/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            c(c(a(a(y,0()),x))) -> c(y)
            c(c(b(c(y),0()))) -> a(0(),c(c(a(y,0()))))
            c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
        - Signature:
            {c/1} / {0/0,a/2,b/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(0) = [8]                  
            p(a) = [1] x1 + [1] x2 + [0]
            p(b) = [1] x1 + [1] x2 + [0]
            p(c) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          c(c(a(a(y,0()),x))) = [1] x + [1] y + [8]
                              > [1] y + [0]        
                              = c(y)               
          
          
          Following rules are (at-least) weakly oriented:
          c(c(b(c(y),0()))) =  [1] y + [8]          
                            >= [1] y + [16]         
                            =  a(0(),c(c(a(y,0()))))
          
            c(c(c(a(x,y)))) =  [1] x + [1] y + [0]  
                            >= [1] x + [1] y + [0]  
                            =  b(c(c(c(c(y)))),x)   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            c(c(b(c(y),0()))) -> a(0(),c(c(a(y,0()))))
            c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
        - Weak TRS:
            c(c(a(a(y,0()),x))) -> c(y)
        - Signature:
            {c/1} / {0/0,a/2,b/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(0) = [7]                  
            p(a) = [1] x1 + [1] x2 + [1]
            p(b) = [1] x1 + [1] x2 + [0]
            p(c) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          c(c(c(a(x,y)))) = [1] x + [1] y + [1]
                          > [1] x + [1] y + [0]
                          = b(c(c(c(c(y)))),x) 
          
          
          Following rules are (at-least) weakly oriented:
          c(c(a(a(y,0()),x))) =  [1] x + [1] y + [9]  
                              >= [1] y + [0]          
                              =  c(y)                 
          
            c(c(b(c(y),0()))) =  [1] y + [7]          
                              >= [1] y + [16]         
                              =  a(0(),c(c(a(y,0()))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            c(c(b(c(y),0()))) -> a(0(),c(c(a(y,0()))))
        - Weak TRS:
            c(c(a(a(y,0()),x))) -> c(y)
            c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
        - Signature:
            {c/1} / {0/0,a/2,b/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        MI {miKind = Automaton Nothing, miDimension = 4, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton Nothing:
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(0) = [0]                                
                 [1]                                
                 [0]                                
                 [0]                                
          p(a) = [1 0 0 1]       [1 0 1 0]       [0]
                 [0 0 0 0] x_1 + [0 0 0 0] x_2 + [0]
                 [0 1 1 0]       [0 0 1 0]       [0]
                 [0 0 0 0]       [0 1 0 1]       [0]
          p(b) = [1 0 0 0]       [1 0 0 0]       [0]
                 [0 0 0 0] x_1 + [0 0 0 0] x_2 + [0]
                 [0 0 0 1]       [0 1 0 0]       [0]
                 [0 0 0 1]       [0 1 0 0]       [0]
          p(c) = [1 0 0 1]       [0]                
                 [0 0 0 0] x_1 + [0]                
                 [0 0 1 0]       [0]                
                 [0 1 1 0]       [0]                
        
        Following rules are strictly oriented:
        c(c(b(c(y),0()))) = [1 2 2 1]     [2]    
                            [0 0 0 0] y + [0]    
                            [0 1 1 0]     [1]    
                            [0 1 1 0]     [1]    
                          > [1 2 2 1]     [1]    
                            [0 0 0 0] y + [0]    
                            [0 1 1 0]     [1]    
                            [0 1 1 0]     [0]    
                          = a(0(),c(c(a(y,0()))))
        
        
        Following rules are (at-least) weakly oriented:
        c(c(a(a(y,0()),x))) =  [1 1 2 1]     [1 1 1 1]     [1]
                               [0 0 0 0] x + [0 0 0 0] y + [0]
                               [0 0 1 0]     [0 1 1 0]     [0]
                               [0 0 1 0]     [0 1 1 0]     [0]
                            >= [1 0 0 1]     [0]              
                               [0 0 0 0] y + [0]              
                               [0 0 1 0]     [0]              
                               [0 1 1 0]     [0]              
                            =  c(y)                           
        
            c(c(c(a(x,y)))) =  [1 2 2 1]     [1 1 3 1]     [0]
                               [0 0 0 0] x + [0 0 0 0] y + [0]
                               [0 1 1 0]     [0 0 1 0]     [0]
                               [0 1 1 0]     [0 0 1 0]     [0]
                            >= [1 0 0 0]     [1 1 3 1]     [0]
                               [0 0 0 0] x + [0 0 0 0] y + [0]
                               [0 1 0 0]     [0 0 1 0]     [0]
                               [0 1 0 0]     [0 0 1 0]     [0]
                            =  b(c(c(c(c(y)))),x)             
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            c(c(a(a(y,0()),x))) -> c(y)
            c(c(b(c(y),0()))) -> a(0(),c(c(a(y,0()))))
            c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
        - Signature:
            {c/1} / {0/0,a/2,b/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))