/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: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> d(c(x1))
            a(b(x1)) -> c(c(c(x1)))
            b(b(x1)) -> a(c(c(x1)))
            c(c(x1)) -> b(x1)
            c(d(x1)) -> a(a(x1))
            d(d(x1)) -> b(a(c(x1)))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1] x1 + [2]
          p(b) = [1] x1 + [2]
          p(c) = [1] x1 + [1]
          p(d) = [1] x1 + [3]
        
        Following rules are strictly oriented:
        a(b(x1)) = [1] x1 + [4]
                 > [1] x1 + [3]
                 = c(c(c(x1))) 
        
        d(d(x1)) = [1] x1 + [6]
                 > [1] x1 + [5]
                 = b(a(c(x1))) 
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [4]
                 >= [1] x1 + [4]
                 =  d(c(x1))    
        
        b(b(x1)) =  [1] x1 + [4]
                 >= [1] x1 + [4]
                 =  a(c(c(x1))) 
        
        c(c(x1)) =  [1] x1 + [2]
                 >= [1] x1 + [2]
                 =  b(x1)       
        
        c(d(x1)) =  [1] x1 + [4]
                 >= [1] x1 + [4]
                 =  a(a(x1))    
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> d(c(x1))
            b(b(x1)) -> a(c(c(x1)))
            c(c(x1)) -> b(x1)
            c(d(x1)) -> a(a(x1))
        - Weak TRS:
            a(b(x1)) -> c(c(c(x1)))
            d(d(x1)) -> b(a(c(x1)))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1] x1 + [20]
          p(b) = [1] x1 + [23]
          p(c) = [1] x1 + [12]
          p(d) = [1] x1 + [28]
        
        Following rules are strictly oriented:
        b(b(x1)) = [1] x1 + [46]
                 > [1] x1 + [44]
                 = a(c(c(x1)))  
        
        c(c(x1)) = [1] x1 + [24]
                 > [1] x1 + [23]
                 = b(x1)        
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [40]
                 >= [1] x1 + [40]
                 =  d(c(x1))     
        
        a(b(x1)) =  [1] x1 + [43]
                 >= [1] x1 + [36]
                 =  c(c(c(x1)))  
        
        c(d(x1)) =  [1] x1 + [40]
                 >= [1] x1 + [40]
                 =  a(a(x1))     
        
        d(d(x1)) =  [1] x1 + [56]
                 >= [1] x1 + [55]
                 =  b(a(c(x1)))  
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> d(c(x1))
            c(d(x1)) -> a(a(x1))
        - Weak TRS:
            a(b(x1)) -> c(c(c(x1)))
            b(b(x1)) -> a(c(c(x1)))
            c(c(x1)) -> b(x1)
            d(d(x1)) -> b(a(c(x1)))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 66] x1 + [109]
                 [0  1]      [2]  
          p(b) = [1 68] x1 + [111]
                 [0  1]      [2]  
          p(c) = [1 35] x1 + [38] 
                 [0  1]      [1]  
          p(d) = [1 97] x1 + [214]
                 [0  1]      [3]  
        
        Following rules are strictly oriented:
        a(a(x1)) = [1 132] x1 + [350]
                   [0   1]      [4]  
                 > [1 132] x1 + [349]
                   [0   1]      [4]  
                 = d(c(x1))          
        
        c(d(x1)) = [1 132] x1 + [357]
                   [0   1]      [4]  
                 > [1 132] x1 + [350]
                   [0   1]      [4]  
                 = a(a(x1))          
        
        
        Following rules are (at-least) weakly oriented:
        a(b(x1)) =  [1 134] x1 + [352]
                    [0   1]      [4]  
                 >= [1 105] x1 + [219]
                    [0   1]      [3]  
                 =  c(c(c(x1)))       
        
        b(b(x1)) =  [1 136] x1 + [358]
                    [0   1]      [4]  
                 >= [1 136] x1 + [352]
                    [0   1]      [4]  
                 =  a(c(c(x1)))       
        
        c(c(x1)) =  [1 70] x1 + [111] 
                    [0  1]      [2]   
                 >= [1 68] x1 + [111] 
                    [0  1]      [2]   
                 =  b(x1)             
        
        d(d(x1)) =  [1 194] x1 + [719]
                    [0   1]      [6]  
                 >= [1 169] x1 + [528]
                    [0   1]      [5]  
                 =  b(a(c(x1)))       
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> d(c(x1))
            a(b(x1)) -> c(c(c(x1)))
            b(b(x1)) -> a(c(c(x1)))
            c(c(x1)) -> b(x1)
            c(d(x1)) -> a(a(x1))
            d(d(x1)) -> b(a(c(x1)))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))