may be wholly or partly embodied. The computer  1600  according to the present embodiment includes a CPU  1612 , a RAM  1614 , a graphics controller  1616 , and a display device  1618 , which are mutually connected by a host controller  1610 . The computer  1600  also includes input/output units such as a communication interface  1622 , a hard disk drive  1624 , and a DVD-ROM drive  1626 , which are connected to the host controller  1610  via an input/output controller  1620 . The computer also includes legacy input/output units such as a ROM  1630  and a keyboard  1642 , which is connected to the input/output controller  1620  through an input/output chip  1640 . 
     The host controller  1610  connects the RAM  1614  with the CPU  1612  and the graphics controller  1616 , which access the RAM  1614  at a high transfer rate. The CPU  1612  operates according to programs stored in the ROM  1630  and the RAM  1614 , thereby controlling each unit. The graphics controller  1616  obtains image data generated by the CPU  1612  on a frame buffer or the like provided in the RAM  1614 , and causes the image data to be displayed on the display device  1618 . Alternatively, the graphics controller  1616  may contain therein a frame buffer or the like for storing image data generated by the CPU  1612 . 
     The input/output controller  1620  connects the host controller  1610  with the communication interface  1622 , the hard disk drive  1624 , and the DVD-ROM drive  1626 , which are relatively high-speed input/output units. The communication interface  1622  communicates with other electronic devices via a network. The hard disk drive  1624  stores programs and data used by the CPU  1612  within the computer  1600 . The DVD-ROM drive  1626  reads the programs or the data from the DVD-ROM  1601 , and provides the hard disk drive  1624  with the programs or the data via the RAM  1614 . 
     The ROM  1630  and the keyboard  1642  and the input/output chip  1640 , which are relatively low-speed input/output units, are connected to the input/output controller  1620 . The ROM  1630  stores therein a boot program or the like executed by the computer  1600  at the time of activation, a program depending on the hardware of the computer  1600 . The keyboard  1642  inputs text data or commands from a user, and may provide the hard disk drive  1624  with the text data or the commands via the RAM  1614 . The input/output chip  1640  connects the keyboard  1642  to the input/output controller  1620 , and may connect various input/output units via a parallel port, a serial port, a keyboard port, a mouse port, and the like to the input/output controller  1620 . 
     A program to be stored on the hard disk drive  1624  via the RAM  1614  is provided by a recording medium such as the DVD-ROM  1601  or an IC card. The program is read from the recording medium, installed into the hard disk drive  1624  within the computer  1600  via the RAM  1614 , and executed in the CPU  1612 . 
     A program that is installed in the computer  1600  can cause the computer  1600  to function as an apparatus such as the apparatus  100  of  FIG. 1 . Such a program may act on the CPU  1612  to cause the computer  1600  to function as some or all of the sections, components, elements, databases, etc. of the apparatus  100  of  FIG. 1  (e.g., the detecting section  130 , the clause simplifying section  140 , etc.). 
     A program that is installed in the computer  1600  can also cause the computer  1600  to perform an operational flow such as the operational flow of  FIG. 2 . Such a program may act on the CPU  1612  to cause the computer  1600  to perform some or all of the steps of  FIG. 2  (e.g., apply rules S 220 , determine optimal assignment S 230 , etc.). 
     The information processing described in these programs is read into the computer  1600 , resulting in the cooperation between a program and the above-mentioned various types of hardware resources. An apparatus or method may be constituted by realizing the operation or processing of information in accordance with the usage of the computer  1600 . 
     For example, when communication is performed between the computer  1600  and an external device, the CPU  1612  may execute a communication program loaded onto the RAM  1614  to instruct communication processing to the communication interface  1622 , based on the processing described in the communication program. 
     The communication interface  1622 , under control of the CPU  1612 , reads transmission data stored on a transmission buffering region provided in a recording medium such as the RAM  1614 , the hard disk drive  1624 , or the DVD-ROM  1601 , and transmits the read transmission data to a network or writes reception data received from a network to a reception buffering region or the like provided on the recording medium. In this way, the communication interface  1622  may exchange transmission/reception data with a recording medium by a DMA (direct memory access) method or by a configuration in which the CPU  1612  reads the data from the recording medium or the communication interface  1622  of a transfer destination and writes the data into the communication interface  1622  or the recording medium of the transfer destination, so as to transfer the transmission/reception data. 
