/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: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(a(x1))) -> b(a(a(b(x1))))
            b(a(x1)) -> a(b(x1))
            b(b(b(b(x1)))) -> a(x1)
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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 + [6]
            p(b) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          a(a(a(x1))) = [1] x1 + [18] 
                      > [1] x1 + [14] 
                      = b(a(a(b(x1))))
          
          
          Following rules are (at-least) weakly oriented:
                b(a(x1)) =  [1] x1 + [7]
                         >= [1] x1 + [7]
                         =  a(b(x1))    
          
          b(b(b(b(x1)))) =  [1] x1 + [4]
                         >= [1] x1 + [6]
                         =  a(x1)       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(a(x1)) -> a(b(x1))
            b(b(b(b(x1)))) -> a(x1)
        - Weak TRS:
            a(a(a(x1))) -> b(a(a(b(x1))))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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 + [8]
            p(b) = [1] x1 + [4]
          
          Following rules are strictly oriented:
          b(b(b(b(x1)))) = [1] x1 + [16]
                         > [1] x1 + [8] 
                         = a(x1)        
          
          
          Following rules are (at-least) weakly oriented:
          a(a(a(x1))) =  [1] x1 + [24] 
                      >= [1] x1 + [24] 
                      =  b(a(a(b(x1))))
          
             b(a(x1)) =  [1] x1 + [12] 
                      >= [1] x1 + [12] 
                      =  a(b(x1))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(a(x1)) -> a(b(x1))
        - Weak TRS:
            a(a(a(x1))) -> b(a(a(b(x1))))
            b(b(b(b(x1)))) -> a(x1)
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        MI {miKind = Automaton Nothing, miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton Nothing:
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 1 1]       [0]
                 [0 1 0] x_1 + [2]
                 [0 0 1]       [2]
          p(b) = [1 0 1]       [0]
                 [0 0 1] x_1 + [1]
                 [0 1 0]       [0]
        
        Following rules are strictly oriented:
        b(a(x1)) = [1 1 2]      [2]
                   [0 0 1] x1 + [3]
                   [0 1 0]      [2]
                 > [1 1 2]      [1]
                   [0 0 1] x1 + [3]
                   [0 1 0]      [2]
                 = a(b(x1))        
        
        
        Following rules are (at-least) weakly oriented:
           a(a(a(x1))) =  [1 3 3]      [12]
                          [0 1 0] x1 + [6] 
                          [0 0 1]      [6] 
                       >= [1 3 3]      [10]
                          [0 1 0] x1 + [5] 
                          [0 0 1]      [5] 
                       =  b(a(a(b(x1))))   
        
        b(b(b(b(x1)))) =  [1 2 2]      [2] 
                          [0 1 0] x1 + [2] 
                          [0 0 1]      [2] 
                       >= [1 1 1]      [0] 
                          [0 1 0] x1 + [2] 
                          [0 0 1]      [2] 
                       =  a(x1)            
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(a(x1))) -> b(a(a(b(x1))))
            b(a(x1)) -> a(b(x1))
            b(b(b(b(x1)))) -> a(x1)
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))