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


--------------------------------------------------------------------------------


WORST_CASE(?,O(n^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(a(x1))) -> b(b(x1))
            b(b(b(x1))) -> c(d(x1))
            c(x1) -> a(a(x1))
            d(x1) -> 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 + [4] 
          p(b) = [1] x1 + [6] 
          p(c) = [1] x1 + [8] 
          p(d) = [1] x1 + [10]
        
        Following rules are strictly oriented:
        d(x1) = [1] x1 + [10]
              > [1] x1 + [8] 
              = c(x1)        
        
        
        Following rules are (at-least) weakly oriented:
        a(a(a(x1))) =  [1] x1 + [12]
                    >= [1] x1 + [12]
                    =  b(b(x1))     
        
        b(b(b(x1))) =  [1] x1 + [18]
                    >= [1] x1 + [18]
                    =  c(d(x1))     
        
              c(x1) =  [1] x1 + [8] 
                    >= [1] x1 + [8] 
                    =  a(a(x1))     
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(a(x1))) -> b(b(x1))
            b(b(b(x1))) -> c(d(x1))
            c(x1) -> a(a(x1))
        - Weak TRS:
            d(x1) -> 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 + [7] 
          p(b) = [1] x1 + [10]
          p(c) = [1] x1 + [14]
          p(d) = [1] x1 + [14]
        
        Following rules are strictly oriented:
        a(a(a(x1))) = [1] x1 + [21]
                    > [1] x1 + [20]
                    = b(b(x1))     
        
        b(b(b(x1))) = [1] x1 + [30]
                    > [1] x1 + [28]
                    = c(d(x1))     
        
        
        Following rules are (at-least) weakly oriented:
        c(x1) =  [1] x1 + [14]
              >= [1] x1 + [14]
              =  a(a(x1))     
        
        d(x1) =  [1] x1 + [14]
              >= [1] x1 + [14]
              =  c(x1)        
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(x1) -> a(a(x1))
        - Weak TRS:
            a(a(a(x1))) -> b(b(x1))
            b(b(b(x1))) -> c(d(x1))
            d(x1) -> 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 = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 2] x1 + [0]
                 [0 0]      [2]
          p(b) = [1 2] x1 + [2]
                 [0 0]      [2]
          p(c) = [1 2] x1 + [5]
                 [0 0]      [2]
          p(d) = [1 2] x1 + [5]
                 [0 0]      [2]
        
        Following rules are strictly oriented:
        c(x1) = [1 2] x1 + [5]
                [0 0]      [2]
              > [1 2] x1 + [4]
                [0 0]      [2]
              = a(a(x1))      
        
        
        Following rules are (at-least) weakly oriented:
        a(a(a(x1))) =  [1 2] x1 + [8] 
                       [0 0]      [2] 
                    >= [1 2] x1 + [8] 
                       [0 0]      [2] 
                    =  b(b(x1))       
        
        b(b(b(x1))) =  [1 2] x1 + [14]
                       [0 0]      [2] 
                    >= [1 2] x1 + [14]
                       [0 0]      [2] 
                    =  c(d(x1))       
        
              d(x1) =  [1 2] x1 + [5] 
                       [0 0]      [2] 
                    >= [1 2] x1 + [5] 
                       [0 0]      [2] 
                    =  c(x1)          
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(a(x1))) -> b(b(x1))
            b(b(b(x1))) -> c(d(x1))
            c(x1) -> a(a(x1))
            d(x1) -> 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^1))