/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^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> b(c(x1))
            b(x1) -> a(x1)
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(f(x1))
            d(x1) -> b(x1)
            d(d(x1)) -> f(f(f(x1)))
            f(f(x1)) -> g(a(x1))
            g(g(x1)) -> a(x1)
        - Signature:
            {a/1,b/1,c/1,d/1,f/1,g/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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 + [121]
          p(b) = [1] x1 + [127]
          p(c) = [1] x1 + [114]
          p(d) = [1] x1 + [137]
          p(f) = [1] x1 + [91] 
          p(g) = [1] x1 + [61] 
        
        Following rules are strictly oriented:
        a(a(x1)) = [1] x1 + [242]
                 > [1] x1 + [241]
                 = b(c(x1))      
        
           b(x1) = [1] x1 + [127]
                 > [1] x1 + [121]
                 = a(x1)         
        
        b(b(x1)) = [1] x1 + [254]
                 > [1] x1 + [251]
                 = c(d(x1))      
        
           d(x1) = [1] x1 + [137]
                 > [1] x1 + [127]
                 = b(x1)         
        
        d(d(x1)) = [1] x1 + [274]
                 > [1] x1 + [273]
                 = f(f(f(x1)))   
        
        g(g(x1)) = [1] x1 + [122]
                 > [1] x1 + [121]
                 = a(x1)         
        
        
        Following rules are (at-least) weakly oriented:
        c(c(x1)) =  [1] x1 + [228]
                 >= [1] x1 + [228]
                 =  d(f(x1))      
        
        f(f(x1)) =  [1] x1 + [182]
                 >= [1] x1 + [182]
                 =  g(a(x1))      
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(c(x1)) -> d(f(x1))
            f(f(x1)) -> g(a(x1))
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            b(x1) -> a(x1)
            b(b(x1)) -> c(d(x1))
            d(x1) -> b(x1)
            d(d(x1)) -> f(f(f(x1)))
            g(g(x1)) -> a(x1)
        - Signature:
            {a/1,b/1,c/1,d/1,f/1,g/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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 + [122]
          p(b) = [1] x1 + [129]
          p(c) = [1] x1 + [115]
          p(d) = [1] x1 + [138]
          p(f) = [1] x1 + [92] 
          p(g) = [1] x1 + [61] 
        
        Following rules are strictly oriented:
        f(f(x1)) = [1] x1 + [184]
                 > [1] x1 + [183]
                 = g(a(x1))      
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [244]
                 >= [1] x1 + [244]
                 =  b(c(x1))      
        
           b(x1) =  [1] x1 + [129]
                 >= [1] x1 + [122]
                 =  a(x1)         
        
        b(b(x1)) =  [1] x1 + [258]
                 >= [1] x1 + [253]
                 =  c(d(x1))      
        
        c(c(x1)) =  [1] x1 + [230]
                 >= [1] x1 + [230]
                 =  d(f(x1))      
        
           d(x1) =  [1] x1 + [138]
                 >= [1] x1 + [129]
                 =  b(x1)         
        
        d(d(x1)) =  [1] x1 + [276]
                 >= [1] x1 + [276]
                 =  f(f(f(x1)))   
        
        g(g(x1)) =  [1] x1 + [122]
                 >= [1] x1 + [122]
                 =  a(x1)         
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(c(x1)) -> d(f(x1))
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            b(x1) -> a(x1)
            b(b(x1)) -> c(d(x1))
            d(x1) -> b(x1)
            d(d(x1)) -> f(f(f(x1)))
            f(f(x1)) -> g(a(x1))
            g(g(x1)) -> a(x1)
        - Signature:
            {a/1,b/1,c/1,d/1,f/1,g/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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 + [24]
          p(b) = [1] x1 + [25]
          p(c) = [1] x1 + [23]
          p(d) = [1] x1 + [27]
          p(f) = [1] x1 + [18]
          p(g) = [1] x1 + [12]
        
        Following rules are strictly oriented:
        c(c(x1)) = [1] x1 + [46]
                 > [1] x1 + [45]
                 = d(f(x1))     
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [48]
                 >= [1] x1 + [48]
                 =  b(c(x1))     
        
           b(x1) =  [1] x1 + [25]
                 >= [1] x1 + [24]
                 =  a(x1)        
        
        b(b(x1)) =  [1] x1 + [50]
                 >= [1] x1 + [50]
                 =  c(d(x1))     
        
           d(x1) =  [1] x1 + [27]
                 >= [1] x1 + [25]
                 =  b(x1)        
        
        d(d(x1)) =  [1] x1 + [54]
                 >= [1] x1 + [54]
                 =  f(f(f(x1)))  
        
        f(f(x1)) =  [1] x1 + [36]
                 >= [1] x1 + [36]
                 =  g(a(x1))     
        
        g(g(x1)) =  [1] x1 + [24]
                 >= [1] x1 + [24]
                 =  a(x1)        
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            b(x1) -> a(x1)
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(f(x1))
            d(x1) -> b(x1)
            d(d(x1)) -> f(f(f(x1)))
            f(f(x1)) -> g(a(x1))
            g(g(x1)) -> a(x1)
        - Signature:
            {a/1,b/1,c/1,d/1,f/1,g/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))