/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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(x1) -> c(b(x1))
            b(b(x1)) -> a(c(x1))
            b(c(x1)) -> a(x1)
            c(c(c(x1))) -> b(x1)
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
            innermost 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 + [3]
          p(b) = [1] x1 + [2]
          p(c) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        c(c(c(x1))) = [1] x1 + [3]
                    > [1] x1 + [2]
                    = b(x1)       
        
        
        Following rules are (at-least) weakly oriented:
           a(x1) =  [1] x1 + [3]
                 >= [1] x1 + [3]
                 =  c(b(x1))    
        
        b(b(x1)) =  [1] x1 + [4]
                 >= [1] x1 + [4]
                 =  a(c(x1))    
        
        b(c(x1)) =  [1] x1 + [3]
                 >= [1] x1 + [3]
                 =  a(x1)       
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(x1) -> c(b(x1))
            b(b(x1)) -> a(c(x1))
            b(c(x1)) -> a(x1)
        - Weak TRS:
            c(c(c(x1))) -> b(x1)
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
            innermost 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 + [13]
          p(b) = [1] x1 + [9] 
          p(c) = [1] x1 + [4] 
        
        Following rules are strictly oriented:
        b(b(x1)) = [1] x1 + [18]
                 > [1] x1 + [17]
                 = a(c(x1))     
        
        
        Following rules are (at-least) weakly oriented:
              a(x1) =  [1] x1 + [13]
                    >= [1] x1 + [13]
                    =  c(b(x1))     
        
           b(c(x1)) =  [1] x1 + [13]
                    >= [1] x1 + [13]
                    =  a(x1)        
        
        c(c(c(x1))) =  [1] x1 + [12]
                    >= [1] x1 + [9] 
                    =  b(x1)        
        
* Step 3: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(x1) -> c(b(x1))
            b(c(x1)) -> a(x1)
        - Weak TRS:
            b(b(x1)) -> a(c(x1))
            c(c(c(x1))) -> b(x1)
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c}
    + 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 + [0]
            p(b) = [1] x1 + [4]
            p(c) = [1] x1 + [2]
          
          Following rules are strictly oriented:
          b(c(x1)) = [1] x1 + [6]
                   > [1] x1 + [0]
                   = a(x1)       
          
          
          Following rules are (at-least) weakly oriented:
                a(x1) =  [1] x1 + [0]
                      >= [1] x1 + [6]
                      =  c(b(x1))    
          
             b(b(x1)) =  [1] x1 + [8]
                      >= [1] x1 + [2]
                      =  a(c(x1))    
          
          c(c(c(x1))) =  [1] x1 + [6]
                      >= [1] x1 + [4]
                      =  b(x1)       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(x1) -> c(b(x1))
        - Weak TRS:
            b(b(x1)) -> a(c(x1))
            b(c(x1)) -> a(x1)
            c(c(c(x1))) -> b(x1)
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
            innermost 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 4] x1 + [6]
                 [0 1]      [6]
          p(b) = [1 3] x1 + [1]
                 [0 1]      [4]
          p(c) = [1 1] x1 + [0]
                 [0 1]      [2]
        
        Following rules are strictly oriented:
        a(x1) = [1 4] x1 + [6]
                [0 1]      [6]
              > [1 4] x1 + [5]
                [0 1]      [6]
              = c(b(x1))      
        
        
        Following rules are (at-least) weakly oriented:
           b(b(x1)) =  [1 6] x1 + [14]
                       [0 1]      [8] 
                    >= [1 5] x1 + [14]
                       [0 1]      [8] 
                    =  a(c(x1))       
        
           b(c(x1)) =  [1 4] x1 + [7] 
                       [0 1]      [6] 
                    >= [1 4] x1 + [6] 
                       [0 1]      [6] 
                    =  a(x1)          
        
        c(c(c(x1))) =  [1 3] x1 + [6] 
                       [0 1]      [6] 
                    >= [1 3] x1 + [1] 
                       [0 1]      [4] 
                    =  b(x1)          
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(x1) -> c(b(x1))
            b(b(x1)) -> a(c(x1))
            b(c(x1)) -> a(x1)
            c(c(c(x1))) -> b(x1)
        - Signature:
            {a/1,b/1,c/1} / {}
        - Obligation:
            innermost 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))