/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)) -> b(b(b(x1)))
            b(x1) -> c(c(d(x1)))
            b(c(x1)) -> c(b(x1))
            b(c(d(x1))) -> a(x1)
            c(x1) -> d(d(d(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 + [220]
          p(b) = [1] x1 + [144]
          p(c) = [1] x1 + [64] 
          p(d) = [1] x1 + [16] 
        
        Following rules are strictly oriented:
           a(a(x1)) = [1] x1 + [440]
                    > [1] x1 + [432]
                    = b(b(b(x1)))   
        
        b(c(d(x1))) = [1] x1 + [224]
                    > [1] x1 + [220]
                    = a(x1)         
        
              c(x1) = [1] x1 + [64] 
                    > [1] x1 + [48] 
                    = d(d(d(x1)))   
        
        
        Following rules are (at-least) weakly oriented:
           b(x1) =  [1] x1 + [144]
                 >= [1] x1 + [144]
                 =  c(c(d(x1)))   
        
        b(c(x1)) =  [1] x1 + [208]
                 >= [1] x1 + [208]
                 =  c(b(x1))      
        
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(x1) -> c(c(d(x1)))
            b(c(x1)) -> c(b(x1))
        - Weak TRS:
            a(a(x1)) -> b(b(b(x1)))
            b(c(d(x1))) -> a(x1)
            c(x1) -> d(d(d(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 + [48]
          p(b) = [1] x1 + [32]
          p(c) = [1] x1 + [14]
          p(d) = [1] x1 + [2] 
        
        Following rules are strictly oriented:
        b(x1) = [1] x1 + [32]
              > [1] x1 + [30]
              = c(c(d(x1)))  
        
        
        Following rules are (at-least) weakly oriented:
           a(a(x1)) =  [1] x1 + [96]
                    >= [1] x1 + [96]
                    =  b(b(b(x1)))  
        
           b(c(x1)) =  [1] x1 + [46]
                    >= [1] x1 + [46]
                    =  c(b(x1))     
        
        b(c(d(x1))) =  [1] x1 + [48]
                    >= [1] x1 + [48]
                    =  a(x1)        
        
              c(x1) =  [1] x1 + [14]
                    >= [1] x1 + [6] 
                    =  d(d(d(x1)))  
        
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(c(x1)) -> c(b(x1))
        - Weak TRS:
            a(a(x1)) -> b(b(b(x1)))
            b(x1) -> c(c(d(x1)))
            b(c(d(x1))) -> a(x1)
            c(x1) -> d(d(d(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 24] x1 + [176]
                 [0  1]      [24] 
          p(b) = [1 16] x1 + [48] 
                 [0  1]      [16] 
          p(c) = [1 6] x1 + [0]   
                 [0 1]      [8]   
          p(d) = [1 2] x1 + [0]   
                 [0 1]      [0]   
        
        Following rules are strictly oriented:
        b(c(x1)) = [1 22] x1 + [176]
                   [0  1]      [24] 
                 > [1 22] x1 + [144]
                   [0  1]      [24] 
                 = c(b(x1))         
        
        
        Following rules are (at-least) weakly oriented:
           a(a(x1)) =  [1 48] x1 + [928]
                       [0  1]      [48] 
                    >= [1 48] x1 + [912]
                       [0  1]      [48] 
                    =  b(b(b(x1)))      
        
              b(x1) =  [1 16] x1 + [48] 
                       [0  1]      [16] 
                    >= [1 14] x1 + [48] 
                       [0  1]      [16] 
                    =  c(c(d(x1)))      
        
        b(c(d(x1))) =  [1 24] x1 + [176]
                       [0  1]      [24] 
                    >= [1 24] x1 + [176]
                       [0  1]      [24] 
                    =  a(x1)            
        
              c(x1) =  [1 6] x1 + [0]   
                       [0 1]      [8]   
                    >= [1 6] x1 + [0]   
                       [0 1]      [0]   
                    =  d(d(d(x1)))      
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> b(b(b(x1)))
            b(x1) -> c(c(d(x1)))
            b(c(x1)) -> c(b(x1))
            b(c(d(x1))) -> a(x1)
            c(x1) -> d(d(d(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))