     In addition, the CPU  1612  may cause all or a necessary portion of a file or a database to be read into the RAM  1614  such as by DMA transfer, the file or the database having been stored in an external recording medium such as the hard disk drive  1624 , the DVD-ROM drive  1626  (DVD-ROM  1601 ) and perform various types of processing on the data on the RAM  1614 . The CPU  1612  may then write back the processed data to the external recording medium by means of a DMA transfer method or the like. In such processing, the RAM  1614  can be considered to temporarily store the contents of the external recording medium, and so the RAM  1614 , the external recording apparatus, and the like are collectively referred to as a memory, a storage section, a recording medium, a computer readable medium, etc. 
     Various types of information, such as various types of programs, data, tables, and databases, may be stored in the recording apparatus to undergo information processing. Note that the CPU  1612  may also use a part of the RAM  1614  to perform reading/writing thereto on a cache memory. In such an embodiment, the cache is considered to be contained in the RAM  1614 , the memory, and/or the recording medium unless noted otherwise, since the cache memory performs part of the function of the RAM  1614 . 
     The CPU  1612  may perform various types of processing on the data read from the RAM  1614 , which includes various types of operations, processing of information, condition judging, search/replace of information, etc., as described throughout this disclosure and designated by an instruction sequence of programs, and writes the result back to the RAM  1614 . For example, when performing condition judging, the CPU  1612  may judge whether each type of variable is larger, smaller, no smaller than, no greater than, or equal to the other variable or constant, and when the condition judging results in the affirmative (or in the negative), the process branches to a different instruction sequence or calls a subroutine. 
     In addition, the CPU  1612  may search for information in a file, a database, etc., in the recording medium. For example, when a plurality of entries, each having an attribute value of a first attribute is associated with an attribute value of a second attribute, are stored in a recording apparatus, the CPU  1612  may search for an entry matching the condition whose attribute value of the first attribute is designated, from among the plurality of entries stored in the recording medium, and reads the attribute value of the second attribute stored in the entry, thereby obtaining the attribute value of the second attribute associated with the first attribute satisfying the predetermined condition. 
     The above-explained program or module may be stored in an external recording medium. Exemplary recording mediums include a DVD-ROM  1601 , as well as an optical recording medium such as a Blu-ray Disk or a CD, a magneto-optic recording medium such as a MO, a tape medium, and a semiconductor memory such as an IC card. In addition, a recording medium such as a hard disk or a RAM provided in a server system connected to a dedicated communication network or the Internet can be used as a recording medium, thereby providing the program to the computer  1600  via the network. 
     The present invention may be a system, a method, and/or a computer program product. The computer program product may include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of the present invention. 
     The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium may be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. 
     A non-exhaustive list of more specific examples of the computer readable storage medium includes the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media (e.g., light pulses passing through a fiber-optic cable), or electrical signals transmitted through a wire. 
     Computer readable program instructions described herein can be downloaded to respective computing/processing devices from a computer readable storage medium or to an external computer or external storage device via a network, for example, the Internet, a local area network, a wide area network and/or a wireless network. The network may comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers, and/or edge servers. A network adapter card or network interface in each computing/processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing/processing device. 
