/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> b(b(b(x1)))
            b(x1) -> d(d(x1))
            b(b(x1)) -> c(c(c(x1)))
            c(c(x1)) -> d(d(d(x1)))
            c(d(d(x1))) -> a(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 + [39]
          p(b) = [1] x1 + [26]
          p(c) = [1] x1 + [17]
          p(d) = [1] x1 + [11]
        
        Following rules are strictly oriented:
           b(x1) = [1] x1 + [26]
                 > [1] x1 + [22]
                 = d(d(x1))     
        
        b(b(x1)) = [1] x1 + [52]
                 > [1] x1 + [51]
                 = c(c(c(x1)))  
        
        c(c(x1)) = [1] x1 + [34]
                 > [1] x1 + [33]
                 = d(d(d(x1)))  
        
        
        Following rules are (at-least) weakly oriented:
           a(a(x1)) =  [1] x1 + [78]
                    >= [1] x1 + [78]
                    =  b(b(b(x1)))  
        
        c(d(d(x1))) =  [1] x1 + [39]
                    >= [1] x1 + [39]
                    =  a(x1)        
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> b(b(b(x1)))
            c(d(d(x1))) -> a(x1)
        - Weak TRS:
            b(x1) -> d(d(x1))
            b(b(x1)) -> c(c(c(x1)))
            c(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 + [27]
          p(b) = [1] x1 + [18]
          p(c) = [1] x1 + [12]
          p(d) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        c(d(d(x1))) = [1] x1 + [28]
                    > [1] x1 + [27]
                    = a(x1)        
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [54]
                 >= [1] x1 + [54]
                 =  b(b(b(x1)))  
        
           b(x1) =  [1] x1 + [18]
                 >= [1] x1 + [16]
                 =  d(d(x1))     
        
        b(b(x1)) =  [1] x1 + [36]
                 >= [1] x1 + [36]
                 =  c(c(c(x1)))  
        
        c(c(x1)) =  [1] x1 + [24]
                 >= [1] x1 + [24]
                 =  d(d(d(x1)))  
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> b(b(b(x1)))
        - Weak TRS:
            b(x1) -> d(d(x1))
            b(b(x1)) -> c(c(c(x1)))
            c(c(x1)) -> d(d(d(x1)))
            c(d(d(x1))) -> a(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 + [151]
          p(b) = [1] x1 + [98] 
          p(c) = [1] x1 + [65] 
          p(d) = [1] x1 + [43] 
        
        Following rules are strictly oriented:
        a(a(x1)) = [1] x1 + [302]
                 > [1] x1 + [294]
                 = b(b(b(x1)))   
        
        
        Following rules are (at-least) weakly oriented:
              b(x1) =  [1] x1 + [98] 
                    >= [1] x1 + [86] 
                    =  d(d(x1))      
        
           b(b(x1)) =  [1] x1 + [196]
                    >= [1] x1 + [195]
                    =  c(c(c(x1)))   
        
           c(c(x1)) =  [1] x1 + [130]
                    >= [1] x1 + [129]
                    =  d(d(d(x1)))   
        
        c(d(d(x1))) =  [1] x1 + [151]
                    >= [1] x1 + [151]
                    =  a(x1)         
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> b(b(b(x1)))
            b(x1) -> d(d(x1))
            b(b(x1)) -> c(c(c(x1)))
            c(c(x1)) -> d(d(d(x1)))
            c(d(d(x1))) -> a(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^1))