Denition 3.          *is directional quantied k-consistent iff for every k-tupl* 
   
 

  
  
     *, one of the following cases must be satised*
 
     **              **        
  
    
   
 
     **           
       

*represents either quantier in*  
In particular, directional quantied 2-consistency is usually called directional quantied arc consistency. A binar  
strong, directional quantied k-consistent if it is directional quantied          
quantied globally consistent if it is strong, directional quantied Gent et al. (2008) have presented a quantied arc consistenc        
  
). An algorithm for strong directional quantied path consi      
    

                  
     
      ) is directional quantied globally consistent, a solution         
                               
                           
                      
  

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         ies based on the BTP, which is rst introduced by Cooper, Jeav 
                   
           ass of QCSPs, we rst dene the BTP for QCSPs.

Denition 4.          satises the broken-triangle property (QBTP for short), if The denition of the QBTP is similar to that of the BTP for clas          
QBTP must be same as that of the prex in the QCSP.

         *be a binary QCSP. If P satises the QBTP, for all triples of var*     
                    
                     

         *be a binary QCSP that satises the QBTP, if the reduced proble*   
quantied arc consistency on P is not empty, P*satises the QBTP.*
                     
required in the QAC algorithm. According to the denition of              
            
also satises the QBTP if it is not empty.

         be a binary QCSP. If P satises the QBTP, the satisability of      
 
. We rst enforce QAC on               is unsatisable.

        
     
     
satises the QBTP.

     
is directional quantied globally consistent. Suppose is directional quantied (
                
   
  
  
  
  
     
    
is quantied arc consistent, all values in 
  
  
          
         
   
   

                     
     
   
  
is directional quantied           
  
is directional quantied 2-consistent, it is directional quantied globally consistent. Therefore, is satisable, so is
. To sum up, the satisability of           
                     **            
satises the QBTP.

        ple of ordered variables satises the QBTP requires 
 
        ses the QBTP can be done by an algorithm in 

  
satises the QBTP, the satisability of such problem can be specied in 
   