     Computer readable program instructions for carrying out operations of the present invention may be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like, and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The computer readable program instructions may execute entirely on the user&#39;s computer, partly on the user&#39;s computer, as a stand-alone software package, partly on the user&#39;s computer and partly on a remote computer or entirely on the remote computer or server. 
     In the latter scenario, the remote computer may be connected to the user&#39;s computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA) may execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the present invention. 
     Aspects of the present invention are described herein with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer readable program instructions. 
     These computer readable program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     These computer readable program instructions may also be stored in a computer readable storage medium that can direct a computer, a programmable data processing apparatus, and/or other devices to function in a particular manner, such that the computer readable storage medium having instructions stored therein comprises an article of manufacture including instructions which implement aspects of the function/act specified in the flowchart and/or block diagram block or blocks. 
     The computer readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function(s). 
     In some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions. 
     While the embodiment(s) of the present invention has (have) been described, the technical scope of the invention is not limited to the above described embodiment(s). It is apparent to persons skilled in the art that various alterations and improvements can be added to the above-described embodiment(s). It is also apparent from the scope of the claims that the embodiments added with such alterations or improvements can be included in the technical scope of the invention. 
     The operations, procedures, steps, and stages of each process performed by an apparatus, system, program, and method shown in the claims, embodiments, or diagrams can be performed in any order as long as the order is not indicated by “prior to,” “before,” or the like and as long as the output from a previous process is not used in a later process. Even if the process flow is described using phrases such as “first” or “next” in the claims, embodiments, or diagrams, it does not necessarily mean that the process must be performed in this order. 
     APPENDIX 
     Preliminaries. 
     First, we introduce some notations. For any literal l, var (l) represents the variable to which literal l refers to. This means that var(x i )=var( x i   )=x i  for any variable x i . A weighted clause (c, w) is a pair of a clause c and its weight w, where w is a non-negative integer or ∞. A clause c that does not appear in a formula Φ is sometimes regarded as a part of Φ with weight zero. That is, we sometimes assume that (c, 0)∈Φ even if clause c is not contained in Φ. A weighted clause with weight ∞ is often referred to as a hard clause, and the other clauses are called soft. Empty clause □ cannot be satisfied, and tautology clause T is always satisfied. An assignment I for a formula Φ is a function I: X→{TRUE, FALSE} n , where X is the set of variables. Given a formula Φ and its assignment I, cost (Φ, I) denotes the sum of the weights of the clauses unsatisfied by L. (Note that it is customary to regard the MAX SAT problem as minimizing the sum of the weights of the unsatisfied clauses rather than maximizing the sum of the weights of the satisfied clauses.) Hence our objective is to find an assignment I that minimizes cost(Φ, I). A unit clause is a clause that consists of exactly one literal, and a binary clause is a clause that consists of exactly two literals. For a literal in formula Φ, Φ(l) denotes the subset of the clauses in Φ that contains literal l. 
     For any clause c in formula Φ, w all (c) denotes the sum of the weights of all clauses that contain all of the literals in c, and w unit (c) denotes the weight of clause c. For example, given a formula Φ={(x 1 ∨x 2 ∨x 3 , 1), (x 1 , 2)}, we have w all (x 1 )=3, w all (x 1 ∨ 63  x 2 )=1, (note here that the two literals x 1  and x 2  are contained in clause x 1 ∨x 2 ∨x 3 ), w unit (x 1 )=2, and w unit (x 1 ∨x 2 )=0. 
     We say that two formulas Φ and Φ′ are equivalent Φ≡Φ′ if the costs of the optimum assignments of these formulas are equal. Given a formula Φ with literals l 1  and l 2 , Φ| l     1    denotes the formula obtained by setting l 1 =TRUE, and Φ| l     1     =l     2    denotes the formula obtained by setting l 1 =l 2 . 
     Lemma 8 (Degree-2 Not-Equal Soft Clause Rule). 
     If an input formula Φ can be represented as
 
