/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(a(a(x1)))) -> b(c(x1))
            b(c(x1)) -> c(b(x1))
            c(b(x1)) -> a(a(a(x1)))
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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 + [44] 
          p(b) = [1] x1 + [64] 
          p(c) = [1] x1 + [112]
        
        Following rules are strictly oriented:
        c(b(x1)) = [1] x1 + [176]
                 > [1] x1 + [132]
                 = a(a(a(x1)))   
        
        
        Following rules are (at-least) weakly oriented:
        a(a(a(a(x1)))) =  [1] x1 + [176]
                       >= [1] x1 + [176]
                       =  b(c(x1))      
        
              b(c(x1)) =  [1] x1 + [176]
                       >= [1] x1 + [176]
                       =  c(b(x1))      
        
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(a(a(x1)))) -> b(c(x1))
            b(c(x1)) -> c(b(x1))
        - Weak TRS:
            c(b(x1)) -> a(a(a(x1)))
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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 + [19]
          p(b) = [1] x1 + [34]
          p(c) = [1] x1 + [33]
        
        Following rules are strictly oriented:
        a(a(a(a(x1)))) = [1] x1 + [76]
                       > [1] x1 + [67]
                       = b(c(x1))     
        
        
        Following rules are (at-least) weakly oriented:
        b(c(x1)) =  [1] x1 + [67]
                 >= [1] x1 + [67]
                 =  c(b(x1))     
        
        c(b(x1)) =  [1] x1 + [67]
                 >= [1] x1 + [57]
                 =  a(a(a(x1)))  
        
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(c(x1)) -> c(b(x1))
        - Weak TRS:
            a(a(a(a(x1)))) -> b(c(x1))
            c(b(x1)) -> a(a(a(x1)))
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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 1] x1 + [10] 
                 [0 1]      [48] 
          p(b) = [1 1] x1 + [48] 
                 [0 1]      [32] 
          p(c) = [1 2] x1 + [130]
                 [0 1]      [128]
        
        Following rules are strictly oriented:
        b(c(x1)) = [1 3] x1 + [306]
                   [0 1]      [160]
                 > [1 3] x1 + [242]
                   [0 1]      [160]
                 = c(b(x1))        
        
        
        Following rules are (at-least) weakly oriented:
        a(a(a(a(x1)))) =  [1 4] x1 + [328]
                          [0 1]      [192]
                       >= [1 3] x1 + [306]
                          [0 1]      [160]
                       =  b(c(x1))        
        
              c(b(x1)) =  [1 3] x1 + [242]
                          [0 1]      [160]
                       >= [1 3] x1 + [174]
                          [0 1]      [144]
                       =  a(a(a(x1)))     
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(a(a(x1)))) -> b(c(x1))
            b(c(x1)) -> c(b(x1))
            c(b(x1)) -> a(a(a(x1)))
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))