Φ={( a,w   0 ),( a∨b,w   11 ),(   a   ∨   b   ,w 12 ),( a∨c,w   21 ),(   a   ∨   c   ,w 22 )}∪Φ 0 ,
 
Φ 0  does not contain var(a), and
 
min{ w   11   ,w   12   }≥w   0 +max{ w   21   ,w   22 }
 
then we have Φ≡Φ| a≠b . This means that
 
     
       
         
           
             
               
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     Proof. 
     Let Φ′=Φ\Φ 0 . We consider the two cases (i) b≠c and (ii) b=c separately. Suppose that (i) b≠c. Let I a≠c * be the optimum assignment under the condition that a≠c for Φ. Having that a≠c and b≠c yield a=b, we have cost(Φ′, I a≠c *)≥min{w 11 ,w 12 } since one of the two clauses (a∨b, w 11 ) and (ā∨ b , w 12 ) is unsatisfied. Let I a≠b  be the assignment obtained by flipping the value of literal a from I a≠c *. Since a≠b≠c means that a=c, we have cost(Φ′, I a≠b )≥w 0 +max{w 21 , w 22 }. From the assumption min{w 11 , w 12 }≥w 0 +max{w 21 , w 22 }, we have cost(Φ′, I a≠b )≤cost(Φ′, I a≠c *). By the definition of I a≠c * and I a≠b , the costs associated with the clauses in Φ 0  are the same for I a≠c * and I a≠b  because literal a does not appear in any clause in Φ 0  by the assumption. Hence we have cost(Φ′, I a≠c *)−cost(Φ′,I a≠b )=cost(Φ, I a≠c *)−cost(Φ, I a≠b ). Therefore we have cost(Φ, I a≠c *)≥cost(Φ, I a≠b )≥cost(Φ, I a≠b *) where I a≠b * is the optimum assignment under the condition that a≠b for Φ. This inequality means that Φ≡Φ| a∜b  since we assume (i). 
     Suppose that (ii) b=c. Then we have cost(Φ′, I a=b *)≥min{w 11 ,w 12 }+min{w 21 , w 22 }=w 12 1+(w 21 +w 22 )−max{w 21 , w 22 }, where I a=b * is the optimum assignment under the condition that a=b for Φ and the last equality is due to the fact w 21 +w 22 =min{w 21 , w 22 }+max{w 21 , w 22 }. We also have cost(Φ′, I a≠b )≤w 0 , where I a≠b  is the assignment obtained by flipping the value of literal a from I a=b . From the assumption min{w 11 , w 12 }≥w 0 +max{w 21 , w 22 }, we know that cost(Φ′, I a=b *)≥cost(Φ′, I a≠b ). By the definition of I a=b * and I a≠b , the costs associated with the clauses in Φ 0  are the same for I a=b * and I a≠b  because literal a does not appear in any clause in Φ 0 . Hence we know that cost(Φ′, I a=b *)−cost(Φ′, I a≠b )=cost(Φ, I a=b *)−cost(Φ, I a≠b ). Therefore we know that cost(Φ, I a=b *)≥cost(Φ,I a≠b )≥cost(Φ, I a≠b *) where I a≠b * is the optimum assignment under the condition that a≠b for Φ. This inequality means that Φ≡Φ| a≠b  as desired. 
     Lemma 10 (Degree-2 Hard Binary Clause Rule). 
     Let Φ ∨ (l 1 ,l 2 ) be the set of the clauses that contain at least one of the literals l 1  or l 2 , and let Φ hard (l) be the set of the hard clauses that contain literal l in Φ. If |Φ hard  (u)|=2 and all of the clauses in Φ hard (u) are binary (that is, we can represent Φ hard  (u)={(u∨v 1 , ∞), (u∨v 2 , ∞)} by using two literals v 1  and v 2 ) and formula Φ satisfies 
     
       
         
           
             
               
                 
                   
                     
                       
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     Proof. 
     To prove this lemma, it is enough to show cost(I v     1     =v     2   )≤cost(I v     2     ≠v     2   ), where I v     1     =v     2   * and I v     1     ≠v     2   * are the optimum assignments for Φ under the constraints that v 1 =v 2  and v 1 ≠v 2 , respectively. To do so, we consider two cases: (i) (v 1 , v 2 )=(FALSE, TRUE) and (ii) (v 1 , v 2 )=(TRUE, FALSE) in assignment I v     1     ≠v     2   *. Here, we show cost(I v     1     =v     2   *)≤cost(I v     1     ≠v     2   ) only for Case (i) by using Inequality (1), and we omit the proof for Case (ii) because Case (ii) can be proved similarly using Inequality (2). In Case (i), v 1 =FALSE means that u=TRUE in l v     1     ≠v     2   * because P includes the hard clause (u∨v 1 , ∞). Let I v     1     =v     2    be the assignment obtained by flipping the assignments for v 1  and u from l v     1     ≠v     2   *. Specifically, v 1 =v 2 =TRUE and u=FALSE in I v     1     =v     2    and the assignment for the other variables are same in I v     1     ≠v     2   * and I v     1     =v     2   . By the construction of I v     1     =v     2   , the unsatisfied unit clauses (ū, w ū ) and (v 1 , w v     1   ) (if any) in I v     1     ≠v     2   * become satisfied in I v     1     =v     2   , and that the satisfied clauses associated with at least one of the two literals u and  v 1    in I v     1     =v     2   * might become unsatisfied in I v     1     =v     2   . These facts mean that 
                   cost   ⁡     (     Φ   ,     I       v   1     ≠     v   2       *       )       -     cost   ⁡     (     Φ   ,     I       v   1     =     v   2           )         ≥         w   unit     ⁡     (     u   _     )       +       w   unit     ⁡     (     v   1     )       -       ∑       (     c   ,   w     )     ∈         Φ   ⩔     ⁡     (     u   ,       v   1     _       )       ⁢   \   ⁢       Φ   hard     ⁡     (   u   )                     ⁢   w       ≥   0     ,         
where we use Inequality (1) for the last inequality. Hence we have cost(Φ, I v     1     =v     2   )≥cost(Φ,I v     1     =v     2   )≥cost(Φ,I* v     1     =v     2   *), where the last inequality is immediate from the definition of I v     1     =v     2   *.